The canonical viewpoint of space-time is taken as fundamental by requiring the existence of a global time function $t$ corresponding to a global foliation of space-time into a set of spatial hypersurfaces. Moreover it is required that space-time can be foliated into a set of timelike curves corresponding to a family of fundamental observers and that these two foliations are orthogonal to each other. It is shown that this leads to a ``quasi-metric'' space-time geometry if $t$ is taken to represent one extra degenerate time dimension. The quasi-metric framework then consists of a differentiable manifold equipped with a one-parameter family of Lorentzian 4-metrics ${g}_t$ parameterized by the global time function $t$, in addition to a non-metric connection compatible with the family ${g}_t$.
doctor9.tex
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\begin{document}
\begin{center}
{\Huge\bf A Non-Metric Approach to Space,\\\ \\ Time and Gravitation}
\end{center} 
{\vspace*{4cm}}
\begin{center}
\bf by
\end{center}
{\vspace*{0.5cm}}
\begin{center}
\bf Dag {\O}stvang
\end{center}
{\vspace*{2cm}}
\begin{center}
{\Large\bf Thesis submitted for the degree of \\\ \\
DOCTOR SCIENTIARUM}
\end{center}
{\vspace*{2.5cm}}
\begin{center}\bf
NORWEGIAN UNIVERSITY OF SCIENCE \\ AND TECHNOLOGY, TRONDHEIM
\end{center}
{\vspace*{0.5cm}}
\begin{center}\bf
Faculty of Physics, Informatics and Mathematics \\
Department of Physics
\end{center}
{\vspace*{5mm}}
\begin{center}
\bf 2001
\end{center}

\newpage                                        
\thispagestyle{empty}
\vspace*{5cm}
\newpage
\thispagestyle{empty}
\begin{center}
{\Large {\bf Forord}}
\end{center}
$\\ [2mm]$

Dette arbeidet har blitt utf{\o}rt ved Institutt for fysikk, NTNU med professor
K{\aa}re Olaussen som hovedveileder og professor Kjell Mork som formell 
veileder. Avhandlingen er resultatet av flere {\aa}rs utvikling fra noe 
som startet som temmelig vage, kvalitative ideer. Veien fram til en 
konkretisering av disse ideene har v{\ae}rt temmelig t{\o}ff og blindsporene 
har v{\ae}rt mange. Sikkert er det i alle fall at uten st{\o}tte fra min 
faglige veileder professor K{\aa}re Olaussen, som kom til unnsetning da 
utsiktene var som m{\o}rkest, ville dette arbeidet aldri ha blitt fullf{\o}rt. 
Jeg synes det var modig gjort av ham {\aa} satse tid og prestisje p{\aa} noe 
som kunne fortone seg som temmelig usikkert, og jeg er ham stor takk skyldig. 

Ellers har en rekke personer vist interesse for prosjektet. Av disse vil 
jeg spesielt nevne Amir Ghaderi, Terje R. Meisler, John Fredrik Moxnes og 
Tommy {\O}verg{\aa}rd.

Det skal dessuten nevnes at professor Steve Carlip har v{\ae}rt s{\aa} vennlig 
{\aa} komme med detaljkritikk via e-post. Jeg vil ogs{\aa} takke mine foreldre 
for moralsk st{\o}tte.

{\vspace*{1cm}}
\begin{center}
Trondheim, april 2001
\end{center}
{\vspace*{5mm}}

\begin{center}
Dag {\O}stvang
\end{center}

\newpage
\thispagestyle{empty}
\vspace*{5cm}
\newpage
\thispagestyle{empty}
\contentsline{section}
{\numberline \ Notations}{1}
\vspace{\baselineskip}
\contentsline{section}
{\numberline \ Prologue}{5}
\vspace{\baselineskip}
\contentsline{section}
{\numberline \ Preliminaries: Some Basic Mathematical Concepts}{9}
\contentsline{subsection}
{\numberline {1}Introduction}{11}
\contentsline{subsection}
{\numberline {2}From 3-geometry to 4-geometry}{11}
\contentsline{subsection}
{\numberline {3}Connection and curvature}{14}
\contentsline{subsection}
{\numberline {4}Two simple examples}{15}
\contentsline{subsection}
{\numberline \ References}{16}
\vspace{\baselineskip}
\contentsline{section}
{\numberline \ Chapter One: A Non-Metric Approach to Space, Time and 
Gravitation}{17}
\contentsline{subsection}
{\numberline {1}Introduction}{19}
\contentsline{subsection}
{\numberline {2}An alternative framework of space and time}{23}
\contentsline{subsection}
{\numberline {3}The role of gravity}{28}
\contentsline{subsection}
{\numberline {4}Kinematics and non-kinematics of the FHSs}{36}
\contentsline{subsection}
{\numberline {5}Test particle motion revisited}{39}
\contentsline{subsection}
{\numberline {6}Discussion}{41}
\contentsline{subsection}
{\numberline \ References}{42}
\vspace{\baselineskip}
\contentsline{section}
{\numberline \ Chapter Two: On the Non-Kinematical Evolution of the 3-Geometry}
{43}
\contentsline{subsection}
{\numberline {1}Introduction}{45}
\contentsline{subsection}
{\numberline {2}The quasi-metric framework}{46}
\contentsline{subsubsection}
{\numberline {2.1}Uniqueness and existence of the global time function}{55}
\contentsline{subsubsection}
{\numberline {2.2}The question of general covariance}{58}
\contentsline{subsubsection}
{\numberline {2.3}The metric approximation}{59}
\contentsline{subsection}
{\numberline {3}Equations of evolution}{64}
\contentsline{subsubsection}
{\numberline {3.1}Projections}{67}
\contentsline{subsubsection}
{\numberline {3.2}The field equations}{72}
\contentsline{subsubsection}
{\numberline {3.3}The quasi-metric initial value problem}{78}
\contentsline{subsubsection}
{\numberline {3.4}Equations of motion}{81}
\contentsline{subsection}
{\numberline {4}Spherically symmetric vacua}{83}
\contentsline{subsubsection}
{\numberline {4.1}The conformally Minkowski case}{85}
\newpage
\thispagestyle{empty}
\contentsline{subsubsection}
{\numberline {4.2}Testing the NKE-paradigm}{87}
\contentsline{subsection}
{\numberline {5}Conclusion}{92}
\contentsline{subsection}
{\numberline \ References}{93}
\contentsline{subsection}
{\numberline \ Appendix A: Derivation of a static line element}{94}
\contentsline{subsection}
{\numberline \ Appendix B: The Newtonian limit}{95}
\contentsline{subsection}
{\numberline \ Appendix C: Isotropic cosmological models with matter}{97}
\contentsline{subsection}
{\numberline \ Appendix D: Symbols and acronyms}{98}
\vspace{\baselineskip}
\contentsline{section}
{\numberline \ Chapter Three:  The Gravitational Field outside a Spherically\\ 
Symmetric, Isolated Source in a Quasi-Metric Theory of Gravity}{101}
\contentsline{subsection}
{\numberline {1}Introduction}{103}
\contentsline{subsection}
{\numberline {2}General equations of motion}{104}
\contentsline{subsubsection}
{\numberline {2.1}Special equations of motion}{107}
\contentsline{subsection}
{\numberline {3}The dynamical problem}{110}
\contentsline{subsection}
{\numberline {4}Discussion}{113}
\contentsline{subsubsection}
{\numberline {4.1}Expanding space and the solar system}{113}
\contentsline{subsection}
{\numberline \ References}{118}
\vspace{\baselineskip}
\contentsline{section}
{\numberline \ Chapter Four:  An Assumption of a Static Gravitational Field\\ 
Resulting in an Apparently Anomalous Force}{119}
\contentsline{subsection}
{\numberline {1}Introduction}{121}
\contentsline{subsection}
{\numberline {2}A quasi-metric model}{123}
\contentsline{subsubsection}
{\numberline {2.1}Cosmic expansion and the PPN-formalism}{128}
\contentsline{subsection}
{\numberline {3}Conclusion}{129}
\contentsline{subsection}
{\numberline \ References}{130}
\vspace{\baselineskip}
\contentsline{section}
{\numberline \ Chapter Five:  On the Non-Kinematical Expansion of\\ 
Gravitationally Bound Bodies}{131}
\contentsline{subsection}
{\numberline {1}Introduction}{133}
\contentsline{subsection}
{\numberline {2}A brief survey of the quasi-metric theory}{134}
\contentsline{subsection}
{\numberline {3}Metrically static, spherically symmetric interiors}{136}
\contentsline{subsubsection}
{\numberline {3.1}Analytical calculations and a numerical recipe}{137}
\contentsline{subsection}
{\numberline {4}Discussion}{140}
\contentsline{subsection}
{\numberline \ References}{142}

\newpage

\setcounter{page}{1}

\section*{Notations}
This is a table of symbols and acronyms used in the thesis. Sign conventions 
and notations are as in the classic book {\em Gravitation} by C.W. Misner, 
K.S. Thorne and J.A. Wheeler if otherwise not stated. In particular
Greek indices may take values $0-3$ and Latin indices may take values $1-3$.

\begin{table}[h]
\begin{tabular}{lp{11.1cm}}
{\bf SYMBOL} & {\bf NAME} \\ 
KE/NKE & kinematical/non-kinematical evolution (of the FHSs) \\
FOs/FHSs & fundamental observers/fundamental hypersurfaces \\
NKR/NKCE & non-kinematical redshift/non-kinematical coupling equation \\
GTCS & global time coordinate system \\
HOCS & hypersurface-orthogonal coordinate system \\
WEP/EEP & weak/Einstein equivalence principle \\
GWEP/SEP & gravitational weak/strong equivalence principle \\
LLI/LPI & local Lorentz/positional invariance \\
SR/GR & special/general relativity \\
SC & Schiff's conjecture \\
$t=t(x^{\mu})$ & global time function, conventionally we define $t=x^{0}/c$
when using a GTCS (where $x^0$ is a global time coordinate) \\
${\cal M}$/${\overline{\cal M}}$ & metric space-time manifolds obtained by 
holding $t$ constant \\
${\cal N}$/${\overline{\cal N}}$ & quasi-metric space-time manifolds,
foliated into the set of FHSs \\
${\bf g}_t$ or $g_{(t){\mu}{\nu}}$ & one-parameter family of
space-time metric tensors on ${\cal N}$ \\
${\bf {\bar g}}_t$ or ${\bar g}_{(t){\mu}{\nu}}$ & one-parameter family of
space-time metric tensors on ${\overline{\cal N}}$, solution of field
equations \\
${\bf {\tilde g}}_t$ or ${\tilde g}_{(t){\mu}{\nu}}$ & 
one-parameter family of auxiliary space-time metric tensors 
on ${\overline{\cal N}}$ \\
${\bf n}_t$ or $n_{(t)}^{\mu}$ & unit vector fields normal to the FHSs 
in the metrics ${\bf g}_t$ \\
${\bf {\bar n}}_t$ or ${\bar n}_{(t)}^{\mu}$ & unit vector fields normal to 
the FHSs in the metrics ${\bf {\bar g}}_t$ \\
${\bf h}_t$ or $h_{(t)ij}$ & spatial metric tensors of the FHSs obtained from
${\bf g}_t$ \\
${\bf {\bar h}}_t$ or ${\bar h}_{(t)ij}$ & spatial metric tensors of the FHSs 
obtained from ${\bf {\bar g}}_t$ \\
$N(x^{\mu})$ & lapse function field in the metrics ${\bf g}_t$ \\
${\bar N}_t(x^{\mu},t)$ & lapse function field family in the metrics 
${\bf {\bar g}}_t$ 
\end{tabular}
\end{table}
\clearpage
\begin{table}[t]
\begin{tabular}{lp{10.8cm}}
$N^j(x^{\mu})$ & components of the shift vector field in the metrics 
${\bf g}_{t_0}$ \\
${\bar N}^j(x^{\mu})$ & components of the shift vector field in the metrics 
${\bf {\bar g}}_{t_0}$ \\
${\Gamma}^{\alpha}_{(t){\beta}{\gamma}}$ & components of the metric connections
${\nabla}_{t}$ compatible with ${\bf g}_{t}$\\
${\bar {\Gamma}}^{\alpha}_{(t){\beta}{\gamma}}$ & components of the metric 
connections ${\bar {\nabla}}_{t}$ compatible with ${\bf {\bar g}}_{t}$\\
${\tilde {\Gamma}}^{\alpha}_{(t){\beta}{\gamma}}$ & components of the metric 
connections ${\tilde {\nabla}}_{t}$ compatible with ${\tilde {\bf g}}_{t}$\\
${\topstar{\Gamma}}^{\alpha}_{t{\gamma}}$,
${\topstar{\Gamma}}^{\alpha}_{{\beta}{\gamma}}$ & components of the degenerate
5-dimensional connection ${\topstar{\nabla}}$ associated with the
family ${\bf g}_{t}$\\
${\topstar{\bar {\Gamma}}}^{\alpha}_{t{\gamma}}$,
${\topstar{\bar {\Gamma}}}^{\alpha}_{{\beta}{\gamma}}$ & components of the 
degenerate 5-dimensional connection ${\stackrel{\star}{\bf {\bar {\nabla}}}}$ 
associated with the family ${\bf {\bar g}}_{t}$ \\
${\bf a}_t$ or ${a}_{(t)}^{\mu}$ & family of metric 4-accelerations 
(any observer)\\
$()_{{\bar *}{\alpha}}$ & coordinate expression for degenerate covariant 
derivation compatible with ${\bf {\bar g}}_t$ \\
$()_{{\tilde *}{\alpha}}$ & coordinate expression for degenerate covariant 
derivation compatible with ${\bf {\tilde g}}_t$\\
$()_{{\mid}j}$ & coordinate expression for spatial covariant derivation 
compatible with ${\bf h}_t$ or ${\bf {\bar h}}_t$ as appropriate \\
$()_{;{\alpha}}$ & coordinate expression for metric covariant derivation 
compatible with a member of ${\bf g}_t$ or ${\bf {\bar g}}_t$ as appropriate \\
${\perp}$/${\bar {\perp}}$ & projection symbols (projection 
on the normal direction to the FHSs) \\
{\pounds}$_{\bf X}$ & Lie derivative with respect to {\bf X} in 
${\cal M}/{\overline{\cal M}}$ \\
${\topstar{\pounds}}_{\bf X}$ & Lie derivative with respect to {\bf X} in 
${\cal N}/{\overline{\cal N}}$ \\
${\cal L}_{\bf X}$ & projected Lie derivative with respect to {\bf X} 
in ${\cal M}/{\overline{\cal M}}$ \\
${\topstar{\cal L}}_{\bf X}$ & projected Lie derivative with respect to 
{\bf X} in ${\cal N}/{\overline{\cal N}}$ \\
${\bf a}_{\cal F}$/${\bf {\bar a}}_{\cal F}$ or 
$a^i_{\cal F}$/${\bar a}^i_{\cal F}$ & four-acceleration of the FOs
in $\cal M$/${\overline{\cal M}}$ \\
${\bf b}_{\cal F}$ or $b^i_{\cal F}$ & ``local distance vector from the
center of gravity'' generalized from the spherically symmetric case \\
${\bf {\bar e}}_b$/${\bf {\bar e}}^b$ or ${\bar e}_b^i$/${\bar e}^b_i$ & unit 
3-vector/covector field in the ${\bf b}_{\cal F}$-direction \\
${\bf w}_t$ or $w^i_{(t)}$ & coordinate 3-velocity of test particle with 
respect to the FOs \\
${\bf u}_t$ or $u^{\mu}_{(t)}$ & 4-velocity of test particle in $\cal N$ \\
${\bf {\bar u}}_t$ or ${\bar u}^{\mu}_{(t)}$ & 4-velocity of test particle 
in ${\overline{\cal N}}$ \\
${\bf v}_t$ or $v^{\mu}_{(t)}$ & 3-vector fields giving the motion of FOs
in ${\overline{\cal N}}$ compared to FOs in ${\cal N}$ 
\end{tabular}
\end{table}
\newpage
\begin{table}[t]
\begin{tabular}{lp{9.9cm}}
${\stackrel{\star}{\bf a}}$ or ${\topstar{a}}^{\mu}$ & degenerate 
4-acceleration in $\cal N$ (any observer) \\
$G_{(t){\mu}{\nu}}$/${\bar G}_{(t){\mu}{\nu}}$ & Einstein tensors in 
$\cal M$/${\overline{\cal M}}$ \\
$R_{(t){\mu}{\nu}}$/${\bar R}_{(t){\mu}{\nu}}$ & Ricci tensors in 
$\cal M$/${\overline{\cal M}}$ \\
$H_{(t)ij}$/${\bar H}_{(t)ij}$ & spatial Einstein tensors of the FHSs in
$\cal M$/${\overline{\cal M}}$ \\
$P_{(t)ij}$/${\bar P}_{(t)ij}$ & spatial Ricci tensors of the FHSs in
$\cal M$/${\overline{\cal M}}$ \\
$P_t$/${\bar P}_t$ & spatial Ricci scalars of the FHSs in
$\cal M$/${\overline{\cal M}}$ \\
${\bf K}_t$ or $K_{(t)ij}$ & family of extrinsic curvature tensors of the FHSs 
in $\cal M$ \\
${\bf {\bar K}}_t$ or ${\bar K}_{(t)ij}$ & family of extrinsic curvature 
tensors of the FHSs in ${\overline{\cal M}}$ \\
${\bf {\hat Z}}_t$ & the space-time tensor family ${\bf Z}_t$ projected into 
the FHSs \\
${\bar H}_t$/${\bar y}_t$ & local measures of the NKE \\
${\bar x}_t$ & local measure of the KE \\
$d{\tau}_t$/$\overline{d{\tau}}_t$ & proper time interval (any observer) in 
$\cal N$/${\overline{\cal N}}$ \\
$d{\tau}_{\cal F}$/$\overline{d{\tau}}_{\cal F}$ {\hspace*{2.3cm}} & proper 
time interval for a FO in $\cal N$/${\overline{\cal N}}$ \\
${\bf T}_t$ or $T^{{\mu}{\nu}}_{(t)}$ {\hspace*{2.3cm}} & 
stress-energy tensor as an active source of gravitation (in 
${\overline{\cal N}}$) \\
${\cal T}_t$ or ${\cal T}^{{\mu}{\nu}}_{(t)}$ {\hspace*{2.3cm}} & 
passive stress-energy tensor (in ${\cal N}$) \\
$G/{\kappa}$ {\hspace*{2.3cm}} & Newton/Einstein gravitational constant \\
$m_t/M_t$ {\hspace*{2.3cm}} & active masses (scalar fields measured 
dynamically) \\
${\bar F}_t{\equiv}{\bar N}_tct$ {\hspace*{2.3cm}} & local scale factors of 
the FHSs in ${\overline{\cal N}}$ \\
${\Psi}_t{\equiv}{\bar F}_t^{-1}$ {\hspace*{2.3cm}} & a scalar field 
describing how atomic time units vary in space-time \\
$-U$ {\hspace*{2.3cm}} & Newtonian potential \\
${\rho}_m$ {\hspace*{2.3cm}} & passive mass density in local rest frame of 
the source \\
$p$ {\hspace*{2.3cm}} & passive pressure \\
${\tilde {\rho}}_m$ {\hspace*{2.3cm}} & active mass density in local rest 
frame of the source \\
${\tilde p}$ {\hspace*{2.3cm}} & active pressure \\
${\bar {\rho}}_m$ {\hspace*{2.3cm}} & the amount of active mass per 
coordinate volume \\
${\bar p}$ {\hspace*{2.3cm}} & pressure corresponding to ${\bar {\rho}}_m$ \\
\vbox{\vspace{5.25cm}}
\end{tabular}
\end{table}
\clearpage
\hspace*{1cm}

\newpage

\setcounter{page}{5}

\section*{Prologue}

$\\ [2mm]$
Observationally the Hubble law in its most famous form may be found by 
measuring spectral shifts of ``nearby" galaxies and thereby inferring their 
motion; it relates the average recessional velocity $v{\ll}c$ of 
such galaxies to their distance $d$ via the equation
\eqa
v=Hd, {\nonumber}
\ena
where $H$ is the Hubble parameter. We see that the Hubble law does not depend 
on direction. This is merely a consequence of the fact that it is an empirical 
law; had the observations suggested the existence of anisotropic recessional 
velocities the Hubble law could still be formulated but in an anisotropic 
version. 

The question now is if this apparent lack of anisotropy may follow from some 
hitherto undiscovered physical principle rather than being a consequence of 
some rather special kinematical initial conditions. Once one suspects this it 
is natural to assume that the potential new fundamental law is local in nature.
That is, by inspecting the behaviour of the Hubble law for small distances $d$ 
one may get a hint of how it may be understood as a special case of a local 
law.

We see that the particular form of the Hubble law implies that recessional 
velocities may be made to vanish to arbitrary accuracy by considering
sufficiently small distances. This observation may be significant since there 
in fact exists a local geometrical quantity the effects of which have just the 
property that they may be made to vanish to arbitrary accuracy in a 
sufficiently small region. This geometrical quantity is {\em curvature}. Hence 
one may get the idea that a hypothetical local version of the Hubble law has 
something to do with curvature. Furthermore the natural way of relating the 
Hubble law to curvature is via the Hubble parameter. Since this is a scalar 
and not a tensor (as it would have to be in an anisotropic version of the 
Hubble law) one may suspect that the Hubble parameter in fact may be expressed
as some function of curvature scalars.

In standard cosmology the Hubble law applies only to a particular set of 
observers associated with the smeared-out motion of the galaxies. This 
suggests that a hypothetical local version of the Hubble law should also be 
associated with a privileged class of observers. 
At least parts of the interrelationship between nearby privileged 
observers should consist of an ``expansion". For reasons explained in the 
thesis this kind of expansion is called ``non-kinematical". Furthermore 
one may guess that the Hubble parameter should depend on curvature scalars 
calculated from the 3-dimensional space defined by the 
privileged observers taken at constant values of some privileged time 
coordinate. This privileged time coordinate then acts as a global notion of
``simultaneity'' and should be a basic element of the space-time geometry.

It should be clear from the above that any attempt to construct a local version
of the Hubble law and implementing this into a general geometrical structure 
in fact necessitates the construction of a new framework of space and time. We 
show in this thesis that this new framework disposes of the space-time metric 
as a global field. This means that the mathematical structure of the new 
framework is not based on semi-Riemannian geometry. Thus not all aspects of 
metric theory will hold in the new framework. On the other hand, important 
physical principles, previously thought to hold only in metric theory, also 
hold in the new framework. For this reason we call the new framework 
``quasi-metric". Since the quasi-metric framework must be relativistic one 
may always construct a {\em local} space-time metric on the tangent space of 
each event by identifying the local inertial frames with local Lorentz frames.
But rather than demanding that the set of local metrics constitutes a global 
space-time metric field we require the less stringent condition that the set 
of local metrics constitutes a semi-global metric field. That is, the domain 
of validity for the semi-global metric field is not the entire space-time 
manifold but rather a 3-dimensional submanifold defined by some constant value
of the privileged time coordinate. The quasi-metric space-time manifold may be 
thought of as consisting of a family of such submanifolds each of being 
equipped with a semi-global space-time metric field. The crucial fact is that 
the set of such semi-global space-time metric fields does not necessarily 
constitute a single global space-time (non-degenerate) metric field. The 
reason for this is that the affine connection compatible with the set of 
semi-global metric fields depends directly on the existence of a privileged 
time coordinate. That is, unlike the Levi-Civita connection the affine 
connection compatible with the set of semi-global metrics cannot in general 
be derived from any single space-time (non-degenerate) metric field.

In this thesis it is shown that the absence of a global Lorentzian metrical
field and the existence of a privileged time coordinate does not make it 
necessary to give up important physical principles such as e.g. the various 
versions of the principle of equivalence (but the strong principle of 
equivalence does not hold in its most stringent interpretation). On the other 
hand the generalization of non-gravitational physical laws to curved 
space-time is more complicated in the quasi-metric framework than in the 
metric framework. This should not be surprising since non-gravitational fields
may possibly couple to fields characterizing the quasi-metric geometrical
structure in a way not possible in metric geometry, but such that terms 
representing such couplings vanish in the local inertial frames. (That is, in 
the local inertial frames the non-gravitational physics take their standard 
special relativistic form just as in metric theory.) Thus, even if there is a 
preferred notion of space and time, this does not mean that there must exist
a preferred coordinate frame in the sense that the outcomes of local 
experiments (gravitational and non-gravitational) depend on the velocity with
respect to it. And this is easily justified since the geometric
structure of quasi-metric theory does not depend on the existence of any
particular coordinate frame. On the other hand there exists a {\em set} of 
``preferred'' coordinate systems especially well adapted to the geometric
structure of quasi-metric space-time. This is a consequence of the fact
that the preferred time coordinate represents an ``absolute'' geometrical 
element. However, the existence of non-dynamical fields is possible in metric 
theory also. Thus there should not be any {\em a priori} objections to 
constructing theories of gravity based on the quasi-metric framework. 
Consequently, in this thesis it is shown how to construct such a theory. This 
theory corresponds with General Relativity in particular situations and may 
possibly be viable. 

When it comes to comparison of testable models based on the quasi-metric 
theory to experiment, it still remains to develop a suitable weak field 
expansion analogous to the parameterized post-Newtonian expansion valid for 
the metric framework. Obviously this is a subject for further work. However, 
even if a suitable weak field expansion is missing one may still try to 
construct specific models for idealized situations. In the thesis this is done
for certain spherically symmetric systems and it is shown that the classical
solar system tests come in just as for General Relativity. But one important 
difference is that the quasi-metric theory predicts that the gravitational 
field of the solar system expands with the Universe whereas General Relativity
predicts no such thing. Moreover, gravitationally bound bodies made of ideal
gas are predicted to expand in this manner also. As discussed in chapter 3, 
chapter 4 and chapter 5 of the thesis, observational evidence for expanding
gravitational fields so far seems to favour the quasi-metric theory rather 
than General Relativity. Thus the observational evidence seems to indicate 
that the recessional velocities of galaxies are non-kinematical in nature and 
moreover that space is not flat. If the observational evidence holds up one 
must have in mind that the quasi-metric framework in general and the 
predictions of expanding gravitational fields and certain gravitationally 
bound bodies in particular, are based on a reinterpretation of the Hubble law.

It may be surprising that a reinterpretation of the simple empirical Hubble 
law leads to nothing less than the construction of a new geometrical framework
as the basis for relativistic physics. However, from a philosophical point of
view one should prefer a theoretical framework having the property that 
general empirical laws follow from first principles. Thus seen the existence 
of general empirical laws which do {\em not} follow from first principles may
be interpreted as a sign of incompleteness for any theoretical framework and 
may potentially lead to the demise of that framework.

Now that the reader has had a foretaste of what this thesis is all about,
it is useful to get a overview of the mathematical concepts used in the
thesis before starting on its main parts. 

\newpage


{\vspace*{5cm}}

\begin{center}
{\huge {\bf Preliminaries: \\\ \\Some Basic Mathematical Concepts}}
\end{center}

\newpage
{\hspace*{1cm}}

\newpage

\begin{center}
{\large {\bf Some Basic Mathematical Concepts}}
\end{center}
\begin{center}
by
\end{center}
\begin{center}
Dag {\O}stvang \\
{\em Institutt for Fysikk, Norges teknisk-naturvitenskapelige universitet, 
NTNU \\
N-7491 Trondheim, Norway}
\end{center}
\begin{abstract}
The canonical viewpoint of space-time is taken as fundamental by requiring the
existence of a global time function $t$ corresponding to a global foliation of 
space-time into a set of spatial hypersurfaces. Moreover it is required that
space-time can be foliated into a set of timelike curves corresponding to a
family of fundamental observers and that these two foliations are orthogonal to
each other. It is shown that this leads to a ``quasi-metric'' space-time 
geometry if $t$ is taken to represent one extra degenerate time dimension.
The quasi-metric framework then consists of a differentiable manifold equipped
with a one-parameter family of Lorentzian 4-metrics ${\bf g}_t$ parameterized 
by the global time function $t$, in addition to a non-metric connection 
compatible with the family ${\bf g}_t$.
\end{abstract}
\topmargin 0pt
\oddsidemargin 5mm
\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\section{Introduction}
In this thesis we construct a new relativistic theory of gravity. This theory
is compatible with a geometric framework that differs from the usual metric
framework. While there is nothing remarkable about the mathematical description
of the new ``quasi-metric'' framework (it is just ordinary differential 
geometry), the mathematics and its applications may seem unfamiliar to the 
traditional relativist who is used to working only with metric geometry. For 
this reason we give an intuitive introduction to the basic mathematical 
concepts of the quasi-metric geometry; the physical motivation for introducing
it will have to wait until the main part of the thesis.
\section{From 3-geometry to 4-geometry}
The basic premise defining metric space-time geometry is that space-time can be
described as a 4-dimensional semi-Riemannian manifold. In particular, in metric
geometry the affine geometry follows uniquely from the space-time metric
(barring torsion). Thus, since the affine geometry does not depend on any 
preferred time coordinate, the viewpoint is taken as fundamental, that there 
is no need to specify any particular splitting of space-time into space and 
time except for reasons of convenience. And certainly no such splitting can 
have anything to do with fundamental physics.

Yet one may wish to describe the geometry of space-time via the evolution with 
time of fields defined on a spatial hypersurface as in the ADM formalism [1] 
(this is called a canonical description of the space-time geometry). However, 
in metric theory the choice of said spatial hypersurface is arbitrary, and its 
deformation along some arbitrary timelike vector field is totally dependent on 
the exact form of the space-time metric tensor field ${\bf g}$. In particular 
this implies that in metric theory, one cannot impose general requirements 
that certain canonical structures should exist without simultaneously imposing 
restrictions on the form of ${\bf g}$. And from a physical point of view this 
is unacceptable since in metric theory, the affine structure follows directly 
from the space-time metric, so any restrictions on that will interfere with 
the dynamics of any potential theory of gravity.

Suppose then, that for some reason one wishes to take the canonical viewpoint 
of space-time as fundamental but such that space-time needs not be metric. 
Then there is no reason to believe that the requirement that certain canonical 
structures exist should restrict the affine geometry unduly. In particular,
one may require that a preferred time coordinate should exist. In this case 
the affine geometry does not necessarily depend critically on a space-time 
metric. We illustrate this in the next section. Meanwhile we describe the 
canonical structures that are required to exist within our quasi-metric 
framework.

Mathematically, to justify the view that a canonical description of space-time 
is fundamental, such a description must involve canonical structures that are 
taken as basic. Thus such structures must always exist, and they should in 
some way be ``simpler" than canonical structures in metric theory. 
One should regard these criteria as fulfilled if one requires that the 
space-time manifold $\cal N$ always can be foliated into a particular set of 
spatial fundamental hypersurfaces (FHSs) ${\cal S}_t$ parameterized by the 
global time function $t$. To ensure that the parameterization in terms of $t$
exists independent of any space-time metric we let $t$ represent one extra
degenerate time dimension. Besides we require that $\cal N$ can always be 
foliated into a family of timelike curves corresponding to a set of 
fundamental observers (FOs), and such that these two foliations are orthogonal
to each other, see figure 1. 

\begin{figure}
\begin{center}
\epsfig{file=nonmetric2.eps,height=8cm}
\caption{{\small Basic canonical structure: space-time accepting two 
mutually orthogonal foliations.}}
\end{center}
\end{figure}

If this is to make sense, there must be a space-time metric tensor available 
at each tangent space such that scalar products can be taken. To be sure that
such a space-time tensor does exist we require that on ${\cal S}_t$, there 
must exist a family of scalar fields $N_t$ and a family of intrinsic spatial 
metrics ${\bf h}_t$. Moreover there must exist a set of preferred coordinate 
systems where the time coordinate $x^0$ can be identified with the global time
function $t$. We call such a coordinate system a {\em global time coordinate 
system} (GTCS). The coordinate motion of the FOs with respect to a GTCS 
defines a family ${\bf N}_t$ of spatial covector fields tangential to the 
FHSs. Now $N_t$, ${\bf N}_t$ and ${\bf h}_t$ uniquely determine a family of 
Lorentzian space-time metrics ${\bf g}_t$. This metric family may be thought 
of as constructed from measurements available to the FOs if the functions 
$N_t$ are 
identified with the lapse functions describing the proper time elapsed as 
measured by the FOs, and if the covectors ${\bf N}_t$ are identified with the 
shift covector fields of the FOs in a GTCS. With the help of $N_t$ and 
${\bf N}_t$ we may then construct the family of normal vector fields 
${\bf n}_t$ being tangent vector fields to the world lines of the FOs and such
that ${\bf h}_t({\bf n}_t,{\cdot})=0$. Thus by construction the foliation of 
$\cal N$ into the set of world lines of the FOs is orthogonal to the foliation
of $\cal N$ into the set of FHSs, as asserted. To illustrate this we may write 
\eqa
{\bf g}_t=-{\bf g}_t({\bf n}_t,{\cdot}){\otimes}
{\bf g}_t({\bf n}_t,{\cdot})+{\bf h}_t.
\ena
Note that there is no reason to believe that any affine connection compatible
with the above construction must be metric. Thus the point is to illustrate 
that it may be possible to create a geometrical structure where a space-time 
metric tensor is available at every event but where the affine geometry does 
not necessarily follow directly from a space-time metric. The reason for this 
is that the existence of the global time function $t$ may be crucial when 
determining the affine connection; this possibility would not exist if the 
space-time geometry were required to be metric.
\section{Connection and curvature}
The existence of a Lorentzian space-time metric ${\bf g}{\in}{\bf g}_t$ for 
each FHS ${\cal S}{\in}{\cal S}_t$ implies that each FHS can be viewed as a 
spatial submanifold of a {\em metric} space-time manifold $\cal M$. We now
perform a mathematical trick inasmuch as whenever we take this viewpoint we
may define a global time coordinate $x^0$ on $\cal M$ such that $x^0=ct$ on 
each FHS; any coordinate system where this relationship holds is by definition
a GTCS. We may then regard $x^0$ and $t$ as representing separate time 
dimensions such that $t=$constant on $\cal M$. It is thus possible to view 
$\cal N$ as a 4-dimensional submanifold of ${\cal M}{\times}{\bf R}$. The 
point with this is that on the FHSs we may take scalar products of fields 
defined on space-time in $\cal M$ whereas time derivatives of such fields 
depend on $t$ as well as on $x^0$ in $\cal N$. Thus the existence of extra 
time derivatives with respect to $t$ of ${\bf g}_t$ means that we can find an
affine connection that is not fully determined by any single ${\bf g}{\in}
{\bf g}_t$. See figure 2 for visualization of quasi-metric space-time
geometry.


\begin{figure}
\begin{center}
\epsfig{file=Separation.eps,height=8cm}
\caption{The geometrical structure of quasi-metric space-time, visualized.}
\end{center}
\end{figure}

Since each ${\bf g}{\in}{\bf g}_t$ is defined on a spatial submanifold only, 
there is in general no way to extend it to all of $\cal N$. However, the 
Levi-Civita connection corresponding to each ${\bf g}{\in}{\bf g}_t$
can be calculated on each FHS by holding $t$ constant. But this family of
metric connections does not fully determine the connection compatible with
the family ${\bf g}_t$. To find the full connection compatible with the 
family ${\bf g}_t$ it is convenient to view ${\bf g}_t$ as a single 
5-dimensional degenerate metric on ${\cal M}{\times}{\bf R}$. One may then 
construct a linear, symmetric and torsion-free connection 
${\,}{\topstar{\nabla}}{\,}$ on ${\cal M}{\times}{\bf R}$ by requiring that 
this connection is compatible with the non-degenerate part of the metric 
family ${\bf g}_t$ in addition to the condition that
${\,}{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf n}_t$
vanishes. For obvious reasons this connection is called ``degenerate", and it 
should come as no surprise that it is non-metric. One may then restrict the 
degenerate connection to $\cal N$ by just considering the submanifold 
$x^0=ct$ in a GTCS. For the exact form of the connection coefficients 
determining ${\,}{\topstar{\nabla}}{\,}$, see e.g. [2]. 

At this point a natural question would be if there are physical phenomena 
which are naturally modelled by the non-metric part of quasi-metric geometry. 
And as we shall see in the main part of this thesis this is indeed the case.
\section{Two simple examples}
We finish this mathematical introduction by giving two simple examples. The
first example illustrates how a family of space-time metrics can be represented
by a family of line elements, whereas the second example illustrates how 
affinely parameterized curves are represented.
\\

{\em Example 1} \\
A family of Minkowski metrics can be represented by the family of line elements
\eqa
ds_t^2=-(dx^0)^2+{\Xi}^2(t){\Big(}dx^2+dy^2+dz^2{\Big)}, 
\ena
where ${\Xi}(t)$ is a scale factor, $x^0=ct$ and $x,y,z$ are ordinary 
Cartesian coordinates. Note that the $t$-direction is treated as degenerate. 
It is easy to show that the free fall curves obtained from the equations of 
motion (derived in chapter 1, section 5 of this thesis) and applied to (2) 
are different than their counterparts obtained by setting $dx^0=cdt$ in (2) 
and using the geodesic equation compatible with the resulting single metric.
\\

{\em Example 2} \\
Let ${\{}t(\lambda),x^{\alpha}(\lambda){\}}$ represent an affinely 
parameterized curve, where $x^{\alpha}$ are space-time coordinates in a
GTCS. Then the tangent vector field along the curve may be represented as
\eqa
{\frac{\partial}{{\partial}{\lambda}}}={\frac{dt}{d{\lambda}}}
{\frac{\partial}{{\partial}t}}+{\frac{dx^{\mu}}{d{\lambda}}}
{\frac{\partial}{{\partial}x^{\mu}}}.
\ena
Now the length of ${\frac{\partial}{{\partial}{\lambda}}}$ may be calculated
along the curve, e.g. for a timelike curve 
${\bf g}_t({\frac{\partial}{{\partial}{\lambda}}},
{\frac{\partial}{{\partial}{\lambda}}})<0$ along the curve. On the other hand
the degenerate connection taken along ${\frac{\partial}{{\partial}{\lambda}}}$ 
is
\eqa
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}{\lambda}}}}=
{\frac{dt}{d{\lambda}}}{\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}+
{\frac{dx^{\mu}}{d{\lambda}}}
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}x^{\mu}}}}.
\ena
Thus we have illustrated that the degenerate part of 
${\frac{\partial}{{\partial}{\lambda}}}$ does not contribute to its length
(since ${\bf g}_t({\frac{\partial}{{\partial}t},{\cdot}}){\equiv}0$).
However, as can be seen from equation (4), the degenerate part of
${\frac{\partial}{{\partial}{\lambda}}}$ does influence parallel-transport of 
space-time objects.
\\

Hopefully, anyone who has understood the basic mathematical concepts 
underlying the quasi-metric framework will not be so easily distracted by 
mathematical hurdles and will be in a better position to follow the derivation
of the main results presented in this thesis. Accordingly, having read this 
mathematical introduction, the reader should now be ready for the main parts 
of the thesis.
\\ [4mm]
{\bf References} \\ [1mm]
{\bf [1]} C.W. Misner, K.S. Thorne, J.A. Wheeler,
{\em Gravitation}, W.H. Freeman ${\&}$ Co. (1973). \\
{\bf [2]} D. {\O}stvang, chapter 1, section 2 of this thesis (2001). 

\newpage

{\vspace*{5cm}}

\begin{center}
{\huge {\bf Chapter One: \\\ \\ A Non-Metric Approach to
\vspace{0.5cm}\\Space, Time and Gravitation}}
\end{center}
\newpage
{\hspace*{1cm}}

\newpage

\setcounter{equation}{0}

\begin{center}
{\large {\bf A Non-Metric Approach to Space, Time and Gravitation}}
\end{center}
\begin{center}
by
\end{center}
\begin{center}
Dag {\O}stvang \\
{\em Institutt for fysikk, Norges teknisk-naturvitenskapelige universitet, 
NTNU \\ N-7491 Trondheim, Norway}
\end{center}

\begin{abstract}
This document surveys the basic concepts underlying a new geometrical framework
of space and time developed elsewhere [2]. Moreover it is shown how this
framework is used to construct a new relativistic theory of gravity.
\end{abstract}

\section*{1{\hspace*{9mm}} Introduction}
A proper description of Nature's fundamental forces involves their formulation 
as dynamical laws subject to appropriate initial conditions. This defines
the initial value problem for whatever dynamical system one wishes to study.

The initial value problem for gravitation generally involves what one may 
loosely call ``evolution of space with time". More precisely, given a 
one-parameter family of 3-dimensional manifolds ${\cal S}_t$ each of which is 
identified as ``space" at ``time" $t$ (such that $t=$constant on each 
${\cal S}{\in}{\cal S}_t$), the family defines an evolution of the 3-manifolds
${\cal S}_t$ with $t$. However, every conceivable theory of gravity
includes an overlying 4-dimensional geometrical structure called ``space-time"
with which the given family must be compatible. Thus it is not possible 
to analyze a given family ${\cal S}_t$ in isolation; the evolution with $t$ of 
fields that determine the relationship between the family and the overlying 
space-time must be analyzed simultaneously. When this is done one recovers the 
space-time geometry.

Naturally the structure of any space-time framework should be more general than
any particular theory of gravity formulated within that framework. This implies
that in general there should exist geometrical relations between the space-time
framework and any given family ${\cal S}_t$ that do not depend on the
detailed form of the dynamical laws one uses. A particular set of such 
relations are called {\em kinematical} relations. The kinematical relations of
different space-time frameworks are recognized by their sharing of a common 
feature; that they are not sufficient by themselves to determine solutions of 
the initial value problem. That is, in addition to the initial conditions one 
must supply extra fields coming from the dynamics at each step of the 
evolution of ${\cal S}_t$ if one wants to recover the space-time geometry. Thus
the kinematical relations explicitly exhibit the dependence on the existence of
some coupling between geometry and matter fields, even if the detailed nature
of the coupling is not specified. This is what one should expect, since any 
relations defined as kinematical always should give room for appropriate 
dynamics.

As an example we take the case of the stratified non-relativistic space-time
framework of Newton-Cartan [4]. Here there exists a unique family
${\cal S}_t$ consisting of Euclidean spaces (and left intact by Galilean
transformations), where $t$ is Newton's absolute 
time. The overlying geometrical structure belonging to this space-time is the
space-time covariant derivative ${\nabla}$ represented by the only non-zero
connection coefficients ${\Gamma}^{i}_{tt}{\equiv}{\Phi},_i$ (using a Cartesian
coordinate system ${\{}x^i{\}}$). Here ${\Phi}(t,x^i)$ is a potential function
and the comma denotes partial derivation. The equation of motion for inertial
test particles is the geodesic equation
\eqa
{\frac{d^2x^i}{dt^2}}+{\Gamma}^{i}_{tt}=0,
\ena
so to calculate the world lines of such test particles one must know
${\Gamma}^{i}_{tt}$ for every member of the family ${\cal S}_t$. Now we notice 
that there is a general relationship between the only non-zero components of 
the Riemann tensor and the potential function given by [4]
\eqa
R_{itjt}={\frac{{\partial}^2}{{\partial}x^i{\partial}x^j}}{\Phi}
={\frac{\partial}{{\partial}x^i}}{\Gamma}^{j}_{tt}.
\ena
That this relation is kinematical we see from the fact that to determine the 
${\Gamma}^{i}_{tt}$ for all times and thus all information relevant for the 
motion of inertial test particles, one must explicitly supply the tensor 
$R_{itjt}$ for each time step. Furthermore, equation (2) is valid
regardless of the detailed coupling between $R_{itjt}$ and matter fields.

As a second example we take the more realistic case of a metric relativistic 
space-time. In this case the choice of family ${\cal S}_t$ is not unique, 
so the kinematical relations should be valid for any such choice. Furthermore
the overlying geometrical structure belonging to space-time is the space-time
metric field, and this field induces relations between different choices of
${\cal S}_t$. Analogous to the first example the kinematical relations are
given by (using a general spatial coordinate system and applying Einstein's 
summation convention)
\eqa
R_{i{\perp}j{\perp}}=N^{-1}{\cal L}_{N{\bf n}}K_{ij} + {K_i}^kK_{kj}
+c^{-2}a_{i{\mid}j}+c^{-4}a_ia_j,
\ena
see e.g. [1] for a derivation. Here {\bf a} is the 4-acceleration field of 
observers moving orthogonally to each ${\cal S}{\in}{\cal S}_t$, $N$ is their
associated lapse function field and `$\perp$' denotes projection onto the 
normal direction of ${\cal S}_t$. Furthermore ${\cal L}_{\bf n}$ denotes the 
projected Lie derivative in the normal direction and `${\mid}$' denotes 
spatial covariant derivation intrinsic to each ${\cal S}{\in}{\cal S}_t$. The 
extrinsic curvature tensor {\bf K} describes the curvature of each 
${\cal S}{\in}{\cal S}_t$ relative to the overlying space-time curvature. 
There exists a well-known relationship between {\bf K} and the evolution in 
the normal direction of the intrinsic geometry of each 
${\cal S}{\in}{\cal S}_t$, namely (see e.g. [1], [4])
\eqa
K_{ij}=-{\frac{1}{2}}{\pounds}_{\bf n}h_{ij},
\ena
where {\pounds}$_{\bf n}$ denotes Lie derivation in the normal direction and
${\bf h}$ denotes the intrinsic metric of each ${\cal S}{\in}{\cal S}_t$.
Another, geometrically equivalent expression of the kinematical relations (3) 
is found (see e.g. [1]) by projecting the Einstein space-time tensor field 
${\bf G}$ into respective normal to the spatial hypersurfaces ${\cal S}_t$. 
The result is ($K{\equiv}{K^k}_k$)
\eqa
G_{{\perp}{\perp}}&=&{\frac{1}{2}}(P+K^2-K_{kl}K^{kl}), {\nonumber} \\
G_{{\perp}j}&=&(K^k_j-K{\delta}^k_j)_{{\mid}k}, {\nonumber} \\
G_{ij}&=&-N^{-1}{\cal L}_{N{\bf n}}(K_{ij}-Kh_{ij})-2{K_i}^kK_{kj}+3KK_{ij}
-{\frac{1}{2}}(K_{kl}K^{kl}+K^2)h_{ij} {\nonumber} \\
&-&c^{-2}(a_{i{\mid}j}-{a^k}_{{\mid}k}h_{ij})-c^{-4}(a_ia_j-a^ka_kh_{ij})
+H_{ij},
\ena
where ${\bf H}$ and $P$ are respectively the Einstein tensor and the Ricci 
scalar intrinsic to the hypersurfaces. The equation (4) in combination with 
equation (3) (or equivalently the totally spatial projection $G_{ij}$ shown in 
(5), the other projections representing constraints on initial data) 
may be recognized as kinematical equations. This follows from the fact that in
addition to initial values of $h_{ij}$ and $K_{ij}$ at an initial hypersurface,
one needs to specify the tensor field $G_{ij}$ or equivalently the tensor 
field $R_{i{\perp}j{\perp}}$ at each step of the evolution of ${\cal S}_t$ if 
one wishes to recover the space-time metric field and thus all information 
about the geometry of space-time as predicted at subsequent hypersurfaces.
Notice that this does not depend on Einstein's equations being valid (see e.g. 
[1] for a further discussion).

In light of the two above examples we may call the evolution represented by a 
family ${\cal S}_t$ a {\em kinematical evolution} (KE) if it depends on the
supplement of extra fields coming from an overlying geometrical structure at 
each step of the evolution. The question now is if this type of evolution
exhausts all possibilities. The reason for this concern is that one may 
possibly imagine a type of evolution of a spatial hypersurface that does 
{\em not} depend on the supplement of extra fields at each step of the 
evolution. It would be natural to call such a kind of evolution 
{\em non-kinematical} (NKE). Since by construction the NKE does not give room 
for explicit dynamics, any geometrical framework having the capacity of 
accommodating both KE and NKE must be different from the standard metric 
framework. 

As illustrated by the above examples we see from equations (2) and (3)
that the extra fields one must supply at each time step to get the KE working,
take the form of curvature tensors. Thus we may expect that to get the NKE 
working, it too must have something to do with curvature. However this 
curvature must somehow be available without referring to an overlying 
geometrical structure. The only possibility of realizing this is if the wanted 
curvature is obtainable as an intrinsic property of each member ${\cal S}$ of 
a {\em unique} family ${\cal S}_t$. Since we want our alternative space-time 
framework to be relativistic, $t$ must represent a ``preferred'' 
time coordinate and any member of the unique family ${\cal S}_t$ must
represent a ``preferred'' notion of space. Observers moving normally 
to any ${\cal S}{\in}{\cal S}_t$ get the status of preferred observers. 
However, the existence of the unique family ${\cal S}_t$ does {\em not} imply 
that there exists a preferred coordinate frame in the sense that the outcomes 
of local experiments depend on it [2]. On the other hand it will exist a
{\em class} of ``preferred'' coordinate systems especially well adapted to the
geometry of space-time. One may expect that equations take special forms
in any such a coordinate system.

Now to the questions; are there physical phenomena which may be suspected to 
have something to do with NKE and if so, do we get any hint of which form it 
should take? The fact that our Universe seems to be compatible with a 
preferred time coordinate at large scales makes us turn to cosmology in an 
attempt to answer these questions. With the possibility of a rather dramatic 
reinterpretation of the Hubble law in mind we guess that the NKE should take 
the form of a local increase in scale with time as measured by the above
defined preferred observers. Moreover, analogous to the coupling between
the KE and curvature as shown in the above examples, there should exist a
connection between the NKE and some aspects of the intrinsic geometry of each 
member ${\cal S}$ of the preferred family ${\cal S}_t$. That is, we guess that
the Hubble law has nothing to do with kinematics; rather the Hubble law should
be the basic empirical consequence of the NKE associated with the global 
curvature of space.

The above guess represents the key to finding a new framework of space and time
which incorporates both KE and NKE in a natural way. We call this new
framework ``quasi-metric'' since it is not based on semi-Riemannian geometry
and yet the new framework is compatible with important physical 
principles previously thought to hold in metric theory only. In this thesis we
explore some of the possibilities of the quasi-metric framework; among other 
things we derive a new theory of gravity and show how this theory helps to 
define the exact relationship between curvature and the NKE. The thesis is 
organized as follows: In chapter 1 we define the quasi-metric framework and 
list the important results without going through all the gory details while 
chapters 2, 3, 4 and 5 are self-contained articles containing the detailed 
calculations underlying the results listed in chapter 1.
\section*{2{\hspace*{9mm}} An alternative framework of space and time}
Our first problem is to arrive at a general framework describing space and time
in a way compatible with both KE and NKE, as discussed in the introduction.
Since at least parts of the KE of any member ${\cal S}$ of the preferred family
${\cal S}_t$ should be described by a Lorentzian space-time metric ${\bf g}$ 
we may assume that the KE represented by ${\bf g}$ is discernible at each 
time step. That is, at least in a (infinitesimally) small time interval 
centered at the hypersurface $t=$constant it should be possible to evolve this 
hypersurface purely kinematically in the hypersurface-orthogonal direction 
with the help of the 4-metric ${\bf g}$ without explicitly taking the NKE into 
account. But since this should be valid for every ${\cal S}{\in}{\cal S}_t$, 
to be compatible with the NKE different ${\cal S}{\in}{\cal S}_t$ should be 
associated with different metrics ${\bf g}{\in}{\bf g}_t$. Thus the 
preferred family ${\cal S}_t$ of 3-dimensional spatial manifolds should define 
a one-parameter family of 4-dimensional Lorentzian metrics ${\bf g}_t$, the 
domain of validity of each being exactly the hypersurface $t=$constant, but 
such that one may extrapolate each metric at least in a (infinitesimally) 
small interval centered at this hypersurface.

We may think of the parameter $t$ as representing an extra time dimension,
since fixing $t$ is equivalent to fixing a space-time metric 
${\bf g}{\in}{\bf g}_t$ on (at least part of) a space-time 4-manifold. This 
indicates that the mathematically precise definition of the quasi-metric 
framework should involve a 5-dimensional product manifold 
${\cal M}{\times}{\bf R}$ where ${\cal M}$ is a Lorentzian space-time manifold 
and ${\bf R}$ is the real line. We may then interpret $t$ as a global 
coordinate on ${\bf R}$ and the preferred family ${\cal S}_t$ as representing a
foliation into spatial 3-manifolds of a 4-dimensional submanifold ${\cal N}$ of
${\cal M}{\times}{\bf R}$. ${\cal N}$ is thus by definition a 4-dimensional
space-time manifold equipped with a one-parameter family of Lorentzian metrics
${\bf g}_t$ in terms of the global time function $t$. This is the mathematical
definition of the quasi-metric space-time framework we want to use as the 
basis for constructing a new relativistic theory of gravity.

Henceforth we refer to the members of ${\cal S}_t$ as the {\em fundamental
hypersurfaces} (FHSs). Observers always moving orthogonally to the FHSs are 
called {\em fundamental observers} (FOs). Note that the FOs are in general
non-inertial. When doing calculations it is often necessary to define useful 
coordinate systems. A particular class of coordinate systems especially well 
adapted to the existence of the FHSs is the set of {\em global time coordinate 
systems} (GTCSs). A coordinate system ${\{}x^{\mu}{\}}$ is a GTCS if and only
if the relationship in $\cal N$ between the time coordinate $x^0$ and the 
global time function $t$ is explicitly given by the equation $x^0=ct$. Since 
the local hypersurface-orthogonal direction in $\cal M$ by definition is 
physically equivalent to the direction along the world line of the local FO in
$\cal N$, the one-parameter family of metrics is most conveniently expressed 
in a GTCS. That is, in a GTCS the spatial line elements of the FHSs evolve 
along the world lines of the FOs in $\cal N$; there is no need to worry about 
the explicit evolution of the spatial geometry along alternative world lines. 
Note that there exist infinitely many GTCSs due to the particular
structure of quasi-metric space-time but that in general it is not possible to
find one where the FOs are at rest. However, in particular cases further
simplification is possible if we can find a {\em comoving} coordinate system. 
A comoving coordinate system is by definition a coordinate system where the 
spatial coordinates are constants along the world lines of the FOs. A comoving
coordinate system which is also a GTCS we conventionally call a 
{\em hypersurface-orthogonal coordinate system} (HOCS). Expressed in a HOCS 
the 3-vector piece $g_{0i}$ of each 4-metric vanishes on its domain FHS. Thus 
the proper time along the world lines of the FOs in $\cal N$ may be found 
directly from the scalar piece $g_{00}$ of each 4-metric in $\cal M$ if 
each 4-metric is expressed in a HOCS. But in general it is not possible to 
find a HOCS; that is possible for particular cases only.

The condition that the global time function should exist everywhere implies 
that in ${\cal M}$, each FHS must be a Cauchy hypersurface for every allowable
metric ${\bf g}{\in}{\bf g}_t$. Furthermore each FHS must be a member of a 
family of hypersurfaces foliating ${\cal M}$ equipped with any allowable 
metric ${\bf g}{\in}{\bf g}_t$ when this metric is extrapolated off the FHS. 
Note that the hypothetical observers moving orthogonal to this family of 
hypersurfaces may be thought of as FOs with the explicit NKE ``turned off" and
that the normal curves of these hypersurfaces are in general not geodesics of 
${\bf g}{\in}{\bf g}_t$. 

Actually it may not be possible to eliminate all the effects of the NKE just 
by holding $t$ constant. The reason for this is the possibility that some 
of the NKE is not ``realized''. That is, it is possible that the
FOs move in a way such that some of the ``expansion" representing the NKE is 
exactly cancelled out. In other words, the FOs may be thought of as 
``falling'' with a 3-velocity $-{\bf v}_t$ and simultaneously expanding with 
a 3-velocity ${\bf v}_t$ due to some of the NKE, the net result being 
``stationariness''. But this type of stationariness is not equivalent to being
stationary with no NKE; for example one typically gets time dilation factors 
associated with the extra motion. Thus some non-kinematical features may be 
disguised as kinematical features belonging to the metric structure of each 
member of the family ${\bf g}_t$. This represents a problem since the family 
${\bf g}_t$ should be found from appropriate field equations. But due to the 
implicit effects of the NKE, ${\bf g}_t$ cannot represent a {\em solution} of 
these field equations. To overcome this problem we introduce another metric 
family ${\bf {\bar g}}_t$ on a corresponding product manifold 
${\overline{\cal M}}{\times}{\bf R}$ and consider a 4-dimensional submanifold 
${\overline{\cal N}}$ just as for the family ${\bf g}_t$. The crucial 
difference is that the family ${\bf {\bar g}}_t$ by definition represents a 
solution of appropriate field equations. Thus when we construct such field 
equations we will study the general properties of ${\bf {\bar g}}_t$ rather 
than those of ${\bf g}_t$. Of course it must be possible to construct 
${\bf g}_t$ from ${\bf {\bar g}}_t$ and the vector field family ${\bf v}_t$, 
see [2] for more details.

The family of hypersurfaces foliating ${\cal M}$ (or ${\overline{\cal M}}$) 
equipped with any allowable metric ${\bf g}{\in}{\bf g}_t$ (or ${\bf {\bar g}}
{\in}{\bf {\bar g}}_t$) defines a one-parameter family of 3-manifolds in terms 
of a suitably defined time coordinate function $x^0$. A sufficient condition 
to ensure the existence and uniqueness of the global time coordinate function 
$x^0$ on ${\cal M}$ (or ${\overline{\cal M}}$) such that the gradient field of
this coordinate is everywhere timelike, is to demand that each FHS is 
{\em compact} (without boundaries) [3]. Furthermore, since the NKE of the FHSs
by hypothesis should depend on aspects of their intrinsic curvature, it seems
reasonable that the requirement of compact FHSs implies some sort of prior
3-geometry on the FHSs. That is, we guess that any global curvature associated
with the compactness of the FHSs should not depend on the dynamics of
${\bf {\bar g}}_t$. Thus, in order to construct gravitational field equations 
we must investigate which restrictions should be laid on ${\bf {\bar g}}_t$ 
to ensure that any prior 3-geometry associated with the compact FHSs does not 
interfere with the dynamics of ${\bf {\bar g}}_t$. It turns out that a
sufficient condition for this is that the dynamics of ${\bf {\bar g}}_t$ is
represented by one scalar field. In fact, this condition means that in a GTCS 
we may write the general form of the family ${\bf {\bar g}}_t$ on the form [2]
\eqa
{\overline{ds}}^2_t={\Big [}{\bar N}_s(x^{\mu},t){\bar N}^s(x^{\mu})
-{\bar N}_t^2(x^{\mu},t){\Big ]}(dx^0)^2+2{\frac{t}{t_0}}{\bar N}_idx^idx^0
+{\frac{t^2}{t_0^2}}{\bar N}_t^2S_{ik}(x^{\mu})dx^idx^k,
\ena
where $S_{ik}dx^i dx^k$ is the metric of the 3-sphere (with radius $ct_0$)
and $t_0$ is an arbitrary epoch setting the scale of the spatial coordinates.
Furthermore ${\bar N}_t$ is the lapse function family of the FOs in
${\bf {\bar g}}_t$ and ${\frac{t}{t_0}}{\bar N}_idx^i$ is the family of shift 
covector fields of the FOs in ${\bf {\bar g}}_t$.

Compared to the metric framework, the restriction that each metric must be 
of the particular form compatible with equation (6) represents a
reduction in the possible explicit dynamical degrees of freedom for any theory
of gravity formulated within the quasi-metric framework. Besides, within the 
metric framework any such restriction automatically implies the existence
of ``prior geometry", i.e., any dynamically relevant aspect of the geometry 
that cannot be changed by changing gravitating source terms. However in the 
quasi-metric framework the form (6) of ${\bf {\bar g}}_t$ merely indicates 
that an explicit dynamical degree of freedom has been abandoned in order to 
accommodate the NKE and that this is valid for all theories of gravity 
consistent with the quasi-metric framework. This is realized by the fact that 
in quasi-metric theory there exists a ``basic" 3-geometry of 
${\overline{\cal N}}$ in the sense that the geometry 
${\bf {\bar h}}{\in}{\bf {\bar h}}_t$ of each FHS is conformal to this basic 
3-geometry but such that this does not interfere with the dynamically relevant 
aspects of the {\em space-time} geometry. Furthermore we may expect that the
abandoned dynamical degree of freedom will be put back implicitly via the NKE
when constructing ${\bf g}_t$ from ${\bf {\bar g}}_t$. It turns out that this 
will work and that we get a consistent theory if we choose the basic 
3-geometry to be ${\bf S}^3$, as we have done in equation (6).

Next an obvious question is which sort of affine structure is induced on
${\cal N}$ by the family of metrics ${\bf g}_t$ (and analogously which affine
structure is induced on ${\overline{\cal N}}$ by ${\bf {\bar g}}_t$). To 
answer this question it is convenient to perceive ${\bf g}_t$ as a single 
degenerate 5-dimensional metric on ${\cal M}{\times}{\bf R}$ and construct a 
torsion-free connection ${\ }{\topstar{\nabla}}{\ }$ (almost) compatible with 
this degenerate metric. But in addition to metric-compatibility we should also
require that the unit normal vector field family of the FHSs does not change 
when it is parallel-transported in the $t$-direction. That is, we should 
require that ${\,}{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf n}_t$
vanishes. When a connection satisfying these requirements is found we can
restrict it to ${\cal N}$ and we have the wanted affine structure. 

In order to construct ${\,}{\,}{\topstar{\nabla}}{\,}{\,}$ on 
${\cal M}{\times}{\bf R}$, the metric-preserving condition
\eqa
{\frac{\partial}{{\partial}t}}{\bf g}_t({\bf y}_t,{\bf z}_t)=
{\bf g}_t({\,}{\,}{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}
{\bf y}_t,{\bf z}_t)+
{\bf g}_t({\bf y}_t,{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}
{\bf z}_t),
\ena
involving arbitrary families of vector fields ${\bf y}_t$ and ${\bf z}_t$
in ${\cal M}$, may be used to find a candidate connection where the connection
coefficients are determined from ${\bf g}_t$ alone. It turns out that such a
candidate connection is unique and and that its difference from the usual
Levi-Civita connection is determined from the only non-zero connection
coefficients containing $t$. These are given by 
${\topstar{\Gamma}}^{\alpha}_{{\mu}t}$ and are equal to
${\frac{1}{2}}g_{(t)}^{{\alpha}{\sigma}}{\frac{\partial}{{\partial}t}}
g_{(t){\mu}{\sigma}}$. (Here we have introduced the coordinate notation 
$g_{(t){\mu}{\sigma}}$ for the family ${\bf g}_t$.) But since the explicit
$t$-dependence of ${\bf g}_t$ is equal to that of ${\bf {\bar g}}_t$ shown
in equation (6) (by construction $N$ does not depend on $t$), this candidate 
connection does in general not satisfy the criterion 
${\,}{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf n}_t=0$. But for 
the particular case when the shift vector field 
$N^i{\frac{\partial}{{\partial}x^i}}$ vanishes it does. We may thus choose
the connection on the grounds that it should reproduce the results of the
above-mentioned candidate connection for the case when $N^i$ vanishes.
We then find the unique connection coefficients
\eqa
{\topstar{\Gamma}}^{\alpha}_{{\mu}t}={\frac{1}{t}}{\delta}_i^{\alpha}
{\delta}^i_{\mu},
{\qquad} {\topstar{\Gamma}}^{\alpha}_{t{\mu}}{\equiv}{\ }
{\topstar{\Gamma}}^{\alpha}_{{\mu}t},
\ena
valid in a GTCS. All the other connection coefficients than (8) are 
found by requiring that they should be identical to those found from the 
family of Levi-Civita connections explicitly defined by the ${\bf g}_t$.
It may be readily shown that equation (8), together with an explicit
$t$-dependence of ${\bf g}_t$ similar to that of ${\bf {\bar g}}_t$ shown in 
equation (6) yield
\eqa
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf g}_t=0, \qquad
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf n}_t=0, \qquad
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf h}_t=0,
\ena
thus the connection ${\,}{\topstar{\nabla}}{\,}$ has the desired properties.
The restriction of ${\,}{\,}{\topstar{\nabla}}{\,}{\,}$ to $\cal N$ is trivial 
inasmuch as the same formulae are valid in $\cal N$ as in 
${\cal M}{\times}{\bf R}$, the only difference being the restriction 
$x^0=ct$ in a GTCS.

Now we want to use the above defined affine structure on ${\cal N}$ to find 
equations of motion for test particles. Let ${\lambda}$ be an affine parameter 
along the world line of an arbitrary test particle. (In addition 
to the affine parameter $\lambda$, $t$ is also a (non-affine) parameter along 
any non-spacelike curve in $\cal N$.) If we define coordinate 
vector fields ${\frac{\partial}{{\partial}x^{\alpha}}}$,
${\frac{dt}{d{\lambda}}}{\frac{\partial}{{\partial}t}}+
{\frac{dx^{\alpha}}{d{\lambda}}}{\frac{\partial}{{\partial}x^{\alpha}}}$ is 
the coordinate representation of the tangent vector field 
${\frac{\partial}{{\partial}{\lambda}}}$ along the curve. We then define the 
covariant derivative along the curve as
\eqa
{\topstar{\nabla}}_{\frac{\partial}{{\partial}{\lambda}}}{\equiv}
{\frac{dt}{d{\lambda}}}{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}
+{\frac{dx^{\alpha}}{d{\lambda}}}
{\topstar{\nabla}}_{\frac{\partial}{{\partial}x^{\alpha}}}.
\ena
A particularly important family of vector fields is the 4-velocity tangent
vector field family ${\bf u}_t$ of a curve. That is, by definition we have
\eqa
{\bf u}_t{\equiv}u_{(t)}^{\alpha}{\frac{\partial}{{\partial}x^{\alpha}}}
{\equiv}{\frac{dx^{\alpha}}{d{\tau}_t}}
{\frac{\partial}{{\partial}x^{\alpha}}},
\ena
where ${\tau}_t$ is the proper time as measured along the curve. 

The equations of motion are found by calculating the covariant derivative of 
4-velocity tangent vectors along themselves using the connection in $\cal N$. 
According to the above, this is equivalent to calculating 
${\,}{\,}{\topstar{\nabla}}{\,}{\,}{\bf u}_t$ along 
${\frac{\partial}{{\partial}{\tau}_t}}$. Using the coordinate representation 
of ${\frac{\partial}{{\partial}{\tau}_t}}$ in an arbitrary coordinate system 
we may thus define the vector field ${\stackrel{\star}{\bf a}}$ by
\eqa
{\stackrel{\star}{\bf a}}{\equiv}{\stackrel{\star}{\nabla}}_
{\frac{\partial}{{\partial}{\tau}_t}}{\bf u}_t={\Big (}
{\frac{dt}{d{\tau}_t}}{\stackrel{\star}{\nabla}}_{\frac{\partial}{{\partial}t}}
+{\frac{dx^{\alpha}}{d{\tau}_t}}
{\nabla}_{\frac{\partial}{{\partial}x^{\alpha}}}{\Big )}{\bf u}_t{\equiv}
{\frac{dt}{d{\tau}_t}}{\stackrel{\star}{\nabla}}_{\frac{\partial}{{\partial}t}}
{\bf u}_t+{\bf a}_t.
\ena
We call this vector field the ``degenerate" 4-acceleration. It may be shown 
that ${\stackrel{\star}{\bf a}}$ is orthogonal to ${\bf u}_t$. 

The coordinate expression for ${\stackrel{\star}{\bf a}}$ shows that this 
yields equations of motion, namely
\eqa
{\frac{d^2x^{\alpha}}{d{\lambda}^2}}+{\Big(}
{\topstar{\Gamma}}^{\alpha}_{t{\sigma}}{\frac{dt}{d{\lambda}}}+
{\Gamma}^{\alpha}_{(t){\beta}{\sigma}}{\frac{dx^{\beta}}{d{\lambda}}}{\Big)}
{\frac{dx^{\sigma}}{d{\lambda}}}=
{\Big(}{\frac{d{\tau}_t}{d{\lambda}}}{\Big)}^2{\stackrel{\star}{a}}^{\alpha}.
\ena
To get these equations to work the degenerate 4-acceleration must be specified 
independently for each time step; thus they are kinematical equations.
But ${\stackrel{\star}{\bf a}}$ cannot be chosen freely.
That is, given an initial tangent vector of the world line of an arbitrary 
test particle, as we can see from equation (12) ${\stackrel{\star}{\bf a}}$ 
is uniquely determined by ${\bf a}_t$ and ${\frac{dt}{d{\tau}_t}}
{\stackrel{\star}{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf u}_t$.
However, in a subsequent section we will see that 
${\stackrel{\star}{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf u}_t$ vanishes 
so we in fact have ${\stackrel{\star}{\bf a}}$=${\bf a}_t$.

Now ${\bf a}_t$ has the usual interpretation as
an expression for the inertial forces experienced by the test particle. The 
standard coordinate expression for ${\bf a}_t$ is
\eqa
{\frac{d^2x^{\alpha}}{d{\lambda}^2}}+{\Gamma}^{\alpha}_{(t){\sigma}{\rho}}
{\frac{dx^{\rho}}{d{\lambda}}}{\frac{dx^{\sigma}}{d{\lambda}}}{\equiv}
{\Big(}{\frac{d{\tau}_t}{d{\lambda}}}{\Big)}^2a^{\alpha}_{(t)}.
\ena
However, to find the curves of test particles equations (13) will be 
solved rather than equations (14); the latter are insufficient for this task
since they do not take into account the dependence of ${\bf u}_t$ on $t$, 
i.e., they hold only in $\cal M$.
\section*{3{\hspace*{9mm}} The role of gravity}
We are now able to set up the postulates which must be satisfied for every
theory of gravity compatible with the quasi-metric framework. These are:
(postulates for the metric framework [5] are stated in parenthesis for 
comparison)
\begin{itemize}
{\item Space-time is equipped with a metric family ${\bf g}_t$ as described
in the previous section, (metric theory: space-time is equipped with one single
Lorentzian metric),}
{\item Inertial test particles follow curves for which ${\bf a}_t=0$ 
calculated from (13), (metric theory: the curves are geodesics calculated from
(14)),}
{\item In the local inertial frames of ${\bf g}_t$ the non-gravitational 
physics is as in special relativity. (Metric theory: in the local Lorentz 
frames of the metric the non-gravitational physics is as in special 
relativity.)}
\end{itemize}
The third postulate means that we may always find a local coordinate system
such that the connection coefficients in equation (13) vanish. Later we will 
see that in fact ${\stackrel{\star}{\bf a}}$=${\bf a}_t$, meaning that 
inertial test particles follow geodesics of ${\,}{\topstar{\nabla}}{\,}$, see
equations (48), (49) and the subsequent discussion for verification. Thus 
the local inertial frames of ${\bf g}_t$ can be identified with local Lorentz 
frames, just as in metric theory. This means that there can be no local 
consequences of curved space-time on non-gravitational physics. But it is 
important to note that this applies strictly locally. That is, at any finite 
scale the geodesics of ${\,}{\topstar{\nabla}}{\,}$ do not coincide with the 
geodesics of any single space-time metric as long as there is a dependence on 
$t$. This means that the non-gravitational physics at any finite scale may vary
in a way incompatible with any metric geometry. Despite this any allowable 
theory does possess local Lorentz invariance (LLI), and furthermore
it follows that we do have local positional invariance (LPI), but that
deviations from metric theory should be detectable at any finite scale. (LPI 
is by definition that the outcome of any local nongravitational test 
experiment is independent of where or when it is performed.) Note that
quasi-local deviations from LLI and LPI typically occur explicitly
in differential laws such as conservation laws. That is, such laws may be 
different from their counterparts in metric theory since non-gravitational
fields may couple to fields characterizing the FHSs in such a way
that these couplings vanish in the local inertial frames.

The second postulate ensures that the WEP is valid (i.e., that the free fall
curves traced out by test particles are independent of the particles'
internal structure). The sum of LLI, LPI and the WEP is called the Einstein 
equivalence principle (EEP) [5] and is satisfied by any metric theory. 
Note that the EEP was previously thought to hold for metric theories only. 
Nevertheless, despite the differences in mathematical structure, the EEP 
should also hold for the class of quasi-metric theories according to the 
arguments made above. 

Now we want to construct a theory of gravity satisfying the above postulates. 
This particular theory should contain no extra independent gravitational 
fields other than ${\bf {\bar g}}_t$ and also no prior 4-geometry. Since the 
quasi-metric framework fulfills LLI and LPI we might hope to construct a 
theory which embodies the strong equivalence principle (SEP) (that the EEP is 
valid for self-gravitating bodies and local gravitational test experiments as 
well as for test particles). But as we shall see, the quasi-metric framework 
predicts scale variations in space-time between non-gravitational and
gravitational systems, and this is not compatible with the SEP in its most
stringent interpretation. So we cannot ask for more than that the 
gravitational weak equivalence principle (GWEP) is satisfied (that the WEP is 
valid for self-gravitating bodies as well as for test particles). However, if 
the GWEP is not violated, this is sufficient to eliminate the most serious 
effects resulting from the violation of the SEP.

One distinct feature of our framework is the existence of a unique set of 
FHSs. This means that we can split up the metrics
${\bf {\bar g}}_t$ into space and time in a preferred way; i.e., we have a
preferred family of unit normal vector fields ${\bf {\bar n}}_t$. In addition 
we have the 3-metrics ${\bf {\bar h}}_t$ intrinsic to the FHSs. We may 
then define a family of 4-acceleration fields ${\bf {\bar a}}_{\cal F}$ by
\eqa 
c^{-2}{\bar a}_{\cal F}^{\mu} {\equiv}  
{\bar n}_{(t);{\nu}}^{\mu}{\bar n}_{(t)}^{\nu}, 
\ena
where `$;$' denotes covariant derivation defined by the metric connection 
${\bf {\bar {\nabla}}}$ compatible with any single metric
${\bf {\bar g}}{\in}{\bf {\bar g}}_t$.

It is useful to denote a scalar product of any space-time object with
$-{\bf {\bar n}}_t$ by the symbol `${\bar {\perp}}$'; this operation defines a 
normal projection. Analogously any projection into a FHS may be performed by 
taking scalar products with ${\bf {\bar h}}_t$. To avoid confusion about 
whether we are dealing with a space-time object or a space object intrinsic to
the FHSs we will label space objects with a ``hat'' if necessary. That is, the
space objects ${\bf {\hat A}}_t$ are the result of a maximal number of spatial
projections of the space-time object ${\bf A}_t$ with ${\bf {\bar h}}_t$. 

By repeated projections with $-{\bf {\bar n}}_t$ or ${\bf {\bar h}}_t$ any 
space-time object eventually reduces to a space object. For example the 
space-time vector field ${\bf {\bar a}}_{\cal F}$ has the normal projection 
${\bar a}_{{\cal F}{\bar {\perp}}}{\equiv}-
{\bar a}^{\alpha}_{\cal F}{\bar n}_{(t){\alpha}}$ and the components of the 
horizontal projection are just ${\bar a}_{{\cal F}}^j$. In particular one may 
easily show that
\eqa
{\bar a}_{{\cal F}{\bar {\perp}}}=0, {\qquad} 
c^{-2}{\bar a}_{{\cal F}j}={\frac{\partial}{{\partial}x^j}}{\ln}{\bar N}_t.
\ena
Furthermore we define another useful quantity
\eqa
c^{-2}{\bar x}_t+c^{-1}{\bar y}_t
{\equiv}-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}+
{\frac{{\bar N}_t,_t}{c{\bar N}_t^2}},
\ena
which takes into consideration how ${\bar N}_t$ changes in the direction normal
to the FHSs. The reason why the left hand side of (17) consists of two terms 
is that the first term determines the effect on ${\bar N}_t$ through the KE of 
the FHSs and the second term determines the effect on ${\bar N}_t$ through the 
NKE of the FHSs. We also note that the family of extrinsic curvature tensors
corresponding to the metric family ${\bf {\bar g}}_t$ of the type given in 
equation (6) has the form
\eqa
{\bar K}_{(t)ij}={\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
-{\frac{t_0}{t}}c^{-2}{\bar a}_{{\cal F}k}{\frac{{\bar N}^k}{{\bar N}_t}}
{\Big )}{\bar h}_{(t)ij}+{\frac{t}{2{\bar N}_tt_0}}({\bar N}_{i{\mid}j}+
{\bar N}_{j{\mid}i}).
\ena
Now we want to postulate what the nature of gravity should be in our theory. 
To arrive at that, we notice that the lapse function ${\bar N}_t$ in (6) also 
plays the role of a spatial scale factor. This means that we may think of 
${\bar N}_t^2$ as a {\em conformal} factor in (6). We may then interpret 
${\bar N}_t$ as resulting from a position-dependent units transformation. This 
indicates that the scale factor ${\bar F}_t{\equiv}{\bar N}_tct$ of the FHSs
may be thought of as representing a difference in scale between atomic and
gravitational systems. That is, ${\bar F}_t$ may be thought of as 
representing a gravitational scale measured in atomic units. An equivalent
way of interpreting ${\bar F}_t$ is to postulate that fixed operationally 
defined ``atomic'' units are considered formally variable throughout 
space-time. Since the intercomparison of operationally defined units only has 
meaning locally for objects at rest with respect to each other, any non-local 
intercomparison is purely a matter of definition [7]. 

The variation in scale between gravitational and atomic systems means that 
gravitational quantities get an extra formal variation when measured in atomic
units (and vice versa); in particular the dimensional constants of nature may 
depend on this formal variation. That is, the ``constants" of nature may in 
principle depend on whether they are expressed in atomic or gravitational 
units. We notice that when performing a units transformation we are free to 
{\em define} as constants physical constants that cannot be related to each 
other by dimensionless numbers [6]. We exploit this fact by defining $c$ and 
Planck's constant ${\hbar}$ not to be formally variable. This means that 
measured in atomic units, the variation of gravitational lengths and times 
are identical and inverse to that of gravitational energy (or mass). 

The requirement that $c$ and ${\hbar}$ are not formally variable yields no 
restrictions on any possible relationship between the variation of units and 
physical fields. On the other hand we may combine the unit charge $e$ with
${\hbar}$ and $c$ to form the dimensionless number $e^2/{\hbar}c$. Thus by 
{\em defining} $e$ not to be formally variable we deny any possible direct 
connection between gravitation and electromagnetism. 

Formally variable atomic units represent changes of scale between 
non-gravitational and gravitational systems, and such changes of scale should
be described by the metric family ${\bf {\bar g}}_t$. Moreover
${\bf {\bar g}}_t$ should be found from field equations. We now define the
scalar field ${\Psi}_t$; by definition ${\Psi}_t$ tells how atomic time units
vary in space-time. Any gravitational quantity is associated with ${\Psi}_t$ 
to some power describing how the quantity changes due to the difference in 
scale between gravitational and atomic systems. In particular we find from 
dimensional analysis that Newton's gravitational coupling parameter $G_t$ has 
the dimension of time units squared. Since $G_t$ is a gravitational quantity, 
measured in atomic units it will then vary as a factor ${\Psi}_t^{-2}$. 
But since $G_t$ usually occurs in combination with masses it is convenient to
define $G_t$ to take a constant value $G$ and rather separate between
{\em active mass} $m_t$ which is measured dynamically and {\em passive mass} 
$m$ (i.e., passive gravitational mass or inertial mass). This means that any 
active mass $m_t$ is not a constant but a scalar field. That is, unlike $c$, 
${\hbar}$ and $e$, $m_t$ gets a formal variation when measured in atomic 
units. Since the formal variability of $G$ should be included into $m_t$ we 
find that the formal variation of active mass goes as ${\Psi}_t^{-1}$. Another
example of a gravitational quantity is the active stress-energy tensor
${\bf T}_t$ which plays the role as a source of gravitation. One may show that
${\bf T}_t$ gets an extra formal variability as ${\Psi}_t^2$. The extra 
variability must be taken care of in relevant differential laws, e.g. in local
conservation laws.

The difference between active mass and passive mass represents a violation 
of the SEP. To see if this is ruled out from experiments testing the constancy
of $G$, one may consider laboratory experiments (e.g. Cavendish experiments) 
and space experiments in the solar system (e.g. lunar laser ranging). In the 
latter class of experiments it is assumed that any variation in $G$ can be 
found simply by replacing the gravitational constant with an independently 
variable $G$ in Newton's equations of motion [5]. But this cannot be done 
within the quasi-metric framework since it would be inconsistent (see equation
(19) below). Moreover, the global NKE of gravitationally bound systems cannot 
be neglected when testing the temporal constancy of $G$. (For an explicit 
example of this, see [10].) Similarly, in laboratory experiments a variable 
$G$ should not simply be put into Newton's force law without taking into 
account the global NKE. Due to the global effects of the NKE one must also 
separate between bodies bound by atomic forces and gravitationally bound 
bodies when calculating the time variation of $G$. In particular, when testing 
whether or not $G$ changes with time, according to quasi-metric theory the 
result should depend on the specific experiment performed. For example in 
Cavendish experiments, where the experimental system is bound by atomic 
forces, one should expect a different result than by trying to measure the 
variation with time in the acceleration of gravity on the surface of the 
Earth, say. (The former experiment is expected to show that $G$ increases
with time whereas the latter experiment is expected to show that $G$ decreases
with time.) Now laboratory experiments testing the constancy of $G$ are 
nowhere near the precision level attained from the corresponding space 
experiments. But since the interpretation of the space experiments is 
explicitly theory-dependent, they are not very useful as direct tests on the 
variability of $G$ as predicted by a quasi-metric theory. And laboratory 
experiments are not yet sufficiently precise to rule out the predicted 
variations of $G$.

We conclude that any difference between active and passive mass-energy is a 
violation of the SEP since there are locally measurable consequences of this. 
But the fact that the locally measured $G$ is variable only via the variation 
of (atomic) units (in contrast to any ``physical" variation in $G$ independent
of the variation of units) is important regarding the interpretation of 
experiments. So far it seems that no experiment has been capable of ruling out
the formal variation of $G$.

The following postulate is consistent with the above considerations:
\begin{itemize}
{\item Gravity is described by the metric family ${\bf g}_t$
(constructible from another metric family ${\bf {\bar g}}_t$) measured in 
atomic units. Measured in atomic units,
gravitational quantities get an extra formal variability 
depending on dimension. By construction $c$, ${\hbar}$ and
$e$ are not formally variable. But to be able to define $G$ as constant it is
necessary to separate between active and passive mass-energy. In particular
the formal variability of active mass-energy is included into
the active stress-energy tensor ${\bf T}_t$ considered as a source of gravity.
The formal variation of atomic units in space-time is by definition 
identical to the variation of the scalar field ${\Psi}_t$. How ${\Psi}_t$ 
varies in space-time depends on the fields ${\bar a}_{{\cal F}j}$, 
${\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}$,
${\frac{{\bar N}_t,_t}{c{\bar N}_t^2}}$ and ${\frac{1}{c{\bar N}_tt}}$.}
\end{itemize}
To get any further we must know the formal dependence of ${\Psi}_t$ on 
${\bar a}_{{\cal F}j}$, ${\frac{{\bar N}_t,_{\bar{\perp}}}{{\bar N}_t}}$,
${\frac{{\bar N}_t,_t}{{\bar N}_t}}$ and ${\frac{1}{t}}$. From equation (6)
we note that the scale factor ${\bar N}_tct$ of the FHSs is measured in atomic
units. This suggests that any change in this scale factor may be explained 
solely by means of the formal variation of units. We thus define 
${\Psi}_t{\equiv}{\frac{1}{c{\bar N}_tt}}$, so we have
\eqa
{\Psi}_t,_t=-{\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big )}
{\Psi}_t, {\qquad}
{\Psi}_t,_{\bar {\perp}}=-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
{\Psi}_t \qquad {\Psi}_t,_j=-c^{-2}{\bar a}_{{\cal F}j}{\Psi}_t, \nonumber \\
m_t,_t={\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\big )}m_t 
\qquad m_t,_{\bar {\perp}}={\frac{{\bar N}_t,_{{\bar {\perp}}}}{{\bar N}_t}}
m_t, {\qquad} m_t,_j=c^{-2}{\bar a}_{{\cal F}j}m_t.
\ena
Note that the Newtonian approximation of equation (6) (which is obtained by 
neglecting the shift covector, setting the spatial scale factor equal to unity
and replacing $S_{ij}$ with an Euclidean 3-metric), is inconsistent with any 
variation in space-time of active mass. That is, in quasi-metric relativity 
$G$ {\em must} be a constant in the Newtonian limit.
 
Then there is the question if the FHSs play any explicit role in the coupling 
of matter to geometry. And indeed they do, but only via ${\bf K}_t$ and
derivatives of ${\bar F}_t$ as we shall see below. The curvature
intrinsic to the FHSs should not couple directly to matter fields.
Before we try to find gravitational field equations we first find how the 
quasi-metric local conservation laws differ from their counterparts in
metric theory. That is, the degenerate covariant divergence
${\ }{\bf {\topstar{\bar {\nabla}}}}{\ }{\bf {\cdot}}{\bf T}_t$
should not be expected to vanish in general due to the postulates
determining the role of gravity within the quasi-metric framework. 

It may seem natural to believe that local conservation of mass-energy and 
momentum is necessarily violated since
${\ }{\bf {\topstar{\bar {\nabla}}}}{\ }{\bf {\cdot}}{\bf T}_t$ may be
non-zero. However, given the possibilities of coupling ${\bf T}_t$ to the 
fields ${\bar a}_{{\cal F}j}$, 
${\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}$,
${\frac{{\bar N}_t,_t}{{\bar N}_t}}$ and ${\frac{1}{t}}$ such that 
LLI and LPI hold, it is in fact possible that local conservation laws within 
the quasi-metric framework do indeed take the form 
${\ }{\bf {\topstar{\bar {\nabla}}}}{\ }{\bf {\cdot}}{\bf T}_t{\neq}0$,
but such that their special relativistic form still holds in the local 
inertial frames. Note that the formal variation of the units is taken care of 
both in derivatives of $m_t$ and in derivatives such as 
${\ }{\bf {\topstar{\bar {\nabla}}}}{\ }{\bf {\cdot}}{\bf T}_t$. 

Proceeding with the task of finding local conservation laws; to separate out 
most effects coming from the conformal factor ${\bar N}_t^2$ in (6) we define
the metric family ${\bf {\tilde g}}_t{\equiv}{\bar N}_t^{-2}{\bf {\bar g}}_t$
and its associated degenerate covariant derivative 
${\ }{\bf {\topstar{\tilde {\nabla}}}}{\ }$. In coordinate notation we denote 
the operations of degenerate covariant derivation 
${\ }{\bf {\topstar{\bar {\nabla}}}}{\ }$ and 
${\ }{\bf {\topstar{\tilde {\nabla}}}}{\ }$ with the symbols `${\bar *}$' and 
`${\tilde *}$', respectively. Using this notation, a short calculation yields 
the following expression, which turns out to be helpful:
\eqa
T_{(t){\mu}{\bar *}{\nu}}^{\nu}{\equiv}
c^{-1}T_{(t){\mu}{\bar *}t}^0+T_{(t){\mu};{\nu}}^{\nu}=
T_{(t){\mu}{\tilde *}{\nu}}^{\nu}+
4{\frac{{\bar N}_t,_{\alpha}}{{\bar N}_t}}T_{(t){\mu}}^{\alpha}
-{\frac{{\bar N}_t,_{\mu}}{{\bar N}_t}}T_{(t){\alpha}}^{\alpha}.
\ena
We now project this expression into respective normal to the FHSs. 
It is possible to figure out expressions for the projections of 
$T_{(t){\mu}{\tilde *}{\nu}}^{\nu}$ by guessing how they should behave for a
null fluid. These calculations can be found in [2], where it is shown that the
local conservation laws in ${\overline{\cal N}}$ read (note that these laws 
take into account the effects of the formally variable units)
\eqa
T_{(t){\bar {\perp}}{\bar *}{\nu}}^{\nu}=
2c^{-2}{\bar a}_{{\cal F}i}{\hat T}^i_{(t){\bar {\perp}}}
-2{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
+(c{\bar N}_t)^{-1}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
{\Big )}T_{(t){\bar {\perp}}{\bar {\perp}}},
\ena
\eqa
T_{(t)j{\bar *}{\nu}}^{\nu}=
2c^{-2}{\bar a}_{{\cal F}i}{\hat T}^i_{(t)j}
-2{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
+(c{\bar N}_t)^{-1}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
{\Big )}T_{(t){\bar {\perp}}j}.
\ena
Note that since $c^{-2}{\bar a}_{{\cal F}j}=-{\bar N}_t
{\topstar{\bar {\Gamma}}}_{j{\bar {\perp}}}^0$, 
${\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}=-{\bar N}_t
{\topstar{\bar {\Gamma}}}_{{\bar {\perp}}{\bar {\perp}}}^{0}$ and 
${\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
={\frac{1}{3}}{\topstar{\bar {\Gamma}}}_{ti}^{i}$ in a GTCS, 
these local conservation laws take their special relativistic form in the 
local inertial frames, as asserted previously. Besides, by applying equations
(21) and (22) to a source consisting of a perfect fluid with no pressure
(i.e., dust) and projecting the resulting equations with the quantity
${\bf {\bar g}}_t+c^{-2}{\bf {\bar u}}_t{\otimes}{\bf {\bar u}}_t$, one may
show that the dust particles move on geodesics of 
${\,}{\,}{\topstar{\bar {\nabla}}}{\,}{\,}$ in ${\overline{\cal N}}$.

Next, the projections of (21) and (22) yield the conditions [2]
(with ${\bar K}_t{\equiv}{\bar K}_{(t)i}^i$) 
\eqa
{\cal L}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}{\bar {\perp}}}={\Big (}
{\bar K}_t-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\Big )}
T_{(t){\bar {\perp}}{\bar {\perp}}}+{\bar K}_{(t)ik}{\hat T}^{ik}_{(t)}-
{\hat T}^i_{(t){\bar {\perp}}{\mid}i},
\ena
\eqa
{\cal L}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}j}&=&
{\Big (}{\bar K}_t-
2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\Big )}T_{(t){\bar {\perp}}j}
-c^{-2}{\bar a}_{{\cal F}j}T_{(t){\bar {\perp}}{\bar{\perp}}} \nonumber \\
&&+c^{-2}{\bar a}_{{\cal F}i}{\hat T}_{(t)j}^i+
c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}{\frac{{\bar N}^i}{{\bar N}_t}}
T_{(t){\bar {\perp}}i}-{\hat T}^i_{(t)j{\mid}i}.
\ena
Equations (23) and (24) determine the evolution in the hypersurface-orthogonal
direction of the matter density and the momentum density in 
${\overline{\cal M}}$. To find these evolutions in ${\overline{\cal N}}$ we
add the dependence on $t$. Using the projected Lie derivative 
${\topstar{\cal L}}_{{\bf {\bar n}}_t}$ valid in ${\overline{\cal N}}$ we find
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}
T_{(t){\bar {\perp}}{\bar {\perp}}}=
{\cal L}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}{\bar {\perp}}}
-{\frac{2}{c{\bar N}_t}}{\Big (}{\frac{1}{t}}+
{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big )}T_{(t){\bar {\perp}}{\bar {\perp}}},
\ena
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}j}=
{\cal L}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}j}
-(c{\bar N}_t)^{-1}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
T_{(t){\bar {\perp}}j}.
\ena
Note that one expects that the global NKE implies local non-conservation of
mass-energy and momentum in special situations; in these cases equations
(25) and (26) will be violated. That is, whenever the global NKE is so strong
that non-gravitational quantum fields cannot be treated as localized to
sufficient accuracy, one should get spontaneous pair production from 
excitations of vacuum fluctuations of such quantum fields, violating equations
(25) and (26). This will be important only in the early universe and possibly 
in extremely deep gravitational wells, though. On the other hand, equations
(23) and (24) are required to hold always.

Since ${\bf T}_t$ is the active stress-energy tensor it is not directly
measurable locally. That is, one must know how it relates to the {\em passive
stress-energy tensor} ${\cal T}_t$ in ${\cal N}$. Now the relationship between 
${\bf T}_t$ and ${\cal T}_t$ depends in principle on the general nature of the
matter source. To illustrate this, consider ${\bf T}_t$ and ${\cal T}_t$ for a
perfect fluid:
\eqa
{\bf T}_t=({\tilde {\rho}}_m+c^{-2}{\tilde p}){\bf {\bar u}}_t{\otimes}
{\bf {\bar u}}_t+{\tilde p}{\bf {\bar g}}_t, \qquad
{\cal T}_t=({\hat{\rho}}_m+c^{-2}{\hat p}){\bf u}_t{\otimes}{\bf u}_t+
{\hat p}{\bf g}_t,
\ena
where ${\tilde {\rho}}_m$, ${\tilde p}$ are the active mass-energy density and
the pressure in the local rest frame of the source. Here we have used the 
definitions ${\rho}_m{\sqrt{{\bar h}_t}}{\equiv}{\hat {\rho}}_m{\sqrt{h_t}}$
and $p{\sqrt{{\bar h}_t}}{\equiv}{\hat p}{\sqrt{h_t}}$, where ${\rho}_m$ and
$p$ are the passive counterparts to ${\tilde {\rho}}_m$ and ${\tilde p}$, 
respectively. Also ${\bar h}_t$ and $h_t$ are defined to be the determinants 
of ${\bf {\bar h}}_t$ and ${\bf h}_t$, respectively. Furthermore the 
relationship between ${\tilde {\rho}}_m$ and ${\rho}_m$ is given by
\eqa
{\tilde {\rho}}_m=
\left\{
\begin{array}{ll}
{\frac{t}{t_0}}{\bar N}_t{\rho}_m & $for a fluid of material point particles,$
\\ [1.5ex]
{\frac{t^2}{t_0^2}}{\bar N}_t^2{\rho}_m & $for a null fluid,$
\end{array}
\right.
\ena
and a similar relationship exists between ${\tilde p}$ and $p$. The reason why
the relationship between ${\rho}_m$ and ${\tilde {\rho}}_m$ is different for 
a null fluid than for other perfect fluids is that spectral shifts of null 
particles influence their passive mass-energy but not their active mass-energy.
Besides, note that more ``physically correct'' local conservation laws can be 
found by calculating ${\,}{\,}{\topstar{\nabla}}{\,}{\cdot}{\cal T}_t$ when 
${\bf g}_t$ is known. But these local conservation laws for passive 
mass-energy do not take any predetermined form.

To find field equations, the requirement that we 
must have metric correspondence with Newtonian theory for weak stationary 
fields leads to the postulate (${\kappa}{\equiv}8{\pi}G/c^4$)
\eqa
2{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}=
2(c^{-2}{\bar a}_{{\cal F}{\mid}i}^i+
c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{\cal F}^i-
{\bar K}_{(t)ik}{\bar K}_{(t)}^{ik}+{\cal L}_{{\bf {\bar n}}_t}{\bar K}_t)
={\kappa}(T_{(t){\bar {\perp}}{\bar {\perp}}}+{\hat T}_{(t)i}^i),
\ena
where ${\bf {\bar R}}_t$ is the Ricci tensor family corresponding to the
metric family (6) and ${\bf {\bar K}}_t$ is the extrinsic curvature tensor 
family of the FHSs in ${\overline{\cal M}}$. Equation (29) represents one 
field equation with the particular property that it does not depend explicitly
on the intrinsic curvature of the FHSs. We postulate a second set of field 
equations having this property, namely
\eqa
{\bar R}_{(t)j{\bar {\perp}}}={\bar K}_{(t)j{\mid}i}^i-{\bar K}_t,_j=
{\kappa}T_{(t)j{\bar {\perp}}}.
\ena
Note that metric counterparts to equations (29) and (30) hold in general 
relativity (GR) for projections on an arbitrary spatial hypersurface.

In particular cases it is possible to choose a convenient coordinate condition
which makes the field equations more easy to handle. That is, the field
equations become less complicated if
\eqa
{\bar N}^i_{{\,}{\,}{\,}{\mid}i}=3c^{-2}{\bar a}_{{\cal F}i}{\bar N}^i
\qquad  \Rightarrow  \qquad
{\bar K}_t=3{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}.
\ena
Note that the geometrical interpretation of the condition (31) is that the 
extrinsic curvature family ${\bf {\tilde K}}_t$ (calculated from the metric 
family ${\bf {\tilde g}}_t$) is traceless. In other words the condition (31) 
implies that the FHSs are maximally embedded in ${\bf {\tilde g}}_t$.

Several comments apply to the field equations (29), (30). Firstly we notice 
that they do not represent a full coupling of ${\bf T}_t$ to space-time
geometry but only a partial one. Secondly, the field equations superficially
look like just a subset of the Einstein field equations in ordinary GR. But
the crucial difference is that the equations (29), (30) are valid only for
projections into the FHSs; for projections into any other hypersurfaces they 
do not hold. Thirdly we observe that the FHSs enter the field equations 
geometrically only via their extrinsic curvature; the intrinsic curvature of 
the FHSs does not couple directly to matter. Fourthly equation (29) is a 
dynamical equation whereas equations (30) are constraints. We thus have one 
dynamical equation and three constraint equations. One may compare this to GR 
where from (5) one can see that the Einstein equations split up into six 
dynamical equations and four constraint equations. Fifthly, the local 
conservation laws do not follow from the field equations as they do in GR. 
Rather the local conservation laws are coupled to the field equations and 
represent independent restrictions on the nature of the matter source. 
Finally we observe that our field equations contain only one explicit 
dynamical field, namely ${\bar K}_t$. Since equations (18) and (31) indicate 
that ${\bar K}_t=3{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}$ at least
for some cases, this means that the true dynamical field is the lapse function
${\bar N}_t$. That is, our field equations have only one explicit dynamical 
degree of freedom whereas GR has two [4]. However, a second {\em implicit} 
dynamical degree of freedom is added to our theory via the NKE when 
constructing ${\bf g}_t$ from ${\bf {\bar g}}_t$ [2].
\section*{4{\hspace*{9mm}} Kinematics and non-kinematics of the FHSs}
In this section we take a closer look at the evolution with time of the 
geometry of the FHSs in ${\overline{\cal N}}$ and in ${\cal N}$. To begin with
(6) yields the evolution equation [2]
\eqa
{\topstar{\pounds}}_{{\bf {\bar n}}_t}{\bar h}_{(t)ij}=
2{\Big (}(c{\bar N}_t)^{-1}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}+
{\frac{t_0}{t}}c^{-2}{\bar a}_{{\cal F}k}{\frac{{\bar N}^k}{{\bar N}_t}}
{\Big )}{\bar h}_{(t)ij}-{\frac{t}{t_0}}{\Big (}
{\frac{{\bar N}_{i{\mid}j}}{{\bar N}_t}}+
{\frac{{\bar N}_{j{\mid}i}}{{\bar N}_t}}{\Big )}.
\ena
On the other hand we may {\em define} the split-up into respectively
kinematical and non-kinematical terms of the time evolution of the
spatial scale factor in (6). We may thus define
\eqa
c^{-2}{\bar x}_t+c^{-1}{\bar H}_t{\equiv}
(c{\bar N}_t)^{-1}{\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
{\Big )}-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}, 
\ena
where ${\bar H}_t$ is a quantity associated with the NKE of the FHSs. (We 
define ${\bar H}_t$ in equation (34) below.) 

In the introduction we defined the NKE as a local increase of scale with time
of the FHSs. To quantify this definition we introduce the local scale factor 
${\bar F}_t{\equiv}{\bar N}_tct$ of the tangent spaces of any FHS in 
${\overline{\cal N}}$. By definition the local change of scale with time as 
defined by nearby FOs is entirely expressible as a change in ${\bar F}_t$. 
Thus we may define the non-kinematical scale rate ${\bar H}_t$ by
\eqa
{\bar H}_t{\equiv}
c{\bar F}_t^{-1}{\topstar{\cal L}}_{{\bf {\bar n}}_t}{\bar F}_t-
c^{-1}{\bar x}_t.
\ena
Note that we have subtracted the kinematical contribution 
$c^{-1}{\bar x}_t$ to the scale change in equation (34). Since ${\bar H}_t$ 
is a purely non-kinematical quantity, it should be available directly from the
geometry intrinsic to the FHSs. We argue in [2] that the correct form of this 
so-called non-kinematical ``coupling" equation is given by
\eqa
{\bar H}_t={\frac{1}{{\bar N}_tt}}+
c^{-1}{\sqrt{{\bar a}_{\cal F}^i{\bar a}_{{\cal F}i}}}.
\ena
Besides we see from (17) that ${\bar y}_t$ represents the non-kinematical part
of the evolution of ${\bar N}_t$ with time. Combining equations (17), (33) 
and (35) we find
\eqa
{\bar y}_t={\bar H}_t-{\frac{1}{{\bar N}_tt}}=
c^{-1}{\sqrt{{\bar a}_{\cal F}^i{\bar a}_{{\cal F}i}}},
\ena
\eqa
c^{-2}{\bar x}_t=-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}+
(c{\bar N}_t)^{-1}{\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
{\Big )}-c^{-1}{\bar H}_t.
\ena
For cases where there is no coupling to non-gravitational fields via the
equations of motion (13) in ${\cal N}$ we now have a well-defined 
initial-value system in ${\overline{\cal N}}$: on an initial FHS we choose 
initial data ${\bar N}_t$, ${\bar N}_t,_{\bar {\perp}}$, ${\bar N}_t,_t$, 
${\bar N}^i$, $({\bar N}^i),_{\bar {\perp}}$ and the various projections of 
${\bf T}_t$ which satisfy the constraint equations (30); then we evolve 
${\bf {\bar h}}_t$ through ${\overline{\cal N}}$ using equations (29), (30), 
(18), (19), (23), (24), (25) and (26). For each step of the evolution we must 
supply the quantity $2{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}$ to be able 
to construct ${\bar N}_t$ and ${\bar N}^j$ for the subsequent steps. Then the 
non-kinematical quantities follow automatically via (35) and (36). Note that
the local conservation laws (23), (24), (25) and (26) are coupled to the
field equations since they represent non-trivial evolution equations of 
$T_{(t){\bar {\perp}}{\bar {\perp}}}$ and $T_{(t){\bar {\perp}}j}$. But to 
know the time evolution of the full stress-energy tensor we need to know the 
equation of state characterizing the matter source; i.e., we need to know how 
the passive mass-energy is related to the passive pressure.

Note that for the initial-value system defined by the equations (4) and (5) it
is possible to choose the lapse function $N$ freely throughout space-time 
since the evolution of $N$ is not determined by the constraint preservation 
[8]. On the other hand, for our initial-value system ${\bar N}_t$ is the lapse
function for a particular set of observers; the evolution of the FHSs tells 
how ${\bar N}_t$ evolves with time and there is no arbitrariness of choice 
involved.

But in general the evolution scheme in ${\overline{\cal N}}$ is coupled to the
evolution in scheme $\cal N$ via non-gravitational fields. This means that 
these evolution schemes are intertwined, and that we need to construct 
${\bf g}_t$ for each step of the evolution. This can be done by finding the 
family of 3-vector fields $-{\bf v}_t$ which tells how much the FOs in 
${\overline{\cal N}}$ ``move'' compared to the FOs in $\cal N$. Expressed in a 
GTCS said vector field may be defined as [2]
\eqa
v^{j}_{(t)}{\equiv}{\bar y}_tb_{\cal F}^j, \qquad
v{\equiv}{\bar y}_t
{\sqrt{{\bar h}_{(t)ik}b_{\cal F}^ib_{\cal F}^k}},
\ena
where ${\bf b}_{\cal F}$ is a 3-vector field representing ``the local distance
from the center of gravity'' generalized from the spherically symmetric case. 
Furthermore ${\bf b}_{\cal F}$ is defined by the equation
\eqa
{\Big [}{\bar a}_{{\cal F}{\mid}k}^k+c^{-2}{\bar a}_{{\cal F}k}
{\bar a}_{\cal F}^k{\Big ]}b_{\cal F}^j-{\Big [}
{\bar a}_{{\cal F}{\mid}k}^j+c^{-2}{\bar a}_{{\cal F}k}{\bar a}_{\cal F}^j
{\Big ]}b_{\cal F}^k-2{\bar a}_{\cal F}^j=0,
\ena
which has the unique solution ${\bf b}_{\cal F}=
r{\frac{\partial}{{\partial}r}}$ for the spherically symmetric case (using a
spherical GTCS). This means that we intuitively expect that ${\bf b}_{\cal F}$
should vanish when ${\bf {\bar a}}_{\cal F}$ does and that 
${\bf {\bar a}}_{\cal F}$ and ${\bf b}_{\cal F}$ should have the same 
direction. 

From equation (38) we see that the construction of ${\bf g}_t$ from 
${\bf {\bar g}}_t$ involves that part of the NKE which is not ``realized'' in 
the evolution equation (34). That is, from equation (36) we see that 
${\bar y}_t=c^{-1}{\sqrt{{\bar a}_{{\cal F}i}{\bar a}_{\cal F}^i}}$, and this 
quantity is not included in the explicit time evolution of the spatial scale 
factor. Note that $v$ should not depend explicitly on $t$. It is also worth 
noticing that there exists only one propagating degree of freedom in 
${\overline{\cal N}}$. However the quantity $v$ represents a second degree of 
freedom so one would expect that the number of propagating degrees of freedom 
in $\cal N$ is two, the same as in GR.

In [2] it is shown how to construct ${\bf g}_t$ (or, equivalently ${\bf h}_t$,
$N$ and $N^j$) algebraically from ${\bf {\bar g}}_t$ (or, equivalently
${\bf {\bar h}}_t$, ${\bar N}_t$ and ${\bar N}^j$) and the vector field family
${\bf v}_t$ for each step of the evolution; thus we also have a well-defined 
initial-value system in $\cal N$. 

Actually, transformations similar to ${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$
apply to any tensor field which norm is required to remain unchanged when
${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$. As an example we list the
formulae valid for the transformation ${\bf {\bar Z}}_t{\rightarrow}{\bf Z}_t$
where ${\bf {\bar Z}}_t$ is a rank 1 tensor field family. These transformation
formulae involve the unit vector field ${\bf {\bar e}}_b{\equiv}
{\frac{t_0}{t}}{\bar e}_b^i{\frac{\partial}{{\partial}x^i}}$ along the 
${\bf b}_{\cal F}$-direction in ${\overline {\cal N}}$ and the corresponding 
covector field ${\bf {\bar e}}^b{\equiv}{\frac{t}{t_0}}{\bar e}^b_idx^i$. The 
transformation formulae read
\eqa
Z_{(t)0}={\Big (}1-{\frac{v^2}{c^2}}{\Big )}{\bar Z}_{(t)0}, \qquad
Z_{(t)}^0={\Big (}1-{\frac{v^2}{c^2}}{\Big )}^{-1}{\bar Z}_{(t)}^0,
\ena
\eqa
Z_{(t)j}={\bar Z}_{(t)j}+{\frac{2{\frac{v}{c}}}{1-{\frac{v}{c}}}}
({\bar e}_b^i{\bar Z}_{(t)i}){\bar e}^{b}_j, \qquad
{\hat Z}_{(t)}^j={\hat {\bar Z}}_{(t)}^j-
{\frac{2{\frac{v}{c}}}{1+{\frac{v}{c}}}}
({\bar e}^{b}_i{\hat {\bar Z}}_{(t)}^i){\bar e}_{b}^j.
\ena
Note that it is also possible to find analogous formulae (involving the unit 
vector field ${\bf e}_b{\equiv}{\frac{t_0}{t}}e_b^i
{\frac{\partial}{{\partial}x^i}}$ in the ${\bf b}_{\cal F}$-direction and its 
associated covector field ${\bf e}^b{\equiv}{\frac{t}{t_0}}e^b_idx^i$ in 
$\cal N$) for the inverse transformation 
${\bf Z}_t{\rightarrow}{\bf {\bar Z}}_t$.

One example of a (co)vector field obeying the rules (40)-(41) is a general 
4-velocity (co)vector family ${\bf {\bar u}}_t$. Another important example
is any 4-acceleration field family ${\bf {\bar a}}_t$ determined from a
non-gravitational force (e.g., if ${\bf {\bar a}}_t$ is due to the Lorentz 
force acting on charged matter). For such cases the norm of
${\bf {\bar a}}_t$ must be invariant under the transformation. In particular 
${\bf {\bar a}}_t$ may vanish, and this means that {\em geodesic motion in 
${\overline{\cal N}}$ implies geodesic motion in ${\cal N}$.} Note that 
${\bf {\bar a}}_{\cal F}$ does not in general transform according to the rules
(40)-(41) since ${\bf {\bar a}}_{\cal F}$ is determined from the requirement 
that the FOs move normal to the FHSs rather from some non-gravitational force.

For completeness we also list the transformation formulae valid for the
transformation ${\bf {\bar Q}}_t{\rightarrow}{\bf Q}_t$, where 
${\bf {\bar Q}}_t$ is a rank 2 tensor field family. We have
\eqa
Q_{(t)00}={\Big (}1-{\frac{v^2}{c^2}}{\Big )}^2{\bar Q}_{(t)00}, \qquad
Q_{(t)}^{00}={\Big (}1-{\frac{v^2}{c^2}}{\Big )}^{-2}{\bar Q}_{(t)}^{00},
\ena
\eqa
Q_{(t)0j}&=&{\Big (}1-{\frac{v^2}{c^2}}{\Big )}{\Big [}{\bar Q}_{(t)0j}+
{\frac{2{\frac{v}{c}}}{1-{\frac{v}{c}}}}
({\bar e}_{b}^i{\bar Q}_{(t)0i}){\bar e}_j^{b}{\Big ]}, \nonumber \\
{\hat Q}_{(t)}^{0j}&=&{\Big (}1-{\frac{v^2}{c^2}}{\Big )}^{-1}{\Big [}
{\hat {\bar Q}}_{(t)}^{0j}-{\frac{2{\frac{v}{c}}}{1+{\frac{v}{c}}}}
({\bar e}^{b}_i{\hat {\bar Q}}_{(t)}^{0i}){\bar e}^j_{b}{\Big ]},
\ena
\eqa
Q_{(t)ij}&=&{\bar Q}_{(t)ij}+{\frac{2{\frac{v}{c}}}{(1-{\frac{v}{c}})^2}}
{\bar e}_b^k({\bar e}^{b}_i{\bar Q}_{(t)kj}+{\bar Q}_{(t)ik}{\bar e}^b_j), 
\nonumber \\
{\hat Q}_{(t)}^{ij}&=&{\hat {\bar Q}}_{(t)}^{ij}-
{\frac{2{\frac{v}{c}}}{(1+{\frac{v}{c}})^2}}
{\bar e}^{b}_k({\bar e}_b^i{\hat {\bar Q}}_{(t)}^{kj}+
{\hat {\bar Q}}_{(t)}^{ik}{\bar e}_b^j).
\ena
The transformation formulae shown in equations (42)-(44) may in particular
be used to find ${\bf g}_t$ from ${\bf {\bar g}}_t$; moreover they may easily 
be generalized to tensor field families of higher rank.
\section*{5{\hspace*{9mm}} Test particle motion revisited}
Once we know ${\bf g}_t$ we may calculate paths of arbitrary test
particles using the equations of motion (13). But before we are able to do 
such calculations we need to have an independent expression for the degenerate
acceleration field ${\stackrel{\star}{{\bf a}}}$. In this section we will see 
that we in fact have ${\stackrel{\star}{\bf a}}$=${\bf a}_t$.

In the following it is convenient to introduce the difference in coordinate 
3-velocity between an arbitrary test particle and the local FO. We call this
quantity ${\bf w}_t$, and we write its square as $w^2$. Thus we define (using 
a GTCS)
\eqa
w_{(t)}^j{\equiv}{\frac{dx^j}{d{\tau}_{\cal F}}}+{\frac{t_0}{t}}
{\frac{N^j}{N}}c, {\qquad}
{\topstar{\gamma}}{\equiv}(1-c^{-2}w_{(t)}^iw_{(t)i})^{-1/2}=
{\frac{d{\tau}_{\cal F}}{d{\tau}_t}},
\ena
where ${\tau}_t$ is the proper time as measured along the curve. Note that 
${\bf w}_t$ is an object intrinsic to the FHSs and that we
may separate out the $t$-dependence of ${\bf w}_t$, namely [2]
\eqa
w_{(t)j}={\frac{t}{t_0}}w_{(t_0)j}{\equiv}{\frac{t}{t_0}}w_j, \qquad
w_{(t)}^j={\frac{t_0}{t}}w_{(t_0)}^j{\equiv}{\frac{t_0}{t}}w^j.
\ena
One may readily show that the tangent 4-velocity can be 
split up into parts respectively normal and tangential to the FHSs, i.e.
\eqa
{\bf u}_t={\topstar{\gamma}}(c{\bf n}_t+{\bf w}_t).
\ena
Since any 4-velocity vector ${\bf {\bar u}}_t$ in ${\overline{\cal N}}$
by construction has constant norm we may locally calculate the transformation
${\bf {\bar u}}_t{\rightarrow}{\bf u}_t$ from equations (40) and (41). We may
then show that the norm ${\bar w}$ of the coordinate 3-velocity 
${\bf {\bar w}}_t$ (and thus ${\topstar{\gamma}}$) is invariant under the 
transformation, i.e., ${\bar w}=w$.

Using equations (46) and (47) it is now possible to calculate the quantity
${\,}{\,}{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf u}_t$ and
thereby the degenerate acceleration ${\stackrel{\star}{\bf a}}$ from
equation (12). That is, using equation (9) it is straightforward to show that
\eqa
{\stackrel{\star}{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf w}_t=0,
\qquad \Rightarrow \qquad
{\stackrel{\star}{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf u}_t=0,
\qquad \Rightarrow \qquad {\stackrel{\star}{\bf a}}={\bf a}_t.
\ena
The coordinate expression (13) for the equations of motion then take the form
(this form is valid in a general coordinate system, but equation (8) is valid 
only in a GTCS)
\eqa
{\frac{d^2x^{\mu}}{d{\lambda}^2}}+{\Big(}
{\topstar{\Gamma}}^{\mu}_{t{\sigma}}{\frac{dt}{d{\lambda}}}+
{\Gamma}^{\mu}_{(t){\beta}{\sigma}}{\frac{dx^{\beta}}{d{\lambda}}}{\Big)}
{\frac{dx^{\sigma}}{d{\lambda}}} 
={\Big(}{\frac{cd{\tau}_t}{d{\lambda}}}{\Big)}^2c^{-2}a_{(t)}^{\mu}.
\ena
For timelike curves one may parameterize the curve by proper time by
choosing ${\lambda}=c{\tau}_t$. For null curves ${\lambda}$ must be chosen as
some other affine parameter such that ${\frac{cd{\tau}_t}{d{\lambda}}}=0$. 

The equations of motions (49) are the geodesic equations obtained from 
${\,}{\topstar{\nabla}}{\,}$. Moreover, it can be readily shown that (49) does
not reduce to the geodesic equation (14) whenever the dependence on $t$ of the
metric components cannot be neglected. That is, if the global part 
${\frac{1}{{\bar N}_tt}}$ of the NKE (see equation (35)) is not negligible 
inertial test particles do not move as if they were following geodesics of any
single space-time metric. 
\section*{6{\hspace*{9mm}} Discussion}
As we have seen in this survey, the realization that physical space should not
necessarily be treated as a purely kinematical field may have important 
theoretical consequences; this idea leads ultimately not merely to a new 
theory of gravity but to a new geometrical framework as the basis for 
relativistic physics. One of the reasons that makes the possible replacement 
of the currently prevailing metric framework with our alternative even 
remotely probable, is the fact that this can be done without sacrificing the 
EEP. 

Even though our theory seems well-posed and self consistent, the
question is still open if it is viable according to experimental criteria.
We show in [2] that it is at least possible to find certain limits where the 
predictions of our theory match those of GR within experimental precision.
Particular cases here are static systems with spherical symmetry; we 
show that by assuming that the effects of a global scale factor are negligible,
it is possible to find a one-parameter family ${\bf {\bar g}}_t$ of identical, 
conformally flat metrics as a vacuum solution of our field equations. We
then include the {\em local} effects of the NKE into the family ${\bf g}_t$ 
which we construct from the conformally flat metrics. That is, when the global 
effects of the NKE can be neglected the metric families ${\bf {\bar g}}_t$ and
${\bf g}_t$ can be represented by single metrics ${\bf {\bar g}}$ and 
${\bf g}$ since there is no explicit dependence on $t$ in this case. Moreover,
for the static, spherically symmetric case it turns out that ${\bf g}$
agrees with the Schwarzschild metric to sufficient order in the quantity 
$r_s/r$, where $r_s$ is the Schwarzschild radius. However, there is 
disagreement for higher orders since neither ${\bf {\bar g}}$ nor ${\bf g}$
possesses an event horizon. This illustrates the fact that the geometrical 
structure of quasi-metric space-time does not allow the formation of black 
holes. Furthermore we easily find a toy cosmological solution of our field 
equations; a family of metrics with the geometry ${\bf S}^3{\times}{\bf R}$ 
constitutes a vacuum solution of the field equations. (In this case the 
families ${\bf {\bar g}}_t$ and ${\bf g}_t$ are identical.) This indicates 
that our theory does not allow any physical singularities since even at the 
cosmological singularity no physical quantities diverge. (But we need a matter
creation mechanism for a Universe with an empty beginning, this may perhaps be
constructed along the lines described in [9].) Moreover no counterparts to the 
Friedmann-Robertson-Walker models exist in quasi-metric theory; besides the 
empty toy model the only allowed isotropic cosmological solutions of the field
equations represent toy universes filled with a null fluid. This means that 
the usual cosmological parameters are not useful in quasi-metric theory. 

Another prediction of our theory is that the use of Newtonian dynamics for 
weak field systems has its limitations; in addition to the weak field criterion
systems should be ``small" so that the {\em global} effects of the NKE (i.e., 
the effects due to a ``real" expansion) can be neglected. (That is, even if 
these global effects are small locally, over large distances or long time 
spans they may not be small.) We explore some of the global effects of the NKE
in [10], where we discuss the possible relevance of the results for explaining
the observed asymptotically non-Keplerian rotation curves of spiral galaxies.
Furthermore, even in the solar system there may be observable consequences of 
the NKE. In [11] we discuss one such effect and compare to data the 
predictions of the model considered in [10]. In addition we show in [12] that 
the global effects of the NKE are predicted to apply inside gravitationally 
bound perfect fluid bodies as well. In particular quasi-metric theory predicts
that the radii of bodies made of ideal gas should not be constant in time but 
rather change similarly to the cosmological scale factor. This model can be 
compared to data on expansion of the Earth. Also possible is comparison of 
the model to data on spin-down of the Earth's rotation rate.

A lot of work remains to be done before one can decide whether or not our 
theory is viable. For example it is necessary to work out if it correctly
models binary pulsars, quasi-stellar objects, cosmological observations etc.
Should our theory turn out to be viable, it is believable that its
possible acceptance hinges on if it is able to explain the workings of the 
Universe from first principles better than competing theories. Only then will
the NKE get the status as a fundamental principle realized by nature.
\\ [4mm]
{\bf References} \\ [1mm]
{\bf [1]} K. Kucha{\v{r}}, {\em J. Math. Phys.} {\bf 17}, 792 (1976). \\
{\bf [2]} D. {\O}stvang, chapter 2 of this thesis (2001). \\
{\bf [3]} C. Gerhardt, {\em Commun. Math. Phys.} {\bf 89}, 523 (1983). \\
{\bf [4]} C.W. Misner, K.S. Thorne, J.A. Wheeler,
{\em Gravitation}, W.H. Freeman ${\&}$ Co. (1973). \\
{\bf [5]} C.M. Will, {\em Theory and Experiment in Gravitational Physics}, 
Cambridge University {\hspace*{7mm}}Press (1993). \\
{\bf [6]} C.H. Brans and R.H. Dicke, {\em Phys. Rev.} {\bf 124}, 925 (1961). \\
{\bf [7]} R.H. Dicke, {\em Phys. Rev.} {\bf 125}, 2163 (1962). \\
{\bf [8]} J. Isenberg and J. Nester, {\em Canonical Gravity}, in {\em General
Relativity and Gravitation, {\hspace*{6.2mm}}Vol. I}, Editor A. Held, Plenum 
Press (1980). \\ 
{\bf [9]} L. Parker, {\em Particle Creation by the Expansion of the Universe}, 
in {\em Handbook of} \\ {\hspace*{5.9mm}}
{\em Astronomy, Astrophysics and Geophysics, Vol. II Galaxies and Cosmology}, 
\\
{\hspace*{7mm}}Editors V.M Canuto and B.G. Elmegreen, Gordon {\&} Breach 
(1988). \\
{\bf [10]} D. {\O}stvang, chapter 3 of this thesis (2001). \\
{\bf [11]} D. {\O}stvang, chapter 4 of this thesis (2001). \\
{\bf [12]} D. {\O}stvang, chapter 5 of this thesis (2001). 

\newpage

{\vspace*{5cm}}

\begin{center}
{\huge {\bf Chapter Two: \\\ \\ On the Non-Kinematical Evolution
\vspace{0.5cm}\\of the 3-Geometry}}
\end{center}
\newpage
{\hspace*{1cm}}

\newpage

\newcounter{seceqn}[section]
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\begin{center}
{\large {\bf On the Non-Kinematical Evolution of the 3-Geometry}}
\end{center}
\begin{center}
by
\end{center}
\begin{center}
Dag {\O}stvang \\
{\em Institutt for fysikk, Norges teknisk-naturvitenskapelige universitet, 
NTNU \\
N-7491 Trondheim, Norway}
\end{center}
\begin{abstract}
We introduce the ``quasi-metric'' framework consisting of a 4-dimensional 
differential space-time manifold equipped with a one-parameter family of 
Lorentzian 4-metrics parameterized by a global time function. Within this 
framework, we define the concept ``non-kinematical evolution of the 
3-geometry" and show how this is compatible with a new relativistic theory of 
gravity. It is not yet clear whether or not this theory is viable. However, 
it does agree with the classic solar system tests.
\end{abstract}
\section{Introduction}
The class of space-time theories defined as metric theories of gravity 
satisfies the following postulates [1]: (1) space-time is equipped with a 
Lorentzian non-degenerate metric, (2) inertial test particles follow 
non-spacelike geodesics of this metric, and (3) in the local Lorentz frames of 
the metric, the non-gravitational physics is as in special relativity. In light
of the highly successful record of the metric framework, and with the lack of 
any obvious experimental motivation, one might wonder why anyone should bother 
to construct a non-metric alternative. The answer to this question is that even
though it is possible to model a wide range of phenomena within the metric 
framework, things do not always follow from first principles. If one within a
non-metric framework can do at least as well as with any metric theory, and if
additionally one gets a better connection to first principles, the non-metric 
framework should be considered as a serious alternative. The key questions 
are, of course, firstly; which physical/theoretical principles should guide 
one towards constructing a non-metric framework, and secondly; which phenomena
should be considered being sufficiently fundamental to motivate the 
formulation of such principles. 
 
In the search for a suitable non-metric framework part of our phenomenological
motivation consists of the fact that the observable large-scale Universe seems 
to take part in a global increase of scale. Even though this observation has a 
natural interpretation within the metric framework, it does not follow from 
first principles. That is, within the metric framework one may model many types
of universe besides the expanding ones, e.g. collapsing universes, static 
universes or universes whose evolution at large scales cannot be modelled by a
global scale factor. Contrary to this we want to construct a non-metric 
framework which explains the above-mentioned observation from first 
principles. 

The above considerations suggest that a central feature of our non-metric 
framework should be the postulated existence of a specific set of 
(hypothetical) observers whose interrelationship consists at least partly of 
an increase of local scale with time. For reasons explained in section 2 this 
increase of local scale with time is called the ``non-kinematical evolution'' 
(NKE) of the 3-geometry spanned by these observers. In this paper we construct 
a particular type of non-metric framework which accommodates this concept as 
part of the basic geometrical structure. 

The introduction of the NKE can be done without sacrificing one of the basic 
principles defining metric theory; namely the Einstein equivalence principle 
(EEP). However, the mathematical structure of the new framework is not based
on semi-Riemannian geometry, so postulate (1) above does not apply. But 
despite that the new framework is so similar to the metric framework that we 
may call it ``quasi-metric''.

This paper is organized as follows: In section 2 we define the quasi-metric
framework in general terms, in section 3 we construct a theory of gravity
consistent with this framework, and in section 4 we find special solutions
and discuss the question of testing. A list of acronyms and mathematical 
symbols can be found in an appendix.
\section{The quasi-metric framework}
We start by briefly describing the well-known concept of ``kinematical
evolution" (KE) of the 3-geometry, also called geometrokinematics, within
the metric framework.

By slicing an arbitrary 3-dimensional (Cauchy) hypersurface ${\cal S}$ out of 
the space-time manifold ${\cal M}$ such that its normal vector field is
everywhere timelike with respect to the metric {\bf g}, one splits space-time 
into space and time. Suppose one furthermore specifies an arbitrary 
``deformation" timelike vector field on ${\cal S}$. Then, from the intrinsic 
metric of ${\cal S}$, the extrinsic curvature tensor field and its rate of 
change along the deformation vector field, one may deduce the metric on the 
deformed spatial hypersurface. This is the essence of geometrokinematics, and 
its detailed relations hold irrespective of any dynamical laws. The reason is 
that these relations describe how a single space-time tensor field, 
sufficiently represented by the Einstein tensor field ${\bf G}$, is projected 
into tangential and normal directions to ${\cal S}$, and this severely 
restricts how the projections behave under deformations of ${\cal S}$, see 
e.g. [2].

The kinematical relations of $\cal M$ have one characteristic defining 
property, namely that they are not sufficient by themselves to determine the
space-time geometry. That is, in addition to the initial data on $\cal S$ one
must supply extra fields coming from the overlying space-time geometry at each
step of the deformation of $\cal S$ if one wants to predict the metric on
$\cal S$ for each time step of the deformation process. This is as it should 
be, since the geometry of space-time should be dependent on the coupling to
geometry of matter fields; only when appropriate dynamics is specified 
should it become possible to calculate the space-time geometry.

Any equations recognized as kinematical must leave room for dynamics without 
specifying the exact nature of the coupling between matter and geometry.
This is realized geometrically by the fact that the extra fields that 
must be supplied take the form of curvature tensors. As noted above it is
sufficient to supply the projection of ${\bf G}$ into $\cal S$ for each step of
the deformation; in itself this does not depend on how ${\bf G}$ is coupled to
matter. See [2] for a detailed discussion.

The kinematical relations hold equally well for the KE of any spatial
3-hypersurface. By contrast, one might think of restricting the KE of some
{\em particular} spatial hypersurface to take some prescribed form along
some particular deformation vector field. This would be equivalent to just 
laying restrictions on the form of the space-time metric, and the KE of the 
special hypersurface up the prescribed foliation would just represent the
kinematical relations applied to this particular foliation.

However, one might imagine that there are aspects of 3-geometry evolution which
do not depend on the continuous supplement of curvature fields coming from an 
overlying space-time metric. Rather, it would seem conceivable to try to 
construct a type of evolution which may be defined in terms of the 
{\em intrinsic} curvature of $\cal S$. Such aspects of 3-geometry evolution 
one may call ``non-kinematical" since they by construction are self-advancing,
i.e., they leave no room for dynamics. It is clearly not possible to make any 
sense of the NKE concept within the metric framework due to the global 
restrictions of the metrical field. However, one may try to get around this by
postulating more than one space-time metric field; thus one may define some 
sort of evolution of the {\em space-time} metric field. Then there is no 
reason to expect that the NKE should have anything to do with the kinematical 
relations defined by any of the single metric fields. Of course it might be 
possible to construct a single metric field containing a particular foliation 
such that the evolution of $\cal S$ up this foliation is indistinguishable 
from the evolution of $\cal S$ resulting from an evolution of space-time 
metric fields. But the evolution of $\cal S$ should be interpreted as 
physically different for these two cases; in principle it should be possible 
to perform test experiments that are capable of showing whether or not the 
evolution of $\cal S$ is purely kinematical.

We want to create a non-metric framework which incorporates the NKE in a 
natural way. Since this framework should include more than one space-time 
metric field and some way to evolve one of those into another, there must 
exist some evolution parameter by means of which this can be defined. This 
parameter must have an existence independent of the details of each single 
metric field. The straightforward way to define such a parameter is to equip 
the space-time manifold with a global time function; this quantity should be a
natural feature of the general geometrical structure on which we base our 
non-metric framework. We proceed by defining such a basic geometrical 
structure.

Consider a 5-dimensional differential manifold with the global structure
${\cal M}{\times}{\bf R}$, where ${\cal M}$ is a Lorentzian space-time 
manifold and ${\bf R}$ is the real line. We may then introduce local 
coordinates ${\{}x^{\epsilon}{\}}$ on ${\cal M}$ (${\epsilon}$ taking the 
values 0,1,2,3 where $x^{0}$ is the time coordinate), and a global coordinate 
$t$ on ${\bf R}$. Suppose next that we slice a 4-dimensional hypersurface 
${\cal N}$ specified by the equation $t=t(x^{\epsilon})$ out of 
${\cal M}{\times}{\bf R}$. On this slice there is required to exist a unique 
invertible functional relationship between $t$ and $x^{0}$ throughout the 
coordinate patch covered by ${\{}x^{\epsilon}{\}}$. Now we let a point with 
coordinate $t_0$ in ${\bf R}$ represent a space-time metric ${\bf g}_{t_0}$ on
${\cal M}$. Then ${\cal N}$ is defined to be a 4-dimensional space-time 
manifold equipped with a one-parameter family of Lorentzian 4-metrics 
${\bf g}_{t}$ parameterized by the global time function $t$. This general 
geometric structure is the basis of the quasi-metric framework, and it has the 
capacity to accommodate both KE and NKE. We come back to questions of 
uniqueness and existence of such a geometrical structure in the next section.

The global time function $t$ singles out of ${\cal N}$ a set of fundamental 
3-dimensional spatial submanifolds ${\cal S}_{t}$. Each of these hypersurfaces 
is ``preferred" in some sense since each of these hypersurfaces constitutes
the domain of validity for one single member of the metric family; there
corresponds exactly one metric field ${\bf g}{\in}{\bf g}_t$ to each 
${\cal S}{\in}{\cal S}_t$. We call the hypersurfaces ${\cal S}_t$ the 
{\em fundamental hypersurfaces} (FHSs). Observers always moving orthogonally 
to the FHSs are called {\em fundamental observers} (FOs). It is important to 
have in mind that each 4-metric applies exactly only on its domain FHS. This 
means that each member of the metric family is relevant to the evolution of 
the FHSs only at the corresponding FHS. Thus it is sufficient to extrapolate 
each 4-metric to an infinitesimally small neighbourhood of the FHSs whereas 
any further extrapolation may be of dubious value physically.

Now the FHSs are subject to two types of evolution with proper time 
${\tau}_{{\cal F}}$ (depending directly on $t$) as measured by the FOs when 
traversing the one-parameter family of 4-metrics. Firstly, it is the KE of the 
FHSs as defined from its dependence on the supplement of curvature information
from $\cal N$; this information should be available from the 4-geometry defined
by the metric family. Secondly it is the NKE of the FHSs; this type of 
evolution is defined by being explicitly dependent on some aspects of the 
intrinsic curvature of the FHSs and not on the space-time curvature calculated
from the metric family. From this it follows that the concept of NKE is 
intimately connected with the existence of ``preferred" hypersurfaces from 
which the relevant intrinsic curvature can be calculated. On the other hand, 
within the metric framework all spatial hypersurfaces are equivalent as far as
the kinematical relations are concerned. This is true even though different 
spatial hypersurfaces in general will have very different intrinsic 
curvatures; that is why any sort of NKE is impossible in metric theory. 

If one considers a metric space-time manifold as given {\em a priori}, there
are in principle two different ways of splitting up space-time into space and 
time. The first one is the foliation of space-time into a family of space-like
hypersurfaces. This corresponds to choosing a time coordinate representing
``surfaces of simultaneity''. Note that the gradient field of this time 
coordinate is in general not orthogonal to the hypersurfaces. The second way 
of splitting up space-time is by foliating space-time into a congruence of 
time-like curves representing a family of observers. This corresponds to
assigning a set of constant spatial coordinates to each observer, i.e., each
curve of the congruence is characterized by a set of three numbers. There is,
however, no natural choice of simultaneity connecting the local rest spaces
of the different observers representing the family of curves. 

In quasi-metric relativity each metric of the metric family is {\em defined}
from the local measurements of the FOs with the global time function serving 
as the (unique) choice of simultaneity. The construction of quasi-metric 
space-time as consisting of two mutually orthogonal foliations means that it 
must be possible to choose a global time coordinate for each metric of the
metric family. Consequently, it should exist a set of ``preferred'' coordinate
systems which adapt to the FHSs in a natural way. A set of such coordinate 
systems is found by first noting that it exists, by definition of the global 
time function, a unique invertible functional relationship between $x^{0}$ and
$t$ throughout the coordinate patch covered by ${\{}x^{\epsilon}{\}}$. We may 
then define a particular type of coordinate system where this relationship is 
given explicitly by $x^{0}=ct$. Such a coordinate system where $x^0$ and $ct$ 
are identified and thus physically equivalent in ${\cal N}$ we call a 
{\em global time coordinate system} (GTCS). It is usually most convenient to 
use a GTCS since such a coordinate system is better adapted to the 
quasi-metric space-time geometry than any alternative coordinate system. In 
particular cases further simplification is possible if one can find a 
{\em comoving} coordinate system, i.e., a coordinate system where the spatial 
coordinates ${\{}x^i{\}}$ assigned to any FO do not vary along its world line.
That is, the spatial coordinates along such a world line do not depend on $t$ 
in ${\cal N}$. A GTCS which is also a comoving coordinate system we call a 
{\em hypersurface-orthogonal coordinate system} (HOCS). Notice that in 
principle infinitely many GTCSs exist, but that a HOCS can be found only in
particular cases.

If we define a GTCS ${\{}x^{\epsilon}{\}}$, each single metric of the 
one-parameter family can be expressed by its components
$g_{{\mu}{\nu}}(x^{\epsilon})$. The fact that we have a one-parameter
family of metrics rather than a single metric on ${\cal N}$ we express by an 
implicit dependence on the coordinates via the global time function 
$t$. In component notation we write this formally as
$g_{(t){\mu}{\nu}}(x^{\epsilon})$, where the dependence on $t$ is put in
a parenthesis not to confuse it with the coordinate indices. By treating
$t$ as a constant, we get the components of each single metric. Note that
the off-diagonal components $g_{(t)0i}{\equiv}g_{(t)i0}$ (where the spatial
indices $i$ take the values 1,2,3) vanish in a HOCS.

By postulating the existence of the family ${\bf g}_t$ we automatically define 
a family of Levi-Civita connections ${\nabla}_t$, where each connection is 
found from the corresponding metric by holding $t$ constant. But we are more
interested in the affine structure induced on $\cal N$ by the metric family 
than in the Levi-Civita connection of any single metric. Thus we want to 
define a unique covariant derivative in ${\cal N}$, given the functions 
$g_{(t){\mu}{\nu}}$. The natural way to do this is by thinking of the metric 
family as one single degenerate 5-dimensional metric on 
${\cal M}{\times}{\bf R}$, where the degeneracy manifests itself via the 
condition ${\bf g}_t({\frac{\partial}{{\partial}t}},{\cdot}){\equiv}0$. We may 
then introduce a torsion-free, metric-compatible 5-dimensional connection
${\,}{\topstar{\nabla}}{\,}$ on ${\cal M}{\times}{\bf R}$ and consider the 
restriction of ${\,}{\topstar{\nabla}}{\,}$ to $\cal N$; in this way we obtain
the wanted affine structure on $\cal N$. We name the connection 
${\,}{\topstar{\nabla}}{\,}$ the ``degenerate" connection since it is (almost)
compatible with which may be perceived as a degenerate 5-dimensional metric. 
The requirement that ${\,}{\topstar{\nabla}}{\,}$ must be metric-preserving 
reads (a comma denotes partial derivation, and we use Einstein's summation 
convention throughout)
\eqa
{\frac{\partial}{{\partial}t}}g_{(t){\mu}{\nu}}&{\equiv}&
{\frac{\partial}{{\partial}t}}{\bf g}_t
({\frac{\partial}{{\partial}x^{\mu}}},{\frac{\partial}{{\partial}x^{\nu}}})
={\bf g}_t(\ {\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}
{\frac{\partial}{{\partial}x^{\mu}}},{\frac{\partial}{{\partial}x^{\nu}}})+
{\bf g}_t({\frac{\partial}{{\partial}x^{\mu}}},
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}
{\frac{\partial}{{\partial}x^{\nu}}}).
\ena
Moreover we define the degenerate connection coefficients 
${\topstar{\Gamma}}^{\alpha}_{{\mu}t}$, ${\topstar{\Gamma}}^{\alpha}_{t{\mu}}$
and ${\topstar{\Gamma}}^{\alpha}_{{\mu}{\nu}}$ by
\eqa
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}
{\frac{\partial}{{\partial}x^{\mu}}}{\equiv}{\ }
{\topstar{\Gamma}}^{\alpha}_{{\mu}t}{\frac{\partial}{{\partial}x^{\alpha}}},
{\qquad} {\topstar{\Gamma}}^{\alpha}_{t{\mu}}{\equiv}{\ }
{\topstar{\Gamma}}^{\alpha}_{{\mu}t}, {\qquad}
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}x^{\nu}}}}
{\frac{\partial}{{\partial}x^{\mu}}}{\equiv}{\ }{\topstar{\Gamma}}^
{\alpha}_{{\mu}{\nu}}{\frac{\partial}{{\partial}x^{\alpha}}}, \qquad
{\topstar{\Gamma}}^{\alpha}_{{\mu}{\nu}}
{\equiv}{\Gamma}^{\alpha}_{(t){\mu}{\nu}},
\ena
where ${\Gamma}^{\alpha}_{(t){\mu}{\nu}}$ are the connection coefficients
obtained from the Levi-Civita connections ${\nabla}_t$. Note that equation 
(2.2) implies that ${\,}{\topstar{\nabla}}{\,}$ does not have torsion since it
is symmetric. This means that any counterpart to equation (2.1) obtained by 
e.g. interchanging the coordinate vector fields 
${\frac{\partial}{{\partial}t}}$ and ${{\frac{\partial}{{\partial}x^{\mu}}}}$
does not hold in general. So ${\,}{\topstar{\nabla}}{\,}$ does not preserve 
the degenerate part of degenerate 5-dimensional metric on 
${\cal M}{\times}{\bf R}$. Furthermore notice that other degenerate connection
coefficients than those given by (2.2) vanish by definition. This implies that
the gradient of the global time function is covariantly constant, i.e., that
${\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}dt=
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}x^{\mu}}}}dt=0$, see [4] for an 
analogy with Newton-Cartan-theory. 

We must now find expressions for the degenerate connection coefficients
${\topstar{\Gamma}}^{\alpha}_{{\mu}t}$ defined in equation (2.2). If the
metric-preserving condition (2.1) were the only restriction one wanted to
lay on the degenerate connection, a natural choice of degenerate connection
coefficients would be ones totally determined from partial derivatives with
respect to $t$ of the functions $g_{(t){\mu}{\nu}}$, i.e., from terms
containing expressions like ${\frac{\partial}{{\partial}t}}g_{(t){\mu}{\nu}}$.
It can be shown that the requirement that the degenerate connection 
coefficients take this form yields that ${\topstar{\Gamma}}^{\alpha}_{{\mu}t}$
must be equal to ${\frac{1}{2}}g_{(t)}^{{\alpha}{\sigma}}
g_{(t){\sigma}{\mu}},_t$. But in addition to the condition (2.1) we also have
the condition that the geometrical structure of quasi-metric space-time
should not be affected by parallel-transport in the $t$-direction. That is,
we should require that the degenerate  connection ensures that the unit normal
vector field family ${\bf n}_t$ (i.e., a vector field family having the 
property ${\bf g}_t({\bf n}_t,{\bf n}_t)=-1$) of the FHSs will be 
parallel-transported in the $t$-direction. We thus require that
\eqa
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf n}_t=0.
\ena
Now it turns out that the degenerate connection defined from the candidate
degenerate connection coefficients shown above does not satisfy equation
(2.3) in general. However, in particular cases it does, and this should be
significant.

Furthermore, it turns out that a degenerate connection which fulfills both 
(2.1) and (2.3) must necessarily be defined from the form the degenerate 
connection coefficients take in a GTCS. We choose such a degenerate connection
on the grounds that it should reproduce the results of the above-mentioned
candidate degenerate connection for those particular cases where it satisfies
(2.3). It may be shown that this criterion yields the unique choice
(valid in a GTCS)
\eqa
{\topstar{\Gamma}}^{\alpha}_{{\mu}t}={\frac{1}{t}}{\delta}_i^{\alpha}
{\delta}^i_{\mu}.
\ena
Note that the spatial indices occurring in equation (2.4) make the
degenerate connection explicitly dependent on the structure of
quasi-metric space-time.

On ${\cal M}{\times}{\bf R}$, the Levi-Civita connections ${\nabla}_t$ define 
covariant derivatives along vectors tangent to $\cal M$. We extend the 
${\nabla}_t$ by applying these covariant derivatives to the vector
${\frac{\partial}{{\partial}t}}$ also. This must be done in a way that meshes 
with the degenerate connection ${\topstar{\nabla}}$ such that the torsion 
vanishes. By definition we then have (to avoid confusion we drop the $t$-label
of ${\nabla}_t$ in expressions such as ${\nabla}_{\bf X}$)
\eqa
{\Gamma}^{\alpha}_{t{\mu}}{\equiv}{\Gamma}^{\alpha}_{{\mu}t}{\equiv}
{\ }{\topstar{\Gamma}}^{\alpha}_{{\mu}t}=
{\frac{1}{t}}{\delta}_i^{\alpha}{\delta}^i_{\mu}, \qquad
{\nabla}_{{\frac{\partial}{{\partial}x^{\mu}}}}
{\frac{\partial}{{\partial}t}}=
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}x^{\mu}}}}
{\frac{\partial}{{\partial}t}}=
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}
{\frac{\partial}{{\partial}x^{\mu}}},
\ena
and this makes the torsion vanish by definition, since coordinate vector fields
commute.

Any vector field ${\bf z}$ on ${\cal M}{\times}{\bf R}$ can be split up into 
pieces, where one piece $z^{\mu}{\frac{\partial}{{\partial}x^{\mu}}}$ is 
tangential to $\cal M$ and the other piece $z^t{\frac{\partial}{{\partial}t}}$ 
is normal to $\cal M$. To take the degenerate connection 
${\,}{\topstar{\nabla}}{\,}$ along a given direction in 
${\cal M}{\times}{\bf R}$ it is sufficient to know 
${\nabla}_{{\frac{\partial}{{\partial}x^{\nu}}}}$ and
${\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}$ since the degenerate 
covariant derivative ${\topstar{\nabla}}_{\bf z}$ is defined by using the 
appropriate Levi-Civita connection along the piece tangent to $\cal M$ 
and using ${\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}$ along the 
remaining piece. In particular, we find by direct calculation that
\eqa
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}{\bf g}_t&=& 
{\Big(}{\frac{\partial}{{\partial}t}}g_{(t){\mu}{\nu}}-
{\topstar{\Gamma}}^{\alpha}_{{\mu}t}g_{(t){\alpha}{\nu}}-
{\topstar{\Gamma}}^{\alpha}_{{\nu}t}g_{(t){\mu}{\alpha}}{\Big)}
dx^{\mu}{\otimes}dx^{\nu}, \nonumber \\
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}{\bf n}_t&=& 
{\Big(}{\frac{\partial}{{\partial}t}}n_{(t)}^{\mu}+
{\topstar{\Gamma}}^{\mu}_{{\alpha}t}n_{(t)}^{\alpha}{\Big)}
{\frac{\partial}{{\partial}x^{\mu}}}.
\ena
The expressions in equation (2.6) vanish provided that $g_{(t)00}$ and
$n_{(t)}^0$ do not depend on $t$, $g_{(t)0j}{\propto}t$,
$n_{(t)}^j{\propto}t^{-1}$ and $g_{(t)ij}{\propto}t^2$ (expressed in a GTCS).
This dependence on $t$ ensures that members of the metric family and the
unit normal vector family are parallel-transported along
${\frac{\partial}{{\partial}t}}$. Thus the degenerate connection is compatible
with the metric family. This guarantees that different metrics at different 
events can be compared unambiguously.

Let $\lambda$ be some affine parameter describing the world line of an 
arbitrary test particle. The tangent vector field 
${\frac{\partial}{{\partial}{\lambda}}}$ along the curve then has the
coordinate representation
${\frac{dt}{d{\lambda}}}{\frac{\partial}{{\partial}t}}+
{\frac{dx^{\alpha}}{d{\lambda}}}{\frac{\partial}{{\partial}x^{\alpha}}}$, 
where the ${\frac{\partial}{{\partial}x^{\alpha}}}$ are coordinate vector 
fields. Using the coordinate representation of 
${\frac{\partial}{{\partial}{\lambda}}}$ we may define 
${\topstar{\nabla}}_{\frac{\partial}{{\partial}{\lambda}}}{\equiv}
{\frac{dt}{d{\lambda}}}{\,}{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}+
{\frac{dx^{\alpha}}
{d{\lambda}}}{\nabla}_{\frac{\partial}{{\partial}x^{\alpha}}}$; this is
the definition of taking the covariant derivative along a curve in 
${\cal M}{\times}{\bf R}$. The restriction to ${\cal N}$ is easily obtained by
only considering curves in ${\cal N}$, i.e., curves where $x^0=ct$ in a GTCS.

As an example we find the degenerate covariant derivative of a family of 
vector fields ${\bf z}_t$ along an affinely parameterized curve 
$x^{\beta}({\lambda})$ in $\cal N$. Expressed in (arbitrary) coordinates 
this is
\eqa
{\topstar{\nabla}}_{{\frac{\partial}{{\partial}{\lambda}}}}{\bf z}_t  =
{\Big [}{\frac{dz_{(t)}^{\mu}}{d{\lambda}}}+{\Big (}
{\topstar{\Gamma}}^{\mu}_{{\alpha}t}{\frac{dt}{d{\lambda}}}+
{\Gamma}^{\mu}_{(t){\alpha}{\beta}}{\frac{dx^{\beta}}{d{\lambda}}}{\Big )}
z^{\alpha}_{(t)}{\Big ]}{\frac{\partial}{{\partial}x^{\mu}}}.
\ena
Whether or not the fields ${\bf z}_t$ are parallel-transported with respect to 
the degenerate connection ${\,}{\topstar{\nabla}}{\,}$ along the above curve 
depends on whether (2.7) vanishes or not. 

An important special case is when a family of vector fields ${\bf u}_t$ is
identified with the 4-velocity tangent vector field of the curve, that is
\eqa
{\bf u}_t{\equiv}u^{\alpha}_{(t)}{\frac{\partial}{{\partial}x^{\alpha}}}
{\equiv}{\frac{dx^{\alpha}}{d{\tau}_t}}{\frac{\partial}{{\partial}x^{\alpha}}},
\ena
where ${\tau}_t$ is the proper time measured along the curve. Then we may 
calculate the degenerate covariant derivative of ${\bf u}_t$ along the tangent
vector field ${\frac{\partial}{{\partial}{\tau}_t}}$; this yields the 
degenerate acceleration vector field ${\stackrel{\star}{\bf a}}$ defined by
\eqa
{\stackrel{\star}{\bf a}}{\equiv}
{\topstar{\nabla}}_{\frac{\partial}{{\partial}{\tau}_t}}{\bf u}_t
={\frac{dt}{d{\tau}_t}}
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf u}_t+
{\frac{dx^{\alpha}}
{d{\tau}_t}}{\nabla}_{\frac{\partial}{{\partial}x^{\alpha}}}{\bf u}_t{\equiv}
{\frac{dt}{d{\tau}_t}}
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf u}_t+{\bf a}_t.
\ena
Using coordinate notation we see that (2.9) yields equations of motion
\eqa
{\frac{d^2x^{\mu}}{d{\lambda}^2}}+{\Big (}{\topstar{\Gamma}}^{\mu}_{{\alpha}t}
{\frac{dt}{d{\lambda}}}+
{\Gamma}^{\mu}_{(t){\alpha}{\beta}}{\frac{dx^{\beta}}{d{\lambda}}}
{\Big )}{\frac{dx^{\alpha}}{d{\lambda}}}
{\equiv}{\Big (}{\frac{d{\tau}_t}{d{\lambda}}}{\Big )}^2{\topstar{a}}^{\mu},
\ena
where ${\frac{d{\tau}_t}{d{\lambda}}}$ is constant along the curve since
${\lambda}$ and ${\tau}_t$ are both affine parameters. We may compare (2.10)
to the standard coordinate expression for the family of 4-acceleration fields
${\bf a}_t{\equiv}{\nabla}_{{\bf u}_t}{\bf u}_t$ compatible with the
Levi-Civita connections ${\nabla}_t$, namely
\eqa
{\frac{d^{2}x^{\beta}}{d{\lambda}^{2}}}+
{\Gamma}_{(t){\sigma}{\rho}}^{\beta}{\frac{dx^{\sigma}}{d{\lambda}}}
{\frac{dx^{\rho}}{d{\lambda}}}={\Big (}{\frac{d{\tau}_t}{d{\lambda}}}{\Big )}^2
a^{\beta}_{(t)}.
\ena
However the equations (2.11) cannot be used as equations of motion in $\cal N$
since they do not include the dependence of ${\bf u}_t$ on $t$. That is, the
equations (2.11) only hold in $\cal M$.

A geodesic for the degenerate connection ${\,}{\topstar{\nabla}}{\,}$ (or 
equivalently for the metric family) in $\cal N$ is defined as a curve along 
which ${\stackrel{\star}{\bf a}}$ vanishes. Clearly we need to specify 
${\stackrel{\star}{\bf a}}$ independently to determine which curves of 
${\cal N}$ are geodesics for ${\,}{\topstar{\nabla}}{\,}$ and which are not.
We will show later (section 3.4) that in fact
${\topstar{\nabla}}_{{\frac{\partial}{{\partial}t}}}{\bf u}_t=0$,
so we have ${\stackrel{\star}{\bf a}}$=${\bf a}_t$ from equation (2.9). Thus,
similar to the Levi-Civita connection the geodesics for the degenerate
connection are identified with world lines of inertial test particles.
However, as we can see from equations (2.10) and (2.11), in quasi-metric
theory the paths of inertial test particles are different from their
counterparts in metric theory.

Notice that since ${\bf u}_{\cal F}=c{\bf n}_t$ is the 4-velocity of the FOs, 
the FOs always move orthogonally to the FHSs. We may also show that we in 
general have
\eqa
{\topstar{a}}^{\mu}{\frac{dx^{\nu}}{d{\lambda}}}g_{(t){{\mu}{\nu}}}=0, 
\ena
for any curve in ${\cal N}$, thus the degenerate acceleration field
is normal to the tangent vector field (see equation (3.73) for direct
verification). This is necessary to ensure that the length of ${\bf u}_t$ is 
constant along the curve.

A simple illustrative example of a one-parameter family of 4-metrics is 
defined by the line elements (given in spherical GTCS
$(x^{0},r,{\theta},{\phi})$)
\eqa
ds^{2}_t = -(dx^{0})^{2} + 
({\frac{t}{t_{0}}})^{2}{\bigg(}{\frac{dr^{2}}
{1-{\frac{r^{2}}{{\Xi}_{0}^{2}}}}}+r^{2}d{\Omega}^{2}{\bigg)},
\ena
where $d{\Omega}^{2}{\equiv}d{\theta}^{2}+{\sin}^{2}{\theta}d{\phi}^{2}$,
and $t_{0}$ and ${\Xi}_{0}$ are constants. We see that this is a one-parameter 
family of static metrics, each one having the geometry of 
${\bf S}^{3}{\times}{\bf R}$, but with different scale factors. When doing
kinematical calculations using (2.13) one must of course apply the equations 
of motion (2.10). Notice the difference between this situation and that one 
gets by applying the equations (2.11) to the single non-static metric one 
obtains by substituting $cdt=dx^{0}$ into (2.13); the geodesics of this single
metric are not equivalent to the geodesics calculated from the metric family 
(2.13). And although the null paths are the same, the role of $t$ as a 
parameter along these paths is different for the two cases. (See section 4.2 
for some calculations.)

One immediate question is how to calculate the metric family ${\bf g}_t$, the 
obvious answer being that it should be found from appropriate field equations.
However, this does not necessarily mean that ${\bf g}_t$ should represent a 
{\em solution} of said field equations. Rather, we should take the more 
general view that it must be possible to construct the family ${\bf g}_t$ from
another family ${\bf {\bar g}}_t$ which {\em does} represent a solution of said
field equations, together with fields obtainable from the family 
${\bf {\bar g}}_t$. Why we should take this view is explained below.

As mentioned earlier the NKE should be coupled to {\em curvature} of some sort.
At this point, we may guess the qualitative form this ``coupling" equation
should take. To do that we remind of two points. Firstly, in contrast to the 
kinematical relations, this equation should not apply to an arbitrary spatial
hypersurface, but only to the FHSs (where the 4-metrics of the family apply 
exactly). Secondly, the coupling equation should not depend on the surrounding
4-metrics via extrinsic curvature etc., but only via the evolution parameter. 
Thus this equation should essentially be scalar and it should look something 
like
\eqa
{\bar H}_t{\equiv}c{\bar F}_t^{-1}
{\topstar{\cal L}}_{{\bf {\bar n}}_t}{\bar F}_t-c^{-1}{\bar x}_t
={\bar f}_t.
\ena
Here ${\bar F}_t$ are the local scale factors of the tangent spaces of the 
FHSs in ${\bf {\bar g}}_t$, ${\topstar{\cal L}}_{{\bf {\bar n}}_t}$ denotes
the Lie derivative in the direction normal to the FHSs and
$c^{-1}{\bar x}_t$ represents a possible kinematical contribution 
to the change with time of ${\bar F}_t$. Furthermore ${\bar f}_t$ are scalars 
constructed from the geometry intrinsic to the FHSs and the only constant of 
nature we have implicitly used this far, namely the velocity of light.

We want the right hand side of (2.14) to be positive everywhere. This means 
that the NKE should take the form of a local ``expansion" (defined by the 
FOs). In particular, doing cosmology within the quasi-metric framework, the 
Hubble law is interpreted as a special case of (2.14), and the cosmic 
expansion thus follows from first principles, in contrast to the situation 
within the metric framework. We will give the exact form of the 
non-kinematical coupling equation (NKCE) in section 3.3.

From the above we expect that $\cal N$ contains the non-kinematical 
``expansion" as part of the geometrical structure. However, it may not be 
possible to eliminate all the effects of this ``expansion" just by holding $t$
constant. This means that in general each metric of the family ${\bf g}_t$ 
should contain some non-kinematical features integrated into the metric 
structure. That is, for each metric of the family some non-kinematical effects 
are treated as kinematical effects. But this means that the family ${\bf g}_t$
should in general not represent a solution of appropriate field equations, 
since such solutions should not contain any features coming from the NKE of 
the FHSs. Rather, as mentioned previously, the family ${\bf g}_t$ should be 
constructed from a solution family ${\bf {\bar g}}_t$ and fields obtainable 
from ${\bf {\bar g}}_t$. In section 2.3 we show that this construction 
involves a family of vector fields.
\subsection{Uniqueness and existence of the global time function}
In this section we investigate which restrictions should be put on each member
of the family of metrics ${\bf {\bar g}}_t$ to ensure that they are acceptable
according to the quasi-metric framework. The reason that we put the 
restrictions on ${\bf {\bar g}}_t$ and not directly on the ``physical" family
${\bf g}_t$ is that the former family by hypothesis represents a solution of 
field equations; if we know any general properties of solutions this may 
be useful when constructing such equations. 

Obviously we want the postulated global time function to exist always. That 
is, as observed earlier, there always should exist a unique invertible 
functional relationship between the coordinate time $x^0$ and $t$ throughout 
the coordinate patch covered by an arbitrary local coordinate system 
${\{}x^{\epsilon}{\}}$. The requirement that the global time function exists 
leads to restrictions on the metric family ${\bf {\bar g}}_t$. In particular 
we show that the existence of the global time function implies that it is 
possible to describe all the time-dependent aspects of the KE of 
${\bf {\bar g}}_t$ with the help of a scalar function plus a spatial vector 
field. Besides, extrapolating ${\bf {\bar g}}_t$ off the domain FHSs, we 
require that each FHS is a member of a unique foliation of 
${\overline{\cal M}}$ (equipped with any allowable metric) into a family of 
spatial 3-hypersurfaces if we hold $t$ constant and propagate the FHS out into
the surrounding 4-geometry of ${\overline{\cal M}}$. (We write 
${\overline{\cal M}}$ instead of ${\cal M}$ and ${\overline{\cal N}}$ instead 
of $\cal N$ when we deal with the family ${\bf {\bar g}}_t$.) 

To ensure the existence of a global time function, firstly it is important to 
ensure that the gradient field of the global time function in 
${\overline{\cal N}}$, i.e., $dt={\frac{{\partial}t}{{\partial}x^{\alpha}}}
dx^{\alpha}$, is indeed timelike everywhere. (If it were not, a unit time-like 
vector field family ${\bf {\bar n}}_t$ would not exist everywhere on the 
FHSs.) Besides the world lines of different FOs should never intersect; this
condition concerns the geometry of each 4-metric of ${\bf {\bar g}}_t$ in the 
sense that they cannot contain event horizons. By hypothesis the family 
${\bf {\bar g}}_t$ should represent a solution field equations. This means 
that if solutions of candidate field equations do indeed contain event 
horizons, such candidates must be discarded as inconsistent with the 
quasi-metric framework.

Secondly, even if the FHSs are spatial Cauchy hypersurfaces, one must ensure 
that each FHS can be seen as a member of a family of spatial 3-hypersurfaces
foliating ${\overline{\cal M}}$ equipped with any allowable 4-geometry 
${\bf {\bar g}}{\in}{\bf {\bar g}}_t$. A sufficient condition to ensure this 
is to demand that the FHSs are {\em compact} (without boundaries) [3]. Thus, 
to ensure that the global time function exists, we will require that the FHSs 
are Cauchy and compact. Also this will be sufficient to ensure that the global
time function is unique in the sense that it splits space-time into a unique 
family of FHSs. (The above result carries over to the asymptotically Minkowski
case when we replace the condition of compactness with the boundary condition 
that the FHSs asymptotically can be identified with the flat 3-spaces of one 
of the global Lorentz frames of Minkowski space-time, namely the one where the
global time coordinate $x^0=$constant. We feel free to use asymptotically flat
FHSs in particular cases. However, doing this one may leave out crucial 
physical effects; it is important to be aware of that asymptotically flat FHSs
may be used only as an approximation. Thus in principle the FHSs should be 
compact.)

As observed earlier, the existence of a global time function singles out a 
preferred set of coordinate systems in ${\overline{\cal N}}$, namely those
where the identification $x^0=ct$ is made. In the following we specify such a 
GTCS ${\{}x^0,x^{i}{\}}$, where the $x^{i}$ are spatial coordinates for 
$i=1,2,3$. Using the notation of the ADM-formalism [4], an arbitrary metric 
family ${\bf {\bar g}}_t$ can be expressed in a GTCS via the family of line 
elements (we use the signature convention $(-+++)$ throughout):
\eqa
\overline{ds}^{2}_t = {\Big [}{\bar N}_{(t)s}{\bar N}_{(t)}^s-
{\bar N}_t^{2}(x^{\mu},t){\Big ]}(dx^{0})^{2}+
2{\bar N}_{(t)i}(x^{\mu},t)dx^idx^0+
{\bar g}_{(t)ij}(x^{\mu},t)dx^{i}dx^{j}, 
\ena
where ${\bar N}_t$ is the family of lapse functions for the FOs, the 
${\bar N}_{(t)i}$ are the components of the family of shift covectors for the 
FOs and ${\bar g}_{(t)ij}$ are the spatial components of the metric family 
(these are identical to the components of the metric family intrinsic to the 
FHSs). Note that the form (2.15) is too general. That is, one should require 
the spatial part of (2.15) to be conformal to some ``basic'' 3-geometry to 
ensure that there will be no conflict between the NKE and the KE of the FHSs. 
Besides one needs to specify the role of the global time function $t$ 
in (2.15) such that
${\topstar{{\bar {\nabla}}}}_{\frac{\partial}{{\partial}t}}{\bf {\bar g}}_t$
and
${\topstar{{\bar {\nabla}}}}_{\frac{\partial}{{\partial}t}}{\bf {\bar n}}_t$
vanish.

As mentioned above, to ensure that the global time function exists we have 
restricted the 4-manifold ${\overline{\cal N}}$ in a topological sense by 
requiring the existence of compact Cauchy hypersurfaces (without boundaries). 
Besides there must exist a ``basic" 3-geometry in the sense that the 
geometries ${\bar g}_{(t)ij}$ of the FHSs are restricted to be conformal to 
this basic 3-geometry. A natural choice of such a basic 3-geometry is the 
geometry of the 3-sphere ${\bf S}^3$ since it is compact and since its 
positive intrinsic curvature fits well with an equation of type (2.14). We may
take this as a definition, thus any metric family (2.15) should contain
``prior'' 3-geometry that cannot be changed by changing the dynamics of 
gravitating sources. 

Now any dependence on $t$ in (2.15) should reflect non-kinematical aspects of 
the theory whereas any dependence on $x^0$ should reflect kinematical aspects.
Since the non-kinematical evolution should take the form of an ``expansion'', 
it should affect the spatial fields only (if local conservation of energy and 
momentum hold), so we guess that the family of lapse functions should in 
general not depend on $t$ in ${\overline{\cal N}}$. However, even if this 
holds, it is still possible that ${\bar N}_t$ may be a function of the 
quotient ${\frac{x^0}{ct}}$. Besides, in particular cases (in the early 
Universe and in extremely strong gravitational fields) one expects that local 
conservation laws may be violated. That is, when the global NKE is so strong 
that non-gravitational quantum fields cannot be treated as localized to 
sufficient accuracy, one should get spontaneous particle pair creation due to 
excitations of vacuum fluctuations of such quantum fields. But besides these 
possibilities, we guess that the family of lapse functions is equivalent to the
ordinary lapse function. We also guess that the explicit dependence on $t$ of 
the spatial metrics ${\bar g}_{(t)ij}$ enters (2.15) as a conformal factor. As 
justified in section 4, this factor is proportional to $t^2$. But consistency 
then requires that the explicit dependence on $t$ of the family of shift 
covectors must be linear and that the dependence of the family of shift vectors
on $t$ goes as ${\frac{1}{t}}$. Thus we may write 
\eqa
{\bar N}_{(t)}^j={\frac{t_0}{t}}{\bar N}_{(t_0)}^j(x^{\mu}){\equiv}
{\frac{t_0}{t}}{\bar N}^{j}(x^{\mu}), \qquad
{\bar N}_{(t)i}={\frac{t}{t_0}}{\bar N}_{i}(x^{\mu},t){\equiv}
{\bar g}_{(t)ij}{\bar N}^{j}_{(t)},
\ena
so that the family of shift vector fields is adequately represented by one 
ordinary shift vector field. The $t$-dependence in equation (2.15) is now
determined such that
${\topstar{{\bar {\nabla}}}}_{\frac{\partial}{{\partial}t}}{\bf {\bar g}}_t$
and
${\topstar{{\bar {\nabla}}}}_{\frac{\partial}{{\partial}t}}{\bf {\bar n}}_t$
vanish. But since ${\bar N}_t$ may possibly depend on $t$ (see above), the
affine structure on ${\overline{\cal N}}$ will be slightly more general than 
that on ${\cal N}$. That is, the connection coefficients in 
${\overline{\cal N}}$ read (compare to equation (2.5))
\eqa
{\topstar{\bar {\Gamma}}}^0_{0t}={\frac{{\bar N}_t,_t}{{\bar N}_t}}, \qquad
{\topstar{\bar {\Gamma}}}^i_{jt}=({\frac{1}{t}}+
{\frac{{\bar N}_t,_t}{{\bar N}_t}}){\delta}^i_j, \qquad 
{\topstar{\bar {\Gamma}}}^{\alpha}_{{\mu}t}{\equiv}
{\topstar{\bar {\Gamma}}}^{\alpha}_{t{\mu}}, \qquad
{\topstar{\bar {\Gamma}}}^{\alpha}_{{\nu}{\mu}}{\equiv}
{\bar {\Gamma}}^{\alpha}_{(t){\nu}{\mu}}.
\ena
To avoid any restrictions on the dynamically relevant parts of 
${\bf {\bar g}}_t$, the dynamical content of the family ${\bf {\bar g}}_t$ 
should be adequately described by the lapse function family ${\bar N}_t$. 
Thus, to ensure that the existence of a prior 3-geometry should not interfere 
with the dynamics of ${\bf {\bar g}}_t$, it is sufficient that (2.15) takes 
the form
\eqa
\overline{ds}^{2}_t = {\Big [}{\bar N}_{s}(x^{\mu},t){\bar N}^s(x^{\mu})-
{\bar N}_t^{2}(x^{\mu},t){\Big ]}(dx^{0})^{2}+
2{\frac{t}{t_0}}{\bar N}_{i}dx^idx^0+
{\frac{t^2}{t_0^2}}{\bar N}_t^2S_{ij}(x^{\mu})dx^{i}dx^{j}, 
\ena
where $S_{ij}dx^idx^j$ is the metric of the 3-sphere (with constant radius 
equal to $ct_0$, see section 4). Note that the components of the 
inverse metric family ${\bf {\bar g}}_t^{-1}$ are given by [4]
\eqa
{\bar g}_{(t)}^{00}=-{\bar N}_t^{-2}, {\qquad} {\bar g}_{(t)}^{0j}=
{\bar g}_{(t)}^{j0}={\frac{t_0{\bar N}^j}{t{\bar N}_t^2}}, {\qquad}
{\bar g}_{(t)}^{ij}={\bar h}_{(t)}^{ij}-
{\frac{t_0^2{\bar N}^i{\bar N}^j}{t^2{\bar N}_t^2}},
\ena
where ${\bar h}_{(t)}^{ij}={\frac{t_0^2}{t^2}}{\bar N}_t^{-2}S^{ij}$ are the 
components of the inverse spatial metric family ${\bf {\bar h}}_t^{-1}$.

The consequences of (2.18) are rather far-reaching, as it means that the 
dynamical content of the family ${\bf {\bar g}}_t$ should adequately be 
described by the scalar field ${\bar N}_t$ for any theory of gravity 
formulated within the quasi-metric framework. This means that if one specifies
initial conditions on an initial FHS it is sufficient to know the time 
evolution of one scalar field to predict the geometry of all of the 
hypersurfaces, i.e., to recover the space-time geometry. (This should be seen 
in contrast to e.g. General Relativity (GR), where analogously one must know 
the time evolution of the full extrinsic curvature {\em tensor} field since 
it is not restricted a priori.)

We close this section by noticing that the proper time ${\bar {\tau}}_{\cal F}$
of the FOs in ${\overline{\cal N}}$ is given by $d{\bar {\tau}}_{\cal F}
{\equiv}{\bar N}_t(x^{\mu}(t),t)dt$, which may in principle be integrated along
the world line of any FO. We thus see that (1) quasi-metric space-time is
asymmetric; once time-orientation of the FHSs is chosen $t$ must increase when 
${\bar {\tau}}_{\cal F}$ increases, (2) it is convenient to fix the
scale of ${\bar N}_t$ by setting it equal to unity for an isotropic, globally
empty toy universe (section 4), (3) it is possible to scale $t$ (and thus
${\bar {\tau}}_{\cal F}$) with a constant (this corresponds to an ordinary 
units transformation), (4) it is natural to choose $t=0$ for 
${\bar {\tau}}_{\cal F}=0$ as the beginning of time. Once this is done and
the units and the scale of ${\bar N}_t$ are fixed, $t$ is unique.
\subsection{The question of general covariance}
As mentioned above, the existence of a global time function implies that
space-time can be foliated into a set of ``preferred'' spatial hypersurfaces,
that is the FHSs. This geometric structure naturally defines a class of 
``preferred" coordinate systems, namely the GTCSs. The FOs by definition 
always move orthogonally to the FHSs, but the FOs usually have non-zero 
3-velocities in any GTCS. That is, the FOs are in general not stationary with 
respect to any particular coordinate frame. This indicates that there is no 
preferred coordinate frame naturally associated with the quasi-metric 
geometrical framework. (Averaged over cosmic distances however, the FOs are at
rest with respect to a particular set of GTCSs defining the cosmic rest frame.
But this is not a preferred frame in the sense that the outcomes of local 
experiments depend on it.)

It may be shown [15] that the local inertial frames of ${\bf g}_t$ can be 
identified with local Lorentz frames. This means that in quasi-metric theory,
there can be no local consequences of curved space-time for non-gravitational 
physics and that local Lorentz invariance (LLI) holds for non-gravitational 
test experiments. But since there should be no preferred-frame effects 
associated with quasi-metric geometry, there is no reason not to search for a
quasi-metric theory of gravity where LLI holds for {\em all} local experiments
including local gravitational experiments (such as Cavendish experiments). 
Thus there should be no {\em a priori} reasons to prefer metric over 
quasi-metric geometry as the basis for a new theory of gravity.

But another question arises due to the natural existence of GTCSs within 
quasi-metric geometry. The GTCSs represent a set of preferred coordinate 
systems in quasi-metric relativity, so what happens to general covariance?
It was recognized long ago that if one with ``general covariance'' understands
``diffeomorphism-invariance", it is physically vacuous. In this sense the 
quasi-metric framework is, of course, generally covariant. Nor does 
``general covariance" in the sense of invariance of form under general
coordinate transformations contain any general physical principle. (We do not 
expect our framework to be generally covariant in this sense, since (2.18) is
form invariant only for transformations from one GTCS to another.) Rather, the 
modern classification of theories depends on whether or not there exists 
``prior geometry" [1], [4]. This concept refers to any dynamically relevant 
aspect of the geometry that cannot be changed by changing the matter 
distribution.

By construction the quasi-metric framework guarantees the existence of prior 
geometry. After all, we have restricted the 4-metrics of the family 
${\bf {\bar g}}_t$ in such a way that they are of the specific type shown in 
(2.18). But as mentioned in the previous section, the existence of a prior 
3-geometry does not necessarily interfere with the dynamics of the metric 
family ${\bf {\bar g}}_t$. And even if this metric family has only one 
explicit dynamical degree of freedom, as we shall see later a second, 
{\em implicit} dynamical degree of freedom coming from the NKE is present in 
the ``physical'' metric family ${\bf g}_t$. So even if the structure of 
quasi-metric geometry implies the existence of prior geometry there is no 
reason to expect that this prior geometry lays any extra restrictions on the 
dynamics. 

In the next section we try to find for which general cases there exists some 
sort of correspondence with the metric framework.
\subsection{The metric approximation}
In the following we describe how to construct the family ${\bf g}_t$ from the
family ${\bf {\bar g}}_t$ and a family of vector fields. We do this by first 
studying a particular example which we will call a ``metric approximation". A
metric approximation is defined as a metric family ${\bf {\bar g}}_t$ (or
${\bf g}_t$) which does not depend explicitly on the global time function $t$;
thus in effect we have only one metric. This metric necessarily represents an 
approximation since one neglects the effects of a changing global scale 
factor. Note that metric approximations make physically sense for isolated 
systems only. 

Metric approximations cannot work if the FHSs are taken to be compact since 
the NKE associated with the global curvature of space should result in global 
scale changes of the FHSs. And in quasi-metric theory global scale changes
cannot be described by any single metric since the global curvature of space
has nothing to do with gravity. (This is easily seen from the spatial prior 
geometry present in equation (2.18).) However, in particular cases it may 
be a good approximation to relax the requirement of compactness, replacing it
with the requirement of asymptotic flatness. If the FHSs are considered as 
asymptotically flat and embedded in asymptotically Minkowski space-time the 
problem of global scale changes is avoided, since we expect the NKE to vanish 
for flat FHSs. But we still want to identify the FHSs with the
hypersurfaces $x^0=$constant, where $x^0$ is the global time coordinate we
started out with. That is, when making metric approximations we keep the
unique global time coordinate $x^0$ we had to begin with and define the FHSs
from the condition $x^0=$constant. Moreover, we may define a GTCS as a
global time coordinate system where this global time coordinate is used.
But note that after having made a metric approximation, it is possible to
choose other global time coordinates than the one we started out with. 

Metric approximations can be constructed from any one-parameter family
${\bf {\bar g}}_t$ of the form (2.18) by setting ${\frac{t}{t_0}}$ to unity
and replacing $S_{ij}dx^idx^j$ by an Euclidean 3-metric. Thus any metric 
approximation may be written formally as
${\bf {\bar g}}_t{\rightarrow}{\bf {\bar g}}$ (or ${\bf g}_t{\rightarrow}
{\bf g}$); that is, metric families are approximated by single metrics. 
Now we will study the particular case where a GTCS exists in which the family 
${\bf {\bar g}}_t$ does not depend on $x^0$ and the shift field ${\bar N}_j$ 
vanishes; this is called the {\em metrically static} case. Besides, for
reasons explained later, we will require spherical symmetry. In this case the 
metric approximation of the family ${\bf {\bar g}}_t$ is a single {\em static,
spherically symmetric} metric ${\bf {\bar g}}$. That the metric 
${\bf {\bar g}}$ is independent of time means that the FHSs are static, i.e., 
there is no net change of scale anywhere on the FHSs. But this does not mean 
that there are no effects of the NKE. Rather, to construct a new metric 
${\bf g}$ from the metric ${\bf {\bar g}}$ we need to put the NKE of the FHSs 
into the metric structure of ${\bf g}$. This means that if the FOs are 
considered as stationary for the metric ${\bf g}$, then the stationariness of 
the FOs for the metric ${\bf {\bar g}}$ should be seen as a result of both the
non-kinematical expansion and the motion of the FOs taken together. That is, 
the FOs in ${\overline{\cal N}}$ may be perceived as moving with a velocity 
$-{\bf v}$ say, with respect to the FOs in ${\cal N}$. Of course this motion 
is cancelled out when one adds another vector field ${\bf v}$ constructed from
the NKE. Thus one may think of the FOs as ``expanding" and ``falling" 
simultaneously such that the combination results in stationariness. However, 
this way of recovering stationariness is not equivalent to being stationary 
with no NKE in the metric ${\bf {\bar g}}$. We may find the difference between 
${\bf {\bar g}}$ and ${\bf g}$ by quantifying the two different versions of 
stationariness in terms of the vector field ${\bf v}$. As we shall see the 
effect of ${\bf v}$ is that one must introduce correction factors to the 
components of ${\bf {\bar g}}$; these corrections yield the wanted metric 
$\bf g$.

Before proceeding we should choose an appropriate coordinate system. When there
is spherical symmetry with respect to one distinguished point we naturally 
define a spherical GTCS $(x^0,r,{\theta},{\phi})$ with the distinguished point
at the spatial origin. The distinguished point plays the role as a natural 
``center of gravity''. Now, according to (2.14) the NKE takes the form of a 
(in general different) uniform increase in scale for each tangent space of the 
FHSs. Due to the spherical symmetry this increase in scale can be expressed as 
a purely radial, outwards-pointing 3-vector field 
${\bf v}=v^{r}(r){\frac{\partial}{{\partial}r}}$. We thus define ${\bf v}$ by 
applying the ``Hubble law'' in {\em the tangent space} of each point of the 
FHSs. Besides, the distance $\ell$ to the spatial origin in the tangent space 
of each point on the FHSs is given by ${\ell}=r{\sqrt{{\bar g}_{rr}}}$. The 
radial 3-vector field ${\bf v}$ and its speed $v{\equiv}{\sqrt{v^2}}$ are thus
defined by (where ${\bar H}$ is the metric approximation of ${\bar H}_t$ given
in (2.14))
\eqa
v^{r}(r)={\bar H}(r)r, {\qquad} 
v(r)={\bar H}(r){\ell}(r)={\bar H}(r){\sqrt{{\bar g}_{rr}(r)}}r.
\ena
We now calculate the new metric ${\bf g}$ when ${\bf {\bar g}}$ is static and 
spherically symmetric, as assumed above. In this case the spherical GTCS 
defined above is in fact a HOCS and the metric takes the standard form
\eqa
\overline{ds}^{2} = -{\bar B}(r)(dx^0)^{2} + {\bar A}(r)dr^{2} + 
r^{2}d{\Omega}^{2},
\ena 
where ${\bar A}(r)$ and ${\bar B}(r)$ are smooth functions without zeros and 
$d{\Omega}^{2}{\equiv}d{\theta}^{2}+{\sin}^{2}{\theta}d{\phi}^{2}$.

We now try to find a new metric ${\bf g}$ from the metric ${\bf {\bar g}}$ of 
the type (2.21) and the vector field $\bf v$. From the NKE the FOs in 
${\overline{\cal N}}$ get a net 3-velocity 
${\bf v}=v^r{\frac{\partial}{{\partial}r}}$ in the positive radial direction 
of the spherical radial coordinate system according to equation (2.20). To get 
rid of this the FOs in ${\overline{\cal N}}$ must move with a velocity 
$-{\bf v}$. That is, the static condition of the FOs in ${\overline{\cal N}}$ 
may be seen as the result of adding two opposite 3-velocities. Moreover all 
observers in ${\cal N}$ are also static, but in a way that does not involve 
any ``motion'' by definition. We now want to find the time and length 
intervals as expressed by the static FOs in $\cal N$. We do this for each 
tangent space, and taken together the result will be a new metric $\bf g$ 
derived from the metric ${\bf {\bar g}}$ given in equation (2.21). And just as
${\bf {\bar g}}$ is the metric approximation of a spherically symmetric, 
metrically static family ${\bf {\bar g}}_t$, the new metric $\bf g$ may be 
thought of as the metric approximation of a new spherically symmetric,
metrically static family ${\bf g}_t$.

We proceed in two steps, each step giving the same correction factors to the 
metric (2.21). Step one takes into account how the length and time intervals 
of ``static'' FOs are seen by observers moving with 3-velocity $-{\bf v}$ and 
by definition ``at rest'', before taking the NKE into account. That is, the 
``static'' FOs are not ``at rest'' but move outwards with velocity ${\bf v}$ 
compared to the observers ``at rest'' when the NKE is not taken into account. 
In each tangent space this extra motion yields a correction to the radial 
interval due to the radial Doppler effect, the correction factor being 
${\Big(}{\frac{1+{\frac{v}{c}}}{1-{\frac{v}{c}}}}{\Big)}^{1/2}$. There will 
also be an inverse time dilation correction factor 
$(1-{\frac{{v}^2}{c^2}})^{1/2}$ to the time interval. There are no correction 
factors for the angular intervals. 

Step two transforms the effects of the NKE in ${\overline{\cal N}}$ into 
correction factors to the metric coefficients. In 
${\overline{\cal N}}$ the NKE is described as a velocity ${\bf v}$ in the 
positive radial direction for each tangent space, and the effects of $\bf v$
as seen by static FOs, now by definition ``at rest'', are taken care of by 
using a pair of correction factors identical to those in step one.

So the new metric $\bf g$ constructed from the metric ${\bf {\bar g}}$
given in equation (2.21) and $\bf v$ reads
\eqa
ds^{2} = -{\Big(}1-{\frac{{v}^{2}(r)}{c^{2}}}{\Big)}^{2}{\bar B}(r)(dx^0)^{2}+ 
{\bigg(}{\frac{1+{\frac{v(r)}{c}}}{1-{\frac{v(r)}{c}}}}
{\bigg)}^{2}{\bar A}(r)dr^{2} + r^{2}d{\Omega}^{2},
\ena
and it is clear that we must have $v(r)<c$ everywhere if the above expression 
is to make sense. Note that any metric approximation works because equations 
(2.10) reduce to the geodesic equations (2.11) for metric approximations.

It must be stressed that any metric approximation is adequate only if one can
sensibly neglect the NKE coming from a global scale factor. This observation 
leads to the question which vector fields one should use when constructing
${\bf g}_t$ from ${\bf {\bar g}}_t$ for the general case. The answer clearly 
depends on the evolution of the FHSs. That is, the FHSs are not stationary 
as is in the above case but will in general expand. However the FHSs may in 
principle even contract in some places due to gravitation. This means that the
net change of scale for each tangent space at points of each FHS may be 
positive or negative. But to construct the radial vector fields ${\bf v}_t$ 
for the general case we are interested in that part of the evolution of the 
FHSs which is not ``realized'', i.e., not reflected directly in
${\bf {\bar g}}_t$. When a metric approximation is sufficient this 
non-realized part of the evolution is represented by ${\bar H}$. In the 
general case, however, the non-realized part of the evolution of the FHSs is 
not represented by ${\bar H}$ but some other quantity ${\bar y}_t$, say.

This means that for the general case, we should replace ${\bar H}_t$ with 
${\bar y}_t$. But besides this there is a more subtle problem. Namely, that 
there is no obvious choice of ``the center of gravity'' in the absence of 
spherical symmetry (this is why we imposed spherical symmetry when deriving 
equations (2.20) and (2.22)). That is, in the general case one needs to find a
3-vector field family ${\bf b}_{\cal F}$ representing ``a local distance 
vector field'' from some fictitious center of gravity, and such that 
${\bf b}_{\cal F}$ reduces to the radial coordinate vector field
$r{\frac{\partial}{{\partial}r}}$ for the spherically symmetric case. We will
define ${\bf b}_{\cal F}$ in section 3.3. Meanwhile we show how to construct 
${\bf g}_t$ from ${\bf {\bar g}}_t$ and ${\bf v}_t$ when it is assumed that 
${\bf b}_{\cal F}$ is known.

To begin with, using the quantities ${\bar y}_t$ and 
${\bf b}_{\cal F}$, we may define the vector field family ${\bf v}_t$ 
just as we defined equation (2.20); this 3-vector field family represents the 
generalization of the vector field ${\bf v}$ used for spherically symmetric 
metric approximations. Instead of (2.20) we define (using an arbitrary GTCS)
\eqa
v^{j}_{(t)}={\bar y}_tb_{\cal F}^j, \qquad 
v={\bar y}_t{\sqrt{{\bar g}_{(t)ik}b_{\cal F}^ib_{\cal F}^k}}.
\ena
Note that the explicit $t$-dependence should cancel out in the second formula 
and that we can eliminate any possible implicit $t$-dependence via ${\bar N}_t$
by making the substitution $t=x^0/c$. Also note that the NKE in the tangent 
space of each point of the FHSs still arises from ${\bar H}_t$. But the part 
of the NKE coming from a global scale factor must be subtracted from 
${\bar H}_t$ when constructing ${\bf v}_t$; we thus get (2.23) instead of 
(2.20). We will call the effects of the NKE associated with ${\bf v}_t$ the 
{\em local} effects of the NKE.

In the following we will write down the formulae for the components of 
${\bf g}_t$ which must hold for the general case. The main difference from the
spherically symmetric case is that the ``extra motion'' of the FOs in 
${\overline{\cal N}}$ is locally along the ${\bf b}_{\cal F}$-direction
rather than in the radial direction. But besides that, the principles behind 
the construction of ${\bf g}_t$ are similar. To begin with we notice that the 
transformation formula should be identical to the spherically symmetric case 
when no spatial components are involved. That is, as in equation (2.22) we 
define
\eqa
g_{(t)00}={\Big (}1-{\frac{v^2}{c^2}}{\Big )}^2{\bar g}_{(t)00}.
\ena
To guess reasonable transformation formulae when spatial components are 
involved it is convenient to define the unit vector field 
${\bf {\bar e}}_{b}{\equiv}{\frac{t_0}{t}}{\bar e}_{b}^i
{\frac{\partial}{{\partial}x^i}}$ and the corresponding covector field 
${\bf {\bar e}}^{b}{\equiv}{\frac{t}{t_0}}{\bar e}_i^{b}dx^i$ along 
${\bf b}_{\cal F}$. But it is in general not possible or practical to 
construct a GTCS where ${\bf {\bar e}}_{b}$ is parallel to one of the 
coordinate vector fields. For this reason one expects that the general 
transformation formulae for spatial components should involve components of 
${\bf {\bar e}}^{b}$. By trial and error (involving the spherically symmetric
case using non-spherical coordinates) we find the following transformation 
formula for the purely spatial components of ${\bf g}_t$:
\eqa
g_{(t)ij}={\bar g}_{(t)ij}+{\frac{t^2}{t_0^2}}{\frac{4{\frac{v}{c}}}
{(1-{\frac{v}{c}})^2}}{\bar e}^{b}_i{\bar e}^{b}_j.
\ena
The transformation formula (2.25) has the correct correspondence with the
result found in equation (2.22) and it may be shown that this correspondence
holds in any GTCS. Moreover, from equations (2.24) and (2.25) and consistency
requirements we may find the remaining components of ${\bf g}_t$, namely
\eqa
g_{(t)0j}={\Big (}1-{\frac{v^2}{c^2}}{\Big )}{\Big [}{\bar g}_{(t)0j}+
{\frac{t}{t_0}}{\frac{2{\frac{v}{c}}}{1-{\frac{v}{c}}}}
({\bar e}_{b}^i{\bar N}_i){\bar e}_j^{b}{\Big ]}.
\ena
Given equations (2.24), (2.25) and (2.26) it is now straightforward to find
the transformation formulae valid for ${\bf g}_t^{-1}$.
\section{Equations of evolution}
Our goal now is to construct a theory of gravity consistent with the 
quasi-metric framework laid out in section 2. The class of such theories must 
satisfy the postulates: (1) space-time is constructed as a 4-dimensional 
submanifold ${\cal N}$ of a 5-dimensional product manifold 
${\cal M}{\times}{\bf R}$ as described in section 2. This implies that 
space-time is equipped with a global time function, an one-parameter family of 
Lorentzian 4-metrics ${\bf g}_t$ (constructible from another family 
${\bf {\bar g}}_t$ where each metric is restricted by the constraints stated 
in section 2.1), and a degenerate connection ${\topstar{\nabla}}$ compatible 
with the metric family, (2) inertial test particles follow curves calculated 
from equation (2.10) with the condition ${\bf a}_t=0$ and where the relation 
between ${\stackrel{\star}{\bf a}}$ and ${\bf a}_t$ must be specified,
(3) in the local inertial frames of the metric family ${\bf g}_t$, the
non-gravitational physics is as in special relativity (SR).

Postulate (3) means that it should be possible to identify the local inertial 
frames of ${\bf g}_t$ with local Lorentz frames, see [15] for verification. 
This implies that the quasi-metric framework by itself induces no local 
consequences of curved space-time on the non-gravitational physics. Thus both 
LLI and local positional invariance (LPI) should hold within the quasi-metric 
framework. But for any finite scale the free fall curves of ${\bf g}_t$ as 
obtained from from (2.10) do not coincide with the geodesics any single 
space-time metric, see [15] for a further discussion. Furthermore this 
deviation depends on position and on the velocity with respect to the FOs. 
Such dependence implies that non-gravitational laws may appear different from 
their counterparts in any metric theory for any finite-sized observer. 

On the other hand, postulate (2) ensures that the weak equivalence principle 
(WEP) is fulfilled (i.e., that test particles' trajectories do not depend on 
the particles' internal composition). The sum of the WEP, LLI and LPI is 
called the Einstein equivalence principle (EEP); every metric theory of 
gravity satisfies the EEP [1]. The proposition that every consistent theory 
which satisfies the WEP also satisfies the EEP is called Schiff's conjecture 
(SC) [1]. Thus SC continues to hold for any theory consistent with the above 
postulates.

From equation (2.18) we see that how the scale of the FHSs varies from event 
to event is given by the scale factor ${\bar F}_t{\equiv}{\bar N}_tct$. 
One may interpret this scale factor as resulting from a position-dependent 
transformation of length units. Because any dependence on scale of physical 
laws should be experimentally determined only by the variation with position 
of dimensionless numbers, we are free to {\em define} as constants physical 
constants that cannot be related to other such constants by dimensionless 
numbers (see e.g. [5]). Thus, by definition the velocity of light $c$ in 
vacuum and Planck's constant ${\hbar}$ are constants, i.e., independent of 
${\bar F}_t$. But when writing down line elements describing the geometry of 
space-time, some set of units are implicitly assumed used. Usually one does 
not separate between units defined from purely non-gravitational systems and 
other types of units. However, this must be done if one wants to describe 
position-dependent scale changes between non-gravitational and gravitational 
systems.
 
We are thus led to the notion that any fixed ``atomic'' units, i.e., units
operationally defined by means of non-gravitational forces only, may be
considered as formally variable throughout space-time; this applies both to
${\overline{\cal M}}$ and to ${\overline{\cal N}}$. That $c$ and ${\hbar}$ are
not formally variable by definition implies that the variation of length and 
time units are similar and inverse to that of energy (or mass) units. One may 
expect that the formally variable units should represent scale changes between
non-gravitational and gravitational systems, where said scale changes should 
be found from a theory of gravity.

This means that all physical laws are formulated in terms of ``variable''
atomic units. That is, gravitational quantities are considered as variable 
measured in such units. But which quantities do we define as gravitational?
Not unreasonably we define the family ${\bf {\bar g}}_t$ as gravitational
since ${\bf {\bar g}}_t$ is by definition a solution of gravitational field
equations. Moreover the Newtonian gravitational ``constant'' $G_t$ and the
stress-energy tensor ${\bf T}_t$ considered as a source of gravity should be
defined as gravitational quantities. 

It is convenient to introduce a scalar field ${\Psi}_t$ which defines how 
atomic time units vary in space-time. Any gravitational quantity should then 
be associated with ${\Psi}_t$ to some power; this factor should tell how the 
field changes due to the formally variable units. To illustrate this point we 
note that $G_t$ has a formal variation similar to that of time units squared. 
Since $G_t$ is a gravitational quantity this means that the extra function 
associated with $G_t$ is a factor ${\Psi}_t^{-2}$. That is, measured in atomic
units $G_t$ will not be a constant. Now, one obvious consequence of this is to
postulate that $G_t$ varies as ${\Psi}_t^{-2}$. But since $G_t$ always occurs
in combination with masses it is more convenient to {\em define} $G_t$ to 
take a constant value $G$ in atomic units and rather differentiate between 
{\em active mass} and {\em passive mass}. That is, active mass plays the role 
as a source of gravity and varies measured in atomic units. On the other hand 
passive mass (inertial mass or passive gravitational mass) does not vary and 
should be used in atomic interactions. In the following we will denote any 
active mass $m_t$ (corresponding to a passive mass $m$) with an index. 

Mass considered as a gravitational quantity scales via the variation of mass 
units, which vary as ${\Psi}_t^{-1}$. Thus seen active mass is associated with
a factor ${\Psi}_t$. But since we have defined the gravitational coupling 
parameter to be constant we must explicitly include the variation of the 
gravitational coupling into the active mass. This means that {\em active mass 
$m_t$ varies as ${\Psi}_t^{-1}$ measured in atomic units}. Furthermore, since 
matter fields should be coupled to geometry via the active stress-energy 
tensor ${\bf T}_t$ which dimension is (active) mass density, ${\bf T}_t$ 
considered as a source of gravity is associated with an extra factor 
${\Psi}_t^2$. This means that ${\bf T}_t$ experiences extra changes due to the
formally variable units, and this must be taken care of in local conservation 
laws.

The question now is how ${\Psi}_t$ (or equivalently $m_t$) changes from event 
to event in ${\overline{\cal N}}$. Since line elements of the type (2.18)
are measured in atomic units we propose that any change in the scale factor 
${\bar N}_tct$ can be explained solely by means of the formally variable 
units. This will work if
\eqa 
{\Psi}_t{\equiv}{\bar F}_t^{-1}={\frac{1}{{\bar N}_tct}} \qquad \Rightarrow 
\qquad {\Psi}_t={\frac{t_0{\bar N}_0}{t{\bar N}_t}}{\Psi}_0, {\qquad}
{m}_t={\frac{t{\bar N}_t}{t_0{\bar N}_0}}{m}_0,
\ena
where ${\Psi}_0$, $m_0$ etc. refer to values taken at some arbitrary reference
event. By convention we may choose ${\bar N}_0=1$, ${\Psi}_0={\frac{1}{ct_0}}$
which means that the chosen (hypothetical) reference situation is an empty
universe at epoch $t_0$ (see section 4).
Note that the formal variability of gravitational quantities should be 
unaffected by the construction of the family ${\bf g}_t$ from the family 
${\bf {\bar g}}_t$ since this construction only involves the formally 
invariant vector fields ${\bf v}_t$.

The strong equivalence principle (SEP) is essentially a generalization of the 
EEP. That is, for a theory to satisfy the SEP, the EEP must be valid for 
self-gravitating bodies and gravitational test experiments as well as for 
non-gravitating bodies and non-gravitational test experiments, see e.g. [1].
Since theories consistent with postulate (3) satisfy the EEP, one might
hope to find a quasi-metric theory that satisfies the SEP as well. However,
since active mass is not equal to passive mass the SEP in its most stringent
interpretation cannot be satisfied within the quasi-metric framework; all we 
can hope for is that the gravitational weak principle of equivalence (GWEP) 
(that the WEP is valid for self-gravitating bodies as well) holds.

But the result of any laboratory experiment testing the constancy of $G$ 
should reveal only the variation of $G$ via the formally variable units and 
not any extra, independent kind of variability. Thus we should be able to 
investigate the dynamical properties of ${\bf {\bar g}}_t$ without having to 
worry about a ``physically" variable $G$, independent of the formally variable
units. Besides the formal variability of ${G}$ should depend only on 
quantities constructed from pieces of the metric family ${\bf {\bar g}}_t$ and
not on any extra gravitational fields. But this means that we may be able to 
construct a theory of gravity, consistent with the quasi-metric framework, 
which satisfies the GWEP. In fact the GWEP holds since it may be shown that 
active mass dust particles really move on geodesics of ${\bf {\bar g}}_t$ and
of ${\bf g}_t$ [15].
\subsection{Projections}
If we define a GTCS ${\{}x^{\mu}{\}}={\{}x^0,x^{i}{\}}$ this can be used to
find the coordinate representations of the orthonormal vector field basis 
${\{}{\bf {\bar n}}_t,{\bf e}_{i}{\}}$ defining the moving frame field spanned 
by the FOs, where ${\bf e}_{i}{\equiv}{\frac{\partial}{{\partial}x^{i}}}$.
The vector field family ${\bf {\bar n}}_t$ is identified with the unit-length 
vector field family $({\bar n}_{(t){\mu}}{\bar n}_{(t)}^{\mu}=-1)$ tangent to 
the world lines of the FOs, but ${\bf {\bar n}}_t$ is in general not parallel 
to the coordinate vector field ${\frac{\partial}{{\partial}x^0}}$. For
the coordinate representation of ${\bf {\bar n}}_t$ in a GTCS, see below. We 
can use ${\bf {\bar n}}_t$ and its corresponding covector field together 
with the metric tensor family ${\bf {\bar h}}_t$ intrinsic to the FHSs to
project space-time objects into space objects intrinsic to the FHSs. That is, 
repeated scalar products with ${\bf {\bar n}}_t$ eventually reduce space-time 
objects to space scalars, whereas repeated scalar products with 
${\bf {\bar h}}_t$ eventually reduce space-time objects to space objects of 
the same type and rank. Combinations of these operations are possible. To 
avoid confusion about whether we are dealing with a space-time object or its 
spatial projection, we will label space objects with a ``hat'' if necessary. 
That is, the space objects ${\bf {\hat A}}_t$ are the spatial projections of
the space-time objects ${\bf A}_t$. 

Component-wise, we define a label ${\bar {\perp}}$ to indicate a projection by 
$-{\bf {\bar n}}_t$, whereas spatial covariant indices indicate spatial 
projections. Note that contravariant spatial indices do {\em not} indicate 
spatial projections; here one must use the ``hat'' label to differentiate 
between the components of a space-time object and its spatial projection. As 
an example we take the space vectors ${\hat Q}^{j}_{(t){\bar {\perp}}}{\equiv}
-{\bar h}_{(t){\beta}}^jQ^{\beta}_{(t){\alpha}}{\bar n}_{(t)}^{\alpha}$ which
are mixed projections of the the space-time tensors $Q^{\beta}_{(t){\alpha}}$.
The corresponding space covectors are ${\hat Q}_{(t)j{\bar {\perp}}}{\equiv}
-{\bar h}_{(t)j}^{\beta}Q_{(t){\beta}{\alpha}}{\bar n}_{(t)}^{\alpha}$.

The decomposition of ${\bf {\bar g}}_t$ and its inverse with respect to each 
domain FHS is given by
\eqa
{\bar g}_{(t){\mu}{\nu}}&=&-{\bar n}_{(t){\mu}}{\bar n}_{(t){\nu}} + 
{\bar h}_{(t){\mu}{\nu}}, \nonumber \\
{\bar g}_{(t)}^{{\mu}{\nu}}&=&-{\bar n}_{(t)}^{\mu}{\bar n}_{(t)}^{\nu} + 
{\bar h}_{(t)}^{{\mu}{\nu}},
\ena
where ${\bar h}_{(t){\mu}{\nu}}$ are the components of ${\bf {\bar h}}_t$. We
see from (3.2) that ${\bar h}_{(t){\mu}{\nu}}$ represents the restriction to 
the domain FHS of each space-time metric ${\bar g}_{(t){\mu}{\nu}}$. In the 
GTCS, ${\bf {\bar h}}_t$ considered as a space-time tensor field, has the 
potential non-zero mixed or time components
\eqa
{\bar h}_{(t)00}={\bar N}_{k}{\bar N}^k, \qquad
{\bar h}_{(t)j0}={\bar h}_{(t)0j}={\frac{t}{t_0}}{\bar N}_j, \qquad 
{\bar h}_{(t)0}^j={\frac{t_0}{t}}{\bar N}^j=
{\frac{t_0}{t}}{\bar N}^{-2}_tS^{js}{\bar N}_s,
\ena
whereas ${\bf {\bar n}}_t$ has the coordinate representation
\eqa
{\bar n}_{(t)}^0={\bar N}_t^{-1}, \qquad {\bar n}_{(t)}^j=
-{\frac{t_0}{t}}{\frac{{\bar N}^j}{{\bar N}_t}}, \qquad
{\bar n}_{(t)0}=-{\bar N}_t, \qquad {\bar n}_{(t)j}=0.
\ena
Using (3.2), (3.3) and (3.4) we may verify that 
${\bar h}_{(t)}^{{\mu}{\nu}}{\bar n}_{(t){\nu}}=
{\bar h}_{(t){\mu}{\nu}}{\bar n}_{(t)}^{\nu}=0$,
${\bar g}_{(t){\bar {\perp}}{\bar {\perp}}}=-1$, 
${\bar g}_{(t){\bar {\perp}}j}={\bar g}_{(t)j{\bar {\perp}}}=0$, 
${\bar g}_{(t)ij}={\bar h}_{(t)ij}$ and
${\bar g}_{(t)}^{ij}={\bar h}_{(t)}^{ij}-
{\frac{t_0^2}{t^2}}{\frac{{\bar N}^i{\bar N}^j}{{\bar N}_t^2}}$.

Any metric ${\bf {\bar g}}{\in}{\bf {\bar g}}_t$ and 
${\bf {\tilde g}}{\in}{\bf {\tilde g}}_t$ (where 
${\bf {\tilde g}}_t{\equiv}{\bar N}_t^{-2}{\bf {\bar g}}_t$) define symmetric 
linear metric connection fields ${\bf {\bar {\nabla}}}$ and 
${\bf {\tilde {\nabla}}}$ respectively. In component notation, we denote the 
operation of ${\bf {\bar {\nabla}}}$ on space-time objects by the symbol `;'. 
To illustrate this by an example, if ${\bf Y}_t$ are vector fields, the 
components of the tensor fields ${\bf {\bar {\nabla}}Y}_t$ are 
$Y^{\alpha}_{(t);{\beta}}$. The link between the Christoffel symbols 
corresponding to ${\bf {\bar {\nabla}}}$ and ${\bf {\tilde {\nabla}}}$ is
\eqa
{\tilde {\Gamma}}^{\alpha}_{(t){\beta}{\gamma}}=
{\bar {\Gamma}}^{\alpha}_{(t){\beta}{\gamma}}-
{\frac{{\bar N}_t,_{\gamma}}{{\bar N}_t}}{\delta}^{\alpha}_{\beta}
-{\frac{{\bar N}_t,_{\beta}}{{\bar N}_t}}{\delta}^{\alpha}_{\gamma}+
{\frac{{\bar N}_t,_{\epsilon}}{{\bar N}_t}}
{\bar g}_{(t)}^{{\alpha}{\epsilon}}{\bar g}_{(t){\beta}{\gamma}}.
\ena
By projecting the connections into the FHSs, we can define 
hypersurface-intrinsic connection fields ${\bar {\triangle}}_t$ and
${\tilde {\triangle}}_t$ compatible with the hypersurface-intrinsic metrics 
${\bf {\bar h}}_t$ and ${\bf {\tilde h}}_t$ respectively. In component notation
we denote the operation of ${\bar {\triangle}}_t$ on space objects by 
the symbol `${\mid}$'. (Since we never mix fields constructed from the 
different families ${\bf g}_t$ and ${\bf {\bar g}}_t$ we let the symbols `;' 
and `${\mid}$' denote space-time covariant derivation respectively spatial 
covariant derivation also in $\cal M$. Doing otherwise would imply a serious 
risk of running out of appropriate symbols.)

Also, the manifold structure of ${\overline{\cal M}}$ (and of $\cal M$) 
naturally defines a Lie derivative {\pounds}$_{{\bf X}}$ which operates on
space-time objects. We may also define a spatial Lie derivative 
${\bf {\cal L}_{X}}$ which operates only on space objects, where the vector
field ${\bf X}$ is still treated as a space-time vector field. See ref. [7] 
for the detailed projection formulae relating {\pounds}$_{\bf X}$ and
${\bf {\cal L}_X}$. Analogously the manifold structure of 
${\overline{\cal N}}$ (and of $\cal N$) defines the Lie derivative 
${\topstar{${\pounds}$}}_{{\,}{\bf X}}$ and its spatial counterpart 
${\topstar{\cal L}}_{{\,}{\bf X}}$. The only differences between the Lie 
derivatives in ${\overline{\cal M}}$ and ${\overline{\cal N}}$ arise from the
$t$-dependency of the metric family.

Furthermore the metric families ${\bf {\bar g}}_t$ and ${\bf {\tilde g}}_t$ 
are compatible with the degenerate connections 
${\,}{\topstar{\bar {\nabla}}}{\,}$ and ${\,}{\topstar{\tilde {\nabla}}}{\,}$,
respectively. It may be easily shown that the link between the corresponding 
connection coefficients is
\eqa
{\topstar{\tilde {\Gamma}}}^{\alpha}_{{\beta}t}=
{\frac{1}{t}}{\delta}_i^{\alpha}{\delta}^i_{\beta}, \qquad
{\topstar{\tilde {\Gamma}}}^{\alpha}_{{\beta}{\gamma}}=
{\topstar{\bar {\Gamma}}}^{\alpha}_{{\beta}{\gamma}}-
{\frac{{\bar N}_t,_{\beta}}{{\bar N}_t}}{\delta}^{\alpha}_{\gamma}-
{\frac{{\bar N}_t,_{\gamma}}{{\bar N}_t}}{\delta}^{\alpha}_{\beta}+
{\frac{{\bar N}_t,_{\epsilon}}{{\bar N}_t}}
{\bar g}_{(t)}^{{\alpha}{\epsilon}}{\bar g}_{(t){\gamma}{\beta}}.
\ena
In component notation we use the symbols `${\bar *}$' and `${\tilde *}$' to
denote the operations on space-time fields of 
${\,}{\topstar{\bar {\nabla}}}{\,}$ and ${\,}{\topstar{\tilde {\nabla}}}{\,}$,
respectively. Just as for metric space-time geometry one may construct higher 
rank tensor fields by using these operators. As an example of how this is is 
done, take a vector field family ${\bf Y}_t$ and construct 
${\,}{\topstar{\bar {\nabla}}}{\,}{\bf Y}_t$ defined by its components in a 
GTCS; $Y_{(t){\bar *}i}^{\alpha}{\equiv}Y_{(t);i}^{\alpha}$, 
$Y^{\alpha}_{(t){\bar *}0}{\equiv}Y^{\alpha}_{(t);0}+
c^{-1}Y^{\alpha}_{(t){\bar *}t}=Y^{\alpha}_{(t);0}+c^{-1}Y^{\alpha}_{(t)},_t
+c^{-1}{\topstar{\bar {\Gamma}}}^{\alpha}_{t{\beta}}Y_{(t)}^{\beta}$. Since 
the extra terms represent the components of a 4-vector,
${\,}{\topstar{\bar {\nabla}}}{\,}{\bf Y}_t$ is a family of rank 2 tensor 
fields. Similarly the covariant divergence ${\,}{\topstar {\bar {\nabla}}}
{\,}{\cdot}{\bf Y}_t$, or in component notation 
$Y^{\alpha}_{(t){\bar *}{\alpha}}{\equiv}c^{-1}Y^0_{(t){\bar *}t}+
Y^{\alpha}_{(t);{\alpha}}=
(c{\bar N}_t)^{-1}{\frac{\partial}{{\partial}t}}Y_{(t){\bar {\perp}}}+
Y^{\alpha}_{(t);{\alpha}}$, is a scalar field.

Next we define the 4-acceleration field ${\bf {\bar a}}_{\cal F}$ corresponding
to each normal unit vector field ${\bf {\bar n}}_{(t)}$ by
$c^{-2}{\bar a}_{\cal F}^{\mu}{\equiv}
{\bar n}^{\mu}_{(t);{\nu}}{\bar n}_{(t)}^{\nu}$. 
This is a purely spatial vector field since ${\bar a}_{{\cal F}{\bar {\perp}}}=
-{\bar a}_{{\cal F}{\mu}}{\bar n}_{(t)}^{\mu}=
-{\bar a}_{\cal F}^{\mu}{\bar n}_{(t){\mu}}=0$.
Besides, from (3.4) and the definition of ${\bf {\bar a}}_{\cal F}$ we find
\eqa
c^{-2}{\bar a}_{{\cal F}j}={\frac{\partial}{{\partial}x^{j}}}{\ln}{\bar N}_t.
\ena
Furthermore we define a family of extrinsic curvature tensors 
${\bf {\bar K}}_t$ describing how each FHS is embedded in the relevant 
${\bf {\bar g}}{\in}{\bf {\bar g}}_t$. That is, we define
\eqa
{\bar K}_{(t){\mu}{\nu}}{\equiv}
-{\bar h}^{\alpha}_{(t){\mu}}{\bar h}^{\beta}_{(t){\nu}}
{\bar n}_{(t){\alpha};{\beta}},
\ena
which is symmetric since there is no space-time torsion in 
${\overline{\cal M}}$. The extrinsic curvature tensors ${\bf {\tilde K}}_t$ of
the FHSs in ${\bf {\tilde g}}_t$ are defined analogously. Projection shows 
that each ${\bf {\bar K}}_t$, ${\bf {\tilde K}}_t$ is a purely 
hypersurface-intrinsic object, i.e.,
${\bar K}_{(t){\bar {\perp}}{\bar {\perp}}}=
{\bar K}_{(t){\bar {\perp}}j}={\tilde K}_{(t) {\bar {\perp}}{\bar {\perp}}}=
{\tilde K}_{(t){\bar {\perp}}j}=0$. 
It can also be readily shown [2],[4],[7] that ${\bf {\bar K}}_t$,
${\bf {\tilde K}}_t$ may be written as ${\bar K}_{(t){\mu}{\nu}}=
-{\frac{1}{2}}${\pounds}$_{{\bf {\bar n}}_t}{\bar h}_{(t){\mu}{\nu}}$,
${\tilde K}_{(t){\mu}{\nu}}=
-{\frac{1}{2}}${\pounds}$_{{\bf {\tilde n}}_t}{\tilde h}_{(t){\mu}{\nu}}$,
respectively. After some calculations we find the spatial components
\eqa
{\bar K}_{(t)ij}&=&-{\frac{1}{2}}{\pounds}_{{\bf {\bar n}}_t}{\bar h}_{(t)ij}
={\frac{t}{2t_0{\bar N}_t}}({\bar N}_{i{\mid}j}+{\bar N}_{j{\mid}i})
+{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-{\frac{t_0}{t}}
c^{-2}{\bar a}_{{\cal F}k}{\frac{{\bar N}^k}{{\bar N}_t}}{\Big )}
{\bar h}_{(t)ij},
\nonumber \\ 
{\bar K}_{(t)i}^i&{\equiv}&{\bar K}_t=
{\frac{t_0}{t}}{\frac{{\bar N}_{{\,}{\,}{\,}{\mid}i}^i}{{\bar N}_t}}
+3{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-{\frac{t_0}{t}}
c^{-2}{\bar a}_{{\cal F}k}{\frac{{\bar N}^k}{{\bar N}_t}}{\Big )},
\ena
\eqa
{\tilde K}_{(t)ij}&=&{\bar N}_t^{-1}{\Big (}{\bar K}_{(t)ij}-
{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\bar h}_{(t)ij}{\Big )}.
\ena
Note the difference between the spatial components of 
{\pounds}$_{{\bf {\bar n}}_t}{\bar h}_{(t){\mu}{\nu}}$ and
${\cal L}_{{\bf {\bar n}}_t}{\bar h}_{(t)ij}$; the latter Lie derivative acts 
on spatial objects only and this means that ${\bf {\bar h}}_t$ should be 
treated as a tensor on space when using ${\cal L}_{{\bf {\bar n}}_t}$ but as
a tensor on space-time when using {\pounds}$_{{\bf {\bar n}}_t}$. The 
relationship
\eqa
{\cal L}_{{\bf {\bar n}}_t}{\bar h}_{(t)ij}=
{\pounds}_{{\bf {\bar n}}_t}{\bar h}_{(t)ij}+c^{-2}
{\frac{t}{t_0}}({\bar a}_{{\cal F}j}
{\frac{{\bar N}_{i}}{{\bar N}_t}}+{\bar a}_{{\cal F}i}
{\frac{{\bar N}_{j}}{{\bar N}_t}}),
\ena
follows directly from the definitions of the Lie derivatives. Relationships
similar to (3.11) can be found between the different Lie derivatives of any 
tensor field.

From (2.18) we can find the first derivative of ${\bf {\bar h}}_t$ with 
respect to $t$. This is
\eqa
{\frac{\partial}{{\partial}t}}{\bar h}_{(t)ij}=c{\bar N}_t(
{\topstar{${\pounds}$}}_{{\bf {\bar n}}_t}{\bar h}_{(t)ij}+2{\bar K}_{(t)ij})=
2({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}){\bar h}_{(t)ij}, 
\ena
\eqa
{\frac{\partial}{{\partial}t}}{\bar h}_{(t)}^{ij}=c{\bar N}_t
({\topstar{${\pounds}$}}_{{\bf {\bar n}}_t}{\bar h}_{(t)}^{ij}-
2{\bar K}_{(t)}^{ij})=-2({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
{\bar h}_{(t)}^{ij}.
\ena
We now write down how ${\bar a}_{{\cal F}j}$ varies in the direction 
orthogonal to the FHSs. We find
\eqa
{\topstar{${\pounds}$}}_{{\bf {\bar n}}_t}{\bar a}_{{\cal F}j}=
{\topstar{\cal L}}_{{\bf {\bar n}}_t}{\bar a}_{{\cal F}j}-
{\frac{t_0}{t}}[{\bar a}_{{\cal F}i}{\frac{{\bar N}^i}{{\bar N}_t}}]c^{-2}
{\bar a}_{{\cal F}j},
\ena
and similar formulae for other covectors, and moreover
\eqa
{\pounds}_{{\bf {\bar n}}_t}{\bar a}_{{\cal F}j}=
{\bar N}_t^{-1}(c^{-1}{\bar a}_{{\cal F}j},_0-{\frac{t_0}{t}}
[{\bar a}_{{\cal F}k}{\bar N}^k],_j), {\qquad}
c^{-2}{\frac{\partial}{{\partial}x^0}}{\bar a}_{{\cal F}j}=
{\frac{{\bar N}_t,_{j0}}{{\bar N}_t}}-
c^{-2}{\bar a}_{{\cal F}j}{\frac{{\bar N}_t,_0}{{\bar N}_t}},
\ena
\eqa
{\topstar{${\pounds}$}}_{{\bf {\bar n}}_t}{\bar a}_{{\cal F}}^j=
{\bar h}_{(t)}^{ij}{\topstar{${\pounds}$}}_{{\bf {\bar n}}_t}
{\bar a}_{{\cal F}j}-2(c{\bar N}_t)^{-1}
({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}){\bar a}_{\cal F}^{j}
+2{\bar a}_{{\cal F}i}{\bar K}_{(t)}^{ij},
\ena
and still furthermore we find
\eqa
{\topstar{${\pounds}$}}_{{\bf {\bar n}}_t}
({\bar a}_{{\cal F}i}{\bar a}_{\cal F}^i)=
2{\bar a}_{\cal F}^i{\Big [}{\topstar{${\pounds}$}}_{{\bf {\bar n}}_t}
{\bar a}_{{\cal F}i}-(c{\bar N}_t)^{-1}
({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
{\bar a}_{{\cal F}i}+{\bar a}_{{\cal F}k}{\bar K}_{(t)i}^k{\Big ]}.
\ena
We also get
\eqa
{\topstar{${\pounds}$}}_{{\bf {\bar n}}_t}{\bar a}_{{\cal F}{\mid}i}^i=
{\bar N}_t^{-1}{\Big [}({\bar a}_{{\cal F}{\mid}i}^i),_0
+c^{-1}({\bar a}_{{\cal F}{\mid}i}^i),_t{\Big ]}-{\frac{2}{c{\bar N}_t}}
({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
{\bar a}_{{\cal F}{\mid}i}^i-{\frac{t_0}{t}}{\frac{{\bar N}^i}{{\bar N}_t}}
{\bar a}_{{\cal F}{\mid}ki}^k,
\ena
where
\eqa
{\frac{\partial}{{\partial}x^0}}{\bar a}^{i}_{{\cal F}{\mid}i}=
-3{\frac{{\bar N}_t,_0}{{\bar N}_t}}{\bar a}^i_{{\cal F}{\mid}i}+
c^2{\frac{{\bar N}_{t{\mid}{\,}{\,}0i}^{{\,}{\,}{\,}i}}{{\bar N}_t}}
-{\bar a}_{\cal F}^i{\frac{{\bar N}_t,_{i0}}{{\bar N}_t}}.
\ena
The projection of metric space-time covariant derivatives into spatial objects 
expressed in components may be subject to some confusion of notation. The 
important thing to remember is that when a component contains a metric
{\em space-time} derivative, the projectors ${\bf {\bar n}}_t$ and 
${\bf {\bar h}}_t$ are always used {\em outside} the derivative, whereas if 
the component contains a {\em space} derivative, the projectors are used 
{\em inside} the derivative. For example a mixed projection of the rank 2 
space-time tensor fields ${\bf {\bar {\nabla}}}{\bf Y}_t$ yields the 
components 
$Y_{(t){\bar {\perp}};j}{\equiv}-Y_{(t){\mu};j}{\bar n}_{(t)}^{\mu}$ and not 
$-(Y_{(t){\mu}}{\bar n}_{(t)}^{\mu})_{;j}$, whereas the corresponding 
components of the spatial covector fields ${\bf {\bar {\triangle}}y}_t$ are 
$y_{(t){\bar {\perp}}{\mid}j}{\equiv}
-(y_{(t){\mu}}{\bar n}_{(t)}^{\mu})_{{\mid}j}$.

Formulae expressed in component notation for various projections of metric
space-time covariant derivatives of general families of space-time tensors may
be found by finding the projection formulae for ${\bf {\bar {\nabla}}Z}_t$, 
where ${\bf Z}_t$ is a some family of space-time covector fields. One may then
apply these formulae to every covariant index of a general family of totally 
covariant space-time tensors. Such formulae are derived in e.g. [2] 
(see also [7]), and are given by
\eqa
Z_{(t){\bar {\perp}};j} &=& Z_{(t){\bar {\perp}}{\mid}j}-
{\bar K}_{(t)ij}{\hat Z}^{i}_{(t)}, {\nonumber} \\
Z_{(t)i;j}&=&Z_{(t)i{\mid}j}-Z_{(t){\bar {\perp}}}{\bar K}_{(t)ij}, 
{\nonumber} \\
Z_{(t){\bar {\perp}};{\bar {\perp}}} &=& 
-{\cal L}_{{\bf {\bar n}}_t}Z_{(t){\bar {\perp}}}
-c^{-2}{\bar a}_{\cal F}^{i}Z_{(t)i}, {\nonumber} \\
Z_{(t)j;{\bar {\perp}}}&=& 
-{\cal L}_{{\bf {\bar n}}_t}Z_{(t)j}+
c^{-2}{\bar a}_{{\cal F}j}({\frac{t_0}{t}}
{\frac{{\bar N}^i}{{\bar N}_t}}Z_{(t)i}-
Z_{(t){\bar {\perp}}})-{\bar K}_{(t)ij}{\hat Z}^{i}_{(t)}.
\ena
Similarly we may find the projection formulae for the the tensor fields
${\stackrel{\star}{\bf {\bar {\nabla}}}}{\bf Z}_t$ constructed by using the
degenerate connection in ${\overline{\cal N}}$. The result is
\eqa
Z_{(t){\bar {\perp}}{\bar *}j}= Z_{(t){\bar {\perp}};j}, \qquad
Z_{(t)i{\bar *}j}=Z_{(t)i;j}, \qquad
Z_{(t){\bar {\perp}}{\bar *}{\bar {\perp}}}= 
-{\topstar{\cal L}}_{{\bf {\bar n}}_t}Z_{(t){\bar {\perp}}}
-c^{-2}{\bar a}_{\cal F}^iZ_{(t)i}, 
\ena
\eqa
Z_{(t)j{\bar *}{\bar {\perp}}}&=&
-{\topstar{\cal L}}_{{\bf {\bar n}}_t}Z_{(t)j}+
c^{-2}{\bar a}_{{\cal F}j}({\frac{t_0}{t}}
{\frac{{\bar N}^i}{{\bar N}_t}}Z_{(t)i}-Z_{(t){\bar {\perp}}}) \nonumber \\
&&+(c{\bar N}_t)^{-1}({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})Z_{(t)j}
-{\bar K}_{(t)ij}{\hat Z}^{i}_{(t)}.
\ena
Since we will explicitly need the projection formulae for the metric covariant
derivative of tensor fields of rank 2, we list the relevant ones.
If ${\bf Q}_t$ is a family of such fields, then we have [2]
\eqa
Q_{(t){\bar {\perp}}i;j}&=&Q_{(t){\bar {\perp}}i{\mid}j}-
{\bar K}_{(t)ij}Q_{(t){\bar {\perp}}{\bar {\perp}}}-
{\bar K}_{(t)jk}{\hat Q}^{k}_{(t)i}, %{\nonumber} 
\\ Q_{(t)ij;k}&=&Q_{(t)ij{\mid}k}-{\bar K}_{(t)ik}Q_{(t){\bar {\perp}}j}-
{\bar K}_{(t)jk}Q_{(t){\bar {\perp}}i}, %{\nonumber} 
\\ Q_{(t){\bar {\perp}}{\bar {\perp}};{\bar {\perp}}}&=&
-{\cal L}_{{\bf {\bar n}}_t}Q_{(t){\bar {\perp}}{\bar {\perp}}}
-c^{-2}{\bar a}_{{\cal F}i}{\hat Q}^{i}_{(t){\bar {\perp}}}-
c^{-2}{\bar a}_{{\cal F}i}{{\hat Q}_{(t){\bar {\perp}}}{\,}}^i, %{\nonumber} 
\\ Q_{(t)j{\bar {\perp}};{\bar {\perp}}}&=&
-{\cal L}_{{\bf {\bar n}}_t}Q_{(t)j{\bar {\perp}}}
+(c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}{\frac{{\bar N}^i}{{\bar N}_t}}
-{\bar K}_{(t)j}^i)Q_{(t)i{\bar {\perp}}} \nonumber \\
&&-c^{-2}{\bar a}_{{\cal F}i}{{\hat Q}_{(t)j}}^{i}-
c^{-2}{\bar a}_{{\cal F}j}Q_{(t){\bar {\perp}}{\bar {\perp}}}, %{\nonumber} 
\\ Q_{(t)ij;{\bar {\perp}}}&=&-{\cal L}_{{\bf {\bar n}}_t}Q_{(t)ij}
+(c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}{\frac{{\bar N}^k}{{\bar N}_t}}
-{\bar K}_{(t)j}^k)Q_{(t)ik}-c^{-2}{\bar a}_{{\cal F}j}Q_{(t)i{\bar {\perp}}}
{\nonumber} \\
&&+(c^{-2}{\bar a}_{{\cal F}i}{\frac{t_0}{t}}{\frac{{\bar N}^k}{{\bar N}_t}}
-{\bar K}_{(t)i}^k)Q_{(t)kj}-c^{-2}{\bar a}_{{\cal F}i}Q_{(t){\bar {\perp}}j}.
\ena
The analogous formulae for the degenerate covariant derivative of a family
of tensor fields of rank 2 are
\eqa
Q_{(t){\bar {\perp}}i{\bar *}j}=Q_{(t){\bar {\perp}}i;j}, \qquad
Q_{(t)ij{\bar *}k}=Q_{(t)ij;k},
\ena
\eqa
Q_{(t){\bar {\perp}}{\bar {\perp}}{\bar *}{\bar {\perp}}}=
-{\topstar{\cal L}}_{{\bf {\bar n}}_t}Q_{(t){\bar {\perp}}{\bar {\perp}}}
-c^{-2}{\bar a}_{{\cal F}i}{\hat Q}^{i}_{(t){\bar {\perp}}}-
c^{-2}{\bar a}_{{\cal F}i}{{\hat Q}_{(t){\bar {\perp}}}{\,}}^i,
\ena
\eqa
Q_{(t)j{\bar {\perp}}{\bar *}{\bar {\perp}}}&=&
-{\topstar{\cal L}}_{{\bf {\bar n}}_t}Q_{(t)j{\bar {\perp}}}
+(c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}{\frac{{\bar N}^i}{{\bar N}_t}}
-{\bar K}_{(t)j}^i)Q_{(t)i{\bar {\perp}}} \nonumber \\
&&+(c{\bar N}_t)^{-1}({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
Q_{(t)j{\bar {\perp}}}
-c^{-2}{\bar a}_{{\cal F}i}{{\hat Q}_{(t)j}}^{i}-
c^{-2}{\bar a}_{{\cal F}j}Q_{(t){\bar {\perp}}{\bar {\perp}}}, 
\ena
\eqa
Q_{(t)ij{\bar *}{\bar {\perp}}}&=&-{\topstar{\cal L}}_{{\bf {\bar n}}_t}
Q_{(t)ij}+({\bar a}_{{\cal F}i}{\frac{t_0}{t}}
{\frac{{\bar N}^k}{c^2{\bar N}_t}}-
{\bar K}_{(t)i}^k)Q_{(t)kj}+{\frac{2}{c{\bar N}_t}}
({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})Q_{(t)ij}
\nonumber \\
&&+(c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}{\frac{{\bar N}^k}{{\bar N}_t}}
-{\bar K}_{(t)j}^k)Q_{(t)ik}-c^{-2}({\bar a}_{{\cal F}j}Q_{(t)i{\bar {\perp}}}
+{\bar a}_{{\cal F}i}Q_{(t){\bar {\perp}}j}).
\ena
Note that we may show that ${\bar h}_{(t)ij{\bar *}{\bar {\perp}}}=0$ from 
(3.31) by setting ${\bar Q}_{(t)ij}={\bar h}_{(t)ij}$.
\subsection{The field equations}
In this section we use the formalism described in the previous section to
find local conservation laws for the active stress-energy tensor ${\bf T}_t$
and to express these laws in hypersurface language. But as we shall see the 
local conservation laws in quasi-metric theory are different from their metric
counterparts. We also postulate field equations which must hold for the metric
family ${\bf {\bar g}}_t$. Notice that in quasi-metric theory, the 
stress-energy tensor is really a family of rank 2 tensor fields, this is what 
its $t$-label is meant to indicate.

By construction the quasi-metric framework satisfies the EEP. But this does 
not imply that one should automatically use the usual rule ``comma goes to 
semicolon" when generalizing the non-gravitational physics from flat to curved
space-time. The reason for this is that it is possible to couple 
non-gravitational fields to first derivatives of the spatial scale factor 
${\bar F}_t$ such that the EEP still holds, see below. In particular we find 
that the covariant divergence ${\,}{\bf {\topstar{\bar {\nabla}}}}{\,}
{\bf {\cdot}T}_t{\neq}0$ (expressed in terms of formally variable units) in 
the quasi-metric framework. It must be stressed that this does {\em not} 
necessarily imply any violation of local conservation laws in 
${\overline{\cal N}}$; rather it means that the conservation laws take 
unfamiliar forms in quasi-metric space-time. Any direct coupling to curvature 
does of course not occur in the conservation laws; rather there is coupling to
other fields, namely ${\bar a}_{{\cal F}i}$, 
${\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}$,
${\frac{{\bar N}_t,_t}{c{\bar N}_t^2}}$ and ${\frac{1}{{\bar N}_tct}}$. 

Within the metric framework it can be shown [1] that validity of the local 
conservation laws in the form ${\nabla}{\bf {\cdot}T}=0$ follows from 
universal coupling (that gravity couples similarly to all non-gravitational 
fields, this follows directly from the EEP) and invariance of 
non-gravitational actions. Thus in any metric theory based on an invariant
action principle the local conservation laws in their usual form 
${\nabla}{\bf {\cdot}T}=0$ are valid independently of the detailed form of the
gravitational field equations. (Besides it can be shown [1] that the local 
conservation laws applied to the stress-energy tensor for a perfect fluid 
directly yield the Eulerian equations of hydrodynamics in the limit of weak 
gravitational fields and slow motions, establishing a Newtonian limit.) On the
other hand, since $c^{-2}{\bar a}_{{\cal F}j}=-{\bar N}_t
{\topstar{\bar {\Gamma}}}_{{\bar {\perp}}j}^0$,
${\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}=
{\topstar{\bar {\Gamma}}}_{{\bar {\perp}}0}^0+{\frac{t_0}{t}}
{\frac{{\bar N}^i}{{\bar N}_t}}c^{-2}{\bar a}_{{\cal F}i}=-{\bar N}_t
{\topstar{\bar {\Gamma}}}^0_{{\bar {\perp}}{\bar {\perp}}}$ and 
${\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
={\frac{1}{3}}{\topstar{\bar {\Gamma}}}^i_{ti}$ in a GTCS, 
any coupling to these fields may be made to vanish in the local inertial 
frames so the EEP still holds. For this reason, in a GTCS the local 
conservation laws may be expected to take the unfamiliar form 
${\,}{\bf {\topstar{\bar {\nabla}}}}{\,}{\bf {\cdot}T}_t{\neq}0$ for any 
theory compatible with the quasi-metric framework. (Note that one may 
straightforwardly show that the Newtonian limit of the local conservation laws
applied to a perfect fluid yields the Eulerian equations of hydrodynamics even
for the case ${\,}{\bf {\topstar{\bar {\nabla}}}}{\,}{\bf {\cdot}T}_t{\neq}0$,
see appendix B for some details.)

By exploiting the form the local conservation laws take given the coupling 
dependence on ${\bar a}_{{\cal F}i}$, 
${\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}$, 
${\frac{{\bar N}_t,_t}{c{\bar N}_t^2}}$ and ${\frac{1}{{\bar N}_tct}}$, we 
find constraints which must be satisfied for any candidate geometrical field 
to which we want to couple ${\bf T}_t$. To begin with, using (3.6) we find 
that (expressed in atomic units)
\eqa
T^{\nu}_{(t){\mu}{\bar *}{\nu}}&=&T^{\nu}_{(t){\mu}{\tilde *}{\nu}}
+4{\frac{{\bar N}_t,_{\alpha}}{{\bar N}_t}}T_{(t){\mu}}^{\alpha}
-T^{\alpha}_{(t){\alpha}}{\frac{{\bar N}_t,_{\mu}}{{\bar N}_t}} \nonumber \\
&=&T^{\nu}_{(t){\mu}{\tilde *}{\nu}}+
4(c^{-2}{\bar a}_{{\cal F}i}{\hat T}^i_{(t){\mu}}-
{\frac{{\bar N}_t,_{{\bar {\perp}}}}{{\bar N}_t}}T_{(t){\mu}{\bar {\perp}}})+
{\frac{{\bar N}_t,_{\mu}}{{\bar N}_t}}(T_{(t){\bar {\perp}}{\bar {\perp}}}
-{\hat T}_{(t)i}^i),
\ena
where (3.32) compares the covariant divergences 
${\,}{\bf {\topstar{\bar {\nabla}}}}{\,}{\bf {\cdot}T}_t$ and
${\,}{\bf {\topstar{\tilde {\nabla}}}}{\,}{\bf {\cdot}T}_t$, neither of which 
is expected to vanish. 

To get any further we need to know explicitly how the units vary; i.e.
how ${\Psi}_t$ (and $m_t$) in general depends on ${\bar a}_{{\cal F}i}$,
${\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}$,
${\frac{{\bar N}_t,_t}{{\bar N}_t}}$ and ${\frac{1}{t}}$. This is 
easily found from (3.1). We get 
\eqa
{\Psi}_t,_t&=&-({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
{\Psi}_t, {\qquad} {\Psi}_t,_{\bar {\perp}}=
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\Psi}_t, {\qquad}
{\Psi}_t,_j=-c^{-2}{\bar a}_{{\cal F}j}{\Psi}_t, 
{\nonumber} \\ 
m_t,_t&=&({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
m_t, {\qquad} m_t,_{\bar {\perp}}=
{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}m_t, {\qquad}
m_t,_j = c^{-2}{\bar a}_{{\cal F}j}m_t.
\ena
It is also convenient to have expressions telling how ${\Psi}_t$ and 
$m_t$ vary in the direction normal to the FHSs. Such expressions are
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}{\Psi}_t&=&
{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-{\frac{1}{c{\bar N}_t}}
[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]{\Big )}{\Psi}_t
{\equiv}-{\Psi}_t,_{\bar {\perp}}-{\frac{1}{c{\bar N}_t}}
({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}){\Psi}_t, \nonumber \\
{\topstar{\cal L}}_{{\bf {\bar n}}_t}m_t&=&
-{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
-{\frac{1}{c{\bar N}_t}}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
{\Big )}m_t{\equiv}-m_t,_{\bar {\perp}}+
{\frac{1}{c{\bar N}_t}}({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})m_t.
\ena
Note that we can use (3.34) to show that we may write
\eqa
m_{t{\bar *}{\alpha}}={\Big (}m_t,_{\bar {\perp}}-
{\frac{m_t}{c{\bar N}_t}}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
{\Big )}{\bar n}_{(t){\alpha}}
+m_t,_i{\bar h}_{(t){\alpha}}^i, {\qquad} m_{t{\bar *}0}
{\equiv}c^{-1}m_t,_t+m_t,_0,
\ena
and similarly for other scalar fields. Since ${\bf T}_t$ considered as a 
source of gravity has the dimension of (active) mass density, ${\bf T}_t$ has 
an extra variability like ${\Psi}_t^2$ measured in atomic units. For 
convenience we put all effects of this into the term 
$T_{(t){\mu}{\tilde *}{\nu}}^{\nu}$ in (3.32). We now want to project the 
relation (3.32) into the FHSs using the results of the previous section. We 
have 2 different projections; one yielding a spatial scalar field, the other a
spatial vector field. These are $T_{(t){\bar {\perp}}{\bar *}{\nu}}^{\nu}$ and
$T_{(t)j{\bar *}{\nu}}^{\nu}$, respectively. Using the formulae (3.28), (3.29)
and (3.30) we find
\eqa
T^{\nu}_{(t){\bar {\perp}}{\bar *}{\nu}}&=&
-T_{(t){\bar {\perp}}{\bar {\perp}}{\bar *}{\bar {\perp}}}+
{\bar h}^{ik}_{(t)}T_{(t){\bar {\perp}}i{\bar *}k} 
={\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}{\bar {\perp}}} + 
2c^{-2}{\bar a}_{{\cal F}i}{\hat T}^i_{(t){\bar {\perp}}} \nonumber \\
&&+{\hat T}_{(t){\bar {\perp}}{\mid}i}^i
-{\bar K}_tT_{(t){\bar {\perp}}{\bar {\perp}}}-
{\bar K}_{(t)ik}{\hat T}_{(t)}^{ik}, 
\ena
\eqa 
T^{\nu}_{(t)j{\bar *}{\nu}}&=&
-T_{(t)j{\bar {\perp}}{\bar *}{\bar {\perp}}}+
{\bar h}^{ik}_{(t)}T_{(t)ji{\bar *}k}=
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t)j{\bar {\perp}}}+
c^{-2}{\bar a}_{{\cal F}i}{\hat T}^i_{(t)j} 
+c^{-2}{\bar a}_{{\cal F}j}T_{(t){\bar {\perp}}{\bar {\perp}}}
\nonumber \\
&&+{\hat T}_{(t)j{\mid i}}^i-({\bar K}_t+{\frac{1}{{\bar N}_tct}}
+{\frac{{\bar N}_t,_t}{c{\bar N}_t^2}})T_{(t){\bar {\perp}}j}
-c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}
{\frac{{\bar N}^i}{{\bar N}_t}}T_{(t){\bar {\perp}}i}.
\ena
Furthermore, using (3.32), (3.36) and (3.37) we get, after some short 
calculations
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}
T_{(t){\bar {\perp}}{\bar {\perp}}}&=&
T^{\nu}_{(t){\bar {\perp}}{\tilde *}{\nu}}+
{\Big [}{\bar K}_t-
3{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\Big ]}
T_{(t){\bar {\perp}}{\bar {\perp}}}
+{\bar K}_{(t)ik}{\hat T}_{(t)}^{ik} \nonumber \\
&&-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
{\hat T}_{(t)i}^i+2c^{-2}{\bar a}_{{\cal F}i}
{\hat T}^i_{(t){\bar {\perp}}}
-{\hat T}^i_{(t){\bar {\perp}}{\mid}i}, 
\ena
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t)j{\bar {\perp}}}&=&
T^{\nu}_{(t)j{\tilde *}{\nu}}
+{\Big [}{\frac{1}{c{\bar N}_t}}
({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
-4{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
+{\bar K}_t{\Big ]}T_{(t)j{\bar {\perp}}}
+3c^{-2}{\bar a}_{{\cal F}i}{\hat T}^{i}_{(t)j} \nonumber \\
&&+c^{-2}{\bar a}_{{\cal F}j}({\frac{t_0}{t}}{\frac{{\bar N}^i}{{\bar N}_t}}
T_{(t)i{\bar {\perp}}}-{\hat T}^{i}_{(t)i})
-{\hat T}^i_{(t)j{\mid}i}.
\ena
To get any further with (3.38) and (3.39), we must have an idea of how to find
independent expressions for the terms 
$T^{\nu}_{(t){\bar {\perp}}{\tilde *}{\nu}}$ and 
$T^{\nu}_{(t)j{\tilde *}{\nu}}$. To do that we set up the general expressions
(explicitly including the effect of the formally variable units)
\eqa
T^{\nu}_{(t)j{\tilde *}{\nu}}&=&c^{-2}{\Big(}-2{\bar a}_{{\cal F}i}
{\hat T}^i_{(t)j}+{\bar a}_{{\cal F}j}
[{\xi}_{1}T_{(t){\bar {\perp}}{\bar {\perp}}}
+{\xi}_{2}{\hat T}^i_{(t)i}]{\Big )} \nonumber \\
&&+2{\Big [}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-
{\frac{1}{c{\bar N}_t}}({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
{\Big ]}T_{(t)j{\bar {\perp}}},
\ena
\eqa
T^{\nu}_{(t){\bar {\perp}}{\tilde *}{\nu}}=
-2c^{-2}{\bar a}_{{\cal F}i}{\hat T}_{(t){\bar {\perp}}}^{i}+
{\Big [}{\xi}_3{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
-{\frac{2}{c{\bar N}_t}}({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
{\Big ]}T_{(t){\bar {\perp}}{\bar {\perp}}}
+{\xi}_4{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\hat T}^i_{(t)i},
\ena
where ${\xi}_{1}$, ${\xi}_{2}$, ${\xi}_{3}$ and ${\xi}_4$ are constants to be 
determined. As illustrated in appendix B, we must have ${\xi}_1=-1$ to ensure 
that the Newtonian limit of the resulting conservation laws comes out 
correctly. Furthermore we guess that ${\xi}_2=1$, since this choice seems
natural in the case of a null fluid. For consistency it is then reasonable to 
choose ${\xi}_3=1$ and ${\xi}_4=1$. Inserting the above results into (3.32), 
(3.38), (3.39), (3.40) and (3.41) we get
\eqa
T^{{\nu}}_{(t){\bar {\perp}}{\bar *}{\nu}}=
T^{{\nu}}_{(t){\bar {\perp}};{\nu}}-{\frac{2}{c{\bar N}_t}}
({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
T_{(t){\bar {\perp}}{\bar {\perp}}}, 
\ena
\eqa
T^{\nu}_{(t)j{\bar *}{\nu}}=T^{{\nu}}_{(t)j;{\nu}}-
{\frac{2}{c{\bar N}_t}}({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
T_{(t){\bar {\perp}}j},
\ena
\eqa
T^{{\nu}}_{(t){\bar {\perp}};{\nu}}=2c^{-2}{\bar a}_{{\cal F}i}{\hat T}^i_
{(t){\bar {\perp}}}-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
T_{(t){\bar {\perp}}{\bar {\perp}}}, \qquad
T^{{\nu}}_{(t)j;{\nu}}=2c^{-2}{\bar a}_{{\cal F}i}{\hat T}^i_{(t)j}
-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\hat T}_{(t)j{\bar {\perp}}},
\ena
\eqa
{\cal L}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}{\bar {\perp}}}=
{\Big (}{\bar K}_t-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\Big )}
T_{(t){\bar {\perp}}{\bar {\perp}}}
+{\bar K}_{(t)ik}{\hat T}_{(t)}^{ik}-{\hat T}^i_{(t){\bar {\perp}}{\mid}i},
\ena
\eqa
{\cal L}_{{\bf {\bar n}}_t}T_{(t)j{\bar {\perp}}}&=&
{\Big (}{\bar K}_t-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\Big )}
T_{(t)j{\bar {\perp}}}
-c^{-2}{\bar a}_{{\cal F}j}T_{(t){\bar {\perp}}{\bar {\perp}}} \nonumber \\
&&+c^{-2}{\bar a}_{{\cal F}i}{\hat T}^i_{(t)j}+
c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}{\frac{{\bar N}^i}{{\bar N}_t}}
T_{(t)i{\bar {\perp}}}-{\hat T}^i_{(t)j{\mid}i}.
\ena
Notice that we have separated out the dependence on $t$ in (3.42) and (3.43)
since this explicit dependence has no kinematical significance. Similarly, 
when we take into account the explicit dependence on $t$ in the Lie 
derivatives, we find from equations (3.38), (3.39), (3.40), (3.41), (3.45) 
and (3.46)
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}{\bar {\perp}}}=
{\cal L}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}{\bar {\perp}}}
-{\frac{2}{c{\bar N}_t}}({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
T_{(t){\bar {\perp}}{\bar {\perp}}},
\ena
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t)j{\bar {\perp}}}=
{\cal L}_{{\bf {\bar n}}_t}T_{(t)j{\bar {\perp}}}
-{\frac{1}{c{\bar N}_t}}({\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}})
T_{(t)j{\bar {\perp}}}.
\ena
Note that (3.47) and (3.48) tell how the units vary with $t$; in (3.48) the 
effects of this on the components of a spatial 3-covector is also included
(which is why (3.47) and (3.48) depend differently on $t$).

The relations (3.42), (3.43) and (3.44) (or equivalently (3.45) and (3.46) 
together with (3.47) and (3.48)) represent the local conservation laws for 
active mass-energy in ${\overline{\cal N}}$. Note that these laws are 
expressed in atomic units and that the laws take care of the fact that 
gravitational quantities get extra variability measured in such units. Also 
note that equations (3.47) and (3.48) may be violated in the case of local
non-conservation (which is expected to occur in the early universe and in 
extremely deep gravitational wells). On the other hand equations (3.44), 
(3.45) and (3.46) are required to hold always.

Next, the condition that we must have metric correspondence with Newtonian 
theory for weak stationary fields leads to a postulate, namely
(${\kappa}{\equiv}{\frac{8{\pi}G}{c^{4}}}$)
\eqa
2{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}=
2(c^{-2}{\bar a}_{{\cal F}{\mid}i}^i+
c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{\cal F}^i
-{\bar K}_{(t)ik}{\bar K}_{(t)}^{ik}+{\cal L}_{{\bf {\bar n}}_t}{\bar K}_t)=
{\kappa}(T_{(t){\bar {\perp}}{\bar {\perp}}}+{\hat T}^i_{(t)i}),
\ena
where ${\bf {\bar R}}_t$ is the Ricci tensor field family corresponding to
the metric family (2.18). Note that whereas equation (3.49) is postulated
rather than derived, it is by no means arbitrary; in fact equation (3.49) 
follows naturally from a geometrical correspondence with Newton-Cartan theory.
Moreover, it is possible to write equation (3.49) in the form
\eqa
{\bar N}_{t{\mid}{\,}{\,}{\,}i}^{{\,}{\,}{\,}i}={\bar N}_t{\Big (}
{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}
+{\bar K}_{(t)ik}{\bar K}_{(t)}^{ik}-{\cal L}_{{\bf {\bar n}}_t}{\bar K}_t
{\Big )}.
\ena
According to the maximum principle applied to closed Riemannian 3-manifolds
(see reference [16] and references therein), no solutions of equation (3.50)
can exist on each FHS if the right hand side does not change sign on each
FHS. The only exception to this rule is the solution ${\bar N}_t=$constant for 
the particular case when the right hand side of (3.50) vanishes. Note that 
(the metric approximations of) equations (3.49), (3.50) also hold in GR for 
projections on an arbitrary spacelike hypersurface since
\eqa
{\bar G}_{(t){\bar {\perp}}{\bar {\perp}}} +{\bar G}_{(t)ik}{\bar h}_{(t)}^{ik}
=2{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}.
\ena
Here ${\bf {\bar G}}_t$ is the Einstein tensor family corresponding to the
metric family (2.18). 

Equation (3.49) represents one field equation where ${\bf T}_t$ is partially
coupled to space-time geometry. This coupling has the particular property
that it is does not depend explicitly on the intrinsic curvature of the FHSs.
We postulate a second set of field equations having this property, namely
\eqa
{\bar R}_{(t)j{\bar {\perp}}}={\bar G}_{(t)j{\bar {\perp}}}=
{\bar K}_{(t)j{\mid}i}^i-{\bar K}_t,_j={\kappa}T_{(t)j{\bar {\perp}}}.
\ena
The field equations (3.49) and (3.52) superficially look like just a subset of
the Einstein equations in ordinary GR. But one crucial difference is
that (3.49) and (3.52) hold in GR for projections into arbitrary spacelike 
hypersurfaces, whereas they are required to hold only for projections into the
FHSs in quasi-metric theory. Anyway the full coupling between the Einstein 
tensor and the stress-energy tensor does not exist in quasi-metric theory. One
reason for this is that such a coupling would explicitly depend on the 
intrinsic curvature of the FHSs [2], [15], and NKE of the FHSs would no longer
be possible.

We now have a sufficient number of field equations to determine the unknown
quantities in equation (2.18). In order to solve these field equations it
is very convenient to have general expressions for the spatial curvature of
the FHSs. Such expressions can be found directly from the spatial part of
(2.18). We find
\eqa
{\bar H}_{(t)ij}=c^{-2}{\Big (}{\bar a}_{{\cal F}{\mid}k}^k
-{\frac{1}{({\bar N}_tt)^2}}{\Big )}{\bar h}_{(t)ij}-
c^{-2}{\bar a}_{{\cal F}i{\mid}j}-
c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{{\cal F}j},
\ena
where ${\bf {\bar H}}_t$ is the family of spatial Einstein fields intrinsic
to the FHSs. Moreover, contracting (3.53) we can find an expression for the
family of Ricci scalar fields ${\bar P}_t$ intrinsic to the FHSs, namely
\eqa
{\bar P}_t={\frac{6}{(c{\bar N}_tt)^2}}-4c^{-2}{\bar a}_{{\cal F}{\mid}i}^i
+2c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{\cal F}^i.
\ena
We see that given initial conditions on some FHS in ${\overline{\cal N}}$, 
i.e., ${\bar N}_t$, ${\bar N}_t,_{\bar {\perp}}$, ${\bar N}^i$ and
$({\bar N}^i),_{\bar {\perp}}$, equations (3.52) are immediately determined by
these data whereas equation (3.49) is not. This means that to determine the 
time evolution of ${\bar N}_t$ from equation (3.49) one must in addition 
specify ${\kappa}(T_{(t){\bar {\perp}}{\bar{\perp}}}+{\hat T}^i_{(t)i})$ on 
each FHS. Thus equation (3.49) is the dynamical equation and equations (3.52) 
are constraint equations. Note that for illustrative purposes it is often 
useful to study systems where the condition
\eqa
{\bar N}^i_{{\,}{\,}{\,}{\mid}i}=3c^{-2}{\bar a}_{{\cal F}i}{\bar N}^i,
\qquad {\Rightarrow} \qquad 
{\bar K}_t=3{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}},
\ena
holds, and where the second equation follows from (3.9). The geometrical 
interpretation of the condition (3.55) is that the extrinsic curvature tensor 
family ${\bf {\tilde K}}_t$ is traceless. That is, taking the trace of 
equation (3.10) and requiring that it vanishes yields equation (3.55). Note 
that if the condition (3.55) holds, it implies that it is not necessary to 
specify an initial value $({\bar N}^i),_{\bar {\perp}}$ to be able to solve 
(3.49) on an initial FHS.

Equations (3.49) and (3.52) represent one dynamical equation and three
constraint equations. Compared to e.g. GR, where one has six dynamical 
equations and four constraint equations, we see that the field equation (3.49)
yields the dynamical evolution of one scalar field rather than of the full 
unconstrained extrinsic curvature tensor, as in GR. Besides, due to the 
contracted Bianchi identities the local conservation laws are identities in 
GR. In quasi-metric theory, however, the local conservation laws in 
${\overline{\cal N}}$ represent real constraints. Thus quasi-metric theory is 
quite different from GR when it comes to the way matter is explicitly coupled 
to the space-time geometry; in this respect quasi-metric gravity is more 
similar to Newtonian theory than to GR.

In the next section we will see how both KE and NKE are realized in 
${\overline{\cal N}}$ by formulating the dynamical theory as an initial value 
problem. 
\subsection{The quasi-metric initial value problem}
Now we want to use the results of the previous section to carry out a
further investigation of the dynamical theory in ${\overline{\cal N}}$, which 
is not a metric manifold. We do this by aiming at a formulation of the theory 
as an initial value problem for the evolution with time of the geometry of 
FHSs; in contrast to the standard metrical initial value problem our 
non-metric counterpart should involve both KE and NKE of the FHSs. The 
dynamical theory can be extended to be valid in $\cal N$ by adding extra 
effects coming from the NKE as illustrated in section 2.3.

To begin with we split up the time evolutions of ${\bar N}_t$ and
${\bar h}_{(t)ij}$ into kinematical and non-kinematical parts. We thus define
\eqa
c^{-2}{\bar x}_t+c^{-1}{\bar y}_t{\equiv}{\frac{{\bar N}_t,_t}{c{\bar N}_t^2}}
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}},
\ena
where $c^{-2}{\bar x}_t$ and $c^{-1}{\bar y}_t$ respectively, define the KE 
and the NKE of ${\bar N}_t$ with time. Furthermore the time evolution of 
${\bar h}_{(t)ij}$ is given from equations (3.11) and (3.12), namely 
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}{\bar h}_{(t)ij}&=&2{\Big (}
{\frac{1}{c{\bar N}_t}}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
+{\frac{t_0}{t}}c^{-2}{\bar a}_{{\cal F}k}{\frac{{\bar N}^k}{{\bar N}_t}}
{\Big )}{\bar h}_{(t)ij} \nonumber \\
&&+{\frac{t}{t_0}}{\Big (}c^{-2}{\bar a}_{{\cal F}i}
{\frac{{\bar N}_j}{{\bar N}_t}}+c^{-2}{\bar a}_{{\cal F}j}
{\frac{{\bar N}_i}{{\bar N}_t}}-{\frac{{\bar N}_{i{\mid}j}}{{\bar N}_t}}-
{\frac{{\bar N}_{j{\mid}i}}{{\bar N}_t}}{\Big )}.
\ena
On the other hand we {\em define} the time evolution of the spatial scale
factor to be
\eqa
c^{-2}{\bar x}_t+c^{-1}{\bar H}_t{\equiv}
{\frac{1}{c{\bar N}_t}}{\Big [}{\frac{1}{t}}+
{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big ]}
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}},
\ena
where $c^{-2}{\bar x}_t$ represents the KE and $c^{-1}{\bar H}_t$ 
represents the NKE of the spatial scale factor. From equations (3.56) 
and (3.58) one easily finds that
\eqa
{\bar y}_t={\bar H}_t-{\frac{1}{{\bar N}_tt}},
\ena
and
\eqa
c^{-2}{\bar x}_t=-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-
c^{-1}{\bar H}_t+{\frac{1}{c{\bar N}_t}}{\Big [}{\frac{1}{t}}+
{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big ]}.
\ena
To have a well-defined initial value problem in ${\overline{\cal N}}$ (for the
case when there is no coupling to non-gravitational fields) it remains to 
define the specific form of the NKCE. This will be of the type (2.14). To find
the right hand side of the NKCE we first take the particular case of a static 
vacuum in the metric approximation, using a GTCS where the shift vector field 
is required to vanish. (We show in section 4.1 that such a solution can be 
found.) Then, from the metric approximations of equations (3.49) and 
(3.54) we find 
\eqa
c^{-4}{\bar a}^{i}_{\cal F}{\bar a}_{{\cal F}i}={\frac{{\bar P}}{6}}.
\ena
For this particular case we expect that all the information available for
constructing the NKCE should be contained in the 4-acceleration field 
${\bf {\bar a}}_{\cal F}$ via (3.61). Besides, a second particular case is the
empty cosmological toy model found in section 4. For this case the field
equations are vacuous but we can still construct the curvature scalar
${\frac{{\bar P}_t}{6}}={\frac{1}{c^2t^2}}$. Thus it may seem natural to 
construct the right hand side of the NKCE from the scalar 
${\frac{{\bar P}_t}{6}}$ and $c$. But from equations (3.49) and (3.54) we see 
that that ${\bar P}_t$ may take {\em negative} values inside matter, and since
the right hand side of the NKCE should be non-negative everywhere we cannot 
use this scalar explicitly. However, we can try to combine the expressions 
obtained from the two particular cases into one general expression.
That is, we {\em define} the NKCE as the sum of one global term coming from
the global curvature of space and one term coming from the local gravitational
field, i.e.,
\eqa
{\bar H}_t{\equiv}c{\bar F}_t^{-1}
{\topstar{\cal L}}_{{\bf {\bar n}}_t}{\bar F}_t-c^{-1}{\bar x}_t=
{\frac{1}{{\bar N}_tt}}+c^{-1}{\sqrt{{\bar a}_{\cal F}^i{\bar a}_{{\cal F}i}}},
\ena
where ${\bar F}_t={\bar N}_tct$ are the local scale factors.
From (3.62) we see that ${\bar H}_t$ is indeed non-negative everywhere. 
Besides (3.62) becomes vacuous in the Newtonian limit as it should.

Now we are ready to formulate the quasi-metric initial value problem in
${\overline{\cal N}}$ (in absence of coupling to non-gravitational fields, see
below). This is equivalent to formulating an evolution scheme for the geometry
${\bf {\bar h}}_t$ (or equivalently, for ${\bar N}_t$). This may be done as 
follows: On an initial FHS we choose initial data ${\bar N}_t$, 
${\bar N}_t,_{\bar {\perp}}$, ${\bar N}_t,_t$, ${\bar N}^i$, 
$({\bar N}^i),_{\bar {\perp}}$ and the various projections of ${\bf T}_t$ such
that the constraint equations (3.52) hold. Note that the chosen equation of 
state characterizing the matter source on the initial FHS must be compatible 
with the initial data. Also note that ${\bar H}_t$ and ${\bar y}_t$ are 
explicitly given from the initial data on the initial FHS by respectively 
equations (3.62) and (3.59), and that $c^{-1}{\bar x}_t$ is given on 
the initial FHS from the initial data via equation (3.60). 

From the initial data we are able to find ${\bar N}_t$ on the deformed FHS. 
Moreover, from the local conservation laws (3.45), (3.46), (3.47) and (3.48)
we are able to find ${\kappa}T_{(t+dt){\bar {\perp}}{\bar {\perp}}}$ and 
${\kappa}T_{(t+dt)j{\bar {\perp}}}$ on the deformed FHS. (However, as mentioned
earlier, in special situations one expects that the global NKE implies local 
non-conservation. For such cases equations (3.47) and (3.48) do not hold, and
how much they are violated must be specified in addition.) The remaining 
projections of ${\bf T}_t$ on the deformed FHS must be inferred from the 
equation of state. Also the shift vector field ${\bar N}^i$ on the deformed 
FHS must be compatible with equations (3.52) on the deformed FHS. But in order
to predict ${\bar N}_t,_{\bar {\perp}}$ (and ${\bar N}_t,_t$) on the deformed 
FHS, or equivalently ${\bar N}_t$ on yet another deformed FHS, we need to 
solve the dynamical equation (3.49) on the initial FHS. To do this it is 
essential to supply the quantity $2{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}=
{\kappa}(T_{(t){\bar {\perp}}{\bar {\perp}}}+{\hat T}_{(t)i}^i)$ on the 
initial FHS. When we have done this we are able to advance one time step of 
the evolution process. We can then repeat the above described scheme for 
each subsequent time step.

But in general we must also have an evolution scheme in $\cal N$. The reason 
for this is that when there is coupling to non-gravitational fields (e.g.,
electromagnetism), non-gravitational force laws in $\cal N$ determine the
motion of sources and thus ${\bf T}_t$, which enters into the evolution scheme
in ${\overline{\cal N}}$. So when there is coupling to non-gravitational fields
the evolution schemes in ${\overline{\cal N}}$ and $\cal N$ are intertwined.

To have an evolution scheme in $\cal N$ it is necessary to find ${\bar y}_t$
and ${\bf b}_{\cal F}$ for each step of the evolution (see equation (2.23)).
Since ${\bf b}_{\cal F}$ is intended to represent a generalization of
``the local coordinate vector from the center of gravity'' which does have 
meaning for the spherically symmetric case, it is natural to seek for an 
equation which only solution is ${\bf b}_{\cal F}=
r{\frac{\partial}{{\partial}r}}$ for the spherically symmetric case. 
Furthermore the wanted equation should be linear in ${\bf b}_{\cal F}$ to 
ensure unique solutions, and it should involve ${\bf {\bar a}}_{\cal F}$ since 
one expects ${\bf b}_{\cal F}$ to vanish whenever ${\bf {\bar a}}_{\cal F}$ 
does. By inspection of the spherically symmetric case it turns out that it is 
possible to find an equation which has all the desired properties, namely
\eqa
{\Big [}{\bar a}_{{\cal F}{\mid}k}^k+c^{-2}{\bar a}_{{\cal F}k}
{\bar a}_{\cal F}^k{\Big ]}b_{\cal F}^j-{\Big [}
{\bar a}_{{\cal F}{\mid}k}^j+c^{-2}{\bar a}_{{\cal F}k}{\bar a}_{\cal F}^j
{\Big ]}b_{\cal F}^k-2{\bar a}_{\cal F}^j=0.
\ena
Since we may calculate ${\bf b}_{\cal F}$ from equation (3.63) and then
${\bar y}_t$ for each step of the evolution according to the above described
evolution scheme, we are also able to construct ${\bf h}_t$, $N$ and $N^i$ for
each time step along the lines shown in section 2.3, using equations (2.24),
(2.25) and (2.26). Thus with the help of the coupled evolution schemes in 
${\overline{\cal N}}$ and $\cal N$ we are able to find both metric families 
${\bf {\bar g}}_t$ and ${\bf g}_t$. From the relevant equations we see that 
the quasi-metric evolution scheme contains both KE and NKE. The crucial 
difference between the NKE and the KE of the FHSs is that the former can be 
found on subsequent FHSs directly from the spatial metric via the NKCE whereas
the latter depends on the field ${\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}$ 
which must be specified separately for each subsequent FHS.

Now we must emphasize an important point: since ${\bf T}_t$ is the active 
stress-energy tensor and thus not directly measurable locally, one must know 
how it relates to the {\em passive stress-energy tensor} ${\cal T}_t$ in 
${\cal N}$. The relationship between ${\bf T}_t$ and ${\cal T}_t$ depends in 
principle on the general nature of the matter source. To illustrate this, 
consider ${\bf T}_t$ and ${\cal T}_t$ for a perfect fluid:
\eqa
{\bf T}_t=({\tilde {\rho}}_m+c^{-2}{\tilde p}){\bf {\bar u}}_t{\otimes}
{\bf {\bar u}}_t+{\tilde p}{\bf {\bar g}}_t, \qquad
{\cal T}_t=({\hat{\rho}}_m+c^{-2}{\hat p}){\bf u}_t{\otimes}{\bf u}_t+
{\hat p}{\bf g}_t,
\ena
where ${\tilde {\rho}}_m$ is the active mass-energy density in the local
rest frame of the fluid and ${\rho}_m$ is the passive mass-energy density
measured in the local rest frame of the fluid (${\tilde p}$ and $p$ are the
associated pressures). Also, in (3.64) we have used the definitions 
${\rho}_m{\sqrt{{\bar h}_t}}{\equiv}{\hat{\rho}}_m{\sqrt{h_t}}$ and 
$p{\sqrt{{\bar h}_t}}{\equiv}{\hat{p}}{\sqrt{h_t}}$, where 
${\bar h}_t$ and $h_t$ are the determinants of the spatial metric families 
${\bf {\bar h}}_t$ and ${\bf h}_t$, respectively. Now the relationship 
between ${\tilde {\rho}}_m$ and ${\rho}_m$ is given by
\eqa
{\rho}_m=
\left\{
\begin{array}{ll}
{\frac{t_0}{t}}{\bar N}_t^{-1}{\tilde {\rho}}_m & 
$for a fluid of material point particles,$ \\ [1.5ex]
{\frac{t_0^2}{t^2}}{\bar N}_t^{-2}{\tilde {\rho}}_m & 
$for a null fluid,$
\end{array}
\right.
\ena
and a similar relationship exists between ${\tilde p}$ and $p$. The reason why
the relationship between ${\rho}_m$ and ${\tilde{\rho}}_m$ is different for a 
null fluid than for other perfect fluid sources is that spectral shifts of null
particles influence their passive mass-energy but not their active mass-energy.
On the other hand the global NKE does not affect the passive mass-energy of 
material point particles (see section 4.2). Finally, note that more 
``physically correct'' local conservation laws than those shown in equations 
(3.42), (3.43) and (3.44) may be found in $\cal N$ by calculating 
${\,}{\bf {\topstar{\nabla}}}{\,}{\cdot}{\cal T}_t{\,}$ when ${\bf g}_t$ is 
known. But these local ``physical'' conservation laws for passive mass-energy 
do not take any predetermined form.
\subsection{Equations of motion}
In section 2 we derived the equations of motion (2.10) valid for arbitrary test
particles. However these equations are of no practical value as long as we do 
not know the relationship between the 4-acceleration field ${\bf a}_t$ and 
the degenerate 4-acceleration field ${\stackrel{\star}{\bf a}}$. In this 
section we show that in fact ${\stackrel{\star}{\bf a}}$=${\bf a}_t$.

For starters it is convenient to introduce the difference in coordinate
3-velocity between a test particle moving along an arbitrary curve and that of
the local FO. We call this quantity ${\bf w}_t$, and we write its square
as $w^2$. Thus we define (using a GTCS)
\eqa
w_{(t)}^j{\equiv}{\frac{dx^j}{d{\tau}_{\cal F}}}+{\frac{t_0}{t}}
{\frac{N^j}{N}}c, {\qquad}
{\topstar{\gamma}}{\equiv}(1-w^2/c^2)^{-1/2}=
{\frac{d{\tau}_{\cal F}}{d{\tau}_t}},
\ena
where ${\tau}_t$ is the proper time measured along the curve and 
${\topstar{\gamma}}$ is the time dilation factor relating the proper time 
intervals along the curve to the proper time intervals of the local FOs. 
Since ${\bf w}_t$ is required to be a vector field family tangent to the 
FHSs, the space-time scalar product of ${\bf w}_t$ and ${\bf n}_t$ must
vanish. That is, if we view ${\bf w}_t$ as a family of 4-vectors we must have
\eqa
w_{(t)}^0=0, \qquad w_{(t)}^{\mu}n_{(t){\mu}}=
w_{(t){\mu}}n_{(t)}^{\mu}=0, \qquad {\Rightarrow} \qquad
w_{(t)0}={\frac{t_0}{t}}N^iw_{(t)i}.
\ena
Note that since the explicit dependence on $t$ of any scalar gravitational 
quantity is determined from its dimensionality when measured in atomic units, 
we must have
\eqa
{\frac{\partial}{{\partial}t}}w^2{\equiv}
(w_{(t){\mu}}w_{(t)}^{\mu}),_t=
(w_{(t)i}w_{(t)k}h_{(t)}^{ik}),_t=0, \qquad \Rightarrow \qquad
{\frac{\partial}{{\partial}t}}w_{(t)j}={\frac{1}{t}}w_{(t)j}.
\ena
This means that we must have
\eqa
w_{(t)j}={\frac{t}{t_0}}w_{(t_0)j}{\equiv}{\frac{t}{t_0}}w_j, \qquad
w_{(t)}^j={\frac{t_0}{t}}w_{(t_0)}^j{\equiv}{\frac{t_0}{t}}w^j.
\ena
We then split up the 4-velocity field along the curve into tangential and 
normal pieces. This is done as follows:
\eqa
{\bf u}_t{\equiv}{\frac{dx^{\mu}}{d{\tau}_t}}
{\frac{\partial}{{\partial}x^{\mu}}}={\topstar{\gamma}}(c{\bf n}_t+{\bf w}_t).
\ena
From equation (2.9) we see that to find an expression for 
${\stackrel{\star}{\bf a}}$ we need to calculate the quantity
${\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf u}_t$. We will use
equation (3.70) for this purpose. By direct calculation using equations (3.67),
(3.68) and (3.69) we find
\eqa
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}w_{(t){\mu}}=
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}w_{(t)}^{\mu}=0, \qquad
\Rightarrow \qquad
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf w}_t=0,
\ena
and furthermore we calculate
\eqa
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}n_{(t){\mu}}=
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}n_{(t)}^{\mu}=0, \qquad
\Rightarrow \qquad
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf n}_t=0,
\ena
confirming the requirement (2.3). But from equations (2.9),
(3.70), (3.71) and (3.72) we get
\eqa
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf u}_t=0,
\qquad \Rightarrow \qquad
{\stackrel{\star}{\bf a}}={\bf a}_t,
\ena
as asserted previously. Using a GTCS we then get the final form of the 
equations of motion, namely
\eqa
{\frac{d^2x^{\mu}}{d{\lambda}^2}}+{\Big (}
{\topstar{\Gamma}}^{\mu}_{{\nu}t}{\frac{dt}{d{\lambda}}}+
{\Gamma}^{\mu}_{(t){\nu}{\beta}}{\frac{dx^{\beta}}{d{\lambda}}}{\Big )}
{\frac{dx^{\nu}}{d{\lambda}}} 
={\Big(}{\frac{cd{\tau}_t}{d{\lambda}}}{\Big)}^2
c^{-2}a_{(t)}^{\mu}.
\ena
Equations (3.74) are valid for both timelike and null curves. For timelike 
curves one may suitably choose a parameterization in terms of the proper time 
measured along the curve, i.e. one may choose ${\lambda}=c{\tau}_t$. For null 
curves $\lambda$ must be chosen to be some other affine parameter such that 
${\frac{cd{\tau}_t}{d{\lambda}}}=0$. Note that these equations do {\em not} 
violate LLI. See reference [15] for further comments. 
\section{Spherically symmetric vacua}
In this section we solve the field equations for two metrically static, 
spherically symmetric vacuum cases. (We use the term ``metrically static'' to 
refer to cases where ${\bar N}_t$ does not depend on $x^0$ and ${\bar N}_j$ 
vanishes in some GTCS.)

The first case we analyze is the rather trivial one where space-time is 
isotropic and globally free of matter, i.e. ${\bf {\bar a}}_{\cal F}$ vanishes
identically. The field equations (3.52) are vacuous in this case whereas
equation (3.49) has the solution ${\bar N}_t=$constant. However, from
equations (3.53) and (3.54) we get (where ${\bf {\bar P}}_t$ is the family of 
Ricci tensors corresponding to the spatial metrics ${\bf {\bar h}}_t$)
\eqa
{\bar H}_{(t)ij}{\equiv}
{\bar P}_{(t)ij}-{\frac{{\bar P}_t}{2}}{\bar h}_{(t)ij}=
-{\frac{1}{({\bar N}_tct)^2}}{\bar h}_{(t)ij}, {\qquad} 
{\frac{{\bar P}_t}{6}}={\frac{1}{({\bar N}_tct)^2}},
\ena
or equivalently
\eqa
{\bar P}_{(t)ij}={\frac{{\bar P}_t}{3}}{\bar h}_{(t)ij},
\ena
which means that the solution is an Einstein space. It is easy to show that 
{\bf S}$^{3}$ is the only compact spherically symmetric solution of (4.2) with 
${\bar P}_t>0$. Note that ${\bar N}_t$ is just an arbitrary constant, and that
we may set ${\bar N}_t=1$ by making some appropriate choice of scale for the 
spatial coordinates (and simultaneously scale $x^0$ and $t$). This means that 
each space-time metric of the family ${\bf {\bar g}}_t$ is of the type 
{\bf S}$^{3}{\times}${\bf R}. We can find a relationship between these metrics
in terms of the epoch $t$ from the NKCE (3.62). This yields 
${\bar H}_t={\frac{1}{t}}$ which means that $c^{-1}{\bar x}_t={\bar y}_t=0$ 
from (3.59) and (3.60). Besides this yields ${\bf v}_t=0$ from equation 
(2.23). Thus the FOs in ${\overline{\cal N}}$ do not move with respect to the 
FOs in ${\cal N}$, i.e. the metric families ${\bf g}_t$ and ${\bf {\bar g}}_t$
are identical. We may then write
\eqa
N={\bar N}_t=1 {\qquad} {\Rightarrow} {\qquad} {\bar H}_t={\frac{1}{t}}. 
\ena
The one-parameter families of {\bf S}$^{3}{\times}${\bf R} space-time metrics 
consistent with (4.3) take the form (with $t={\frac{x^{0}}{c}}$)
\eqa
ds^{2}_t = \overline{ds}^{2}_t =-(dx^{0})^{2} + 
({\frac{t}{t_{0}}})^{2}{\bigg(}{\frac{dr^{2}}
{1-{\frac{r^{2}}{(ct_{0})^{2}}}}}+r^{2}d{\Omega}^{2}{\bigg)},
\ena
and we notice that (4.4) is of the form (2.18), as required. Furthermore
we see (using equation (3.33)) that (4.4) yields that active mass $m_t$ 
increases with time. We may express $m_t$ and ${\Psi}_t$ by $m_{t_0}$ and
${\Psi}_{t_0}$ valid at the arbitrary epoch $t_0$, getting
\eqa
m_t=m_{t_0}{\frac{t}{t_0}}, {\qquad}
{\Psi}_t={\Psi}_{t_0}{\frac{t_0}{t}}.
\ena
As usual when $m_t$ is variable this is because by definition, gravitational 
quantities get an extra variability when measured in atomic units. In this 
particular case it means that gravitational length scales measured in atomic 
units (and vice versa) are different at different epochs $t$. Note that it is 
just as valid to interpret (4.4) to mean that atomic length units are 
shrinking as the usual interpretation in terms of an increase of spatial 
dimensions with time. This illustrates that the family of line elements (4.4) 
expresses the gravitational scale (represented by the spatial coordinates in a
GTCS) in terms of units determined from the atomic scale. Moreover, since the 
gravitational scale is represented by coordinates belonging to a particular 
class of coordinate systems, this means that no particular {\em intrinsic} 
length scales can be associated with gravitation. In contrast to this 
non-gravitational interactions define intrinsic length (and mass) scales.

It is important to realize that the metrics (4.4) are vacuum metrics, since we 
have neglected matter creation as mentioned in the previous section. Thus 
(4.4) is just an approximate, not realistic toy cosmological model. However we 
may insert ${\bar H}_t={\frac{1}{t}}=H_{0}$ for the value of the Hubble 
parameter for the present epoch into (4.3) to get a realistic value for the 
age of the Universe. This is possible since in contrast to standard cosmology,
one should not expect that inclusion of matter into the toy model must 
necessarily result in ``braking'' of the expansion. In fact one may show that
the only possible cosmological models allowed with a perfectly isotropic matter
distribution are models where the source is a null fluid. That is, according to
our theory no cosmological models with an isotropically distributed material 
source is possible, see appendix C. This indicates that matter density has 
nothing to do with the time evolution of the cosmic expansion. Thus the age 
${\bar H}_t^{-1}$ should be taken as a realistic value and not merely as an 
upper bound. Also one may not neglect time dilation effects in the quantity 
${\frac{1}{{\bar N}_tt}}$ when interpreting redshift measurements. We return 
to the question of interpretation of spectral shifts in section 4.2.
\subsection{The conformally Minkowski case}
The globally matter-free metric family (4.4) found in the previous section 
is a natural candidate for a cosmological toy model since it is isotropic and
thus spherically symmetric about every point. On the other hand we need a 
model for the gravitational field exterior to a spherically symmetric body
in an otherwise empty Universe. In principle we must then find a solution of
the field equations which is spherically symmetric about one particular point 
only and where the boundary conditions for this solution should be 
{\bf S}$^{3}{\times}${\bf R}, i.e. the metric family should be of the form 
(4.4) at the boundary. But such a solution does not exist according to 
equation (3.50) and the subsequent discussion. However, it is possible to 
find a solution if we relax the boundary conditions, see [14]. To avoid the 
problems at the boundary, in the case where we can neglect the NKE due to a 
global scale factor we shall feel free to use a static metric approximation, 
i.e., we assume that the each metric of the metric family is identical and 
asymptotically Minkowski. 

From (4.4) we see that we get a Minkowski background if we let 
$t_0{\rightarrow}{\infty}$ and set $t=t_0$. We may then set 
${\bar N}_t({\infty})=1$. This means that (using (3.33))
\eqa
{\Psi}_t(r)={\Psi}_t({\infty}){\bar N}_t^{-1}(r),
\ena
since the acceleration field is purely radial. Equation (4.6) means that
reference units at infinity are being used for each metric ${\bf {\bar g}}{\in}
{\bf {\bar g}}_t$. However, taking these units as those obtained in the limit 
$t{\rightarrow}{\infty}$ in (3.33) we see that they get singular if one 
compares to the units determined by a ${\bf S}^3{\times}{\bf R}$ background. 
That is, ${\Psi}_t$ vanishes in the limit $t{\rightarrow}{\infty}$. But the
field equations and equations (3.53), (3.54) still work fine for metric
approximations. 

Even if asymptotically Minkowski solutions in principle are the result of 
using singular reference units in the field equations (3.49), (3.52) and in
equations (3.53), (3.54), the discrepancy between these solutions and those 
with acceptable boundary conditions should be small to a good approximation 
for many situations. So we feel free to find an asymptotically Minkowski, 
spherically symmetric static solution of (3.49), and apply it in appropriate 
circumstances. Proceeding with this, firstly we notice that the spherically
symmetric condition makes equations (3.52) vacuous. Secondly, the field 
equation (3.49) together with equation (3.53) give
\eqa
{\bar H}_{(t)ij}+c^{-2}{\bar a}_{{\cal F}i{\mid}j}+
c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{{\cal F}j}+
c^{-4}{\bar a}_{{\cal F}k}{\bar a}_{{\cal F}}^k{\bar h}_{(t)ij}=0,
{\qquad} {\bar a}^{k}_{{\cal F}{\mid}k}+
c^{-2}{\bar a}_{\cal F}^{k}{\bar a}_{{\cal F}k}=0.
\ena
When we apply these equations to the general spherically static line element 
(2.21) we get the unique solution
\eqa
\overline{ds}^{2} = {\bigg(}{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}-
{\frac{r_{s}}{2r}}{\bigg)}^{2}{\bigg(}-(dx^{0})^{2}+
{\Big[}1+({\frac{r_{s}}{2r}})^2{\Big]}^{-1}dr^{2}{\bigg)}+r^{2}d{\Omega}^{2},
\ena
where, analogous to the Schwarzschild solution,
$r_{s}{\equiv}{\frac{2MG}{c^{2}}}$. Some details of the derivation of (4.8) 
are given in an appendix.

Next we calculate the spatial Ricci scalar ${\bar P}$ corresponding to (4.8).
Using (A.4) we find
\eqa
{\bar P}(r)={\frac{3r_{s}^{2}}{2r^{4}}}{\bigg(}{\frac{r_{s}}{2r}}+
{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}{\bigg)}^{2}, {\qquad}
{\bar P}({\rho})={\frac{3r_{s}^{2}}{2{\rho}^{4}
(1-{\frac{r_{s}}{{\rho}}})^{3}}},
\ena
where ${\rho}$ is an isotropic radial coordinate defined in (A.14). 
Furthermore, from the NKCE (3.62) in combination with equation (3.61)
valid for this particular case we find
\eqa
{\bar H}(r)={\frac{r_{s}}{2r^{2}}}{\bigg(}{\frac{r_{s}}{2r}}+
{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}{\bigg)}c, {\qquad}
{\bar H}({\rho})={\frac{r_{s}c}{2{\rho}^{2}(1-{\frac{r_{s}}{{\rho}}})^{3/2}}}.
\ena
As can be readily seen, (4.8) contains no coordinate singularity, thus there
is no event horizon. This is in accordance with the requirement that a global 
time coordinate must exist. Furthermore, there is a curvature singularity at 
the spatial origin. However this singularity is inaccessible due to the fact 
that the NKE in the tangent space increases towards the velocity of light at 
the spatial origin. Using (2.20) and (4.10) we find
\eqa
v(r)={\bar H}(r){\sqrt{{\bar h}_{rr}(r)}}r={\frac{r_{s}}{2r}}
{\frac{c}{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}},
\ena
and from (4.11) we see that $v(r){\rightarrow}c$ in the limit 
$r{\rightarrow}0$. As can be seen from equation (4.13) below, this has the 
effect of removing the curvature singularity present in the intrinsic geometry
of the FHSs when one constructs the geometry ${\bf h}$ from the geometry 
${\bf {\bar h}}$ as described in section 2.3. And although the space-time 
metric (4.13) is still singular at the spatial origin, this is merely a 
consequence of the fact that the spatial origin is inaccessible to material 
particles and thus that the premise of a gravitational point source is 
unphysical. We interpret this to mean that gravitational collapse cannot, even
in principle, form physical singularities when the NKE is taken into account. 

Now we must construct the metric ${\bf g}$ from the solution metric
${\bf {\bar g}}$ shown in (4.8). We find ${\bf g}$ by the method discussed in 
section 2.3. Before we do this we note that equations (3.59) and (3.60) yield
\eqa
c^{-1}{\bar x}_t=-{\bar y}_t=-{\bar H}_t, {\qquad} 
c^{-1}{\bar x}_t,_t={\bar H}_t,_t=0.
\ena
Thus we find that the construction of the metric ${\bf g}$ is consistent with 
the equations of evolution.

We now use equations (2.22), (4.8) and (4.11) to construct the metric 
$\bf g$ and this is sufficient to check if the ``classical'' solar system 
tests come in correctly. We find the unique metric 
\eqa
ds^{2}&=&-{\Big[}1+({\frac{r_{s}}{2r}})^2{\Big]}^{-2}
{\bigg(}{\frac{r_{s}}{2r}}+
{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}{\bigg)}^{-2}(dx^0)^2 {\nonumber} \\
&+&{\Big[}1+({\frac{r_{s}}{2r}})^{2}{\Big]}^{-1}{\bigg(}{\frac{r_{s}}{2r}}+
{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}{\bigg)}^{2}dr^2 + r^{2}d{\Omega}^{2} 
{\nonumber} \\
&=&-{\Big(}1-{\frac{r_{s}}{r}}+{\frac{3r_{s}^{3}}{8r^{3}}}+
{\cdots}{\Big)}(dx^0)^{2}+{\Big(}1+{\frac{r_{s}}{r}}+
{\frac{r_{s}^{2}}{4r^{2}}}+{\cdots}{\Big)}dr^{2}+r^{2}d{\Omega}^{2},
\ena
which agrees with the Schwarzschild metric to sufficient order [1]. Thus the 
metric (4.13) is consistent with the four ``classical" solar system tests 
(radar time delay experiments included), and our theory seems to have the 
correct metric correspondence. (This can be directly confirmed by applying the 
equations of motion (3.74) to the metric (4.13).) We note that since 
(4.8) is conformally flat as can be seen from (A.15), it has vanishing Weyl 
curvature. But since the Schwarzschild metric has vanishing Ricci curvature, 
it is the Weyl curvature of that metric that makes GR agree with the solar 
system tests. This is in contrast to our theory, where the agreement with the
solar system tests at the post-Newtonian level cannot possibly come about
were it not for the fact that the NKE has been integrated into the metric 
structure. Thus the existence of an implicitly coupled dynamical degree of 
freedom is essential in order to make quasi-metric gravity viable.
\subsection{Testing the NKE-paradigm}
To determine whether or not our theory is viable, we must try to confront its
predictions with data from modern relativistic gravitational experiments. But 
first we must show that it has a consistent Newtonian limit. This limit can be
defined in such a way that the FHSs are taken as flat and static. Thus all the
NKE is ignored in the Newtonian limit and we can drop the bar labels where 
this is convenient. In a static GTCS with Cartesian coordinates we have the 
usual Newtonian form of the metrics
\eqa
g_{(t)00}={\bar g}_{(t)00}=-1+2c^{-2}U(x^{\mu}), {\qquad} g_{(t)i0}=
g_{(t)i0}=0, {\qquad} h_{(t)ij}={\bar h}_{(t)ij}={\delta}_{ij},
\ena
where $-U(x^{\mu})$ is the Newtonian potential. Note that (4.14) is consistent
with the general metric family (2.18) since to Newtonian accuracy, we can 
neglect any contribution to ${\bar h}_{(t)ij}$ of order $c^{-2}U(x^{\mu})$. In 
appendix B we use (4.14) to construct a well-defined Newtonian limit for our 
theory by calculating the Newtonian approximation to the evolution equations 
(3.44) applied to a perfect fluid. Besides we show that the field equation 
(3.49) reduces to Newton's gravitational equation in the Newtonian limit and 
that (4.14) is consistent with the equations of motion (3.74) with vanishing 
non-metric terms.

We note that the usual Newtonian assumptions of weak gravitational fields and 
slow motions include the condition that the vector fields ${\bf v}_t$ defined 
in equation (2.23) yield negligible corrections to the metrics (4.14). Also we 
must also assume that the system is ``small" if we wish to have Newtonian 
theory as a limit of our quasi-metric theory; assuming a small system is 
essential to ensure that effects from the ``true" non-kinematical expansion 
coming from a global scale factor are negligible. The reason for this is that 
in the Newtonian approximation the FOs cannot be defined unambiguously, so
we may consider any set of observers static with respect to some arbitrary 
Galilean system as FOs. This is a good approximation for small systems and 
nearly flat FHSs, but for large systems one cannot neglect the ``motion" of 
distant FOs with respect to each other due to the NKE coming from a global 
scale factor. This needs not be small even if the change of the global scale 
factor is small everywhere.

Effects related to that part of the NKE which is integrated into the metric 
structure of ${\bf g}_t$ via ${\bf v}_t$ we refer to as {\em local} effects of 
the NKE, whereas the remainder of the NKE yields the {\em global} effects. In 
essence Newtonian theory should correspond with the {\em metric} part of our 
theory, and not with the non-metric part, which has no Newtonian counterpart. 
Thus even if the local effects of the NKE can be neglected at the Newtonian 
level of precision, there is no reason to think that this also applies to the 
global effects of the NKE.

Most new metric theories of gravity having a satisfactory Newtonian limit can 
be analyzed within the parameterized post-Newtonian (PPN) formalism. This is
constructed to handle the next higher level of precision beyond the Newtonian
approximation for systems with weak fields and slow motions [1]. The result is 
a set of PPN-parameters which must take a set of specific values if the metric 
theory is to be regarded as viable. However, quasi-metric theories are not 
well-suited for a standard PPN-analysis. One reason for this is that the global
effects of the NKE are missed since any existing PPN-formalism does not account
for such effects. This means that a PPN-analysis of quasi-metric theory is 
limited to its metric approximations. But even if metric approximations to
quasi-metric theory can be PPN-analyzed in principle there is another 
complication; namely that the {\em local} effects of the NKE must be taken 
into account at the post-Newtonian level via the construction of ${\bf g}$ 
from ${\bf {\bar g}}$. That is, any PPN-analysis applied to quasi-metric 
theory would result in a post-Newtonian metric ${\bf {\bar g}}$ and not in 
its counterpart ${\bf g}$. This means that any naive attempt to PPN-analyze 
metric approximations to the field equations (3.49), (3.52) and equation 
(2.18) should {\em not} result in an acceptable set of PPN-parameters 
according to metric theory. A particular case confirming this expectation is 
the conformally flat solution (4.8) which disagrees with the accepted values 
for two of the PPN-parameters. It is only when the NKE corrections are 
included that we get a metric with the standard values of these parameters, 
i.e., the metric (4.13). 

Consequently, to analyze our field equations within some sort of weak field 
formalism one might first try to find the PPN-metric ${\bf {\bar g}}$ (which 
does not have an acceptable set of PPN-parameters), and then construct a 
second PPN-metric ${\bf g}$ from ${\bf {\bar g}}$ and ${\bf v}_t$ in the usual
way. But for this to have any chance to work one cannot assume (as is done in 
the standard PPN formalism) that all dynamical degrees of freedom are 
explicitly coupled to ${\bf T}_t$. Taking into account all the complications, 
to construct a weak field expansion suitable for quasi-metric theory would 
seem to be a rather messy task but obviously a subject for further work. 

Even without a standard formalism within which the theory can be compared to
experiment, we can try to pinpoint some of the ``direct" predictions due to the
hypothetical existence of the global NKE. The fact that the NKE takes the form
of an ``expansion" suggests that one of the obvious effects to look for is a 
corresponding {\em non-kinematical redshift} (NKR). As first noted by Synge, 
spectral shifts in GR (and other metric theories) can be analyzed within a 
unified formalism [11]. This means that gravitational, Doppler and standard 
cosmological spectral shifts are of the same nature; we refer to such spectral
shifts as kinematical. Kinematical spectral shifts may be blueshifts or
redshifts depending on the specific circumstances, whereas the NKR is always a
redshift. It should be simple to extend the unified formalism described in
[11] to include the NKR as well by including the non-metric piece of the
connection. This should be so because the NKR essentially arises from the 
non-metric part of the connection just as kinematical spectral shifts arise 
from its metric part.

As a straightforward example, an easy calculation of the NKR can be executed
for the toy cosmological model (4.4). With a change to a convenient radial 
coordinate ${\chi}$ (4.4) reads
\eqa
ds^{2}_t = -(dx^{0})^{2}+c^{2}t^{2}{\Big(}d{\chi}^{2}+{\sin}^{2}{\chi}
d{\Omega}^{2}{\Big)}.
\ena
We want to find the equation of a null geodesic in the ${\chi}$ direction by 
solving (3.74) for the metrics (4.15). To begin with we find that all the 
relevant ordinary connection coefficients vanish, whereas the only relevant 
non-vanishing degenerate connection coefficient is 
${\topstar {\Gamma}}^{\chi}_{t{\chi}}=t^{-1}$. Furthermore the time component
of equation (3.74) yields ${\frac{dx^{0}}{d{\lambda}}}=$constant whereas the 
radial component yields (using $t$ as a time parameter)
\eqa
{\frac{d^{2}{\chi}}{dt^{2}}} + t^{-1}{\frac{d{\chi}}{dt}}=0.
\ena
The solution of (4.16), subject to the condition 
$({\frac{d{\chi}}{dx^0}})_{(t)}=(ct)^{-1}$ (which follows from equations
(3.66) and (3.69)), is 
\eqa
{\chi}(t)={\chi}_0 + {\ln}{\frac{t}{t_0}}.
\ena
Now consider two FOs with radial coordinates ${\chi}_0$ and ${\chi}_1$ 
respectively, and with identical angular coordinates. At epoch $t_0$ two 
pulses of light are emitted in the ${\chi}$-direction by the first FO. The 
pulses are separated by a short time interval ${\Delta}t_0$. At epoch $t_1$ 
the light pulses are received by the second FO, and the separation time 
interval is ${\Delta}t_1$. Using (4.17) we find (neglecting terms of higher 
order in small quantities)
\eqa
{\frac{{\Delta}t_1}{{\Delta}t_0}}={\frac{t_1}{t_0}},
\ena
a result identical to that of metric cosmological models, i.e., representing
a redshift of momentum for decoupled massless particle species. Also, we can 
find the speed
$w{\equiv}{\sqrt{w^2}}=ct({\frac{d{\chi}}{d{\tau}_{\cal F}}})_{(t)}=
c^{2}t({\frac{d{\chi}}{dx^0}})_{(t)}$ with respect to the FOs for some 
inertial material particle emitted in the radial direction. 
Since $({\frac{d{\chi}}{dx^0}})_{(t)}=
{\frac{t_0}{t}}({\frac{d{\chi}}{dx^0}})_{(t_0)}$ (which follows from (3.69))
we have $w=$constant, i.e., material particles do not slow down with respect 
to the FOs. This result is different from that of standard cosmology, and 
illustrates the different nature of the NKE compared to the KE. The curves 
traced out by material inertial test particles moving in the radial direction 
can be found by solving (4.16) subject to the condition 
$({\frac{d{\chi}}{dx^0}})_{(t)}=c^{-2}wt^{-1}$, and the solution is
\eqa
{\chi}(t)={\chi}_0+{\frac{w}{c}}{\ln}{\frac{t}{t_0}}.
\ena
The calculation leading to (4.18) can be generalized, i.e., one can in 
general find the equations of null geodesics connecting pairs of FOs and then 
calculate the frequency shifts of light pulses. The result is a superposition 
of NKR and kinematical spectral shifts. In particular, radiation emitted from 
a FO in a gravitational well should, as seen from a FO far away, show both the 
gravitational redshift and the NKR. A natural question is how this prediction 
of the theory relates to experiments such as measurements of spectral line 
shifts in the solar atmosphere and the Pound-Rebka-Snider measurements of the 
gravitational spectral shift at the Earth's surface [1]. The answer is that 
for these cases the metric approximation (4.13) represents the gravitational 
field to sufficient accuracy, and (4.13) predicts no NKR. (Even if we replace
the Minkowski background present in the metric approximation with a 
${\bf S}^3{\times}{\bf R}$-background the extra NKR due to this background has
only negligible influence on the above-mentioned experiments, see reference 
[14].)

Another direct consequence of the hypothetical existence of the NKE is that
extended systems held together by contact forces should experience extra tidal 
stress due to geodesic deviation. An important special case is that of tidal 
forces experienced by objects moving in the gravitational field of the Earth. 
By using (4.13) we can find the corresponding tidal metric field. The 
difference between this and the analogous tidal field of the Schwarzschild 
metric is of the order ${\frac{r_{s}^{2}c^{2}}{r^{4}}}$ whereas the dominant 
contribution to each tidal field is of order ${\frac{r_{s}c^{2}}{r^{3}}}$. 
Thus it does not seem feasible to directly detect the NKE by measuring tidal 
stresses.

Although it would seem difficult to detect the NKE directly by doing local 
experiments, its relevance for explaining astrophysical observations from 
first principles is quite promising. To have any hope of doing better than 
metric theories in this respect one must focus on situations where the metric 
approximation is not sufficient. In particular, astrophysical systems which 
are large compared to the solar system, but yet non-relativistic in the sense 
of small velocities and weak gravitational fields should be well-suited for 
testing the non-metric aspects of our theory. The most obvious examples of 
such systems are galaxies. Thus one may have some hope that the NKE paradigm 
may possibly explain observed galactic kinematics from first principles, 
a feat that metric theories seem incapable of doing [12]. We return to this 
issue in a followup paper [14], where it is shown that the NKE may also have
been detected in the solar system.

More exotic systems which have been modelled successfully by applying GR are
the binary pulsars [1]. In particular, within GR the decay of orbital periods 
of such objects is naturally explained by assuming that the systems emit
gravitational radiation. To have any hope of competing, this phenomenon must
have a natural explanation within our quasi-metric theory also. Of course, to 
test if this is the case nothing short of a quantitative analysis is 
sufficient.

The fields where the difference in philosophical basis between our 
quasi-metric theory and metric theories shows up most clearly are cosmology 
and large-scale astronomy. That is, the traditional big bang models (and 
alternative models also) describe the Universe as a dynamic system which 
evolution can be deduced from field equations given the initial conditions. 
Using this approach it is not surprising that one of the holy grails of modern
cosmology is to deduce the ``correct" big bang model from observational data. 
In contrast to this, according to our theory the NKE due to a global scale 
factor is the dominant phenomenon on cosmological scales, and thus our 
description of the Universe is not primary that of a dynamic system. In fact, 
a natural beginning of the Universe according to our theory is described by 
the empty Universe toy model (4.4) in the limit $t{\rightarrow}0$. Thus even 
if this model has a curvature singularity at $t=0$ it does not represent a 
{\em physical} singularity since no particle world line actually originates at
the singularity. This removes the vexing question of specifying ``natural" 
initial conditions, a question which cannot be avoided in a purely dynamic 
approach to cosmology. Instead we get the problem of constructing a 
quantitative theory of particle creation based on the NKE and vacuum 
fluctuations of non-gravitational fields. This may perhaps be done along the 
lines described in reference [10]. Thus the possibility exists that one may 
construct a realistic universe model where the Universe has ``expanded from 
a hot dense phase" but yet does not actually belong to the class of big bang 
models.

It remains to be seen if non-kinematically expanding universe models are
capable of giving natural explanations of the plethora of astronomical
observations which exist today; examples are helium abundances, the structure 
of the microwave background etc. In this context one must be aware that
interpretations of observational data may be very different for a 
non-kinematical Universe model than for a big bang model. The main reason for 
this is that the nature of the expansion is different for the two model types. 
In particular this means that in a non-kinematical model one cannot 
uncritically take observed redshifts of objects as indicators of distance. 
That is, any observed spectral shift includes not only the NKR due to the 
expansion on large scales of the Universe, but also the NKR due to the 
intrinsic expansion of gravitational fields. (See [14] for an idealized model
of an expanding gravitational field.) If the latter is not negligible 
compared to the former, interpretation within standard cosmology may lead to 
peculiar results. And for a sufficiently strong gravitational field the NKR
due to the intrinsic expansion will not be negligible since the global part
of the NKE goes as ${\frac{1}{{\bar N}_tt}}$. That is, for an observer far away
${\frac{1}{{\bar N}_tt}}$ far down into the gravitational well appears larger 
than the local value of ${\frac{1}{{\bar N}_tt}}$ due to gravitational time 
dilation. Thus, if an observer far away registers radiation escaping from the 
gravitational well, this radiation gets an extra NKR mimicking the 
cosmological redshift. Thus it is at least possible that massive expanding
objects may systematically show excess redshift which should not necessarily 
be interpreted as cosmological. A well-known example of this is the ``redshift 
controversy" regarding the nature of quasi-stellar objects [13]. Also 
measurements of the Hubble parameter may be sensitive to the NKE intrinsic 
to e.g. galaxy clusters.
\section{Conclusion}
In this paper we have introduced the so-called ``quasi-metric'' framework;
this is a non-metric space-time framework based on the NKE concept. 
Furthermore we have constructed a theory of gravity compatible with this 
framework. The vital question remains; is the theory viable according to 
basic theoretical and experimental criteria?

At least our theory is relativistic, and it displays the correct Newtonian
limit for a large class of situations. However, for ``large" systems 
non-metric effects may not be neglected, and for such systems Newtonian theory 
does not correspond with our theory. On a higher level of precision 
our theory corresponds with GR through metric approximations; we have 
illustrated this by constructing one for the static spherically symmetric 
case. This means that our theory should agree with GR for the classical solar 
system tests (to post-Newtonian accuracy). However, due to the fact that our 
theory is not suitable for a standard PPN analysis, a formal weak field
comparison to GR is not available, see the previous section.

The theoretical structure of our theory makes it plausible that its equations 
are well-posed and self consistent. This also indicates that its predictions 
are unique, i.e., that they do not depend on the method of calculation. 
Finally the theory should not be regarded as complete, i.e. all aspects of
it are not yet fully developed.

So we do not yet know if our theory is viable or not. However, there should
be a fair chance that the NKE concept may be relevant for a better 
understanding from first principles of the behaviour of the physical world. 
This should inspire to further work along the direction we have laid out in 
this paper. \\ [4mm]
{\bf References} \\ [1mm]
{\bf [1]} C.M. Will, {\em Theory and Experiment in Gravitational Physics},
Cambridge University \\ {\hspace*{5.5mm}} Press (1993). \\
{\bf [2]} K. Kucha{\v{r}}, {\em Journ. Math. Phys.} {\bf 17}, 792 (1976). \\
{\bf [3]} C. Gerhardt, {\em Commun. Math. Phys.} {\bf 89}, 523 (1983). \\
{\bf [4]} C.W. Misner, K.S. Thorne, J.A. Wheeler,
{\em Gravitation}, W.H. Freeman ${\&}$ Co. (1973). \\
{\bf [5]} C.H. Brans and R.H. Dicke, {\em Phys. Rev. }{\bf 124}, 925 (1961). \\
{\bf [6]} R.H. Dicke, {\em Phys. Rev. }{\bf 125}, 2163 (1962). \\
{\bf [7]} J. Isenberg and J. Nester, {\em Canonical Gravity}, in
{\em General Relativity and Gravitation,  \\  {\hspace*{5.5mm}} Vol. I}, 
Editor A. Held, Plenum Press (1980). \\
{\bf [8]} K. Kucha{\v{r}}, {\em Journ. Math. Phys.} {\bf 17}, 777 (1976). \\
{\bf [9]} S. Weinberg, {\em Gravitation and Cosmology}, John Wiley ${\&}$ Sons,
Inc. (1972). \\
{\bf [10]} L. Parker, {\em Particle Creation by the Expansion of the 
Universe}, in {\em Handbook of \\
{\hspace*{7.5mm}}Astronomy, Astrophysics and Geophysics, Vol. II Galaxies and 
Cosmology}, \\ {\hspace*{7.5mm}}editors V.M. Canuto and B.G. Elmegreen,
Gordon $\&$ Breach (1988). \\
{\bf [11]} J.V. Narlikar, {\em Am. J. Phys.} {\bf 62}, 903 (1994). \\
{\bf [12]} V.V. Zhytnikov and J.M. Nester, {\em Phys. Rev. Lett.} {\bf 73},
2950 (1994). \\
{\bf [13]} H. Arp, {\em Quasars, Redshifts and Controversies}, Interstellar
Media (1987). \\
{\bf [14]} D. {\O}stvang, chapter 3 of this thesis (2001). \\
{\bf [15]} D. {\O}stvang, chapter 1 of this thesis (2001). \\
{\bf [16]} J. Isenberg, {\em Class. Quantum Grav.} {\bf 12}, 2249 (1993).
\appendix
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\section{Derivation of a static line element}
\setcounter{equation}{0}
In this appendix we give some details of the derivation of the solution 
(4.8). To do this, we insert the general line element (2.21) into (4.7) to 
determine the functions ${\bar A}(r)$ and ${\bar B}(r)$. Connection 
coefficients for the line element (2.21) can be found in e.g. [9]. The 
components of the acceleration field are (using the notation 
$'{\equiv}{\frac{\partial}{{\partial}r}}$)
\eqa
{\bar a}_{{\cal F}r}=c^{2}{\frac{{\bar B}'}{2{\bar B}}}, {\qquad} 
{\bar a}_{{\cal F}{\theta}}={\bar a}_{{\cal F}{\phi}}=0,
\ena
and the spatial covariant derivative of the acceleration field has the 
non-zero components
\eqa
{\bar a}_{{\cal F}r{\mid}r}={\frac{c^{2}}{2}}{\bigg[}
{\frac{{\bar B}''}{{\bar B}}}-({\frac{{\bar B}'}{{\bar B}}})^{2}-
{\frac{{\bar A}'{\bar B}'}{2{\bar A}{\bar B}}}{\bigg]}, {\qquad} 
{\sin}^{-2}{\theta}{\bar a}_{{\cal F}{\phi}{\mid}{\phi}}=
{\bar a}_{{\cal F}{\theta}{\mid}{\theta}}={\frac{c^{2}}{2}}
{\frac{{\bar B}'r}{{\bar A}{\bar B}}}.
\ena
The spatial covariant divergence of the acceleration field is
\eqa
{\bar a}^{k}_{{\cal F}{\mid}k}=c^{2}{\bigg(}
{\frac{{\bar B}''}{2{\bar A}{\bar B}}}-
{\frac{{\bar B}'^{2}}{2{\bar A}{\bar B}^{2}}}-
{\frac{{\bar A}'{\bar B}'}{4{\bar A}^{2}{\bar B}}}+
r^{-1}{\frac{{\bar B}'}{{\bar A}{\bar B}}}{\bigg)}.
\ena
Next a standard calculation yields the spatial Ricci tensor ${\bf {\bar P}}$ 
for the line element (2.21), which we use to find the spatial Einstein tensor 
${\bf {\bar H}}$. The result is
\eqa
{\bar P}&=&{\frac{2}{r}}{\bigg[}r^{-1}(1-{\bar A}^{-1})+
{\frac{{\bar A}'}{{\bar A}^{2}}}{\bigg]}, 
\ena
\eqa
{\bar H}_{rr}&{\equiv}&{\bar P}_{rr}-{\frac{{\bar P}}{2}}{\bar h}_{rr}=
r^{-2}(1-{\bar A}), {\nonumber} \\ {\sin}^{-2}{\theta}{\bar H}_{{\phi}{\phi}}=
{\bar H}_{{\theta}{\theta}}&{\equiv}&
{\bar P}_{{\theta}{\theta}}-{\frac{{\bar P}}{2}}{\bar h}_{{\theta}{\theta}}=
-{\frac{r{\bar A}'}{2{\bar A}^{2}}}.
\ena
Inserting the above relations into equations (4.7) yields
\eqa
{\frac{{\bar B}''}{{\bar B}}}-
{\frac{{\bar A}'{\bar B}'}{2{\bar A}{\bar B}}}+2r^{-2}(1-{\bar A})=0,
\ena
\eqa
{\frac{{\bar B}''}{{\bar B}}}-{\frac{{\bar B}'^2}{2{\bar B}^2}}-
{\frac{{\bar A}'{\bar B}'}{2{\bar A}{\bar B}}}+
{\frac{2}{r}}{\frac{{\bar B}'}{{\bar B}}}=0,
\ena
\eqa
{\frac{{\bar A}'}{{\bar A}}}-{\frac{{\bar B}'}{{\bar B}}}-
r{\frac{{\bar B}'^2}{2{\bar B}^2}}=0. 
\ena
We can separate the variables ${\bar A}(r)$ and ${\bar B}(r)$ by inserting 
equation (A.8) into equations (A.6) and (A.7). The result is
\eqa
{\frac{{\bar A}''}{{\bar A}}}-{\frac{3{\bar A}'^{2}}{2{\bar A}^{2}}}+
r^{-1}{\bar A}'(3{\bar A}^{-1}-1)=0, 
\ena
\eqa
{\frac{{\bar B}''}{{\bar B}}}-{\frac{{\bar B}'^{2}}{{\bar B}^{2}}}-
r{\frac{{\bar B}'^{3}}{4{\bar B}^{3}}}+r^{-1}{\frac{2{\bar B}'}{{\bar B}}}=0, 
\ena
\eqa
{\bar A}=1-r{\frac{{\bar B}'}{{\bar B}}}+
r^{2}{\frac{{\bar B}'^{2}}{4{\bar B}^{2}}}.
\ena
One way of solving these equations is to introduce 
${\bar C}(r){\equiv}r{\frac{{\bar B}'}{{\bar B}}}$. One may then show that 
(A.10) is integrable in this new variable. This yields ${\bar B}(r)$, and 
(A.11) gives ${\bar A}(r)$ directly when ${\bar B}(r)$ is known. The result 
is the metric (4.8), which is unique. An alternative way of writing (4.8) can 
be found by using the identity
\eqa
{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}-{\frac{r_{s}}{2r}}{\equiv}
{\bigg(}{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}+{\frac{r_{s}}{2r}}{\bigg)}^{-1}.
\ena
That (4.8) is indeed conformally flat, we can show by introducing an isotropic
radial coordinate ${\rho}$ replacing the Schwarzschild coordinate $r$:
\eqa
{\rho}{\equiv}{\exp}{\bigg(}{\int}r^{-1}
{\Big[}{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}-{\frac{r_{s}}{2r}}{\Big]}
{\Big[}1+({\frac{r_{s}}{2r}})^{2}{\Big]}^{-1/2}dr{\bigg)}.
\ena
Evaluating the integral and choosing an appropriate constant of integration,
we find
\eqa
{\rho}=r{\bigg(}{\sqrt{1+({\frac{r_{s}}{2r}})^{2}}}+{\frac{r_{s}}{2r}}{\bigg)},
{\qquad} r={\rho}{\sqrt{1-{\frac{r_{s}}{\rho}}}},
\ena
and (4.8) expressed in isotropic coordinates reads
\eqa
\overline{ds}^{2}={\bigg(}1-{\frac{r_{s}}{\rho}}{\bigg)}
{\bigg(}-(dx^{0})^{2}+d{\rho}^{2}+{\rho}^{2}d{\Omega}^{2}{\bigg)},
\ena
which is conformally flat.
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\section{The Newtonian limit}
\setcounter{equation}{0}
To find Newtonian approximations to the local conservation laws (3.44)
in ${\overline{\cal N}}$ or equivalently, to the evolution laws (3.45) and 
(3.46); in a static GTCS we can use the Newtonian form (4.14) of each metric 
with Cartesian coordinates, i.e., ${\bar g}_{00}{\approx}-1+2c^{-2}U$, 
${\bar h}_{ij}{\approx}{\delta}_{ij}$, where $-U$ is the Newtonian potential. 
Furthermore, to Newtonian order the active and passive stress-energy tensors 
are equal. For a perfect fluid we then have, to Newtonian order
\eqa
T^{00}={\rho}_{m}c^{2}, {\qquad} {T^{0}}_{j}={\rho}_{m}cw_{j},
{\qquad}{T^{i}}_{j}={\rho}_{m}w^{i}w_{j}+p{{\delta}^{i}}_{j}.
\ena
Also, to Newtonian accuracy $c^{-2}U$, $c^{-2}w^{2}$, $c^{-2}p/{\rho}_{m}$,
$c^{-4}{\bar a}_{\cal F}^i{\bar a}_{{\cal F}i}/{\mid}
{\frac{\partial}{{\partial}x^0}}{\mid}^2$,
$c^{-2}{\mid}{\bar a}_{\cal F}^i{\mid}/{\mid}
{\frac{\partial}{{\partial}x^i}}{\mid}$ and
${\mid}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\mid}/{\mid}
{\frac{\partial}{{\partial}x^0}}{\mid}$ are all small, of second order in the 
small quantity ${\epsilon}$, say. To Newtonian order one uses the 
approximations 
\eqa
{T^{0{\nu}}}_{{\bar *}{\nu}}{\approx}
{T^{0{\nu}}}_{;{\nu}}{\approx}{T^{00}},_0+{T^{0i}},_i
+{\Gamma}^0_{00}T^{00}, \qquad
{T^{j{\nu}}}_{{\bar *}{\nu}}{\approx}{T^{j{\nu}}}_{;{\nu}}{\approx}
{T^{j0}},_0+{T^{ji}},_i+{\Gamma}^j_{00}T^{00}, 
\ena
this is equivalent to retaining only terms of not higher than first order in 
${\epsilon}$. The approximations made in (B.2) yield the same Newtonian
limit as in the metric case.

Keeping only lowest order terms and terms not higher than one extra order in 
$\epsilon$ in (3.44) yields ($t$ and the ordinary coordinate time are 
equivalent in the Newtonian approximation)
\eqa
{\frac{\partial}{{\partial}t}}{\rho}_{m}+({\rho}_{m}w^{i}),_{i}&=&
O({\epsilon}^2),
{\nonumber} \\ p,_{j}+{\rho}_{m}(w_{j},_{t}+w_{j},_{i}w^{i})&=&
-{\rho}_{m}{\bar a}_{{\cal F}j}+O({\epsilon}^{2})=
{\rho}_mU,_j + O({\epsilon}^2).
\ena
The equations (B.3) are identical to the Euler equations describing 
non-relativistic hydrodynamics. Here we have used the fact that in the 
Newtonian limit ${\bar a}_{{\cal F}i}=-U,_i$ which follows from (4.14). 

Next, use the Newtonian form (4.14) of the metric in combination with (B.1)
to get the Newtonian limit of the field equation (3.49). Neglecting terms of 
$O({\epsilon}^2)$ equation (3.49) yields (using a Cartesian coordinate system)
\eqa
4{\pi}G{\rho}_m={\bar a}_{{\cal F}{\mid}k}^k=-{U_,}^k_{{\ }k},
\ena
which is Newton's gravitational equation as asserted. Note that the field
equations (3.52) become vacuous in the Newtonian limit. This also applies to
the Newtonian limit of equations (3.53) and (3.54).

Finally, to have a well-defined Newtonian limit we must show that our equations
of motion (3.74) reduce to the Newtonian equations of motion in the Newtonian 
limit. But since equations (3.74) reduce to the usual geodesic equation in the
metric approximation (since $t{\rightarrow}{\infty})$, they certainly must 
reduce to the Newtonian equations of motion in the Newtonian limit. With this 
result we have shown that our theory has a consistent Newtonian limit.

When deciding whether or not it is sufficient to apply the Newtonian 
approximation for any particular system we should always have in mind that in 
our theory, the Newtonian limit is a strictly {\em local} limit. That is, even 
if the gravitational field of the system is weak everywhere and the motion of 
of its constituents is always non-relativistic we must also require that the
system is ``small" so that the global effects of the NKE may be neglected if
Newtonian theory is to be applied with confidence. This is one of the crucial
differences between our theory and any metric alternative.
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\section{Isotropic cosmological models with matter}
\setcounter{equation}{0}
Rather than the vacuum toy cosmological model considered in section 4 we may
try to construct non-vacuum toy models where the source is a perfect fluid
comoving with the FOs. That is, as a generalization of equation (4.4), each 
toy model may be described by a metric family ${\bf {\bar g}}_t$ of the form
\eqa
{\overline{ds}}_t^2=-{\bar N}_t^2(dx^0)^2+({\bar N}_tct)^2(d{\chi}^2
+{\sin}^2{\chi}d{\Omega}^2).
\ena
As a counterpart to the Friedmann models in GR it is reasonable to try
to construct models where the matter distribution is isotropic. That is, we
consider models where ${\bar N}_t$ does not depend on spatial coordinates. We 
show in this appendix that the only isotropic models possible in quasi-metric
gravity are models where the source is a null fluid.

It is convenient to express the stress-energy tensor for a perfect fluid 
comoving with the FOs via the coordinate volume density of active mass-energy 
${\bar {\rho}}_m$ and the corresponding pressure ${\bar p}$, i.e.
\eqa
T_{(t){\bar {\perp}}{\bar {\perp}}}{\equiv}
{\Big (}{\frac{t_0}{{\bar N}_tt}}{\Big )}^2{\bar {\rho}}_mc^2, {\qquad}
T_{(t){\chi}}^{\chi}=T_{(t){\theta}}^{\theta}=T_{(t){\phi}}^{\phi}
{\equiv}{\Big (}{\frac{t_0}{{\bar N}_tt}}{\Big )}^2{\bar p}.
\ena
Note that ${\bar {\rho}}_m$ does not depend on $t$ if there is no particle
creation (i.e., if the local conservation laws hold). Moreover, assuming no
particle creation, the local conservation laws in ${\overline{\cal N}}$ are
sufficiently represented by equations (3.45) and (3.46), which yield
\eqa
{\cal L}_{{\bar {\bf n}}_t}{\bar {\rho}}_m=
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}{\Big (}
{\bar {\rho}}_m-3c^{-2}{\bar p}{\Big )}=0.
\ena
One reason why ${\cal L}_{{\bar {\bf n}}_t}{\bar {\rho}}_m$ must vanish is that
no other possibilities yield solutions of the field equations with a proper 
vacuum limit (i.e., the empty model found in section 4). This means that a 
null fluid is the only nontrivial possibility satisfying (C.3). {\em That is, 
in our theory no cosmological model with a perfectly isotropic material fluid 
source can exist.} Thus in our theory, except for the vacuum model found in 
section 4, the only allowable cosmological models with isotropic FHSs are 
models where the source is a null fluid.

To confirm that cosmological models with an isotropic null fluid source are
indeed possible in quasi-metric gravity, we must solve the field equations.
The field equation (3.49) yields, after some straightforward manipulations 
using equation (3.9)
\eqa
{\cal L}_{{\bf {\bar n}}_t}{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}
{{\bar N}_t}}{\Big )}-{\Big (}{\frac{{{\bar N}_t,_{\bar {\perp}}}}{{\bar N}_t}}
{\Big )}^2=
{\Big (}{\frac{t_0}{{\bar N}_tt}}{\Big )}^2{\frac{\kappa}{6}}
({\bar {\rho}}_mc^2+3{\bar p}),
\ena
or equivalently, for a null fluid
\eqa
{\Big (}{\frac{{\bar N}_t,_0}{{\bar N}_t}}{\Big )},_0=
-{\frac{t_0^2}{t^2}}{\frac{\kappa}{3}}{\bar {\rho}}_mc^2,
\ena
where the right hand side of (C.5) is a constant (for $t$ constant). 
Integrating equation (C.5) twice and requiring continuity with the empty model
found in section 4 in the vacuum limit ${\bar {\rho}}_m{\rightarrow}0$, we 
find the solution
\eqa
{\bar N}_t={\exp}{\Big [}-{\frac{\kappa}{6}}{\frac{(ct_0x^0)^2}{(ct)^2}}
{\bar {\rho}}_mc^2{\Big ]}.
\ena
We see from (C.6) that ${\bar N}_t$ is constant in ${\overline{\cal N}}$. Note
that it is also possible to find isotropic null fluid models where 
${\bar {\rho}}_m$ depends on $t$, i.e., where there is local creation of null 
particles. For such models ${\bar N}_t$ will depend on $t$ in
${\overline{\cal N}}$. Also note that more general cosmological models with 
homogeneous but non-isotropic matter distributions should exist in quasi-metric
gravity. In particular models where the matter density on each FHS fluctuates 
around some mean density should be possible according to our theory since the 
FHSs cannot be isotropic in such cases.

The result of this appendix confirms the assertion that in quasi-metric
theory, the rate of the expansion of the Universe should not depend on 
dynamical gravitational parameters such as matter density. Rather the cosmic
expansion must be purely non-kinematical and thus a consequence of the global
curvature of space. Compared to traditional cosmology, this represents a
radical difference in philosophy.
\section{Symbols and acronyms}
In this appendix we give a table of symbols and acronyms used in the text. 
Sign conventions and notations follow reference [4] if otherwise not stated.
In particular Greek indices may take values $0-3$ and Latin indices may take
values $1-3$.
\newpage
%\vspace{1.5cm}
\begin{tabular}{lp{10.8cm}}
{\bf SYMBOL} & {\bf NAME} \\ 
\ & \\
KE/NKE & kinematical/non-kinematical evolution (of the FHSs) \\
FOs/FHSs & fundamental observers/fundamental hypersurfaces \\
NKR/NKCE & non-kinematical redshift/non-kinematical coupling equation \\
GTCS & global time coordinate system \\
HOCS & hypersurface-orthogonal coordinate system \\
WEP/EEP & weak/Einstein equivalence principle \\
GWEP/SEP & gravitational weak/strong equivalence principle \\
LLI/LPI & local Lorentz/positional invariance \\
SR/GR & special/general relativity \\
SC & Schiff's conjecture \\
$t=t(x^{\mu})$ & global time function, conventionally we define $t=x^{0}/c$
when using a GTCS \\
${\cal M}$/${\overline{\cal M}}$ & metric space-time manifolds obtained by 
holding $t$ constant \\
${\cal N}$/${\overline{\cal N}}$ & quasi-metric space-time manifolds 
(foliated by the FHSs) \\
${\bf g}_t$ or $g_{(t){\mu}{\nu}}$ & one-parameter family of
space-time metric tensors on ${\cal N}$ \\
${\bf {\bar g}}_t$ or ${\bar g}_{(t){\mu}{\nu}}$ & one-parameter family of
space-time metric tensors on ${\overline{\cal N}}$ \\
${\bf {\tilde g}}_t$ or ${\tilde g}_{(t){\mu}{\nu}}$ & 
one-parameter family of auxiliary space-time metric tensors 
on ${\overline{\cal N}}$ \\
${\bf n}_t$ or $n_{(t)}^{\mu}$ & unit vector fields normal to the FHSs 
in the metrics ${\bf g}_t$ \\
${\bf {\bar n}}_t$ or ${\bar n}_{(t)}^{\mu}$ & unit vector fields normal to 
the FHSs in the metrics ${\bf {\bar g}}_t$ \\
${\bf h}_t$ or $h_{(t)ij}$ & spatial metric tensors of the FHSs obtained from
${\bf g}_t$ \\
${\bf {\bar h}}_t$ or ${\bar h}_{(t)ij}$ & spatial metric tensors of the FHSs 
obtained from ${\bf {\bar g}}_t$ \\
$N(x^{\mu})$ & lapse function in the metrics ${\bf g}_t$ \\
${\bar N}_t(x^{\mu},t)$ & lapse function family in the metrics 
${\bf {\bar g}}_t$ \\
$N^j(x^{\mu})$ & components of the shift vector field in the metric
${\bf g}_{t_0}$ \\
${\bar N}^j(x^{\mu})$ & components of the shift vector field in the metric
${\bf {\bar g}}_{t_0}$ \\
${\Gamma}^{\alpha}_{(t){\beta}{\gamma}}$ & components of the metric connections
${\nabla}_{t}$ compatible with ${\bf g}_{t}$ \\
${\bar {\Gamma}}^{\alpha}_{(t){\beta}{\gamma}}$ & components of the metric 
connections ${\bar {\nabla}}_{t}$ compatible with ${\bf {\bar g}}_{t}$ \\
${\tilde {\Gamma}}^{\alpha}_{(t){\beta}{\gamma}}$ & components of the metric 
connections ${\tilde {\nabla}}_{t}$ compatible with ${\tilde {\bf g}}_{t}$ \\
${\topstar{\Gamma}}^{\alpha}_{t{\gamma}}$,
${\topstar{\Gamma}}^{\alpha}_{{\beta}{\gamma}}$ & components of the degenerate
5-dimensional connection ${\topstar{\nabla}}$ associated with the
family ${\bf g}_{t}$ \\
${\topstar{\bar {\Gamma}}}^{\alpha}_{t{\gamma}}$,
${\topstar{\bar {\Gamma}}}^{\alpha}_{{\beta}{\gamma}}$ & components of the 
degenerate 5-dimensional connection ${\stackrel{\star}{\bf {\bar {\nabla}}}}$ 
associated with the family ${\bf {\bar g}}_{t}$ \\
${\bf a}_t$ or ${a}_{(t)}^{\mu}$ & family of metric 4-accelerations 
(any observer) 
\end{tabular}
\newpage
\begin{tabular}{lp{10.8cm}}
$()_{;{\alpha}}$ & coordinate expression for metric covariant derivation 
compatible with a single member of ${\bf g}_t$ or ${\bf {\bar g}}_t$ as 
appropriate \\
$()_{{\bar *}{\alpha}}$ & coordinate expression for degenerate covariant 
derivation compatible with the family ${\bf {\bar g}}_t$ \\
$()_{{\mid}j}$ & coordinate expression for spatial covariant derivation 
compatible with ${\bf h}_t$ or ${\bf {\bar h}}_t$ as appropriate \\
${\perp}$/${\bar {\perp}}$ & projection symbols (projection 
on the normal direction to the FHSs) \\
{\pounds}$_{\bf X}$ & Lie derivative with respect to {\bf X} in
${\overline{\cal M}}/{\cal M}$ \\
${\topstar{\pounds}}_{\bf X}$ & Lie derivative with respect to {\bf X} in
${\overline{\cal N}}/{\cal N}$ \\
${\cal L}_{\bf X}$ & Lie derivative of space objects with respect to {\bf X} 
in ${\overline{\cal M}}/{\cal M}$ \\
${\topstar{\cal L}}_{\bf X}$ & Lie derivative of space objects with respect to 
{\bf X} in ${\overline{\cal N}}/{\cal N}$ \\
${\bf a}_{\cal F}$/${\bf {\bar a}}_{\cal F}$ or 
$a^{i}_{\cal F}$/${\bar a}^{i}_{\cal F}$ & four-acceleration of the FOs
in $\cal M$/${\overline{\cal M}}$ \\
${\bf b}_{\cal F}$ or $b^{i}_{\cal F}$ & 3-vector field family representing 
the local ``generalized distance vector to the center of gravity'' \\
${\bf {\bar e}}_b$/${\bf {\bar e}}^b$ or 
${\bar e}^{i}_b$/${\bar e}_{i}^b$ & unit vector/covector along 
${\bf b}_{\cal F}$ \\
${\bf w}_t$ or $w^i_{(t)}$ & the difference in coordinate 3-velocity
between a test particle and the local FO \\
${\bf u}_t$ or $u^{\mu}_{(t)}$ & 4-velocity of test particle \\
${\bf v}_t$ or $v^{i}_{(t)}$ & 3-vector fields giving the motion of FOs
in $\cal N$ compared to FOs in ${\overline{\cal N}}$ \\
${\stackrel{\star}{\bf a}}$ or ${\topstar{a}}^{\mu}$ & degenerate 
4-acceleration in $\cal N$ (any observer) \\
%$G_{(t){\mu}{\nu}}$/${\bar G}_{(t){\mu}{\nu}}$ & Einstein tensors in 
%$\cal M$/${\overline{\cal M}}$ \\
$R_{(t){\mu}{\nu}}$/${\bar R}_{(t){\mu}{\nu}}$ & Ricci tensors in 
$\cal M$/${\overline{\cal M}}$ \\
$H_{(t)ij}$/${\bar H}_{(t)ij}$ & spatial Einstein tensors of the FHSs in
$\cal M$/${\overline{\cal M}}$ \\
$P_{(t)ij}$/${\bar P}_{(t)ij}$ & spatial Ricci tensors of the FHSs in
$\cal M$/${\overline{\cal M}}$ \\
$P_t$/${\bar P}_t$ & spatial Ricci scalars of the FHSs in
$\cal M$/${\overline{\cal M}}$ \\
${\bf K}_t$ or $K_{(t)ij}$ & family of extrinsic curvature tensors of the FHSs 
in $\cal M$ \\
${\bf {\bar K}}_t$ or ${\bar K}_{(t)ij}$ & family of extrinsic curvature 
tensors of the FHSs in ${\overline{\cal M}}$ \\
${\bar H}_t$/${\bar y}_t$& local measures of the NKE \\
${\bar x}_t$ & local measure of the KE \\
$d{\tau}_t$/$\overline{d{\tau}}_t$ & proper time interval (any observer) in 
$\cal N$/${\overline{\cal N}}$ \\
$d{\tau}_{\cal F}$/$\overline{d{\tau}}_{\cal F}$ {\hspace*{1.3cm}} 
& proper time interval for a FO in $\cal N$/${\overline{\cal N}}$ \\
${\bf T}_t$ or $T^{{\mu}{\nu}}_{(t)}$ %{\hspace*{2.3cm}} 
& stress-energy tensor as an active source of gravitation \\
$G/{\kappa}$ %{\hspace*{2.3cm}} 
& Newton/Einstein gravitational constant \\
$m_t$, $M_t$ & active masses (scalar fields measured gravitationally) \\
${\bar F}_t{\equiv}{\bar N}_tct{\equiv}{\Psi}_t^{-1}$ %{\hspace*{2.3cm}} 
& local scale factors of the FHSs in ${\overline{\cal N}}$ 
%${\Psi}_t{\equiv}{\bar F}_t^{-1}$ %{\hspace*{2.3cm}} 
%& a scalar field defining the difference between atomic and
%gravitational scales  \\
%$-U$ %{\hspace*{2.3cm}} 
%& Newtonian potential 
%${\bar {\rho}}_m$ %{\hspace*{2.3cm}} 
%& the amount of active mass per coordinate volume 
%${\bar p}$ %{\hspace*{2.3cm}} 
%& pressure corresponding to ${\bar {\rho}}_m$
\end{tabular}

\newpage

{\vspace*{5cm}}

\begin{center}
{\huge {\bf Chapter Three: \\\ \\The Gravitational Field outside a\\
\vspace{0.5cm}Spherically Symmetric,}}
\end{center}
\vspace{-0.8cm}
\begin{center}
{\huge {\bf Isolated Source in a\\
\vspace{0.5cm}Quasi-Metric Theory of Gravity}}
\end{center}
\newpage
{\hspace*{1cm}}

\newpage

\renewcommand{\theequation}{\arabic{equation}}

\renewcommand{\thesection}{\arabic{section}}

\setcounter{equation}{0}

\setcounter{section}{0}

\begin{center}
{\large {\bf The Gravitational Field outside a Spherically Symmetric, Isolated
Source in a Quasi-Metric Theory of Gravity}}
\end{center}
\begin{center}
by
\end{center}
\begin{center}
Dag {\O}stvang \\
{\em Institutt for Fysikk, Norges teknisk-naturvitenskapelige universitet, 
NTNU \\
N-7491 Trondheim, Norway}
\end{center}
\begin{abstract}
Working within the ``quasi-metric'' framework (QMF) described elsewhere [3],
we find an approximate expression for an isotropic vacuum gravitational field 
in a ${\bf S}^3{\times}{\bf R}$-background and set up equations of motion 
applying to inertial test particles moving in this field. It is found that 
such a gravitational field is not static with respect to the cosmic expansion;
i.e., distances between circular orbits increase according to the Hubble law. 
Furthermore it is found that the dynamically measured mass of the source 
increases with cosmic scale; this is a consequence of the fact that in the 
QMF, the cosmic expansion is not a kinematical phenomenon. The relevance of 
this to the observed asymptotically non-Keplerian behaviour of distant 
orbiters in spiral galaxies is discussed. Also it is shown that if this model 
of an expanding gravitational field is taken to represent the gravitational 
field of the solar system, this has no serious consequences for observational 
aspects of planetary motion.
\end{abstract}
\section{Introduction}
In metric theory the nature of the cosmological expansion is in principle
not different from other types of motion; this follows from the most basic 
postulate in metric theory, namely that space-time can be modelled as a
semi-Riemannian manifold. Furthermore, it is well-known that metric theory 
predicts that local systems are hardly affected at all by the cosmological 
expansion (its effect is at best totally negligible, see e.g. [8] and 
references therein). However, when analyzing the influence of the 
cosmological expansion on local systems there should be no reason to expect
that predictions made within the metric framework should continue to hold in 
a theory where the structure of space-time is non-metric.

Recently a new type of non-metric space-time framework, the so-called
``quasi-metric'' framework, was presented in [3]. Also presented was an
alternative relativistic theory of gravity formulated within this framework. 
This theory correctly predicts the ''classic'' solar system tests in the case 
where an asymptotically Minkowski background is invoked as an approximation 
and the cosmological expansion can be neglected. However, for reasons 
explained in [3], the theory predicts a ${\bf S}^3{\times}{\bf R}$-background 
rather than a Minkowski background for the gravitational field outside a 
spherically symmetric, isolated source. According to the quasi-metric 
framework, the nature of the cosmological expansion is a direct consequence of
the existence of this ${\bf S}^3{\times}{\bf R}$-background. Thus, to find how
the cosmological expansion affects local systems according to the quasi-metric 
theory, one must first calculate the spherically symmetric gravitational field
with this background and then calculate how test particles move in this field.

Of particular interest is if the cosmological expansion has any effects on the
correspondence with Newton's theory of gravity in the non-relativistic limit.
If this is the case one may hope to explain from first principles the
asymptotically non-Keplerian rotational curves of spiral galaxies as inferred
from observations of Doppler shifted 21-cm spectral lines of neutral hydrogen.
(See reference [1] for detailed information.)

It has been shown that it is impossible to construct a realistic {\em metric} 
theory of gravity (subject to some rather general criteria of structure) such 
that the solar system tests come in and in addition galactic observations can 
be explained without dark matter [2]. But one remaining alternative to the 
introduction of dark matter may be the possibility of constructing a 
{\em quasi-metric} relativistic theory of gravity having the desirable 
properties. In this paper we explore if non-metric aspects of our theory 
correctly predict galactic observations from first principles, without 
introducing extra {\em ad hoc} parameters.
\section{General equations of motion}
We start by briefly summarizing the basics of our quasi-metric theory, see 
reference [3] for details. 

The geometric foundation of the quasi-metric framework consists of a 
4-dimensional space-time manifold ${\cal N}$ equipped with a one-parameter 
family of Lorentzian 4-metrics {\bf g}$_t$ parameterized by the global time 
function $t$. The global time function slices out a family ${\cal S}_t$ of 
preferred spatial submanifolds; we call these the fundamental hypersurfaces 
(FHSs). Observers always moving orthogonally to the FHSs are called 
fundamental observers (FOs). Each member of the family {\bf g}$_t$ applies 
exactly on the fundamental hypersurface $t=$constant only, but it can be 
extrapolated at least to a (infinitesimally) small normal interval around this
hypersurface. (Other spatial hypersurfaces than the FHSs cannot be associated 
with the domains of applicability of single members of the metric family 
{\bf g}$_t$.) In component notation we formally write ${\bf g}_t$ as 
$g_{(t){\mu}{\nu}}$, where the dependence on $t$ is put in a parenthesis not 
to confuse it with the coordinate indices. The components of each single 
metric are obtained by treating $t$ as a constant. Since the theory is 
concerned primarily with the evolution of the geometry of the the FHSs along 
the world lines of the FOs we consider only a particular set of coordinate 
systems; namely the {\em global time coordinate systems} (GTCSs). By 
definition a coordinate system is a GTCS if and only if the relation $ct=x^0$ 
exists in $\cal N$ between the global time function and a global time 
coordinate $x^0$. 

The evolution of the scale factor of the FHSs may be thought of as consisting 
of a combination of two different contributions; we call these parts the
non-kinematical evolution (NKE) and the kinematical evolution (KE), 
respectively. We define these terms in the following.

A peculiarity of the quasi-metric framework is that the kinematical metric 
family ${\bf g}_t$ in general does not represent a solution of field 
equations. The reason for this is that ${\bf g}_t$ may contain some features 
inherently coming from the NKE. To avoid having these features as an explicit 
part of the dynamics we introduce a second family of metrics 
${\bf {\bar g}}_t$ which by definition {\em does} represent a solution of 
field equations. The first family ${\bf g}_t$ may then be constructed from the 
second family in a way described in [3].

In a GTCS the general form of the family ${\bf {\bar g}}_t$ may be represented
by the family of line elements
\eqa
{\overline{ds}}^2_t={\Big [}{\bar N}_s{\bar N}^s-{\bar N}_t^2{\Big ]}(dx^0)^2
+2{\frac{t}{t_0}}{\bar N}_idx^idx^0+
{\frac{t^2}{t_0^2}}{\bar N}_t^2S_{ij}dx^idx^j,
\ena
where ${\bar N}_t$ is the lapse function field family of the FOs, 
${\frac{t_0}{t}}{\bar N}^k{\frac{\partial}{{\partial}x^k}}$ is the family of 
shift vector fields of the FOs and where $S_{ij}dx^idx^j$ denotes the metric of
the 3-sphere (with a radius equal to $ct_0$). Note that 
${\bar N}_i{\equiv}{\bar N}_t^2S_{ik}{\bar N}^k$.

The NKE of the FHSs is defined to take the form of an increase with time of 
the local scale factor ${\bar F}_t{\equiv}{\bar N}_tct$ of the FHSs as 
determined by the FOs using the metric family ${\bf {\bar g}}_t$. The 
non-kinematical coupling equation (NKCE) of our theory yields a relation 
between the rate of increase of local scale and the global curvature of 
the FHSs in addition to a term ${\bar y}_t$ coming from the gravitational 
field; this second term is not ``realized'' globally. That is, it can be 
argued [3] that the NKCE should read
\eqa
{\bar H}_t{\equiv}c{\bar F}_t^{-1}{\topstar{\cal L}}_{{\bf {\bar n}}_t} 
{\bar F}_t-c^{-1}{\bar x}_t
={\frac{1}{{\bar N}_tt}}+{\bar y}_t, \qquad
{\bar y}_t{\equiv}{\frac{c}{{\bar N}_t}}
{\sqrt{{\bar N}_t,_i{\bar N}_t,_k{\bar h}_{(t)}^{ik}}},
\ena
where ${\topstar{\cal L}}_{{\bf {\bar n}}_t}$ denotes the Lie derivative
in the direction normal to the FHSs and ${\bar h}_{(t)}^{ik}$ are the
components of the inverse spatial metric family ${\bf {\bar h}}_t^{-1}$.
There is also a kinematical evolution (KE) of the ${\bar F}_t$; this is 
described by the quantity $c^{-1}{\bar x}_t$. The overall evolution of 
${\bar F}_t$ is given by (note that a global term ${\frac{1}{{\bar N}_tct}}$ 
still occurs explicitly here but that there is no term $c^{-1}{\bar y}_t$)
\eqa
c^{-2}{\bar x}_t+c^{-1}{\bar H}_t{\equiv}
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}+
(c{\bar N}_t)^{-1}{\Big (}{\frac{1}{t}}+
{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big )},
\ena 
where the projection symbol ${\bar {\perp}}$ denotes a scalar product with
$-{\bf {\bar n}}_t$, which is a family of unit normal vector fields to the 
FHSs. From equation (1) we see that the spatial metric family 
${\bf {\bar h}}_t$ is conformal to a ``basic" 3-geometry, namely ${\bf S}^3$; 
this implies that the FHSs are compact (without boundaries). Furthermore we 
can find a set of field equations whereof ${\bf {\bar g}}_t$ must be a 
solution. The detailed form of the field equations can be found in [3]. Note 
the characteristic property of our theory that there are systematic scale 
changes between gravitational and atomic systems; this causes gravitational 
quantities to exhibit an extra formal variation when measured in atomic units 
(and vice versa). This may be thought of formally as if fixed operationally 
defined atomic units vary in space-time. Since $c$ and Planck's constant 
${\hbar}$ by definition are not formally variable the formal variation of time
units is equal to that of length units and inverse to that of mass units. We 
represent the formal variation of atomic time units by the scalar field 
${\Psi}_t{\equiv}{\bar F}_t^{-1}$. This means that measured in atomic units, 
the formal variability of gravitational quantities with the dimension of time 
or length goes as ${\Psi}_t^{-1}$. By definition the gravitational constant 
$G$ does not vary in space-time, but we then have to differentiate between 
{\em active mass} $m_t$ as an active source of gravitation and {\em passive 
mass} $m$ (passive gravitational mass or inertial mass). Active mass then 
varies formally as ${\Psi}_t^{-1}$ measured in atomic units.

The formal evolutions of ${\Psi}_t$ and $m_t$ in the normal direction to the
FHSs are defined by [3]
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}{\Psi}_t=
{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-
{\frac{1}{c{\bar N}_t}}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
{\Big )}{\Psi}_t, {\qquad}
{\topstar{\cal L}}_{{\bf {\bar n}}_t}m_t=
-{\Big (}{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-
{\frac{1}{c{\bar N}_t}}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
{\Big )}m_t.
\ena
Furthermore, on $\cal N$ there applies a linear, symmetric ``degenerate" 
connection ${\,}{\,}{\topstar{\nabla}}{\,}{\,}$. This connection is called 
degenerate due to the fact that the family ${\bf g}_t$ may be perceived as one
single degenerate 5-dimensional metric on a product manifold 
${\cal M}{\times}{\bf R}$ whereof $\cal N$ is a 4-dimensional submanifold. 
(Here $\cal M$ is a Lorentzian space-time manifold and $\bf R$ is the real 
line.) We may then introduce a torsion-free, metric-compatible 5-dimensional 
connection ${\,}{\,}{\topstar{\nabla}}{\,}{\,}$ with the property that
\eqa
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf g}_t=0, \qquad
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf n}_t=0, \qquad
{\topstar{\nabla}}_{\frac{\partial}{{\partial}t}}{\bf h}_t=0,
\ena
on ${\cal M}{\times}{\bf R}$ and consider the restriction of 
${\,}{\,}{\topstar{\nabla}}{\,}{\,}$ to $\cal N$. It can be shown [3] that 
in a GTCS the components which do not vanish identically of the degenerate 
connection field are given by (a comma denotes partial derivation, and we use 
Einstein's summation convention throughout)
\eqa
{\topstar{\Gamma}}^{\alpha}_{t{\mu}}{\equiv}{\frac{1}{t}}
{\delta}^{\alpha}_i{\delta}^i_{\mu}, {\qquad}
{\topstar{\Gamma}}^{\alpha}_{{\nu}{\mu}}{\equiv}{\frac{1}{2}}
g_{(t)}^{{\alpha}{\sigma}}{\Big(}g_{(t){\sigma}{\mu}},_{\nu}
+g_{(t){\nu}{\sigma}},_{\mu}-g_{(t){\nu}{\mu}},_{\sigma}{\Big)}{\equiv}
{\Gamma}^{\alpha}_{(t){\nu}{\mu}}.
\ena
The general equations of motion for test particles are identical to the
geodesic equation obtained from ${\,}{\,}{\topstar{\nabla}}{\,}{\,}$. In a 
GTCS they take the form (see [3] for a derivation)
\eqa
{\frac{d^2x^{\mu}}{d{\lambda}^2}}+{\Big(}
{\topstar{\Gamma}}^{\mu}_{t{\nu}}{\frac{dt}{d{\lambda}}}+
{\Gamma}^{\mu}_{(t){\beta}{\nu}}{\frac{dx^{\beta}}{d{\lambda}}}{\Big)}
{\frac{dx^{\nu}}{d{\lambda}}} 
={\Big(}{\frac{cd{\tau}_t}{d{\lambda}}}{\Big)}^2c^{-2}a_{(t)}^{\mu},
\ena
where $d{\tau}_t$ is the proper time as measured along the curve, ${\lambda}$ 
is some general affine parameter and ${\bf a}_t$ is the 4-acceleration as 
measured along the curve.
\subsection{Special equations of motion}
In this paper we analyze the equations of motion (7) in the case of a 
uniformly expanding isotropic gravitational field in vacuum exterior to an 
isolated spherically symmetric source in an isotropic, compact spatial 
background. We also require that the source is at rest with respect to some 
GTCS. It turns out that for an isolated system, also the FOs must be at rest
with respect to this GTCS and consequently the shift vector field vanishes. 
Furthermore we require that ${\bar N}_t$ is independent of $x^0$ and $t$; 
i.e., that the only explicit time dependence is via $t$ in the spatial scale 
factor (using the chosen GTCS). We call this a ``metrically static'' case. 
This scenario may be taken as a generalization of the analogous case with a 
Minkowski background (this case is analyzed in [3]) since the Minkowski 
background is not a part of our theory but rather invoked as an approximation 
being useful in particular cases.

We start by making a specific {\em ansatz} for the form of ${\bf {\bar g}}_t$.
Introducing a spherically symmetric GTCS ${\{}x^0,r,{\theta},{\phi}{\}}$, 
where $r$ is a Schwarzschild radial coordinate, we assume that the 
metric families ${\bf {\bar g}}_t$ and ${\bf g}_t$ can be written in the form 
(compatible with equation (1))
\eqa
c^2{\overline{d{\tau}}}^2_t={\bar{B}}(r)(dx^0)^2-({\frac{t}{t_0}})^2{\Big(}
{\bar{A}}(r)dr^2+r^2d{\Omega}^2{\Big)}, {\nonumber} \\
c^2d{\tau}^2_t=B(r)(dx^0)^2-({\frac{t}{t_0}})^2{\Big(}
A(r)dr^2+r^2d{\Omega}^2{\Big)},
\ena
where $d{\Omega}^2{\equiv}d{\theta}^2+{\sin}^2{\theta}d{\phi}^2$ and $t_0$ is
some arbitrary reference epoch. Note that the spatial coordinate system covers
only half of ${\bf S}^3$, thus the range of the radial coordinate is $r<ct_0$ 
only. The functions ${\bar{A}}(r)$ and ${\bar{B}}(r)$ may be calculated from 
the field equations; we tackle this problem in the next section. The functions
$A(r)$ and $B(r)$ may be found from ${\bf {\bar g}}_t$ and ${\bar y}_t$ as 
shown in [3].

We now calculate the metric and the degenerate connection coefficients from
the metric family ${\bf g}_t$ given in equation (8). A straightforward 
calculation yields (using the notation $'{\hspace*{2mm}}{\equiv}
{\frac{\partial}{{\partial}r}}$) 
\eqa
{\Gamma}^r_{(t)rr}&=&{\frac{A'(r)}{2A(r)}}, {\qquad}
{\Gamma}^r_{(t){\theta}{\theta}}=-{\frac{r}{A(r)}}, {\qquad}
{\Gamma}^r_{(t){\phi}{\phi}}={\Gamma}^r_{(t){\theta}{\theta}}{\sin}^2{\theta},
{\nonumber} \\
{\Gamma}^r_{(t)00}&=&({\frac{t_0}{t}})^2{\frac{B'(r)}{2A(r)}}, 
{\qquad}{\Gamma}^{\theta}_{(t)r{\theta}}={\Gamma}^{\theta}_{(t){\theta}r}=
{\frac{1}{r}}, {\qquad} {\Gamma}^{\theta}_{(t){\phi}{\phi}}=
-{\sin}{\theta}{\cos}{\theta}, {\nonumber} \\
{\Gamma}^{\phi}_{(t)r{\phi}}&=&{\Gamma}^{\phi}_{(t){\phi}r}={\frac{1}{r}},
{\qquad}{\Gamma}^{\phi}_{(t){\phi}{\theta}}=
{\Gamma}^{\phi}_{(t){\theta}{\phi}}={\cot}{\theta}, {\qquad}
{\Gamma}^0_{(t)0r}={\Gamma}^0_{(t)r0}={\frac{B'(r)}{2B(r)}},
\ena
\eqa
{\topstar{\Gamma}}^r_{tr}=
{\topstar{\Gamma}}^{\theta}_{t{\theta}}=
{\topstar{\Gamma}}^{\phi}_{t{\phi}}={\frac{1}{t}}.
\ena
In the following we use the equations of motion (7) to find the paths of 
inertial test particles moving in the metric family ${\bf g}_t$. Since 
${\bf a}_t$ vanishes for inertial test particles we get the relevant equations 
by inserting the expressions (9) and (10) into (7). This yields
(making explicit use of the fact that $cdt=dx^0$ in a GTCS)
\eqa
{\frac{d^{2}r}{d{\lambda}^{2}}} + 
{\frac{A'(r)}{2A(r)}}{\Big(}{\frac{dr}{d{\lambda}}}{\Big)}^2 -
{\frac{r}{A(r)}}{\bigg[}{\Big(}{\frac{d{\theta}}{d{\lambda}}}{\Big)}^{2} + 
{\sin}^{2}{\theta}{\Big(}{\frac{d{\phi}}{d{\lambda}}}{\Big)}^{2}{\bigg]}
{\nonumber} \\
+({\frac{t_0}{t}})^2
{\frac{B'(r)}{2A(r)}}{\Big(}{\frac{dx^0}{d{\lambda}}}{\Big)}^{2} + 
{\frac{1}{ct}}{\frac{dr}{d{\lambda}}}{\frac{dx^0}{d{\lambda}}}=0,
\ena
\eqa
{\frac{d^{2}{\theta}}{d{\lambda}^{2}}} +
{\frac{2}{r}}{\frac{d{\theta}}{d{\lambda}}}{\frac{dr}{d{\lambda}}} -
{\sin}{\theta}{\cos}{\theta}{\Big(}{\frac{d{\phi}}{d{\lambda}}}{\Big)}^{2} + 
{\frac{1}{ct}}{\frac{d{\theta}}{d{\lambda}}}{\frac{dx^0}{d{\lambda}}} = 0,
\ena
\eqa
{\frac{d^{2}{\phi}}{d{\lambda}^{2}}} +
{\frac{2}{r}}{\frac{d{\phi}}{d{\lambda}}}{\frac{dr}{d{\lambda}}} +
2{\cot}{\theta}{\frac{d{\phi}}{d{\lambda}}}{\frac{d{\theta}}{d{\lambda}}} + 
{\frac{1}{ct}}{\frac{d{\phi}}{d{\lambda}}}{\frac{dx^0}{d{\lambda}}} = 0,
\ena
\eqa
{\frac{d^{2}x^0}{d{\lambda}^{2}}} +
{\frac{B'(r)}{B(r)}}{\frac{dx^0}{d{\lambda}}}{\frac{dr}{d{\lambda}}} =0.
\ena
If we restrict the motion to the equatorial plane (12) becomes vacuous, and
(13) reduces to
\eqa
{\frac{d^{2}{\phi}}{d{\lambda}^{2}}} +
{\frac{2}{r}}{\frac{d{\phi}}{d{\lambda}}}{\frac{dr}{d{\lambda}}} +
{\frac{1}{ct}}{\frac{d{\phi}}{d{\lambda}}}{\frac{dx^0}{d{\lambda}}} = 0.
\ena
Dividing equation (15) by ${\frac{d{\phi}}{d{\lambda}}}$ we find (assuming
${\frac{d{\phi}}{d{\lambda}}}{\neq}0$)
\eqa
{\frac{d}{d{\lambda}}}{\bigg[}{\ln}{\Big(}{\frac{d{\phi}}{d{\lambda}}}{\Big)} 
+ {\ln}{\Big(}r^{2}{\frac{t}{t_0}}{\Big)}{\bigg]} = 0.
\ena
We thus have a constant of the motion, namely
\eqa
J{\equiv}{\frac{t}{t_0}}r^2{\frac{d{\phi}}{d{\lambda}}}.
\ena
Dividing equation (14) by ${\frac{dx^0}{d{\lambda}}}$ yields
\eqa
{\frac{d}{d{\lambda}}}{\bigg[}{\ln}{\Big(}{\frac{dx^0}{d{\lambda}}}{\Big)} + 
{\ln}B(r){\bigg]}=0.
\ena
Equation (18) yields a constant of the motion which we can absorb into the 
definition of ${\lambda}$ such that a solution of (18) is [4]
\eqa
{\frac{dx^0}{d{\lambda}}}={\frac{1}{B(r)}}.
\ena
Multiplying equation (11) by 
${\frac{2t^2A(r)}{t_0^2}}{\frac{dr}{d{\lambda}}}$ and 
using the expressions (17), (19) we find
\eqa
{\frac{d}{d{\lambda}}}{\bigg[}
{\frac{t^2A(r)}{t_0^2}}{\Big(}{\frac{dr}{d{\lambda}}}{\Big)}^2-
{\frac{1}{B(r)}}+{\frac{J^2}{r^2}}{\bigg]}=0,
\ena
thus a constant $E$ of the motion is defined by
\eqa
{\frac{t^2A(r)}{t_0^2}}{\Big(}{\frac{dr}{d{\lambda}}}{\Big)}^2-
{\frac{1}{B(r)}}+{\frac{J^2}{r^2}}{\equiv}-E.
\ena
The equation (21) may be compared to an analogous expression obtained for the 
static isotropic gravitational field in the metric framework [4]. Inserting 
the formulae (17), (19) and (21) into (8) and using the fact that in a GTCS we
can formally write $dx^0=cdt$ when traversing the family of metrics, we find
\eqa
c^2d{\tau}_t^2=Ed{\lambda}^2.
\ena
Thus our equations of motion (7) force $d{\tau}_t/d{\lambda}$ to be
constant, quite similarly to the case when the total connection is metric, as 
in the metric framework. From equation (22) we see that we must have
$E=0$ for photons and $E>0$ for material particles.

We may eliminate the parameter ${\lambda}$ from (17), (19), (21) and 
(22) and alternatively use $t$ as a time parameter. This yields
\eqa
{\frac{t}{t_0}}r^2{\frac{d{\phi}}{cdt}}=B(r)J,
\ena
\eqa
({\frac{t}{t_0}})^2A(r)B^{-2}(r){\Big(}{\frac{dr}{cdt}}{\Big)}^2-
{\frac{1}{B(r)}}+{\frac{J^2}{r^2}}{\equiv}-E,
\ena
\eqa
d{\tau}_t^2=EB^2(r)dt^2.
\ena
We may integrate equations (23) and (24) to find the time history 
$(r(t),{\phi}(t))$ along the curve if the functions $A(r)$ and $B(r)$ 
are known.

For the static vacuum metric case with no NKE, one can solve the geodesic
equation for particles orbiting in circles with different radii, and from
this find the asymptotically Keplerian nature of the corresponding rotational
curve [4]. In our case we see from (23) and (24) that we can find circle 
orbits as solutions; such orbits have the property that the orbital speed 
$B^{-1/2}(r){\frac{t}{t_0}}r{\frac{d{\phi}}{dt}}$ is independent of $t$.
\section{The dynamical problem}
In this section we write down equations which must be solved to find the 
unknown functions ${\bar A}(r)$ and ${\bar B}(r)$. (These equations follow
directly from the field equations.) We then solve these equations 
approximately and use this to find approximations to the functions $A(r)$ and 
$B(r)$ as well.

The field equations determining the metric family ${\bf {\bar g}}_t$ of the
type shown in (8) may be set up exactly as for the corresponding case with a
Minkowski background treated in [3]. The main difference between a Minkowski
background case and the case of a compact spatial background is the explicit 
time dependence present for the latter case. However this time dependence is 
only via the scale factor. Now, from equations (1), (2) and (8) we get (since 
${\bar N}_t={\sqrt{{\bar B}(r)}}$ is independent of $t$)
\eqa
{\frac{{\partial}{\bar h}_{(t)ij}}{{\partial}t}}=
{\frac{2}{t}}{\bar h}_{(t)ij}, {\qquad} {\bar{y}}_t=
{\frac{t_0}{t}}{\frac{c{\bar B}'(r)}{2{\bar B}(r){\sqrt{{\bar A}(r)}}}}.
\ena
After some tedious but straightforward calculations the field equations yield
\eqa
{\Big(}1-{\frac{r^2}{{\bar B}(r)c^2t_0^2}}{\Big)}
{\frac{{\bar B}''(r)}{{\bar B}(r)}}&-&
{\Big(}1-{\frac{2r^2}{{\bar B}(r)c^2t_0^2}}{\Big)}
{\frac{{\bar B}'^2(r)}{{\bar B}^2(r)}} {\nonumber} \\ 
&+&{\frac{2}{r}}{\Big(}1-{\frac{3r^2}{2{\bar B}(r)c^2t_0^2}}{\Big)}
{\frac{{\bar B}'(r)}{{\bar B}(r)}}-{\frac{r}{4}}
{\frac{{\bar B}'^3(r)}{{\bar B}^3(r)}}=0,
\ena
\eqa
{\bar A}(r)={\Big(}1-{\frac{r^2}{{\bar B}(r)c^2t_0^2}}{\Big)}^{-1}
{\Big[}1-r{\frac{{\bar B}'(r)}{{\bar B}(r)}}+{\frac{r^2}{4}}
{\frac{{\bar B}'^2(r)}{{\bar B}^2(r)}}{\Big]}.
\ena
Before we try to solve equation (27), it is important to notice that no
solution of it can exist on a whole FHS (except the trivial solution 
${\bar B}=$constant), according to the maximum principle applied to a closed 
Riemannian 3-manifold. The reason for this is the particular form of the field
equations, see reference [3] and references therein for justification. This 
means that {\em in quasi-metric theory, isolated systems cannot exist except 
as an approximation.} 

Even if a non-trivial solution of equation (27) does not exist on a whole FHS 
we may try to find a solution valid in some finite region of a FHS. That is, 
we want to find a solution in the region ${\cal R}<r{\ll}ct_0$, where 
${\cal R}$ is the coordinate radius of the source. We do not specify any 
particular boundary conditions at the boundary ${\Xi}_0{\equiv}ct_0$ since the
approximation made by assuming an isolated system is physically reliable only 
if ${\frac{r}{{\Xi}_0}}{\ll}1$. Furthermore it turns out to be difficult 
to find a useful exact solution of equation (27). Thus we may rather try to 
find an approximative solution in the form of a series expansion, i.e., a 
perturbation around the analogous problem in a Minkowski background. That is, 
in contrast to the analogous case with a Minkowski background there exists the
extra scale ${\Xi}_0$ in addition to the Schwarzschild radius $r_{s0}$ 
determined by the mass of the central object. (The extra index used in 
$r_{s0}$ refers to the time $t_0$ and is necessary since in our theory 
gravitational quantities get an extra formal variability measured in atomic 
units.) Since we try to model the gravitational field exterior to 
galactic-sized objects we may assume that the typical scales involved are 
determined by ${\frac{r}{{\Xi}_0}}\raisebox{-2pt}{\,$\stackrel{>}{\sim}$\,}
{\frac{r_{s0}}{r}}$; this criterion tells how to compare the importance of 
the different terms of the series expansion.

One may straightforwardly show that an approximative solution of equation (27) 
is
\eqa
{\bar B}(r)&=&1-{\frac{r_{s0}}{r}}+{\frac{r_{s0}^2}{2r^2}}+
{\frac{r_{s0}r}{2{\Xi}_0^2}}-{\frac{r_{s0}^3}{8r^3}}+{\cdots} 
{\qquad} {\Rightarrow} {\nonumber} \\
{\bar A}(r)&=&1-{\frac{r_{s0}}{r}}+
{\frac{r_{s0}^2}{4r^2}}+{\frac{r^2}{{\Xi}_0^2}}+{\cdots},
\ena
and furthermore this yields, using equations (2) and (26)
\eqa
{\bar y}_t={\frac{t_0}{t}}
{\frac{r_{s0}c}{2r^2}}[1+{\frac{r_{s0}}{2r}}+O({\frac{r^2_{s0}}{r^2}})],
\qquad {\bar H}_t={\bar y}_t+{\frac{1}{{\sqrt{{\bar B}(r)}}t}}.
\ena
To construct the family ${\bf g}_t$ as described in [3] we need the quantity
(${\bf b}_{\cal F}=r{\frac{\partial}{{\partial}r}}$)
\eqa
v(r){\equiv}{\bar y}_t{\sqrt{{\bar h}_{(t)ik}b_{\cal F}^ib_{\cal F}^k}}=
{\bar y}_tr{\sqrt{{\bar h}_{(t)rr}}}=
{\frac{r_{s0}c}{2r}}[1+O({\frac{r_{s0}^2}{r^2}})].
\ena
We note that $v(r)$ does not depend on $t$. The functions $A(r)$ and $B(r)$ 
are found from the following relations, valid for the spherically symmetric 
case [3] 
\eqa
A(r)={\Big(}{\frac{1+{\frac{v(r)}{c}}}{1-{\frac{v(r)}{c}}}}{\Big)}^2
{\bar A}(r), {\qquad}
B(r)={\Big(}1-{\frac{v^2(r)}{c^2}}{\Big)}^2{\bar B}(r).
\ena
From equations (29) and (32) we may then write
\eqa
ds^2_t&=&-{\bigg(}1-{\frac{r_{s0}}{r}}
+{\frac{r_{s0}r}{2{\Xi}_0^2}}+{\frac{3r_{s0}^3}{8r^3}}+{\cdots}
{\bigg)}(dx^0)^2 \nonumber \\
&&+({\frac{t}{t_0}})^2{\bigg(}{\Big\{}1+{\frac{r_{s0}}{r}}+
{\frac{r^2}{{\Xi}_0^2}} + {\frac{r^2_{s0}}{4r^2}}+{\cdots}
{\Big\}}dr^2+r^2d{\Omega}^2{\bigg)}.
\ena
This expression represents the wanted metric family as a series expansion. 
Note in particular the fact that all spatial dimensions expand whereas the 
corresponding Newtonian potential $-U=-{\frac{c^2r_{s0}}{2r}}$ (to Newtonian
order) remains constant for a fixed FO. This means that the true radius of any
circle orbit (i.e., with $r$ constant) increases but such that the orbit 
velocity remains constant. That is, {\em the mass of the central object as 
measured by distant orbiters increases to exactly balance the effect on circle
orbit velocities of expanding circle radii.} This is not as outrageous as it 
may seem due to the extra formal variation of atomic units built into our 
theory, rather it is a consequence of the fact that the coupling between 
matter and geometry depends directly on this formal variation [3]. 

What is measured by means of distant orbiters is not the ``bare'' mass $M_t$ 
itself but rather the combination $M_tG$. We have, however, {\em defined} $G$ 
to be a constant. And as might be expected, it turns out that the variation of
$M_tG$ with $t$ as inferred from equation (33) is exactly that found directly 
from the formal variation of the active mass $M_t$ with $t$ by using equation 
(4). This means that the dynamically measured mass increase should not be 
taken as an indication of actual particle creation but that the general 
dynamically measured mass scale should be taken to change via a linear 
increase of $M_t$ with $t$, and that this is directly reflected in the 
gravitational field of the source. That is, measured in atomic units, active 
mass increases linearly with epoch.

The dynamical measurement of the mass of the central object by means of 
distant orbiters does not represent a local test experiment. Nevertheless the 
dynamically measured mass increase thus found is just as ``real" as the 
expansion in the sense that neither should be neglected on extended scales.
This must be so since in quasi-metric relativity, the global scale increase 
and the dynamically measured mass increase are two different aspects of the 
same basic phenomenon.

We finish this article by exploring which kinds of free-fall orbits we get 
from equation (33) and the equations of motion. We also discuss observable
consequences thereof.
\section{Discussion}
To begin with we find the shape of the rotational curve as defined by the
orbital velocities of the circle orbits; since equation (24) has no time 
dependence for such orbits we can do a standard calculation [4] and the 
result is 
\eqa
w(r){\equiv}B^{-1/2}(r)
{\frac{t}{t_0}}r{\frac{d{\phi}}{dt}}={\sqrt{\frac{B'(r)r}{2B(r)}}}c.
\ena
However when we apply (34) to the metrics (33) we get a result essentially
identical to the standard Keplerian rotational curve; the only effect of the
dynamically measured mass increase and the non-kinematical expansion is to 
increase the scale but such that the shape of the rotational curve is 
unaffected. It is true that $B(r)$ as found from (33) contains a term linear 
in $r$ in addition to terms falling off with increasing $r$; in reference [5] 
it is shown that such a linear term may be successfully used to model the 
asymptotically non-Keplerian rotational curves of spiral galaxies. 
Unfortunately the numerical value of the linear term found from (33) is too 
small by a factor of order $10^{10}$ to be able to match the data. So at least
the simple model considered in this paper is unable to obtain the 
asymptotically non-Keplerian rotational curves of spiral galaxies from first 
principles. 

Another matter is how the time dependence in the equations of motion will
affect the time histories and shapes of more general orbits than the circle 
orbits. Clearly time histories will be affected as can be seen directly from
(24). However to see if this is valid for shapes as well we may insert
equation (23) into equation (24) to obtain $r$ as a function of ${\phi}$.
This yields
\eqa
{\frac{A(r)}{r^4}}{\Big(}{\frac{dr}{d{\phi}}}{\Big)}^2+{\frac{1}{r^2}}-
{\frac{1}{J^2B(r)}}=-{\frac{E}{J^2}},
\ena
and this is identical to the equation valid for the case of a single 
spherically symmetric static metric [4]. Thus the shapes of free-fall orbits
are unaffected by the non-kinematical expansion present in the metric family
(33). 
\subsection{Expanding space and the solar system}
If one neglects the gravitational effects of the galaxy one may try to apply 
the metric family (33) to the solar system (by using it to describe the 
gravitational field of the Sun). But the solar system is not at rest with 
respect to the cosmic rest frame; this follows from the observed dipole in the 
cosmic microwave background radiation. Besides, if we identify the cosmic 
rest frame with a GTCS, the FOs will be at rest with respect to this GTCS only 
in the limit where the motions of local gravitational sources can be neglected.
Consequently, if we use a GTCS identified with the cosmic rest frame it is 
necessary to introduce a non-vanishing shift vector field family taking into
account the motion of the solar system barycenter. And expressed in such a GTCS
the metric family describing the solar system gravitational field cannot be 
metrically static since the metric coefficients will now depend on $x^0$. 
However, as a good approximation we may neglect the gravitational effects of 
the galaxy and treat the solar system as an isolated system moving with 
constant velocity with respect to the cosmic rest frame. In this case we are 
always able to find a GTCS in which the solar system barycenter is at rest and 
where the metric coefficients do not depend on $x^0$ by construction 
(neglecting gravitational fields of bodies other than the Sun). Furthermore it
turns out that the FOs must be at rest with respect to this new GTCS, so the
shift vector field vanishes and all calculations we have made in this paper are
still valid. Thus, if we treat the solar system as approximately isolated we 
can neglect its motion with respect to the cosmic rest frame and use the metric
family (33) to describe the gravitational field of the Sun. Moreover, the solar
system is so small that we can neglect any dependence on ${\Xi}_0$. The errors 
made by neglecting terms depending on ${\Xi}_0$ in equation (33) are 
insignificant since the typical scales involved for the solar system are 
determined by 
${\frac{r}{{\Xi}_0}}\raisebox{-2pt}{\,$\stackrel{<}{\sim}$\,}
{\frac{r^3_{s0}}{r^3}}$. Equation (33) then takes the form
\eqa
ds^2_t=-{\Big(}1-{\frac{r_{s0}}{r}}+O({\frac{r^3_{s0}}{r^3}}){\Big)}(dx^0)^2
+({\frac{t}{t_0}})^2{\Big(}{\{}1+{\frac{r_{s0}}{r}}+O({\frac{r^2_{s0}}{r^2}})
{\}}dr^2+r^2d{\Omega}^2{\Big)}.
\ena
Equation (35) shows that the shapes of orbits are unaffected by the expansion; 
this means that all the classical solar system tests come in just as for the 
analogous case of a Minkowski background [3]. However, we get at least one 
extra prediction (irrespective of whether the galactic gravitational field 
can be neglected or not); from equation (36) we see that the effective 
distance between the sun and any planet has been smaller in the past. That is,
the spatial coordinates are comoving rather than static, thus the solar system
is not static (measured in atomic units). For example the distance between the
Sun and the Earth at the time of its formation may have been about 50{\%} less
than today. But since gravitationally bound bodies made of an ideal gas are 
predicted to expand according to quasi-metric theory [10], a small Earth-Sun 
distance should not be incompatible with palaeo-climatic data since the Sun is
expected to have been smaller and thus dimmer in the past. Actually, since 
neither the temperature at the center of the Sun (as estimated from the virial
theorem) nor the radiation energy gradient times the mean free path of a 
photon depends on $t$, the cosmic luminosity evolution of the Sun should be 
determined from the cosmic expansion of its surface area as long as the ideal
gas approximation is sufficient. And this luminosity evolution exactly 
balances the effects of an increasing Earth-Sun distance on the effective 
solar radiation received at the Earth.

However, an obvious question is if the predicted effect of the expansion on 
the time histories of non-relativistic orbits is compatible with the observed 
motions of the planets. In order to try to answer this question it is 
illustrating to calculate how the orbit period of any planet depends on $t$. 
For simplicity consider a circular orbit $r=R=$constant. Equation (23) then 
yields
\eqa
{\frac{d{\phi}}{dt}}={\frac{t_0}{t}}B(R)R^{-2}Jc.
\ena
Now integrate (37) one orbit period $T<<t$ (i.e., from $t$ to $t+T$). The
result is
\eqa
T(t)=t({\exp}{\Big [}{\frac{T_{GR}}{t_0}}{\Big ]}-1)={\frac{t}{t_0}}T_{GR}(1+
{\frac{T_{GR}}{2t_0}}+{\cdots}), {\qquad} 
T_{GR}{\equiv}{\frac{2{\pi}R^2}{cJB(R)}},
\ena
where $T_{GR}$ is the orbit period as predicted from GR. From (38) we see that
(sidereal) orbit periods are predicted to increase linearly 
with cosmic scale, i.e. 
\eqa
T(t)={\frac{t}{t_0}}T(t_0), {\qquad} {\frac{dT}{dt}}={\frac{T(t_0)}{t_0}},
\ena
and such that any ratio between periods of different orbits remains constant.
In particular (39) predicts that the (sidereal) year $T_E$ should be
increasing with about 2.5 ms per year and the martian (sidereal) year $T_M$
should be increasing with about 4.7 ms per martian year at the present epoch. 
This should be consistent with observations since the observed difference in
the synodical periods of Mars and the Earth is accurate to about 5 ms.

To compare predictions coming from equation (36) against timekeeping data one 
must also take into account the cosmological contribution to the spin-down of 
the Earth. If one assumes that the gravitational source of the exterior field 
(36) is stable with respect to internal collapse (as for a source made of
ideal gas), i.e., that possible instabilities generated by the expansion can 
be neglected, one may model this source as a uniformly expanding sphere [10].
Due to the cosmic expansion of gravitational fields, the angular momenta of 
test particles moving in the exterior field (36) increase linearly with cosmic 
scale, and this should also apply to the angular momentum $L_s$ of a passive
mass source, that is
\eqa
L_s(t)={\frac{t}{t_0}}L_s(t_0), {\qquad} {\frac{dL_s}{dt}}={\frac{1}{t}}L_s=
(1+O(2))HL_s,
\ena
where the term $O(2)$ is of post-Newtonian order and where the locally 
measured Hubble parameter $H$ is defined by $H{\equiv}{\frac{1}{Nt}}$, or
equivalently (${\tau}_{\cal F}$ is the proper time of the local FO)
\eqa
H{\equiv}{\frac{t_0}{t}}{\frac{d}{d{\tau}_{\cal F}}}({\frac{t}{t_0}})=
{\frac{ct_0}{t}}{\Big (}{\sqrt{B(r)}}{\Big )}^{-1}{\frac{d}{dx^0}}
({\frac{t}{t_0}})={\Big (}{\sqrt{B(r)}}t{\Big )}^{-1}.
\ena
Since the moment of inertia $I{\propto}MR_s^2$, where $M$ is the passive mass 
and $R_s$ is the measured radius of the sphere, we must have (neglecting terms 
of post-Newtonian order)
\eqa
{\frac{dR_s}{dt}}=HR_s, {\qquad}
{\frac{d{\omega}_s}{dt}}=-H{\omega}_s, {\qquad} {\frac{dT_s}{dt}}=HT_s,
\ena
where ${\omega}_s$ is the spin circle frequency and $T_s$ is the spin period
of the sphere. (To show (42), use the definition $L_s{\equiv}I{\omega}_s$.) 
This means that the spin period of a sphere made of ideal gas increases 
linearly with $t$ due to the cosmic expansion. Does this apply to the Earth as
well? The Earth is not made of ideal gas, so the cosmic expansion may induce 
instabilities, affecting its spin period $T_{sE}$. However, here we assume that
the Earth's mantle is made of a material which may be approximately modelled
as a perfect fluid obeying an equation of state close to linear. Then, if this
assumption holds, the effects of instabilities averaged over long time spans 
should be negligible to a good approximation. We may also assume that there is 
no significant tidal friction since given the cosmic contribution, this 
would be inconsistent with the observed mean acceleration of the Moon 
${\dot n}_m$ (see equation (46) below). We then get
\eqa
{\frac{dT_{sE}(t)}{dt}}=HT_{sE}(t_0), \qquad \Rightarrow \qquad
T_{sE}(t)={\frac{t}{t_0}}T_{sE}(t_0).
\ena
From equation (43) we may estimate a cosmic spin-down of the Earth at the 
present epoch of about 0.68 ms per century (using $H{\sim}2.5{\times}10^{-18}
$ s$^{-1}$). To see if this is consistent with the assumption that the dominant
contribution is due to cosmic effects we may compare to historical 
observations indicating a lengthening of the day of about 1.4 ms per century 
averaged over the last 1000 years [12] and of about 1.7 ms per century averaged
over the last 2700 years [13]. But the historical observations depend on 
an assumed value for ${\dot n}_m$ obtained from fitting lunar laser ranging 
(LLR) data to a Newtonian model, making these values theory dependent. This 
theory dependence also affects the comparison of equation (43) with 
palaeo-tidal records obtained from sedimentary tidal rhythmities [11]. However,
if the quasi-metric prediction of ${\dot n}_m$ (see equation (46) below) rather
than the LLR value is used as an input to the historical observations, one gets
a result close to the quasi-metric prediction of the spin-down of the Earth
obtained from equation (43).

Another quantity that can be calculated from equation (43) is the number of 
days $N_y$ in one (sidereal) year $T_E$. This is found to be constant since
\eqa
T_E=N_yT_{sE}={\frac{t}{t_0}}T_E(t_0), {\qquad} \Rightarrow \qquad
{\frac{dN_y}{dt}}=0.
\ena
A similar calculation applies to the number of the days $N_m$ in one 
(sidereal) month $T_m$, i.e.
\eqa
T_m=N_mT_{sE}={\frac{t}{t_0}}T_m(t_0), {\qquad}  {\Rightarrow} \qquad
{\frac{dN_m}{dt}}=0.
\ena
That $N_y$ and $N_m$ are predicted to be constant is not in agreement with 
standard (theory dependent) interpretations of palaeo-geological data [7], 
[11]. (The predicted constancy of the ratio $N_y/N_m$ agrees well with a 
standard interpretation of the data, though.) To see why, the assumption
that $T_E$ is constant is routinely used in the interpretation of tidal 
rhythmities and fossil coral growth data; in particular this applies to [11].
Another assumption used in the data interpretation is that active masses do 
not vary with time.

The apparent constancy of the sidereal year (as indicated by astronomical 
observations of the Sun and Mercury since about 1680 [9]) represents the 
observational basis for adopting the notion that ephemeris time (i.e., the time
scale obtained from the observed motions of the Sun and the planets) is equal 
to atomic time (plus a conventional constant), but different from so-called 
universal time (any time scale based on the rotation of the Earth). However, 
from equations (39) and (43) we see that according to quasi-metric relativity, 
ephemeris time should be scaled with a factor ${\frac{t}{t_0}}$ compared to 
atomic time, and averaged over long time spans it should be equal to universal 
time. But within the Newtonian framework, any secular changes in the Earth-Moon
system are interpreted in terms of tidal friction (and external perturbations),
so seemingly secular inconsistencies between different time scales may be 
blamed on the variable rotation of the Earth. In practice this means 
introducing leap seconds. Given the fact that leap seconds are routinely used 
to adjust the length of the year, the predicted relationship between the 
different time scales should be consistent with observations. In particular,
the extra time corresponding to an increasing year as predicted from our model
may easily be hidden into the declining number of days in a year as predicted
from standard theory. 

As mentioned above, the value of the so-called mean acceleration of the Moon,
${\dot n}_m{\equiv}{\frac{d}{dt}}n_m$, (where $n_m$ is the mean geocentric
angular velocity of the Moon as observed from its motion) is a very important
quantity for calculating the evolution of the Earth-Moon system. From equation
(45) we see that quasi-metric theory predicts that
\eqa
n_m(t)={\frac{d{\phi}_m}{dt}}={\frac{t_0}{t}}n_m(t_0), \qquad \Rightarrow
\qquad {\dot n}_m{\equiv}{\frac{d}{dt}}n_m=-Hn_m,
\ena
and inserting the observed value 0.549$^{{\prime}{\prime}}$/s for $n_m$ at 
the present epoch, we get the corresponding cosmological contribution to 
${\dot n}_m$, namely about $-13.6^{{\prime}{\prime}}$ per century$^2$. This
value may be compared to the value $-$13.68$^{{\prime}{\prime}}$ per 
century$^2$ obtained from fitting LLR data to a lunar model [14]. Note that 
this second value is the {\em total} mean acceleration, wherein other modelled
(positive) contributions, such as planetary perturbations, are included. When 
the other contributions are removed one deduces a {\em tidal} contribution 
${\dot n}_{tid}=-$25.8$^{{\prime}{\prime}}$ per century$^2$ to ${\dot n}_m$ 
[14]. Similar results for the tidal contribution to ${\dot n}_m$ inferred from 
LLR data have been found in, e.g., [15] (${\dot n}_{tid}
=-$25.9$^{{\prime}{\prime}}$ per century$^2$). We see that the quasi-metric 
result is smaller (in absolute value) than the tidal term. But as long as 
there are no independent measurements of the non-tidal contributions to 
${\dot n}_m$, a model containing only the cosmic contribution fits the data 
well.

We may also use Hubble's law directly to calculate the secular recession 
${\dot a}_{qmr}$ of the Moon due to the global cosmic expansion; this yields 
about 3.0 cm per year whereas the value ${\dot a}_{tid}$ inferred from LLR is 
about 3.8 cm per year. However, LLR values are found by fitting the LLR data 
to a Newtonian model where active masses are assumed to be constant. From 
Kepler's third law we get the relationship
\eqa
{\dot a}_{tid}={\frac{2}{3}}
{\frac{{\dot n}_{tid}}{{\dot n}_{qmr}}}{\dot a}_{qmr},
\ena
where ${\dot n}_{qmr}$ is the cosmic contribution. By inserting the numerical 
values given above we see that this is quite consistent with the data.

To round off this paper: we have seen that the predicted effects of the cosmic
expansion on the solar system gravitational field have a number of observable
consequences, none of which is shown to be in conflict with observation so
far. That is, it seems that at this time no model independent evidence exists 
that may rule out the possibility that planetary orbits expand according to 
the Hubble law.
\\ [4mm]
{\bf References} \\ [1mm]
{\bf [1]} J. Binney and S. Tremaine, {\em Galactic Dynamics}, Princeton 
University Press (1987). \\
{\bf [2]} V.V. Zhytnikov and J.M. Nester, {\em Phys.\ Rev.\ Lett.} {\bf 73},
2950 (1994).\\
{\bf [3]} D. {\O}stvang, chapter 2 of this thesis (2001). \\
{\bf [4]} S. Weinberg, {\em Gravitation and Cosmology}, John Wiley ${\&}$ Sons,
Inc. (1972). \\
{\bf [5]} P.D. Mannheim, {\em Foundations of Physics} {\bf 24}, 487 (1994). \\
{\bf [6]} C.W. Misner, K.S. Thorne, J.A. Wheeler,
{\em Gravitation} W.H. Freeman ${\&}$ Co. (1973). \\
{\bf [7]} P. Wesson, {\em Cosmology and Geophysics}, Adam Hilger Ltd, (1978).\\
{\bf [8]} F.I. Cooperstock, V. Faraoni, D.N. Vollick, {\em ApJ} {\bf 503}, 61
(1998). \\
{\bf [9]} R.G. Hipkin, in {\em Palaeogeophysics} (editor S.K. Runcorn),
Academic Press (1970). \\
{\bf [10]} D. {\O}stvang, chapter 5 of this thesis (2001). \\
{\bf [11]} G.E. Williams, {\em Geophysical Research Letters} {\bf 24}, 421
(1997). \\
{\bf [12]} L.V. Morrison and F.R. Stephenson, {\em Observations of Secular
and Decade \\ \hspace*{6.3mm} Changes in the Earth's Rotation}, in {\em Earth 
Rotation: Solved and Unsolved \\ \hspace*{6.3mm}
Problems} (editor A. Cazenave) Reidel Publishing Company (1986). \\
{\bf [13]} F.R. Stephenson and L.V. Morrison, {\em Phil. Trans. R. Soc.
Lond.} {\bf A351}, 165 (1995). \\
{\bf [14]} J. Chapront, M. Chapront-Touz{\'{e}}, G. Francou, {\em A${\&}$A}
{\bf 343}, 624 (1999). \\
{\bf [15]} J.O. Dickey {\em et al.,} {\em Science} {\bf 265}, 482 (1994).

\newpage

{\vspace*{5cm}}

\begin{center}
{\huge {\bf Chapter Four: \\\ \\An Assumption of a Static\\
\vspace{0.5cm}Gravitational Field
}}
\end{center}
\vspace{-0.8cm}
\begin{center}
{\huge {\bf Resulting in an\\
\vspace{0.5cm}Apparently Anomalous Force}}
\end{center}
\newpage
{\hspace*{1cm}}

\newpage

\setcounter{equation}{0}

\setcounter{section}{0}

\begin{center}
{\large {\bf An Assumption of a Static Gravitational Field Resulting in an 
Apparently Anomalous Force}}
\end{center}
\begin{center}
by
\end{center}
\begin{center}
Dag {\O}stvang \\
{\em Institutt for Fysikk, Norges teknisk-naturvitenskapelige universitet, 
NTNU \\
N-7491 Trondheim, Norway}
\end{center}
\begin{abstract}
According to a particular ``quasi-metric'' theory of gravity developed 
elsewhere [1], the apparently anomalous force acting on the Pioneer 10/11, 
Galileo and Ulysses spacecraft as inferred from radiometric data [2], [9], may
be naturally explained as resulting from treating comoving coordinates as 
static ones. More precisely, we show that the illusion of an anomalous 
acceleration of the right size and acting towards the observer arises as a 
consequence of the mismodelling of null paths. The apparently anomalous 
acceleration is of order $cH$ [2], [9] (where $H$ is the Hubble parameter) as 
predicted by a simple non-static model of the solar system gravitational 
field [3]. \\
\end{abstract}
\section{Introduction}
Some time ago a detailed analysis of the observed versus the calculated orbits
of the Pioneer 10, Pioneer 11, Ulysses and Galileo spacecraft was published 
[2]. The main result was that the observed radiometric data did not agree with
calculations based on standard theory; rather the data indicated the existence 
of an ``anomalous", constant acceleration towards the Sun.

A short summary of the results presented in [2] is as follows: For the Pioneer
spacecraft the Doppler frequency shift of the radio carrier wave was recorded 
and analyzed to determine the spacecraft's orbits. Two independent analyses of
the raw data were performed. Both showed an anomalous acceleration towards the
Sun, of respectively $(8.09{\pm}0.20){\times}10^{-8}$ cm/s$^2$ and 
$(8.65{\pm}0.03){\times}10^{-8}$ cm/s$^2$ for Pioneer 10. For Pioneer 11 only 
one result is given; an anomalous acceleration of 
$(8.56{\pm}0.15){\times}10^{-8}$ cm/s$^2$ towards the Sun. The acceleration 
did not vary between $40-60$ astronomical units, within a sensitivity of
$2{\times}10^{-8}$ cm/s$^2$.

For the Galileo and Ulysses spacecraft one also got ranging data in addition
to the Doppler data. (Ranging data are generated by cross correlating a phase
modulated signal with a ground duplicate and measuring the time delay.) For
Ulysses one had to model the solar radiation pressure in addition to any
constant anomalous acceleration. By doing this it was found that Ulysses was
influenced by an anomalous acceleration of
$(12{\pm}3){\times}10^{-8}$ cm/s$^2$ towards the Sun, consistent with both
Doppler and ranging data. For Galileo the corresponding result was an
anomalous acceleration of $(8{\pm}3){\times}10^{-8}$ cm/s$^2$ towards the Sun.

Very recently a new comprehensive study of the anomalous acceleration was
published [9], including a total error budget for the Pioneer 10 data 
analysis. The new result reported in [9] is an ``experimental'' anomalous 
acceleration of $(7.84{\pm}0.01){\times}10^{-8}$ cm/s$^2$ towards the Sun, and
including bias and uncertainty terms the final value becomes
$(8.74{\pm}0.94){\times}10^{-8}$ cm/s$^2$. For Pioneer 11 an experimental
value of $(8.55{\pm}0.02){\times}10^{-8}$ cm/s$^2$ was given. Also other new 
results, such as annular and diurnal variations in the anomalous acceleration,
were reported in [9].

An interpretation of these results according to the standard general 
relativistic model indicates the existence of an anomalous, time-independent 
force acting on the spacecraft. However there are problems with this 
interpretation since according to the planetary ephemeris there is no 
indication that such a force acts on the orbits of the planets; the 
hypothetical force thus cannot be of gravitational origin without violating 
the weak principle of equivalence. Thus it is speculated that the effect is 
due to anisotropic radiation of waste heat from the radioactive thermal 
generators aboard the spacecraft; the design of the spacecraft is such that 
waste heat may possibly be scattered off the back of the high gain antennae in
directions preferentially away from the Sun [4]. Moreover, besides possible 
anisotropic scattering, an estimate shows that the specific arrangement of 
waste heat radiators on the surface of the spacecraft may perhaps cause 
sufficient anisotropy in the radiative cooling to explain the data [5].
However, it seems that these explanations have been effectively refuted 
[6], [7], [8], [9]. Other possible explanations, such as gas leaks, have been 
proposed [6], [9], but so far it seems that no satisfactory explanation based
on well-known physics exists.

However, it is an intriguing fact that the size of the anomalous acceleration
is of the order $cH$ for all the spacecraft, where $H$ is the Hubble 
parameter. Since this seems to be too much of a coincidence one may suspect 
that the data indicate the existence of new physics rather than a prosaic 
explanation based on standard theory. This has been duly noted by others,
see [9] and references therein. But to be acceptable, any non-standard 
explanation should follow naturally from a general theoretical framework. In 
this paper we show that such an explanation can be found, thus the data may 
indeed be taken as evidence for new physics.
\section{A quasi-metric model}
In reference [1] we defined the so-called ``quasi-metric'' space-time 
framework; this framework is non-metric since it is not based on 
semi-Riemannian geometry. Moreover, in reference [3] we introduced a model of 
the gravitational field outside a spherically symmetric, isolated source as 
predicted by a particular quasi-metric theory of gravity developed in [1]. 
According to this theory it was found in [3] that at scales similar to the
size of the solar system, such a gravitational field can be expressed by the 
one-parameter family ${\bf g}_t$ of Lorentzian 4-metrics
\eqa
ds_t^2=-B(r)(dx^0)^2+({\frac{t}{t_0}})^2{\Big (}A(r)dr^2
+r^2d{\Omega}^2{\Big )},
\ena
where $r$ is a comoving radial coordinate and 
$d{\Omega}^2{\equiv}d{\theta}^2+{\sin}^2{\theta}d{\phi}^2$ is the squared 
solid angle line element. Furthermore $t$ is the global time function; its 
value is zero at the beginning of the Universe. In equation (1) $t$ is to be 
viewed as a parameter; the relationship between $t$ and the space-time 
coordinates is given by $x^0=ct$ and $t_0$ represents some arbitrary reference
epoch setting the scale of the spatial coordinates. The reason that one must 
separate between $ct$ and $x^0$ in (1) is that the {\em affine connection} 
compatible with the family ${\bf g}_t$ is non-metric. That is, although $ct$ 
and $x^0$ both can be interpreted as time coordinates, the components of the 
affine connection containing $ct$ is not equivalent to their counterparts 
containing $x^0$. See [1] for a further discussion. The locally measured
Hubble parameter $H$ as calculated from (1) reads [3]
\eqa
H(r,t)={\frac{ct_0}{t}}{\Big(}{\sqrt{B(r)}}{\Big)}^{-1}
{\frac{d}{dx^0}}({\frac{t}{t_0}})={\Big(}{\sqrt{B(r)}}t{\Big)}^{-1}.
\ena
General equations of motion are obtained from the geodesic equation using the 
non-metric connection [1]. But these equations of motion {\em cannot} be 
obtained from the geodesic equation using any metric connection. Moreover, as 
shown in [3] special equations of motion for inertial test particles moving in
the particular metric family (1) take the form (due to the spherical symmetry 
we can restrict the motion to the equatorial plane ${\theta}={\pi}/2$)
\eqa
({\frac{t}{t_0}})^2{\frac{A(r)}{B^2(r)}}({\frac{dr}{cdt}})^2-
{\frac{1}{B(r)}}+{\frac{J^2}{r^2}}=-E,
\ena
\eqa
{\frac{t}{t_0}}r^2{\frac{d{\phi}}{cdt}}=B(r)J,
\ena
\eqa
d{\tau}_t^2=-c^{-2}ds_t^2=EB^2(r)dt^2,
\ena
where $J$ and $E$ are constants of the motion. By setting the scale factor 
${\frac{t}{t_0}}=1$ in equations (1), (3), (4) and (5) we recover the 
equations of motion for inertial test particles moving in a spherically 
symmetric, static gravitational field as obtained from General Relativity 
(GR). Note that $E=0$ for photons and $E>0$ for material particles, which may 
readily be seen from equation (5). 

The functions $A(r)$ and $B(r)$ may be found as series expansions by solving
the field equations, this is done approximately in [3]. For our purposes
we include terms to post-Newtonian order but not higher. Then we have
\eqa
A(r)=1+{\frac{r_{s0}}{r}}+O({\frac{r_{s0}^2}{r^2}}), {\nonumber} \\
B(r)=1-{\frac{r_{s0}}{r}}+O({\frac{r_{s0}^3}{r^3}}),
\ena
where $r_{s0}$ is the Schwarzschild radius at the arbitrary epoch $t_0$.

We now explore some of the differences between the non-static system described
by equation (1) and the corresponding static system obtained by setting 
${\frac{t}{t_0}}=1$ in (1) and using GR. To begin with we notice that the 
{\em shapes} of free fall orbits (expressed e.g. as functions of the type 
$r({\phi})$) are identical for the two cases [3]. Moreover, it can be shown 
that the time dependence present in equations (3), (4) and (5) does not lead 
to easily observable perturbations in the paths of non-relativistic particles 
compared to the static case [3]. However, as we now illustrate, if one 
considers null paths potential observable consequences appear if one treats 
$r$ as a static coordinate rather than as a comoving one. To simplify matters 
we consider purely radial motion, i.e., $J=0$ (one may easily generalize to 
$J{\neq}0$). Since $E=0$ for photons we get from equation (3) that radial null
curves are described by the equation
\eqa
{\frac{dr}{dt}}={\pm}{\frac{ct_0}{t}}{\sqrt{\frac{B(r)}{A(r)}}}
={\pm}{\frac{ct_0}{t}}{\sqrt{1-{\frac{2r_{s0}}{r}}+O({\frac{r^2_{s0}}{r^2}}}})
={\pm}{\frac{ct_0}{t}}{\Big(}1-{\frac{r_{s0}}{r}}+O({\frac{r^2_{s0}}{r^2}})
{\Big)},
\ena
the choice of sign depending on whether the motion is outwards or inwards. By 
integrating (7) to lowest order we find an extra delay, as compared to 
standard theory, in the time it takes an electromagnetic signal to travel from 
an object being observed to the observer. To lowest order this extra time 
delay is ${\frac{HR^2}{2c^2}}$, where $R$ is the radial coordinate distance 
between the object and the observer and $H$ is the Hubble parameter as given 
from equation (2). Also, the fact that the scale factor in equation (1) 
increases with time implies that our model predicts an extra redshift, as 
compared to standard static models, in the Doppler data obtained from any 
object emitting electromagnetic signals. To lowest order this extra redshift 
corresponds to a ``Hubble'' redshift ${\frac{HR}{c}}$. 

But the velocity at any given time of an observed object cannot 
model-independently be split up into one ``ordinary" piece and one ``Hubble" 
piece. This means that there is no direct way to identify the predicted extra 
redshift in the Doppler data. Similarly, at any given time there is no direct 
way to sort out the predicted extra time delay when determining the distance 
to the object. Rather, to test whether the gravitational field is static or 
not one should do observations over time and compare the observed motion to a 
model. In a model one uses a coordinate system and to calculate coordinate 
motion one needs coordinate accelerations. Accordingly we construct the 
``properly scaled coordinate acceleration" quantity $a_c$. For photons this is
\eqa
a_c{\equiv}{\frac{t}{t_0}}{\frac{\sqrt{A(r)}}{B(r)}}{\frac{d^2r}{dt^2}}&=&
{\mp}{\frac{c}{t}}+{\frac{t_0}{t}}{\frac{r_{s0}c^2}{r^2}}+
O({\frac{r_{s0}^2c^2}{r^3}}) {\nonumber} \\
&&={\mp}cH+{\frac{t_0}{t}}{\frac{r_{s0}c^2}{r^2}}
+O({\frac{r_{s0}^2c^2}{r^3}}).
\ena
The point with this is to show that by treating the comoving coordinate system 
as a static one and using GR, an ``anomalous" term ${\mp}cH$ will be missed 
when modelling coordinate accelerations of photons. We see that the sign of 
the anomalous term is such that the anomalous acceleration is oriented in the 
opposite direction to that of the motion of the photons. This means that to 
sufficient accuracy, treating the comoving coordinate system as a static one 
is equivalent to introducing a variable ``effective'' velocity of light 
$c_{eff}$ equal to
\eqa
c_{eff}=c(1-{\int}_{{\!}{\!}{\!}t}^{t+T}{\frac{dt'}{t'}})=c(1-HT+O((HT)^2)),
\ena
where $T$ is the light time along the null path. The change of $c_{eff}$
with $T$ then yields an anomalous acceleration
\eqa
a_a={\frac{d}{dT}}c_{eff}=-cH+O(cH^2T),
\ena
along the line of sight of any observed object. That is, if the comoving
coordinate system is treated as a static one the coordinate motion of any 
object will be observed to slow down by an extra amount if the light time, or 
equivalently, the distance to the observer increases and to speed up by an 
extra amount if the distance decreases. Hence, judging from its coordinate 
motion it would seem as if the object were influenced by an anomalous force 
directed towards the observer.

By integrating the anomalous acceleration over the total observation time 
${\cal T}{\ll}t$ we get an ``anomalous" speed
\eqa
w_a={\int}_{{\!}{\!}{\!}t}^{t+{\cal T}}a_adt'=-cH{\cal T}+O(cH^2{\cal T}^2),
\ena
towards the observer compared to a model where the coordinates are static
rather than comoving. That is, the coordinate motion of any object observed 
over time indicates an anomalous blueshift compared to a model where the 
gravitational field is static.

Now, since any observer is typically located at the Earth, the model in fact 
predicts an anomalous acceleration directed towards the Earth rather than 
towards the Sun. But any directional differences will almost average out over 
time if the observed object moves approximately radially and is located well 
beyond the Earth's orbit. However, compared to a model where the anomalous
acceleration is towards the Sun, even if directional differences nearly 
average out there remains a cumulative difference. We will calculate this 
below. If, on the other hand, the line of sight to the observed object (e.g. 
a planet) deviates significantly from the radial direction, observations 
should not be consistent with an anomalous acceleration directed towards the 
Sun. Rather the direction of the anomalous acceleration expressed in 
Sun-centered coordinates would appear to be a complicated function of time.

To estimate the predicted differences between a model where the anomalous 
acceleration is towards the Sun and the result given by equation (10),
for the case where the observed object moves approximately radially in the
ecliptic plane and is located well beyond the Earth's orbit it is convenient 
to define the average anomalous acceleration ${\langle}a_c{\rangle}_{in}$ of 
a photon away from the Sun during the time of flight $T$ from the object to 
the observer. Thus we define
\eqa
{\langle}a_c{\rangle}_{in}{\equiv}{\frac{1}{T}}{\int}_{{\!}{\!}{\!}t}^{t+T}
a_cdt'={\frac{c}{T}}{\Big [}{\int}_{{\!}{\!}{\!}t}^{t+T_1}-
{\int}_{{\!}{\!}{\!}t+T_1}^{t+T}{\Big ]}{\frac{dt'}{t'}}=
cH{\Big (}2{\frac{T_1}{T}}-1+O(HT){\Big )},
\ena
where $T_1$ is the moment when the photon crosses a plane through the center
of the Sun normal to a line connecting the object and the Sun. (If the Earth
is at the same side of this plane as the object, $T_1=T$.) Similarly we can 
define the average ${\langle}a_c{\rangle}_{out}=-{\langle}a_c{\rangle}_{in}$ 
of a photon towards the Sun during the time of flight from the observer to 
the object. We are now able to estimate the predicted difference ${\delta}a$ 
between a model where the anomalous acceleration is towards the Earth and one 
where it is towards the Sun. We get
\eqa
{\delta}a=-cH-{\langle}a_c{\rangle}_{out}=-2cH(1-{\frac{T_1}{T}}+O(HT)).
\ena
This function has a minimum at solar conjunction and vanishes when the observed
object and the Earth are at the same side of the Sun. One may show that
${\delta}a$ can be written approximately as a truncated sine function where 
all positive values are replaced by zero. The period is equal to one year and
the amplitude is approximately $2cH{\frac{R_e}{R_o}}$, where $R_o$ and $R_e$
are the radial coordinates of the object and of the Earth, respectively.

Anderson {\em et al.} [9] (see also [6]) found an annual perturbation on top 
of the anomalous acceleration $a_P$ of Pioneer 10. (They claim to see such a
perturbation for Pioneer 11 also.) Interestingly, the perturbation 
consistently showed minima near solar conjunction [9] (i.e., the absolute 
value of the anomalous acceleration reached maximum near solar conjunction). 
They fitted an annual sine wave to the velocity residuals coming from the 
annular perturbation term, using data from Pioneer 10 when the spacecraft was 
about 60 AU from the Sun. Taking the derivative with respect to time they 
then found the amplitude of the corresponding acceleration; it was found to 
be $a_{a.t.}=(0.215{\pm}0.022){\times}10^{-8}$ cm/s$^2$. We may compare this 
to the corresponding amplitude in ${\delta}a$ calculated above. We find an 
amplitude of about $0.29{\times}10^{-8}$ cm/s$^2$, close enough to be roughly 
consistent with the data. Besides the annual term a diurnal term was also 
found in the velocity residuals [9], where the corresponding acceleration 
amplitude $a_{d.t.}$ is large compared to $a_P$. But over one year the 
contribution from the diurnal term averages out to insignificance (less than 
$0.03{\times}10^{-8}$ cm/s$^2$ [9]).

To find the trajectories of non-relativistic particles we may set 
$E{\equiv}1-{\frac{w^2}{c^2}}$, where ${\frac{w^2}{c^2}}$ is small. 
Then equation (3) yields
\eqa
{\frac{dr}{dt}}={\pm}{\frac{ct_0}{t}}{\sqrt{{\frac{B(r)}{A(r)}}{\Big (}1+
({\frac{w^2}{c^2}}-1)B(r){\Big )}}}=
{\pm}{\frac{ct_0}{t}}{\sqrt{{\frac{r_{s0}}{r}}+{\frac{w^2}{c^2}}
+O({\frac{r_{s0}^2}{r^2}})}},
\ena
and the properly scaled coordinate acceleration for non-relativistic particles
is
\eqa
a_c={\mp}{\frac{c}{t}}{\sqrt{{\frac{r_{s0}}{r}}+{\frac{w^2}{c^2}}
+O({\frac{r_{s0}^2}{r^2}})}}
-{\frac{t_0}{t}}{\frac{r_{s0}c^2}{2r^2}}+O({\frac{r_{s0}^2c^2}{r^3}}).
\ena
We see that for non-relativistic particles the effect on coordinate
accelerations of treating the comoving coordinates as static ones and using 
GR, is a factor ${\sqrt{{\frac{r_{s0}}{r}}+{\frac{w^2}{c^2}}}}$ smaller than
the corresponding effect for photons. This means that the trajectories of 
non-relativistic particles do not depend crucially on the fact that the 
gravitational field is non-static. 

On the other hand the paths of photons depend more significantly on whether 
the gravitational field is static or not and this yields the illusion of an 
anomalous acceleration. That is, if one receives electromagnetic signals from 
some freely falling object located e.g. in the outer parts of the solar 
system, the coordinate acceleration of the object as inferred from the signals
should not agree with the ``real" coordinate acceleration of the object if one
treats the comoving coordinates as static ones. Rather, from equation (10) we 
see that it would seem as if the object were influenced by an attractive 
anomalous acceleration of size $cH$. The relevance of this is apparent when 
modeling the orbits of spacecraft and comparing to data obtained from radio 
signals received from the spacecraft; in particular this applies to the 
analyses performed in [2], [6] and [9]. An extra bonus for the model 
considered in this paper is that it predicts small deviations during the year 
if the data are compared to a model where the anomalous acceleration is 
directed towards the Sun rather than towards the Earth. And as we have
seen, this prediction seems to be consistent with the data.
\subsection{Cosmic expansion and the PPN-formalism}
Orbit analysis of objects moving in the solar system must be based on some
assumptions of the nature of space-time postulated to hold there. The standard
framework used for this purpose is the parameterized post-Newtonian (PPN)
formalism applicable for most metric theories of gravity. But the standard 
PPN-framework does not contain any terms representing expanding space via a
global scale factor as shown in equation (1) since it is inherently assumed 
that the solar system is decoupled from the cosmic expansion. One may try to 
overcome this by inventing some other sense of ``expanding space'' where
the scale factor varies in space rather than in time. But such a model must
necessarily be different from our quasi-metric model, and we show below that
it cannot work. Thus, to illustrate the inadequacy of the PPN-framework to 
model expanding space we now consider a specific model where suitable terms 
are added by hand in the metric. We may then compare to the change in light 
time obtained from our quasi-metric model.

One may try a post-Newtonian metric of the type
\eqa
ds^2=-{\Big(}1-{\frac{r_s}{r}}+{\frac{2H_0r}{c}}+
O({\frac{r_s^3}{r^3}}){\Big)}(dx^0)^2+
{\Big(}1+{\frac{r_s}{r}}-{\frac{2H_0r}{c}}+O({\frac{r_s^2}{r^2}}){\Big)}dr^2+
r^2d{\Omega}^2,
\ena
where $H_0$ is a constant, to describe expanding space within the 
PPN-framework. (To show that this metric yields a spatially variable
scale factor, transform to isotropic coordinates.) It may be readily shown
that the metric (16) yields a constant anomalous acceleration $cH_0$
towards the origin. But the problem with all metrics of this type is that 
they represent a ``real'' anomalous acceleration of gravitational origin, and 
this is observationally excluded from observations of planetary orbits [9].

Anyway we may calculate the change in light time ${\Delta}T$ due to the 
terms containing $H_0$ in (16) by integrating a radial null path from $r_p$ 
to $r_e$ (let $r_p>r_e$, say). This yields
\eqa
{\Delta}T=-c^{-2}H_0(r_p^2-r_e^2)+{\cdots}{\approx}-H_0T^2,
\ena
where the light time $T$ is equal to $c^{-1}(r_p-r_e)$ to first order and
where the last approximation is accurate only if $r_p$ is much larger than
$r_e$. Note that ${\Delta}T$ is negative; this is quite counterintuitive for 
a model representing expanding space. 

Anderson {\em et al.} [9] have considered a phenomenological model 
representing ``expanding space'' by adding a quadratic in time term to the 
light time in order to determine the coefficient of the quadratic by comparing
to data. To sufficient accuracy this model may be represented by the 
transformation
\eqa
T{\rightarrow}(1+a_{quad}t)T{\equiv}T + {\Delta}T,
\ena
where $a_{quad}$ is a ``time acceleration'' term. This model fits both 
Doppler and range very well [9].

If we compare equations (17) and (18) we see that the value
$a_{quad}=-H_0{\frac{T}{t}}$ corresponds to the change in light time
calculated from the metric (16). This is far too small (of order 
$10^{-30}$ s$^{-1}$ for a light time of a few hours) to be found directly from
the tracking data. Thus, the fact that $a_{quad}$ was estimated to be zero 
based on the tracking data alone [9] in no way favours a constant acceleration
model over a time acceleration model. They are in fact equivalent as far as 
the data are concerned.

To compare differences in $a_{quad}$ we may integrate equation (7) to lowest 
order to find the extra delay 
${\frac{H}{2c^2}}(r_p-r_e)^2={\frac{HT^2}{2}}$ in the light time compared
to the static case. This corresponds to a value $a_{quad}={\frac{H^2T}{2}}$
for the time acceleration term. But a determination of $a_{quad}$ directly 
from the tracking data still reflects the model-dependency explicitly 
present in the orbit determination process. This means that a determination 
of $a_{quad}$ directly from the tracking data should in principle be 
consistent with the model (16) and not with our quasi-metric model. However,
the quantity $a_{quad}$ is so small that it is not feasible to check this. 
But the fact that a phenomenological model of the type (18) works so well
should be taken to mean that the explanation of the anomalous acceleration 
given in this paper is sufficient.
\section{Conclusion}
We conclude that a natural explanation of the data is that the gravitational 
field of the solar system is not static with respect to the cosmic expansion.
This also explains why any orbit analysis program based on the PPN-formalism 
is insufficient for the task and how the largest errors arise from the 
mismodelling of null paths. (In fact, using the PPN-formalism is equivalent to
introducing a variable ``effective'' velocity of light as shown in equation 
(9).) But these explanations, while not involving any ad hoc assumptions, are 
based on the premise that space-time is quasi-metric. That is, rather than 
being described by one single Lorentzian metric, the gravitational field of 
the solar system should be modelled (to a first approximation) by the metric 
family shown in equation (1). From a theoretical point of view this premise is
radical; thus it is essential that the subject is further investigated to make
certain that more mundane explanations may be eliminated. However, so far no 
such explanations based on well-known physics have been found. But the facts 
that the model presented in this paper follows from first principles and fits 
the data very well, together with independent observational evidence in favour
of the prediction that the Earth-Moon system is not static with respect to the 
cosmic expansion [3], indicate that explanations based on quasi-metric 
relativity should be taken seriously.
\\ [4mm]
{\bf References} \\ [1mm]
{\bf [1]} D. {\O}stvang, chapter 2 of this thesis (2001). \\
{\bf [2]} J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto,
S.G. Turyshev,  \\
{\hspace*{6.3mm}}{\em Phys. Rev. Lett.} {\bf 81}, 2858 (1998)
(gr-qc/9808081). \\
{\bf [3]} D. {\O}stvang, chapter 3 of this thesis (2001).\\
{\bf [4]} J.I. Katz, {\em Phys. Rev. Lett.} {\bf 83} 1892 (1999) 
(gr-qc/9809070). \\
{\bf [5]} E.M. Murphy, {\em Phys. Rev. Lett.} {\bf 83} 1890 (1999)
(gr-qc/9810015). \\
{\bf [6]} S.G. Turyshev, J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, 
M.M. Nieto, \\
{\hspace*{6.3mm}} gr-qc/9903024 (1999). \\
{\bf [7]} J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto,
S.G. Turyshev, \\
{\hspace*{6.3mm}}{\em Phys. Rev. Lett.} {\bf 83}, 1893 (1999)
(gr-qc/9906112). \\
{\bf [8]} J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto,
S.G. Turyshev, \\
{\hspace*{6.3mm}}{\em Phys. Rev. Lett.} {\bf 83}, 1891 (1999)
(gr-qc/9906113). \\
{\bf [9]} J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto,
S.G. Turyshev,  \\
{\hspace*{6.3mm}} 
gr-qc/0104064 (2001).

\newpage

{\vspace*{5cm}}

\begin{center}
{\huge {\bf Chapter Five: \\\ \\On the Non-Kinematical\\
\vspace{0.5cm}Expansion of Gravitationally
}}
\end{center}
\vspace{-0.5cm}
\begin{center}
{\huge {\bf Bound Bodies}}

\end{center}
\newpage
{\hspace*{1cm}}

\newpage

\setcounter{equation}{0}

\setcounter{section}{0}

\begin{center}
{\large {\bf On the Non-Kinematical Expansion of Gravitationally Bound Bodies}}
\end{center}
\begin{center}
by
\end{center}
\begin{center}
Dag {\O}stvang \\
{\em Institutt for Fysikk, Norges teknisk-naturvitenskapelige universitet, 
NTNU \\
N-7491 Trondheim, Norway}
\end{center}
\begin{abstract}
It is found that one of the predictions of a so-called ``quasi-metric'' theory 
of gravity developed elsewhere [1], is that the gravitational field inside a 
metrically static, spherically symmetric, isolated body modelled as a perfect 
fluid obeying a linear equation of state (i.e., $p{\propto}{\rho}_m$),
should expand according to the Hubble law. That is, for such a body its radius
should increase like the expansion of the Universe; this is the counterpart to
the similar result found for the corresponding exterior gravitational field 
[2]. On the other hand, if the body consists of a perfect fluid obeying
some other equation of state (i.e., $p{\not}{\propto}{\rho}_m$, as for a 
polytrope for example), the expansion will induce instabilities and the body 
cannot be metrically static; such a body will not in general expand.
\end{abstract}
\section{Introduction}
A new type of non-metric space-time framework, the so-called ``quasi-metric''
framework, was presented in [1]. Also presented was a possibly viable 
quasi-metric theory of gravity compatible with the quasi-metric framework.
This theory predicts that the nature of the Hubble expansion is different than
in metric theory, and as a consequence the Hubble expansion should influence 
gravitationally bound systems, according to the new theory. In particular the
theory predicts that the gravitational field outside a spherically symmetric, 
isolated source expands according to the Hubble law [2], and this seems to be 
consistent with observations of the orbits of distant spacecraft [3].

Now a natural question is what happens to the gravitational fields inside
gravitationally bound bodies; does the quasi-metric theory predict that such
gravitational fields may expand, too? We show in this paper that this is 
indeed the case, and that this result may support an interpretation of 
geological data indicating that the Earth is expanding, see reference [4] and 
references cited therein. Moreover an expanding Earth should cause changes in 
its spin rate; we showed in [2] that the main part of the observed secular 
spin-down of the Earth may in fact be of cosmological origin. This also 
applies to the observed recession of the Moon and its observed mean 
acceleration.
\section{A brief survey of the quasi-metric theory}
In this section we summarize the main features of the quasi-metric framework
and the quasi-metric theory. A considerably more extensive discussion can be
found in [1].

The basic  premise of the quasi-metric framework is that the canonical 
description of space-time is taken as fundamental. That is, space-time is
constructed as consisting of two mutually orthogonal foliations: on the one
hand space-time can be sliced up globally into a family of 3-dimensional 
spacelike hypersurfaces (called the fundamental hypersurfaces (FHSs)) by the
global time function $t$, on the other hand space-time can be foliated into
a family of timelike curves everywhere orthogonal to the FHSs. These curves
represent the world lines of a family of hypothetical observers called the
fundamental observers (FOs), and the FHSs taken at $t=$constant represent
a preferred notion of space. The equations of the theory depend on the
geometrical properties of the FHSs.

The FHSs are associated with two families of Lorentzian space-time metric 
tensors ${\bf {\bar g}}_t$ and ${\bf g}_t$ in such a way that different FHSs
correspond to domains of applicability of different members of these families
in an one to one relationship. The metric family ${\bf {\bar g}}_t$ represents
a solution of field equations, and from ${\bf {\bar g}}_t$ one can construct 
the ``physical'' metric family ${\bf g}_t$ which is used when comparing 
predictions to experiments. The theory is not metric since the affine 
connection compatible with any metric family is non-metric.

To be able to compare theory to experiment we have to represent the metric 
families in terms of components with respect to some coordinate system on 
space-time. It is convenient to use a coordinate system ${\{}x^{\mu}{\}}$ 
where the relationship between the time coordinate $x^0$ and the global time
function is given by $x^0=ct$; this ensures that $x^0$ is a global time
coordinate. A coordinate system with a global time coordinate of this
type we call a global time coordinate system (GTCS). Note that there exist
infinitely many GTCSs.

It can be argued [1] that in a GTCS the most general form allowed for the 
family ${\bf {\bar g}}_t$ is represented by the family of line elements
(this may be taken as a definition)
\eqa
{\overline{ds}}_t^2= {\Big [}{\bar N}_s{\bar N}^s-{\bar N}_t^2{\Big ]}(dx^0)^2
+2{\frac{t}{t_0}}{\bar N}_idx^idx^0+{\frac{t^2}{t_0^2}}
{\bar N}_t^2S_{ik}dx^idx^k,
\ena
where ${\bar N}_t$ is the lapse function field family of the FOs and where
${\frac{t}{t_0}}{\bar N}_idx^i$ is the family of shift covector fields of the 
FOs in ${\bf {\bar g}}_t$. Moreover $S_{ik}dx^idx^k$ is the metric of the 
3-sphere (with radius equal to $ct_0$).

The evolution of the spatial scale factor of the FHSs in the 
hypersurface-orthogonal direction may be split up according to the 
definition (where a comma denotes partial derivation and where the symbol 
${\bar {\perp}}$ denotes a scalar product with the unit normal vector field 
family $-{\bf {\bar n}}_t$ of the FHSs)
\eqa
(c{\bar N}_t)^{-1}{\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
{\Big )}-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
{\equiv}
c^{-2}{\bar x}_t+c^{-1}{\bar H}_t,
\ena
where $c^{-2}{\bar x}_t$ represents the kinematical contribution to 
the evolution and $c^{-1}{\bar H}_t$ represents the so-called non-kinematical 
contribution defined by
\eqa
{\bar H}_t={\frac{1}{{\bar N}_tt}}+{\bar y}_t, \qquad
{\bar y}_t{\equiv}c^{-1}{\sqrt{{\bar a}_{{\cal F}k}{\bar a}_{\cal F}^k}}, 
\qquad c^{-2}{\bar a}_{{\cal F}j}{\equiv}{\frac{{\bar N}_t,_j}{{\bar N}_t}}.
\ena
We see that the non-kinematical evolution (NKE) of the spatial geometry 
takes the form of an ``expansion''. Besides we see from equation (2) that
the evolution of ${\bar N}_t$ with time may be written in the form
\eqa
{\frac{{\bar N}_t,_t}{c{\bar N}_t^2}}
-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}=c^{-2}{\bar x}_t+
c^{-1}{\bar y}_t. 
\ena
The split-ups defined in (3) and (4) are necessary to be able to construct 
${\bf g}_t$ from ${\bf {\bar g}}_t$ [1].

A special property of the quasi-metric theory is that operationally defined 
atomic units are defined to vary in space-time. That is, gravitational 
quantities may get an extra formal variability when measured in atomic units 
(and vice versa). To quantify the notion of formally variable units we 
introduce the scalar field ${\Psi}_t$ telling how atomic time units vary in 
space-time. Measured in atomic units gravitational quantities then get a 
formal variability as some power of ${\Psi}_t$. By definition $c$ and $\hbar$ 
are not formally variable; this means that atomic time units vary similarly 
to atomic length units and inversely to atomic mass units. However, for 
convenience we {\em define} the gravitational constant $G$ to be a constant 
measured in atomic units. But if we do this we must separate between 
{\em active mass}, which is measured dynamically, and {\em passive mass} 
(passive gravitational mass or inertial mass). By dimensional analysis it is 
found [1] that active mass $m_t$ varies formally as ${\Psi}_t^{-1}$ measured 
in atomic units (but passive mass does of course not vary). By definition we 
have ${\Psi}_t{\equiv}{\frac{1}{{\bar N}_tct}}$, so the formal variation of 
the units follows directly from the spatial scale factor of the FHSs as seen 
from (1), that is 
\eqa
{\Psi}_t,_t=-{\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big )}
{\Psi}_t, {\qquad}
{\Psi}_t,_{\bar {\perp}}=-{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
{\Psi}_t, \qquad
{\Psi}_t,_j=-c^{-2}{\bar a}_{{\cal F}j}{\Psi}_t, {\nonumber} \\
m_t,_t={\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big )}m_t, 
{\qquad} m_t,_{\bar {\perp}}={\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}
m_t, \qquad m_t,_j=c^{-2}{\bar a}_{{\cal F}j}m_t,
\ena 
where ${\bf {\bar a}}_{\cal F}$ is the 4-acceleration of the FOs in the family
${\bf {\bar g}}_t$. Local conservation laws involving projections into and 
normal to the FHSs of the (active) stress-energy tensor ${\bf T}_t$ take the 
form (in component notation)
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t){\bar {\perp}}{\bar {\perp}}}
={\Big (}{\bar K}_t-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-
{\frac{2}{c{\bar N}_t}}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
{\Big )}T_{(t){\bar {\perp}}{\bar {\perp}}}+{\bar K}_{(t)ik}{\hat T}_{(t)}^{ik}
-{\hat T}^i_{(t){\bar {\perp}}{\mid}i},
\ena
\eqa
{\topstar{\cal L}}_{{\bf {\bar n}}_t}T_{(t)j{\bar {\perp}}}
&=&{\Big (}{\bar K}_t-2{\frac{{\bar N}_t,_{\bar {\perp}}}{{\bar N}_t}}-
(c{\bar N}_t)^{-1}[{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}]
{\Big )}T_{(t)j{\bar {\perp}}}-c^{-2}{\bar a}_{{\cal F}j}
T_{(t){\bar {\perp}}{\bar {\perp}}} \nonumber \\
&&+c^{-2}{\bar a}_{{\cal F}i}{\hat T}_{(t)j}^i
+c^{-2}{\bar a}_{{\cal F}j}{\frac{t_0}{t}}
{\frac{{\bar N}^i}{{\bar N}_t}}T_{(t)i{\bar {\perp}}} 
-{\hat T}^i_{(t)j{\mid}i},
\ena
where ${\topstar{\cal L}}_{{\bf {\bar n}}_t}$ denotes Lie derivative of spatial
objects in the direction normal to the FHSs and $`{\mid}$' denotes spatial 
covariant derivation. (A ``hat'' denotes an object intrinsic to the FHSs.)
See [1] for a derivation of these equations. Also postulated in [1] are the 
field equations (with ${\kappa}{\equiv}{\frac{8{\pi}G}{c^4}}$)
\eqa
2{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}=
2(c^{-4}{\bar a}_{{\cal F}k}{\bar a}_{\cal F}^k+
c^{-2}{\bar a}^k_{{\cal F}{\mid}k}-{\bar K}_{(t)ik}{\bar K}_{(t)}^{ik}
+{\cal L}_{{\bf {\bar n}}_t}{\bar K}_t)=
{\kappa}(T_{(t){\bar {\perp}}{\bar {\perp}}}+{\hat T}_{(t)i}^i),
\ena
\eqa
{\bar R}_{(t)j{\bar {\perp}}}=
{\bar K}_{(t)j{\mid i}}^i-{\bar K}_t,_j={\kappa}T_{(t)j{\bar {\perp}}},
\ena
where ${\bf {\bar R}}_t$ is the Ricci tensor family and ${\bf {\bar K}}_t$ 
is the extrinsic curvature tensor family of the FHSs corresponding to the
metric family (1). (${\bar K}_t$ is the trace of ${\bf {\bar K}}_t$.)

The geometry intrinsic to the FHSs obtained from equation (1) has the curvature
\eqa
{\bar H}_{(t)ij}=c^{-2}{\Big (}{\bar a}^k_{{\cal F}{\mid}k}-
{\frac{1}{({\bar N}_tt)^2}}{\Big )}{\bar h}_{(t)ij}
-c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{{\cal F}j}
-c^{-2}{\bar a}_{{\cal F}i{\mid}j},
\ena
\eqa
{\bar P}_t={\frac{6}{({\bar N}_tct)^2}}
+2c^{-4}{\bar a}_{{\cal F}i}{\bar a}_{{\cal F}}^i
-4c^{-2}{\bar a}_{{\cal F}{\mid}i}^i,
\ena
where ${\bf {\bar h}}_t$ is the metric family intrinsic to the FHSs,
${\bf {\bar H}}_t$ is the Einstein tensor family intrinsic to the FHSs and
${\bar P}_t$ is the Ricci scalar family intrinsic to the FHSs.
\section{Metrically static, spherically symmetric interiors}
As shown in [2], the gravitational field exterior to an isolated, spherically 
symmetric body can be found from the field equations. In this case the metric 
families ${\bf {\bar g}}_t$ and ${\bf g}_t$ are spherically symmetric in a 
${\bf S}^3{\times}{\bf R}$-background. Furthermore they are ``metrically 
static''; by definition this means that neither ${\bar N}_t$ nor $N$ (the 
lapse function field of the FOs in ${\bf g}_t$) depend on $x^0$ (or $t$) and 
that the shift vector field appearing in equation (1) vanishes in a suitably 
chosen GTCS. It would seem reasonable to look for interior gravitational 
fields of this form.

To find the general form of ${\bf {\bar g}}_t$ and ${\bf g}_t$ for the 
spherically symmetric, metrically static case we start with the general 
expression (1) and introduce a spherically symmetric GTCS
${\{}x^0,r,{\theta},{\phi}{\}}$. Then it is straightforward to show that the 
metric families ${\bf {\bar g}}_t$ and ${\bf g}_t$ can be represented
by the families of line elements ($'{\equiv}{\frac{\partial}{{\partial}r}}$) 
\eqa
c^2{\overline{d{\tau}}}_t^2={\bar B}(r)(dx^0)^2-({\frac{t}{t_0}})^2
{\Big (}{\bar A}(r)dr^2+r^2d{\Omega}^2{\Big )}, {\nonumber} \\
c^2d{\tau}^2_t=B(r)(dx^0)^2-({\frac{t}{t_0}})^2
{\Big (}A(r)dr^2+r^2d{\Omega}^2{\Big )},
\ena
\eqa
{\bar A}(r){\equiv}{\frac{{\Big [}1-r{\frac{{\bar B}'(r)}{2{\bar B}(r)}}
{\Big ]}^2}
{1-{\frac{r^2}{{\bar B}(r)c^2t^2_0}}}},
\ena
where $d{\Omega}^2{\equiv}d{\theta}^2+{\sin}^2{\theta}d{\phi}^2$ and $t_0$
represents some arbitrary reference epoch setting the scale of the spatial
coordinates. Furthermore $x^0=ct$. However, $x^0$ and $ct$ must not be
interchanged in (12) due to the particular form of the connection 
coefficients, see [1] for a discussion. 

We now seek non-vacuum solutions of the type (12) of the field equations, and
where the source is modelled as a perfect fluid. For the vacuum case it was 
found in [2] that putting a line element ${\overline{ds}}^2_t$ of the form
(12) into the field equations results in equations separable in ${\bar A}(r)$
and ${\bar B}(r)$. It is then possible to find a series solution. And it turns
out that for non-vacuum cases one still gets separable equations when using 
line elements of the form (12). But we now have the passive pressure $p(r,t)$ 
as a new variable in addition to ${\bar A}(r)$ and ${\bar B}(r)$. (Note that 
the passive mass density ${\rho}_m(r,t)$ is also a new variable but that it is
related to $p(r,t)$ via an equation of state.) The explicit dependence on $p$ 
in the equations makes analytical calculations rather impracticable so the 
equations should be solved numerically. In the following we set up the 
relevant equations.
\subsection{Analytical calculations and a numerical recipe}
In this section we do some analytical calculations in order to write the
equations in a form appropriate for numerical treatment. Proceeding with this, 
from the definitions we get 
\eqa
c^{-2}{\bar a}_{{\cal F}r}&=&{\frac{{\bar B}'}{2{\bar B}}}, {\qquad}
c^{-2}{\bar a}_{{\cal F}r{\mid}r}={\frac{{\bar B}''}{2{\bar B}}}-
{\frac{{\bar B}'^2}{2{\bar B}^2}}-
{\frac{{\bar A}'{\bar B}'}{4{\bar A}{\bar B}}}, \nonumber \\
c^{-2}{\sin}^{-2}{\theta}{\bar a}_{{\cal F}{\phi}{\mid}{\phi}}&=&
c^{-2}{\bar a}_{{\cal F}{\theta}{\mid}{\theta}}=
{\frac{r{\bar B}'}{2{\bar A}{\bar B}}}, {\qquad}
{\bar P}_t=({\frac{t_0}{t}})^2{\frac{2}{\bar A}}{\Big (}r^{-2}({\bar A}-1)
+{\frac{{\bar A}'}{r{\bar A}}}{\Big )}, 
\ena
\eqa
c^{-2}{\bar a}^k_{{\cal F}{\mid}k}&=&({\frac{t_0}{t}})^2{\Big (}
{\frac{{\bar B}''}{2{\bar A}{\bar B}}}-
{\frac{{\bar B}'^2}{2{\bar A}{\bar B}^2}}
-{\frac{{\bar A}'{\bar B}'}{4{\bar A}^2{\bar B}}}
+{\frac{{\bar B}'}{r{\bar A}{\bar B}}}{\Big )}, {\nonumber} \\
{\bar H}_{(t)rr}&=&r^{-2}(1-{\bar A}), {\qquad}
{\sin}^{-2}{\theta}{\bar H}_{(t){\phi}{\phi}}={\bar H}_{(t){\theta}{\theta}}
=-{\frac{r{\bar A}'}{2{\bar A}^2}}.
\ena
Now the active stress-energy tensor ${\bf T}_t$ varies formally as
${\Psi}_t^2$ when measured in atomic units. For reasons of convenience 
we choose to extract this formal variability explicitly. What is left after
separating out the formal variability from the active mass density is by 
definition {\em the coordinate volume density of active mass} 
${\bar {\rho}}_m$. (The corresponding pressure is ${\bar p}$.) To be able to 
have experimental input it is also necessary to define {\em the passive mass 
density} ${\rho}_m$ and the corresponding passive pressure $p$. For the case 
when the perfect fluid is comoving with the FOs (i.e., 
$T_{(t){\bar {\perp}}j}=0$) we may then write (using (5))
\eqa
T_{(t){\bar {\perp}}{\bar {\perp}}}={\tilde{\rho}}_mc^2{\equiv}
{\frac{t_0^2}{t^2}}{\frac{{\bar {\rho}}_mc^2}{{\bar B}}}, \qquad
T_{(t)r}^r=T_{(t){\theta}}^{\theta}=T_{(t){\phi}}^{\phi}=
{\tilde p}{\equiv}{\frac{t_0^2}{t^2}}{\frac{{\bar p}}{{\bar B}}},
\ena
where ${\tilde {\rho}}_m$ is the density of active mass-energy and ${\tilde p}$
is the active pressure as seen in the local rest frame of the fluid.
Furthermore, using (16) the local conservation laws (6), (7) yield
(${\dot {\ }}{\equiv}{\frac{\partial}{{\partial}t}}$)
\eqa
{\dot {\bar {\rho}}}_m={\dot {\bar p}}=0, \qquad
{\bar p}'=-c^{-2}{\bar a}_{{\cal F}r}({\bar {\rho}}_mc^2-3{\bar p})=
-({\bar {\rho}}_mc^2-3{\bar p}){\frac{{\bar B}'}{2{\bar B}}}.
\ena
Equations (17) are valid for any metrically static perfect fluid. But to have 
experimental input we need to specify an equation of state $p=p({\rho}_m)$. 
We see that we need an expression relating ${\bar {\rho}}_m$ (or equivalently,
${\tilde {\rho}}_m$) to the passive mass-energy density ${\rho}_m$. Such a 
relationship is given by [1]
\eqa
{\tilde {\rho}}_m=
\left\{
\begin{array}{ll}
{\frac{t}{t_0}}{\bar N}_t{\rho}_m {\;}{\;}{\;}
$for a fluid of material point particles,$ \\ [1.5ex] 
{\frac{t^2}{t_0^2}}{\bar N}_t^2{\rho}_m {\;}{\;}{\;}
$for a null fluid,$
\end{array}
\right.
\ena
and a similar relationship exists between ${\tilde p}$ and $p$, namely
\eqa
{\tilde p}=
\left\{
\begin{array}{ll}
{\frac{t}{t_0}}{\bar N}_tp {\;}{\;}{\;}
$for a fluid of material point particles,$ \\ [1.5ex] 
{\frac{t^2}{t_0^2}}{\bar N}_t^2p {\;}{\;}{\;}
$for a null fluid.$
\end{array}
\right.
\ena
To find how the active mass $m_t$ varies in space-time, note that we are free 
to choose the background value of the active mass far from the source to be 
$m_0$. We use this to define $G$ as the constant measured in a Cavendish 
experiment far from the source at epoch $t_0$. Then, using equations (5) and 
(14) we find
\eqa
m_t(r,t)={\bar B}^{1/2}(r){\frac{t}{t_0}}m_0.
\ena
Now we can insert the equations (14), (15) and (16) into the field equation 
(8). Since ${\bf {\bar K}}_t$ vanishes identically for the metrically static 
case [1] equations (9) become vacuous and equation (8) yields 
\eqa
{\frac{{\bar B}''}{\bar B}}-{\frac{{\bar B}'^2}{2{\bar B}^2}}
-{\frac{{\bar A}'{\bar B}'}{2{\bar A}{\bar B}}}+
{\frac{2{\bar B}'}{r{\bar B}}}={\kappa}{\frac{{\bar A}}{{\bar B}}}
({\bar {\rho}}_{m}c^2+3{\bar p}).
\ena
A second equation can be found from the equation (10) for the spatial 
curvature. The radial component just gives equation (13) again but the angular
components yield a new equation, which can be combined with equation (21) 
to give
\eqa
{\frac{{\bar A}'}{{\bar A}}}
-{\frac{{\bar B}'}{{\bar B}}}-r{\frac{{\bar B}'^2}{2{\bar B}^2}}-
{\frac{2r{\bar A}}{(ct_0)^2{\bar B}}}=
-r{\kappa}{\frac{{\bar A}}{{\bar B}}}({\bar {\rho}}_{m}c^2+3{\bar p}).
\ena
We may now eliminate ${\bar A}(r)$ and ${\bar A}'(r)$ from equation (21) by 
inserting equations (13) and (22). The result is (with ${\Xi}_0{\equiv}ct_0$)
\eqa
{\Big (}1-{\frac{r^2}{{\bar B}(r){\Xi}_0^2}}{\Big )}
{\frac{{\bar B}''(r)}{{\bar B}(r)}}
&-&{\Big (}1-{\frac{2r^2}{{\bar B}(r){\Xi}_0^2}}{\Big )}
{\frac{{\bar B}'^2(r)}{{\bar B}^2(r)}} 
+{\frac{2}{r}}{\Big (}1-{\frac{3r^2}{2{\bar B}(r){\Xi}_0^2}}{\Big )}
{\frac{{\bar B}'(r)}{{\bar B}(r)}} \nonumber \\
&&-{\frac{r{\bar B}'^3(r)}{4{\bar B}^3(r)}}=
{\frac{{\kappa}}{{\bar B}(r)}}{\Big (}1-
{\frac{r{\bar B}'(r)}{2{\bar B}(r)}}{\Big )}^3
{\Big [}{\bar {\rho}}_{m}(r)c^2+3{\bar p}(r){\Big ]}.
\ena
To solve equation (23) numerically for ${\bar B}(r)$ one may proceed as 
follows. First specify the boundary conditions at the center of the body. 
From (13) we easily see that ${\bar A}(0)=1$, and moreover we must have 
${\bar B}'(0)={\bar p}'(0)=0$, so ${\bar A}'(0)=0$ from (22). Furthermore, 
noting that $r^{-1}{\bar B}'$ must be stationary near the center of the body 
we have
\eqa
{\bar B}''(0)={\lim}_{r{\rightarrow}0}{\Big [}r^{-1}{\bar B}'(r)
{\Big ]}, \qquad \Rightarrow \qquad
{\bar B}''(0)={\frac{{\kappa}}{3}}{\Big (}{\bar {\rho}}_{m}(0)c^2+
3{\bar p}(0){\Big )},
\ena
where the implication follows from equation (21).

To specify any particular model, choose the boundary condition ${\bar p}(0)$ 
at some arbitrary time. Also choose some arbitrary value ${\bar B}(0)$ as an
initial value for iteration. It must be possible to check how well the chosen
${\bar B}(0)$ reproduces the boundary condition for ${\bar B}({\cal R})$ at 
the surface $r={\cal R}$ of the body. That is, to match the exterior solution 
we must have [2]
\eqa
{\bar B}({\cal R})=1-{\frac{r_{s0}}{\cal R}}+{\frac{r_{s0}^2}{2{\cal R}^2}}
+{\frac{r_{s0}{\cal R}}{2{\Xi}_0^2}}-{\frac{r_{s0}^3}{8{\cal R}^3}}+{\cdots},
\ena
where $r_{s0}$ is a constant which may be identified with the Schwarzschild 
radius of the body at the arbitrary time $t_0$. Hence, by definition we set
\eqa
r_{s0}&{\equiv}{\frac{2M_{t_0}G}{c^2}}, \qquad
M_t{\equiv}c^{-2}{\int}{\int}{\int}{\bar N}_t{\Big [}
T_{(t){\bar {\perp}}{\bar {\perp}}}+{\hat T}^i_{(t)i}{\Big ]}d{\bar V}_{t}
=4{\pi}{\frac{t}{t_0}}{\int}_{{\!}{\!}0}^{\cal R}
{\frac{{\sqrt{\bar A}}}{{\sqrt{\bar B}}}}{\Big [}{\bar {\rho}}_{m}+
3{\bar p}/c^2{\Big ]}r^2dr,
\ena
where the integration is taken over the whole body. (The particular form of
$M_{t_0}$ follows directly from the ``metric approximation'' [1] of equation 
(8) applied to the interior of a spherically symmetric, static source when 
extrapolated to the exact exterior solution found in [1].) Note 
that $M_{t_0}$ is defined to be the total mass-energy of the body as measured 
by distant orbiters at epoch $t_0$ and that $M_{t_0}$ depends on the pressure 
as well.

To have a metrically static system it is necessary to specify an equation of 
state of the type $p{\propto}{\rho}_m$ (e.g. an ideal gas), since this 
ensures that ${\bar {\rho}}_m$ and ${\bar p}$ are independent of $t$. Once a
suitable equation of state is specified we may use equations (16), (18) and 
(19) to find ${\bar {\rho}}_m$ and ${\bar p}$ and then integrate (23) 
outwards from $r=0$ using (17) until the pressure vanishes. The surface of the
body is now reached (that is $r={\cal R}$). If the calculated value 
${\bar B}({\cal R})$ does not match the boundary condition (25), add a 
constant to ${\bar B}$ everywhere and repeat the calculation with the new 
value of ${\bar B}(0)$. Iterate until sufficient accuracy is achieved.

Once we have done the above calculations for an arbitrary time, we know the 
time evolution of the system from equations (16), (17), (18) and (19). That 
is, a spherical gravitationally bound body consisting of a perfect fluid 
obeying an equation of state $p{\propto}{\rho}_m$ will expand according
to the Hubble law. But for perfect fluid bodies obeying other equations of 
state the expansion will induce instabilities; mass currents will be set up 
and such systems cannot be metrically static. However, for the metrically 
static case the gravitational field interior to the body will expand along 
with the fluid; this is similar to the expansion of the exterior gravitational
field found elsewhere [2]. We are now able to calculate the family 
${\bf {\bar g}}_t$ inside the body. To find the corresponding family 
${\bf g}_t$ one uses the method described in [1].
\section{Discussion}
The possibility that cosmic expansion may be relevant for gravitationally 
bound bodies is not a new idea [4]. However, it has not been possible to fit 
this concept into the standard framework of metric gravity. In this paper we 
have shown that in quasi-metric theory, gravitationally bound bodies may be 
expected to expand with the Universe whenever they are made of a perfect fluid
with an equation of state of the type $p{\propto}{\rho}_m$. (This is easily 
seen from (12) and the fact that the coordinate radius ${\cal R}$ of the body 
is constant in time.) But if the linear equation of state is not a good
approximation one expects that the expansion should lead to instabilities in 
order to restore hydrostatic equilibrium.

The Earth is not made of perfect fluid. But if its equation of state 
is close to linear, the Earth should be expanding due to cosmic effects. This 
yields a number of observable consequences. One may, e.g., calculate the 
expected spin-down rate of the Earth due to the expansion; this is quite
consistent with observations if these are interpreted within the quasi-metric 
framework [2] and provided that tidal friction is negligible. Furthermore 
palaeo-tidal records, obtained e.g. from sedimentary tidal rhythmities, are in 
general in agreement with predictions calculated from quasi-metric theory 
as long as no extra theory-dependent assumptions, inconsistent with the 
quasi-metric framework, are made. 

Newtonian stars for which the equation of state takes the form
$p{\propto}{\rho}_m^{\gamma}$, ${\gamma}{\geq}{\frac{6}{5}}$, are called
{\em polytropes} [5]. In quasi-metric theory it is possible to model
polytropes by taking Newtonian limits of the relevant equations but such that 
the $t$-dependence remains. However, polytropes made of degenerate matter
will not be metrically static in such a description. But if the hydrodynamical
effects on the gravitational field coming from instabilities can be neglected,
we may solve the field equations for each epoch $t$ assuming that the 
polytrope is at hydrostatic equilibrium. To do this we take the Newtonian 
limits of equations (16), (17) and (23). One may then straightforwardly 
show that equation (23) reduces to
\eqa
{\frac{d}{dr}}{\frac{r^2}{{\rho}_m}}p'=-4{\pi}Gr^2{\rho}_{m0},
\ena
where ${\rho}_{m0}$ is the density field ${\rho}_m$ at epoch $t_0$. 
For polytropes this equation can be used only for the fixed epoch $t_0$. 
However, the equation is invariant under the scale transformation 
$r{\rightarrow}({\frac{t_0}{t}})^{{\frac{1}{3{\gamma}-4}}}r{\equiv}{\ell}$,
${\rho}_{m0}{\rightarrow}({\frac{t}{t_0}})^{\frac{3}{3{\gamma}-4}}{\rho}_{m0}
{\equiv}{\rho}_m$, $G{\rightarrow}{\frac{t}{t_0}}G{\equiv}G_t$. This means 
that by first scale transforming equation (27), it may be applied to 
polytropes and solved for any fixed epoch $t$. It then becomes equivalent to 
its counterpart in Newtonian theory except for a variable $G_t$. Thus the usual
Newtonian analysis of polytropes [5] applies, but with $G_t$ variable. And as 
a consequence we see that the physical radius 
${\cal L}(t)=({\frac{t_0}{t}})^{{\frac{1}{3{\gamma}-4}}}{\cal L}(t_0)$ of a 
polytrope made of degenerate matter will actually {\em shrink} with epoch
(if ${\gamma}>{\frac{4}{3}}$).

Of particular interest are polytropes for which ${\gamma}={\frac{4}{3}}$,
since such stars are models for Chandrasekhar mass white dwarfs (WDs). From 
[5] we easily find that the (passive) mass $m_c$ and radius ${\cal L}_c$ of 
such WDs (with identical central mass densities) depend on epoch such that 
$m_c(t)=({\frac{t_0}{t}})^{3/2}m_c(t_0)$ and 
${\cal L}_c(t)=({\frac{t_0}{t}})^{1/2}{\cal L}_c(t_0)$, respectively. Since
Chandrasekhar mass WDs are believed to be progenitors of type Ia supernovae,
one may expect that any cosmic evolution of Chandrasekhar mass WDs should 
imply a systematic luminosity evolution of type Ia supernovae over cosmic time 
scales. However, such a luminosity evolution would be inconsistent with their 
use as standard candles when determining the cosmological parameters in 
standard cosmology: luminosity evolution could have serious consequences for 
an interpretation of the supernova data in terms of an accelerating cosmic 
expansion indicating a non-zero cosmological constant [6], [7]. 

Now quasi-metric theory predicts that the cosmic expansion does neither 
accelerate nor decelerate [1]. Moreover, according to quasi-metric theory, the
Chandrasekhar mass varies with epoch and this means that type Ia supernovae 
may be generated from cosmologically induced collapse of progenitor WDs.
The consequences for type Ia supernova peak luminosities due to the predicted 
evolution of progenitor WDs are not clear. Since the luminosity of type Ia
supernovae comes from ${\gamma}$-disintegration of unstable nuclear species
(mainly $^{56}$Ni) synthesized in the explosion, this luminosity could depend
critically on the conditions of the nuclear burn. That is, the detailed nuclear
composition synthesized in the explosion and its energetics might depend on 
progenitor mass. Also the presence of more massive ejecta during the explosion 
could have an influence on supernova luminosities and light curves. But since 
there is no way to know exactly how these observables are affected by the 
predicted progenitor evolution without doing detailed numerical simulations, 
it is not possible to say whether or not the predictions from quasi-metric 
theory are consistent with the data.
 
However, what we {\em can} easily do is to see if it is possible to construct 
a simple luminosity evolution which, in combination with the cosmological toy 
model described in [1], yields a reasonable fit to the supernova data. That is,
we may try a luminosity evolution of the form 
$L(t){\propto}({\frac{t}{t_0}})^{\epsilon}$, or equivalently 
$L(z){\propto}(1+z)^{-{\epsilon}}$ and see if the data are well fit for some 
value(s) of ${\epsilon}$. To check this we use the postulated luminosity 
evolution to plot apparent magnitude $m_{QMT}$ versus redshift for type Ia 
supernovae. The easiest way to compare this to data is to calculate the 
difference in apparent magnitude ${\Delta}m$ between our model and that 
predicted using an empty Friedmann model (which is a piece of Minkowski 
space-time). The result is
\eqa
{\Delta}m{\equiv}m_{QMT}-m_{MIN}&=&2.5{\log}_{10}{\Big [}{\sin}^2{\{}{\ln}
(1+z){\}}{\Big ]} \nonumber \\
&&-5{\log}_{10}{\Big [}{\sinh}{\{}{\ln}(1+z){\}}{\Big ]}-
2.5{\log}_{10}{\frac{L_{QMT}}{L_{MIN}}},
\ena
where $L_{QMT}/L_{MIN}$ represents the luminosity evolution of the source
in our quasi-metric model relative to no luminosity evolution in an
empty Friedmann model.

Comparing the fits obtained from equation (28) to data we find that the 
luminosity evolution $L_{QMT}=(1+z)^{-1/2}L_{MIN}$ fits the data very well. 
We conclude that quasi-metric cosmology combined with a simple luminosity
evolution seems to be consistent with the data but that a much more 
detailed model should be constructed to see if the postulated luminosity
evolution has some basis in the physics of type Ia supernovae.
\\ [4mm]
{\bf References} \\ [1mm]
{\bf [1]} D. {\O}stvang, chapter 2 of this thesis (2001). \\
{\bf [2]} D. {\O}stvang, chapter 3 of this thesis (2001).\\
{\bf [3]} D. {\O}stvang, chapter 4 of this thesis (2001). \\
{\bf [4]} P. Wesson, {\em Cosmology and Geophysics}, Adam Hilger Ltd, (1978).
\\
{\bf [5]} S. Weinberg, {\em Gravitation and Cosmology}, {John Wiley ${\&}$ 
Sons, Inc. (1972). \\
{\bf [6]} S. Perlmutter et al., {\em ApJ} {\bf 517}, 565 (1999). \\
{\bf [7]} A.G. Riess et al., {\em AJ} {\bf 116}, 1009 (1998).

\end{document}
