The quasi-metric manifold $\cal N$ is equipped with two one-parameter families of metric tensors ${{\bar g}}_t$ and ${g}_t$, each parametrized by the global time function $t$. Moreover, in $({\cal N},{{\bar g}}_t)$ one must define two different electromagnetic field tensors corresponding to the active field tensor ${{\tilde F}}_t$ and the passive field tensor ${{\bar F}}_t$, respectively. The active electromagnetic field tensor ${{\tilde F}}_t$ couples to gravity. By construction the norm of the passive electromagnetic field tensor ${{\bar F}}_t$ decreases secularly, defining a global cosmic attenuation (not noticeable locally) of the electromagnetic field. Local conservation laws for passive electromagnetism imply that ${{\bar {\nabla}}}{\cdot} {{\bar F}}_t=0$ in electro-vacuum, ensuring that photons move on null geodesics of $({\cal N},{{\bar g}}_t)$. From ${{\bar F}}_t$ one may construct the passive electrodynamic field tensor ${F}_t$ in $({\cal N},{g}_t)$ in the same way as ${g}_t$ is constructed from ${{\bar g}}_t$. This ensures that photons move on null geodesics of $({\cal N},{g}_t)$ as well. As a simple example the (exact) quasi-metric counterpart to the Reissner-Nordstr\"{o}m solution in General Relativity is calculated. Besides it is found that a classical charged test particle electromagnetically bound to a central charge will participate in the cosmic expansion. But since quantum-mechanical states should be unaffected by the expansion, this classic calculation is hardly relevant for quantum-mechanical systems such as atoms, so there is no reason to think that the cosmic expansion applies to them. Finally it is shown that the main results of geometric optics hold in quasi-metric space-time.
gr-qc0303107.tex
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\begin{document}
\begin{center}
{\large {\bf Classical Electrodynamics in Quasi-Metric Space-Time}}
\end{center}
\begin{center}
by
\end{center}
\begin{center}
Dag {\O}stvang \\
{\em Department of Physics, Norwegian University of Science and Technology 
NTNU \\ N-7491 Trondheim, Norway}
\end{center}
\begin{abstract}
The quasi-metric manifold $\cal N$ is equipped with two 
one-parameter families of metric tensors ${\bf {\bar g}}_t$ and ${\bf g}_t$,
each parametrized by the global time function $t$. Moreover, in
$({\cal N},{\bf {\bar g}}_t)$ one must define two different electromagnetic 
field tensors corresponding to the active field tensor ${\bf {\tilde F}}_t$ 
and the passive field tensor ${\bf {\bar F}}_t$, respectively. The active 
electromagnetic field tensor ${\bf {\tilde F}}_t$ couples to gravity. By 
construction the norm of the passive electromagnetic field tensor 
${\bf {\bar F}}_t$ decreases secularly, defining a global cosmic attenuation 
(not noticeable locally) of the electromagnetic field. Local conservation laws 
for passive electromagnetism imply that ${\bf {\bar {\nabla}}}{\cdot}
{\bf {\bar F}}_t=0$ in electro-vacuum, ensuring that photons move on null 
geodesics of $({\cal N},{\bf {\bar g}}_t)$. From ${\bf {\bar F}}_t$ one may 
construct the passive electrodynamic field tensor ${\bf F}_t$ in 
$({\cal N},{\bf g}_t)$ in the same way as ${\bf g}_t$ is constructed from 
${\bf {\bar g}}_t$. This ensures that photons move on null geodesics of 
$({\cal N},{\bf g}_t)$ as well. As a simple example the (exact) quasi-metric
counterpart to the Reissner-Nordstr\"{o}m solution in General Relativity is 
calculated. Besides it is found that a classical charged test particle 
electromagnetically bound to a central charge will participate in the cosmic 
expansion. But since quantum-mechanical states should be unaffected by the 
expansion, this classic calculation is hardly relevant for quantum-mechanical 
systems such as atoms, so there is no reason to think that the cosmic 
expansion applies to them. Finally it is shown that the main results of 
geometric optics hold in quasi-metric space-time.
\end{abstract}
\topmargin 0pt
\oddsidemargin 5mm
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\section{Introduction}
Recently the so-called quasi-metric framework (QMF) as a geometric basis for 
relativistic gravity [1] was introduced as a possible alternative to the usual
metric framework (MF) underlying metric theories of gravity. The QMF is similar
to the MF in some respects (e.g. both are based on the Einstein equivalence 
principle), but one aspect of the QMF having no counterpart in the MF is the
existence of a non-metric sector. Another characteristic property of the QMF
is that it necessarily exists two fundamentally different scales in the
Universe, one gravitational and one atomic. The atomic scale can be 
constructed operationally from physical systems where gravitational 
interactions are insignificant. In practice this means that local measurement 
devices based on non-gravitational physics operationally define 
``atomic units'' and that both gravitational and non-gravitational scales
are expressed in terms of such units. Now the QMF predicts that expressed in 
atomic units, there should exist systematic scale changes between gravitational
and non-gravitational systems. Moreover, one manifestation of said non-metric 
sector is the {\em global} part of these scale changes, which is identified 
with the global cosmic expansion. Thus, within the QMF, the cosmic expansion 
is interpreted differently and has a different mathematical description than 
within the MF. And as a consequence, the cosmic expansion is predicted to 
affect gravitational systems regardless of scale. This prediction has a number 
of observable consequences, some of which have been detected in the solar 
system; see [2, 3] for details. Furthermore, how matter fields couple to 
gravity cannot be independent of said systematic scale changes. Therefore, the 
QMF does not fulfill the strong equivalence principle since it is necessary to 
separate between {\em active mass-energy} as a source of gravitation and 
{\em passive mass-energy} entering the equations of motion. 

No exceptions from these rules are made for the electromagnetic field; in
particular it is necessary to separate between the active and passive aspects 
of electromagnetic mass-energy. This means that a description of classical 
electrodynamics within the QMF must differ from its counterpart within the MF. 
It turns out that there are two crucial differences. Firstly, global scale 
changes between electromagnetism and gravitation should be present explicitly 
as a general time dependence in the norm of the electromagnetic field tensor. 
By construction, this general time dependence takes the form of a global cosmic
``attenuation'' of the electromagnetic field. Secondly, the necessity to 
separate between active and passive aspects of electromagnetism involves the 
introduction of {\em two} fundamentally different electromagnetic field 
tensors. That is, one must separate between {\em the active electromagnetic 
field tensor} which is coupled to gravitation, and {\em the passive 
electromagnetic field tensor} which is not coupled to gravity but is relevant 
for the equations of motion via the Lorentz force law. These features are
characteristic for a formulation of classical electrodynamics in quasi-metric
space-time.

The main goal of this paper is to formulate classical electrodynamics coupled
to gravity in a way consistent with the QMF. In other words, the question is 
how the general features of the QMF dictates the behaviour of classical 
electrodynamics (i.e., Maxwell's equations) in quasi-metric space-time. To 
answer this question, it is shown in section 3 that it is possible to derive 
the usual Maxwell equations in electro-vacuum for the passive electromagnetic 
field in curved space-time from local conservation laws. This motivates the
general definitions of Maxwell's equations within the QMF. Using these 
definitions, the relationship between the passive and active electromagnetic 
field tensors yields an initial value formulation of electrodynamics coupled 
to gravity. A specific example of this is given in section 4, where exact 
solutions are found for the gravitational and electric fields in the 
electro-vacuum outside a spherically symmetric, metrically static charged 
source. Finally, in section 5 it is shown that the main results of geometric 
optics are valid in quasi-metric space-time. That is, it is shown how to derive
the main results of geometric optics (e.g., that light rays are null geodesics)
from Maxwell's equations in curved space-time the way it is done in General 
Relativity (GR).

At the present time it is not known whether or not quasi-metric gravity is 
viable according to observational criteria (the current observational status 
of the QMF is discussed thoroughly elsewhere [1-3]), and a discussion of that 
is outside the scope of this paper. Besides, just as for GR, fact is that the 
active aspects of the electromagnetic field do not have any consequences which 
are not immeasurably small in practice. However, to be viable according to 
theoretical criteria, it is necessary that the QMF accommodates 
electromagnetism in a consistent and natural way; that is what is shown in 
this paper.
\section{Some relevant aspects of the QMF}
Quasi-metric theory and some of its predictions are described in detail
elsewhere [1, 2]; here we merely repeat the basics and the relevant 
formulae. 

The geometrical basis of the QMF consists of a five-dimensional differentiable
manifold with topology ${\cal M}{\times}{\bf R}_1$, where ${\cal M}={\cal S}
{\times}{\bf R}_2$ is a four-dimensional Lorentzian space-time manifold,
${\bf R}_1$ and ${\bf R}_2$ are two copies of the real line and ${\cal S}$ is 
a three-dimensional compact manifold (without boundaries). This geometrical 
structure implies that there exists one extra (degenerate) time dimension
represented by the {\em global time function $t$} as a global coordinate on
${\bf R}_1$ in addition to a ``preferred'' ordinary global time coordinate
$x^0$ on ${\bf R}_2$ with the property that $x^0$ scales like $ct$. That is,
when $t$ is given, so is the ``preferred'' time coordinate $x^0$. Using this
``preferred'' global time coordinate, the four-dimensional quasi-metric 
space-time manifold ${\cal N}$ is constructed by slicing the submanifold 
determined by the equation $x^0=ct$ out of ${\cal M}{\times}{\bf R}_1$. 

Furthermore, by construction ${\cal N}$ is equipped with a one-parameter 
family of Lorentzian 4-metrics ${\bf g}_t$. One may alternatively regard 
${\bf g}_t$ as a degenerate five-dimensional metric on a subset of 
${\cal M}{\times}{\bf R}_1$. A special set of coordinate systems especially 
well adapted to the structure of quasi-metric space-time is the set of 
{\em global time coordinate systems (GTCSs)} defined by the condition that 
$x^0$ is a ``preferred'' time coordinate in ${\cal N}$. The set of spatial 
submanifolds ${\cal S}$ taken at the set of constant $t$-values is called 
{\em the fundamental hypersurfaces (FHSs)} and represents a ``preferred'' 
notion of space. Observers always moving orthogonal to the FHSs are called 
{\em fundamental observers (FOs).}

The main physical role of the degenerate dimension represented by $t$ is to
describe global scale changes between gravitational and non-gravitational
systems. The reason one needs a degenerate dimension to describe this, is that 
such global scale changes should not have anything to do with ``motion'', 
which is why the evolution of quantities with $t$ is called 
``non-kinematical''. In particular this yields an alternative, non-kinematical
description of the global cosmic expansion.

Now a particular property of the QMF is that the metric family ${\bf g}_t$
does not represent solutions of gravitational field equations. Rather
${\bf g}_t$ is constructed from a second metric family ${\bf {\bar g}}_t$. The 
transformation ${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$ takes the form of a 
deformation of ${\bf {\bar g}}_t$ along a 3-vector field ${\bf b}_{\cal F}$
[1]. In fact the transformation ${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$ is 
just a special case of a more general transformation of tensor field families. 
Moreover, this general transformation applies to any tensor field which norm 
is required to be preserved under the transformation 
${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$. Here we list the transformation 
formulae ${\bf {\bar Z}}_t{\rightarrow}{\bf Z}_t$ valid for a covector family 
${\bf {\bar Z}}_t$. We have (using a GTCS) [2]
\eqa
Z_{(t)0}={\Big (}1-{\frac{v^2}{c^2}}{\Big )}{\bar Z}_{(t)0}, \qquad
Z_{(t)j}={\bar Z}_{(t)j}+{\frac{2{\frac{v}{c}}}{1-{\frac{v}{c}}}}({\bar e}^i_b
{\bar Z}_{(t)i}){\bar e}^b_j,
\ena
where ${\bf {\bar e}}_b$ is the unit vector (with corresponding covector 
${\bf {\bar e}}^b$) along ${\bf b}_{\cal F}$ and $v$ is a scalar field. For 
the general form of $v$, see [1, 2]. For completeness we also list the 
transformation formulae ${\bf {\bar Q}}_t{\rightarrow}{\bf Q}_t$ valid for a 
rank 2 tensor family ${\bf {\bar Q}}_t$. These formulae read [1, 2]
\eqa
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}^i_b
{\bar Q}_{(t)0i}){\bar e}^b_j{\Big ]},
\ena
\eqa
Q_{(t)ij}={\bar Q}_{(t)ij}+
{\frac{2{\frac{v}{c}}}{(1-{\frac{v}{c}})^2}}{\bar e}^k_b({\bar e}^b_i
{\bar Q}_{(t)kj}+{\bar Q}_{(t)ik}{\bar e}^b_j).
\ena
The transformation ${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$ may be found
from equations (2)-(4) as a special case.

To make matters as simple as possible, the topology of the FHSs is required to 
be simply connected in addition to being compact. Besides, the geometry of the 
FHSs in $({\cal N},{\bf {\bar g}}_t)$ is postulated to represent a direct 
measure of gravitational scales as measured in atomic units. Therefore, 
said geometry of the FHSs should be conformal to the 3-sphere ${\bf S}^3$, and
to avoid introducing any extra arbitrary scale or parameter, the scale factor 
should be proportional to $ct$. This means that expressed in an arbitrary 
GTCS, ${\bf {\bar g}}_t$ must take a restricted form. With the explicit 
$t$-dependence of ${\bf {\bar g}}_t$ included this form is [1] (using the 
signature ($-+++$))
\eqa
{\bar {ds}}^2_t=-({\bar N}_t^2-{\bar N}_s{\bar N}^s)(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 in 
$({\cal N},{\bf {\bar g}}_t)$, ${\frac{t}{t_0}}{\bar N}_idx^i$ is the family 
of shift covectors in the chosen GTCS and $S_{ik}dx^idx^k$ is the metric of 
${\bf S}^3$ (with radius equal to $ct_0$, where $t_0$ is constant but 
arbitrary). Furthermore, the family ${\bf {\bar g}}_t$ is required to be a 
solution of the field equations (in component notation using a GTCS)
\eqa
2{\bar R}_{(t){\bar {\perp}}{\bar {\perp}}}=
{\kappa}(T_{(t){\bar {\perp}}{\bar {\perp}}}+{\hat T}_{(t)i}^i),
\ena
\eqa
{\bar R}_{(t){\bar {\perp}}j}={\kappa}T_{(t){\bar {\perp}}j},
\ena
where ${\bf T}_t$ is the active stress-energy tensor field family (in which 
active mass-energy is treated as a scalar field) and ${\bf {\bar R}}_t$ is the
Ricci tensor field family calculated from ${\bf {\bar g}}_t$. Moreover
${\kappa}{\equiv}{\frac{8{\pi}G}{c^4}}$, where the gravitational ``constant''
$G$ by convention is set to the value it would have had in an empty Universe 
at the arbitrary epoch $t_0$, and the symbol `${\bar {\perp}}$' denotes a 
projection with the normal unit vector field family $-{\bf {\bar n}}_t$ of the 
FHSs. (A ``hat'' above an object denotes an object projected into the FHSs.)

Local conservation laws of ${\bf T}_t$ in $({\cal N},{\bf {\bar g}}_t)$
read (a comma denotes partial derivation)
\eqa
T_{(t){\mu}{\bar *}{\alpha}}^{\alpha}{\equiv}c^{-1}T_{(t){\mu}{\bar *}t}^0+
T_{(t){\mu};{\alpha}}^{\alpha}=T_{(t){\mu};{\alpha}}^{\alpha}-
{\frac{2}{c{\bar N}_t}}{\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
{\Big )}T_{(t){\bar{\perp}}{\mu}},
\ena
where the symbol `${\bar *}$' denotes covariant derivation found from the 
five-dimensional connection ${\topstar{\bf {\bar {\nabla}}}}$ compatible with 
${\bf {\bar g}}_t$ and a semicolon denotes metric covariant derivation found 
from the family of metric connections ${\bf {\bar {\nabla}}}$ compatible with 
single members of the family ${\bf {\bar g}}_t$. Furthermore the covariant 
divergence ${\bf {\bar {\nabla}}}{\cdot}{\bf T}_t$ takes the form
\eqa
T_{(t){\mu};{\alpha}}^{\alpha}=2{\frac{{\bar N}_t,_{\nu}}{{\bar N}_t}}
T_{(t){\mu}}^{\nu}.
\ena
Note that unlike its counterpart in GR, 
${\bf {\bar {\nabla}}}{\cdot}{\bf T}_t$ does not vanish in general.
Finally, the equations of motion in $({\cal N},{\bf g}_t)$ take the form
(in a GTCS)
\eqa
{\frac{d^2x^{\alpha}}{d{\lambda}^2}}+{\Big (}
{\topstar{\Gamma}}^{\alpha}_{t{\beta}}{\frac{dt}{d{\lambda}}}+
{\topstar{\Gamma}}^{\alpha}_{{\sigma}{\beta}}{\frac{dx^{\sigma}}{d{\lambda}}}
{\Big )}{\frac{dx^{\beta}}{d{\lambda}}}={\Big (}{\frac{d{\tau}_t}{d{\lambda}}}
{\Big )}^2a_{(t)}^{\alpha},
\ena
\eqa
{\topstar{\Gamma}}^{\alpha}_{t{\beta}}{\equiv}
{\topstar{\Gamma}}^{\alpha}_{{\beta}t}{\equiv}
{\frac{1}{t}}{\delta}^{\alpha}_i{\delta}_{\beta}^i, \qquad
{\topstar{\Gamma}}^{\alpha}_{{\sigma}{\beta}}{\equiv}{\frac{1}{2}}
g_{(t)}^{{\alpha}{\rho}}{\Big (}g_{(t){\rho}{\beta},{\sigma}}+
g_{(t){\sigma}{\rho},{\beta}}-g_{(t){\sigma}{\beta},{\rho}}{\Big )},
\ena
where ${\tau}_t$ is the proper time measured along a time-like curve and
${\lambda}$ is a general affine parameter. (Here $a_{(t)}^{\alpha}$ are the
components of the family of 4-accelerations ${\bf a}_t$.)
\section{Active and passive electromagnetic field tensors}
Within the QMF the active representation of some non-gravitational 
field (i.e., the representation describing its coupling to gravity) will not
be equal to the passive representation. That is, the form non-gravitational
fields take in non-gravitational force laws will not be identical to the form 
non-gravitational fields take when contributing to the active stress-energy 
tensor. For electromagnetism in particular, we will illustrate below how these 
representations must be different to be consistent with the equations of 
motion in $({\cal N},{\bf {\bar g}}_t)$ (and in $({\cal N},{\bf g}_t)$).

We proceed to define two different representations of the electromagnetic field
in quasi-metric space-time. Firstly, we define {\em the passive electromagnetic
field tensor family ${\bf {\bar F}}_t$} in $({\cal N},{\bf {\bar g}}_t)$ via a
vector potential family ${\bf {\bar A}}_t$ in $({\cal N},{\bf {\bar g}}_t)$.
That is, for reasons given below, we define ${\bf {\bar F}}_t$ via its 
components expressed in a GTCS
\eqa
{\bar F}_{(t)0j}&{\equiv}&{\frac{t_0}{t}}{\bar F}_{0j}=
{\frac{t_0}{t}}({\bar A}_{j;0}-{\bar A}_{0;j})=
-{\frac{t_0}{t}}{\bar F}_{j0}{\equiv}-{\bar F}_{(t)j0}, 
\nonumber \\
{\bar F}_{(t)ij}&{\equiv}&{\bar F}_{ij}={\bar A}_{j;i}-
{\bar A}_{i;j}=-{\bar F}_{ji}{\equiv}-{\bar F}_{(t)ji},
\ena
where ${\bar F}_{{\mu}{\nu}}$ and ${\bar A}_{\mu}$ do not depend explicitly 
on $t$ (however, ${\bar F}_{{\mu}{\nu}}$ and ${\bar A}_{\mu}$ may depend
{\em implicitly} on $t$ through the dependence on other fields). It remains to 
define the explicit $t$-dependence of ${\bf {\bar A}}_t$, i.e., to relate
the components of ${\bf {\bar A}}_t$ to ${\bar A}_{\mu}$ (in a GTCS). For
reasons given below we set
\eqa
{\bar A}_{(t)0}= {\frac{t_0}{t}}{\bar A}_{0}, \quad
{\bar A}_{(t)j}={\bar A}_{j}, \quad \Rightarrow \quad
{\mid}{\bf {\bar A}}_t{\mid},_t= 
-({\frac{1}{t}}+{\frac{{\bar N}_{t,t}}{{\bar N}_t}})
{\mid}{\bf {\bar A}}_t{\mid}+{\rm implicit{\,}{\,}variation}.
\ena 
Here ${\mid}{\bf {\bar A}}_t{\mid}{\equiv}{\sqrt{{\bar A}_{(t){\mu}}{\bar A}
_{(t)}^{\mu}}}$ denotes the norm of ${\bf {\bar A}}_t$. The reason for the 
specific dependences on $t$ of ${\bf {\bar A}}_t$ and ${\bf {\bar F}}_t$ is 
that within the QMF, the passive electromagnetic field by construction 
experiences a cosmic attenuation due to the global cosmic expansion. This 
cosmic attenuation represents global scale changes between electromagnetism 
and gravitation. Moreover, these global scale changes are physically realized 
via the explicit $t$-dependence of the norm of ${\bf {\bar F}}_t$; we see from 
equations (12) and (13) that its norm attenuates according to the formula 
${\mid}{\bf {\bar F}}_t{\mid},_t{\equiv}{\sqrt{{\bar F}_{(t){\sigma}{\rho}}
{\bar F}_{(t)}^{{\sigma}{\rho}}}},_t=-2({\frac{1}{t}}+{\frac{{\bar N}_{t,t}}
{{\bar N}_t}}){\mid}{\bf {\bar F}}_t{\mid}+$ implicit variation. We notice 
that we can define ${\bf {\bar F}}_t$ directly from ${\bf {\bar A}}_t$. That 
is, inserting equation (13) into equation (12) yields
\eqa
{\bar F}_{(t)0j}&=&{\frac{t_0}{t}}{\bar A}_{(t)j;0}-{\bar A}_{(t)0;j}=
{\frac{t_0}{t}}{\bar A}_{(t)j,0}-{\bar A}_{(t)0,j}, \nonumber \\
{\bar F}_{(t)ij}&=&{\bar A}_{(t)j;i}-
{\bar A}_{(t)i;j}={\bar A}_{(t)j,i}-{\bar A}_{(t)i,j}.
\ena
To predict the passive aspects of electromagnetism according to the QMF, we 
must also construct the ``physical'' passive electromagnetic field tensor 
family ${\bf F}_t$ in $({\cal N},{\bf g}_t)$ from ${\bf {\bar F}}_t$ in exactly
the same way as ${\bf g}_t$ is constructed from ${\bf {\bar g}}_t$ using 
equations (2)-(4). The construction of ${\bf F}_t$ is necessary since it is 
${\bf F}_t$ which enters into the Lorentz force law 
\eqa
ma_{(t)}^{\alpha}=qF_{(t){\nu}}^{\alpha}u_{(t)}^{\nu},
\ena
in $({\cal N},{\bf g}_t)$, where ${\bf u}_t$ is the 4-velocity field of the 
charged matter and $q$ is the passive charge and $m$ the passive mass of a 
test particle.

Secondly, we define {\em the active electromagnetic field tensor family 
${\bf {\tilde F}}_t$} in $({\cal N},{\bf {\bar g}}_t)$ directly from
${\bf {\bar F}}_t$:
\eqa
{\bf {\tilde F}}_t{\equiv}{\frac{t}{t_0}}{\bar N}_t{\bf {\bar F}}_t, \qquad
{\tilde F}_{(t){\alpha}{\beta}}{\equiv}{\frac{t}{t_0}}
{\bar N}_t{\bar F}_{(t){\alpha}{\beta}}.
\ena
The $t$-dependence of ${\bf {\tilde F}}_t$ differs from that of 
${\bf {\bar F}}_t$ since the active electromagnetic field will experience a 
global cosmic increase of active mass-energy which will partly cancel the 
effects of the global cosmic attenuation (i.e.,
${\mid}{\bf {\tilde F}}_t{\mid},_t=-({\frac{1}{t}}+{\frac{{\bar N}_{t,t}}
{{\bar N}_t}}){\mid}{\bf {\tilde F}}_t{\mid}+$ implicit variation).

The point now is that ${\bf {\tilde F}}_t$ is coupled dynamically to gravity
whereas ${\bf {\bar F}}_t$ is not. That is, ${\bf {\tilde F}}_t$ determines 
the active electromagnetic field stress-energy tensor ${\bf T}_t^{(EM)}$ via 
the familiar formula (in component notation)
\eqa
T_{(t){\alpha}{\beta}}^{(EM)}={\frac{1}{4{\pi}}}{\Big (}
{\tilde F}_{(t){\alpha}}^{{\ }{\ }{\ }{\,}{\,}{\nu}}
{\tilde F}_{(t){\beta}{\nu}}-{\frac{1}{4}}{\tilde F}_{(t){\sigma}{\rho}}
{\tilde F}_{(t)}^{{\sigma}{\rho}}{\bar g}_{(t){\alpha}{\beta}}{\Big )}.
\ena
Similarly, ${\bf {\bar F}}_t$ defines the passive electromagnetic field 
stress-energy tensor 
${\bf {\bar {\cal T}}}_t^{(EM)}$ in $({\cal N},{\bf {\bar g}}_t)$:
\eqa
{\bar {\cal T}}_{(t){\alpha}{\beta}}^{(EM)}={\frac{1}{4{\pi}}}{\Big (}
{\bar F}_{(t){\alpha}}^{{\ }{\ }{\ }{\,}{\,}{\nu}}
{\bar F}_{(t){\beta}{\nu}}-{\frac{1}{4}}{\bar F}_{(t){\sigma}{\rho}}
{\bar F}_{(t)}^{{\sigma}{\rho}}{\bar g}_{(t){\alpha}{\beta}}{\Big )}.
\ena
To find electromagnetic field equations for ${\bf {\bar F}}_t$, we first 
consider the special case of an electro-vacuum. In general, the relationship 
between the active electromagnetic stress-energy tensor ${\bf T}^{(EM)}_t$ 
given in equation (17) and the corresponding passive electromagnetic 
stress-energy tensor ${\bf {\bar {\cal T}}}^{(EM)}_t$ in equation (18) is 
given by
\eqa
{\bf T}^{(EM)}_t={\frac{t^2}{t_0^2}}{\bar N}_t^2{\bf {\bar {\cal T}}}^{(EM)}_t,
\qquad \Rightarrow \qquad  
{\bf {\bar {\nabla}}}{\cdot}{\bf {\bar {\cal T}}}_t^{(EM)}=0, \qquad
{\rm (electro{\text -}vacuum)}
\ena
where the implication follows from equation (9) for electro-vacuum. 
%(If there is no local creation, equation (8) implies that $\,\,{\bf {\topstar 
%{\bf {\bar {\nabla}}}}}\,{\cdot}{\bf {\bar {\cal T}}}_t^{(EM)}=0$ as well.) 
Notice that the relationship (19) takes the same form as for a 
perfect fluid of null particles [2]. Using equations (18) and (19) we may now 
perform a standard derivation of the usual electro-vacuum Maxwell equations 
${\bar F}_{(t){\mu};{\nu}}^{\nu}=0$ in $({\cal N},{\bf {\bar g}}_t)$. Just as
in GR, this derivation acts as a motivation for the general definition of the 
usual Maxwell equations in $({\cal N},{\bf {\bar g}}_t)$, namely
\eqa
{\bar F}_{(t){\alpha}{\beta};{\mu}}+{\bar F}_{(t){\mu}{\alpha};{\beta}}+
{\bar F}_{(t){\beta}{\mu};{\alpha}}={\bar F}_{(t){\alpha}{\beta},{\mu}}+
{\bar F}_{(t){\mu}{\alpha},{\beta}}+{\bar F}_{(t){\beta}{\mu},{\alpha}}=0,
\ena
\eqa
{\bar F}_{(t){\mu};{\nu}}^{\nu}=-{\frac{4{\pi}}{c}}{\bar J}_{(t){\mu}}, \qquad 
{\bar J}_{(t){\mu}}{\equiv}{\rho}_{\text c}{\bar u}_{(t){\mu}}, \qquad 
\Rightarrow \qquad {\bar J}^{\mu}_{(t);{\mu}}=0,
\ena
where ${\bf {\bar J}}_t$ is the 4-current of passive charge in $({\cal N},
{\bf {\bar g}}_t)$ and ${\rho}_{\text c}$ is the density of passive charge $q$
as measured in the local inertial rest frame of the fluid. Note that only the
metric part of the connection enters into equation (21). This means that
equation (21) has the same properties as its counterpart in metric gravity. In 
particular it means that equation (21) ensures that passive charge is 
conserved in $({\cal N},{\bf {\bar g}}_t)$ and that we in general have
\eqa
{\bf {\bar {\cal T}}}_{(t){\alpha}{\,}{\,}{\,}{\,}{\,}{\,}{\,}{\,}
{\,};{\beta}}^{(EM){\beta}}=
-{\frac{1}{c}}{\bar F}_{(t){\alpha}{\beta}}{\bar J}_{(t)}^{\beta}, \qquad
{\rm (general{\,}{\,}case)}
\ena
rather than the special case shown in equation (19). Now one important point 
is that no counterparts to equations (20), (21) exist in 
$({\cal N},{\bf g}_t)$. However, it is still possible to define the 4-current 
of passive charge ${\bf J}_t$ in $({\cal N},{\bf g}_t)$:
\eqa
{\bf J}_t{\equiv}{\hat {\rho}}_{\rm c}{\bf u}_t{\equiv}
{\sqrt{{\frac{{\bar h}_t}{h_t}}}}{\rho}_{\rm c}{\bf u}_t, \qquad
\Rightarrow \qquad J_{(t){\perp}}{\sqrt{h_t}}={\bar J}_{(t){\bar {\perp}}}
{\sqrt{{\bar h}_t}},
\ena
where $J_{(t){\perp}}{\equiv}-n_{(t){\mu}}J_{(t)}^{\mu}$ (and ${\bf n}_t$ is 
the unit normal vector field family to the FHSs in $({\cal N},{\bf g}_t)$). 
Moreover ${\bar h}_t$ and $h_t$ are the determinants of the spatial metric 
families ${\bf {\bar h}}_t$ and ${\bf h}_t$, respectively. The square root 
factor is necessary in the definition of ${\bf J}_t$ to ensure that the amount
of passive charge is identical in $({\cal N},{\bf {\bar g}}_t)$ and in 
$({\cal N},{\bf g}_t)$. The implication in equation (23) follows because the 
norm of ${\bf {\bar u}}_t$ (and of ${\bf {\bar n}}_t$) must be invariant under
the transformation ${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$. This means that 
${\bf {\bar u}}_t{\rightarrow}{\bf u}_t$ and 
${\bf {\bar n}}_t{\rightarrow}{\bf n}_t$, i.e., that ${\bf {\bar u}}_t$ and 
${\bf {\bar n}}_t$ transform according to equation (1). It is now 
straightforward to show that equation (21) together with equation (23) imply 
that passive charge is conserved in $({\cal N},{\bf g}_t)$ as well as in 
$({\cal N},{\bf {\bar g}}_t)$.

For weak electromagnetic fields it is a convenient approximation to neglect 
the effects of electromagnetism on space-time geometry. This is equivalent to
considering passive electromagnetism only in an independent quasi-metric 
curved space-time background and the calculations may be done as follows: 
given ${\rho}_{\rm c}$ and ${\bf {\bar u}}_{t_1}$ (found from ${\bf u}_{t_1}$) 
for some initial value $t=t_1$ it is possible to calculate 
${\bf {\bar F}}_{t_1}$ from equations (20) and (21). Then ${\bf F}_{t_1}$ can 
be found from ${\bf {\bar F}}_{t_1}$ just as ${\bf g}_{t_1}$ can be found from 
${\bf {\bar g}}_{t_1}$, using equations (2)-(4). Moreover, the motion of the 
source charges can be found from the geodesic equation (10) in 
$({\cal N},{\bf g}_t)$ (using equation (15)). This again yields 
${\rho}_{\rm c}$ and ${\bf u}_{(t_1+{\Delta t})}$ (from which 
${\bf {\bar u}}_{(t_1+{\Delta}t)}$ can be calculated) for some later time 
$t=t_1+{\Delta}t$. The procedure can then be repeated to find the evolution of 
the electromagnetic field at progressively later times. We thus have a 
well-defined initial value problem for passive electromagnetism in quasi-metric
space-time. Notice that the transformations (2)-(4) preserve norm by 
construction, so if, e.g., ${\bf {\bar F}}_t$ is null, so is ${\bf F}_t$. This 
means that passive electrodynamics in quasi-metric space-time respects the 
local light cone.

However, in general one must also take into account the active aspects of
electromagnetism. To do that we may we use the definition (16) together with 
equation (21) to find field equations for the active electromagnetic field:
\eqa
{\tilde F}_{(t){\mu};{\nu}}^{\nu}=-{\frac{4{\pi}}{c}}{\tilde J}_{(t){\mu}}+
{\frac{{\bar N}_t,_{\nu}}{{\bar N}_t}}{\tilde F}_{(t){\mu}}^{\nu}, \qquad 
{\tilde J}_{(t){\mu}}{\equiv}{\tilde {\rho}}_{\text c}{\bar u}_{(t){\mu}}, 
\qquad \Rightarrow \qquad {\tilde J}^{\mu}_{(t);{\mu}}=
{\frac{{\bar N}_t,_{\nu}}{{\bar N}_t}}{\tilde J}_{(t)}^{\nu}.
\ena
Here the 4-current of active charge ${\bf {\tilde J}}_t$ in
$({\cal N},{\bf {\bar g}}_t)$ is defined by
\eqa
{\bf {\tilde J}}_t{\equiv}{\tilde{\rho}}_{\text c}{\bf {\bar u}}_t=
{\frac{t}{t_0}}{\bar N}_t{\rho}_{\text c}{\bf {\bar u}}_t,
\ena
where the last step follows from equation (26) below, and where
${\tilde{\rho}}_{\text c}$ is the density of active charge $q_t$. Analogous to 
active mass, active charge is a scalar field and it describes how charge 
couples to gravity. How active charge varies in quasi-metric space-time is 
found from dimensional analysis [2]. That is, just as one may infer the 
variability of the active mass $m_t$ from the gravitational quantity 
$Gm_t/c^2$ (which has the dimension of length), one may infer the variability 
of the active charge from the gravitational quantity $Gq_t^2/c^4$ which has 
the dimension of length squared. The result is
\eqa
q_t,_{\mu}= {\frac{{\bar N}_t,_{\mu}}{{\bar N}_t}}q_t, \qquad
q_t,_t={\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big )}q_t, 
\qquad \Rightarrow \qquad q_t={\frac{t}{t_0}}{\bar N}_tq.
\ena
We see from equation (24) that active charge is not necessarily conserved, nor 
is there any reason that it should be. 

Finally we notice that the exterior derivative of ${\bf {\tilde F}}_t$ will 
not vanish in general, so there is no counterpart to equation (20) valid for
${\bf {\tilde F}}_t$. Also notice that equations (9), (24) and (25) yield that
\eqa
T_{(t){\alpha}{\,}{\,}{\,}{\,}{\,}{\,}{\,}{\,}{\,};{\beta}}^{(EM){\beta}}=
2{\frac{{\bar N}_{t,{\beta}}}{{\bar N}_t}}T_{(t){\alpha}}^{(EM){\beta}}
-{\frac{1}{c}}{\tilde F}_{(t){\alpha}{\beta}}{\tilde J}_{(t)}^{\beta}.
\ena
When the active stress-energy tensor of the charged matter source is added to 
equation (27), we get back equation (9) for the total active stress-energy 
tensor.
 
The description of how electromagnetism couples to gravity in quasi-metric 
space-time is now given from the definition (16) and the usual Maxwell 
equations (20) and (21) in curved space-time. These equations are coupled to 
the gravitational field equations (6) and (7) and should be solved 
simultaneously. Given suitable initial conditions it is thus the coupled system
of equations (6), (7), (10), (15), (20) and (21) which must be solved to find 
the fields ${\bf {\bar F}}_t$ and ${\bf {\bar g}}_t$ for each time step. 
Besides, to find these fields at progressively later times, it is also 
necessary to calculate ${\bf g}_t$ and ${\bf F}_t$ for each time step by the 
method described in [1, 2] using equations (2)-(4). It is thus possible to set 
up a well-defined initial value problem for electromagnetism coupled to 
gravitation in quasi-metric space-time. Notice that active electromagnetism in 
quasi-metric space-time is also guaranteed to respect the local light cone 
since ${\bf {\tilde F}}_t$ is null if and only if ${\bf {\bar F}}_t$ is null 
(this follows from equation (16)).
\section{The spherically symmetric, metrically static case}
\subsection{Electro-vacuum outside a charged source}
In this section we set up the equations valid for the gravitational and
electromagnetic fields outside a spherically symmetric, charged source. It is
required that the system is ``metrically static'', i.e., that the only time
dependence is via the effect on the spatial geometry of the global cosmic
expansion. We then find exact solutions of these equations.

Introducing a spherical GTCS ${\{}x^0,r,{\theta},{\phi}{\}}$, 
where $r$ is a Schwarzschild radial coordinate, a metrically static, 
spherically symmetric metric family ${\bf {\bar g}}_t$ may be expressed in 
the form ($'{\equiv}{\frac{{\partial}}{{\partial}r}}$)
\eqa
{\overline{ds}}^2_t=-{\bar B}(r)(dx^0)^2+({\frac{t}{t_0}})^2{\Big (}
{\bar 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){\Xi}_0^2}}}},
\ena
where $d{\Omega}^2{\equiv}d{\theta}^2+{\sin}^2{\theta}d{\phi}^2$,
${\Xi}_0{\equiv}ct_0$ and $t_0$ is some arbitrary reference epoch. The special
form (29) of the function ${\bar A}(r)$ follows from equation (5) since 
${\bar B}(r){\equiv}{\bar N}_t^2(r)$. Moreover, ${\bar A}(r)$ should not be 
confused with components of the vector potential ${\bf {\bar A}}_t$.

We will now find an expression for ${\bf {\tilde F}}_t$. Due to symmetry
its only components in the above defined GTCS are given in terms of
the active electric field ${\bf {\tilde E}}_t$ as seen by the
FOs. That is, ${\bf {\tilde E}}_t$ is defined by its radial component
\eqa
{\tilde E}_{(t)r}{\equiv}
{\tilde F}_{(t){\bar {\perp}}r}{\equiv}-
{\bar n}_{(t)}^{\mu}{\tilde F}_{(t){\mu}r}
=-{\frac{1}{{\bar N}_t}}{\tilde F}_{(t)0r}={\tilde E}_{(t_0)r}.
\ena
Since the components of the active magnetic field vanish in the chosen GTCS,
equation (24) yields
\eqa
{\Big (}{\bar N}_t^{-1}{\tilde F}_{(t)}^{0r}{\Big )}_{;r}={\frac{4{\pi}}{c}}
{\bar N}_t^{-1}{\tilde J}_{(t)}^0,
\ena
or equivalently
\eqa
{\frac{\partial}{{\partial}r}}{\Big (}{\bar N}_t^{-1}{\sqrt{-{\bar g}_t}}
{\tilde F}_{(t)}^{0r}{\Big )}=4{\pi}{\frac{t}{t_0}}{\sqrt{-{\bar g}_t}}
{\bar N}_t^{-1}{\rho}_{\text c},
\ena
where ${\bar g}_t$ is the determinant of ${\bf {\bar g}}_t$. We now integrate 
equation (32) from the origin to some radial coordinate $r>{\cal R}$, where 
${\cal R}$ is the coordinate radius of the source. Assuming that all fields 
are continuous and well-behaved we get
\eqa
r^2{\bar N}_t^{-1}{\bar A}^{-1/2}{\tilde E}_{(t_0)r}=4{\pi}{\frac{t^3}{t_0^3}}
{\int}_{{\!}{\!}{\!}0}^{\cal R}{\bar A}^{1/2}{\rho}_{\text c}r^2dr=
{\int}{\int}{\int}{\rho}_{\text c}{\sqrt{{\bar h}_t}}d^3x{\equiv}Q,
\ena
where the triple integration is taken over the volume of the source. We see
that, since we have integrated the passive charge density over the source 
volume, it is natural to interpret $Q$ as the passive charge of the source, 
which is obviously a constant (since the left hand side of equation (33) has 
no time dependence). From equation (33) we thus have that
\eqa
{\tilde E}_{(t)r}={\frac{{\bar N}_t{\sqrt{{\bar A}(r)}}Q}{r^2}}.
\ena
Next we must find expressions for projections of ${\bf T}^{(EM)}_t$ to put 
into the field equations. Using equations (17), (30) and (34) and doing the 
projections we find
\eqa
T_{(t){\bar {\perp}}{\bar {\perp}}}^{(EM)}={\frac{1}{8{\pi}}}
{\hat{\tilde F}}_{(t){\bar {\perp}}}^r{\tilde F}_{(t)r{\bar {\perp}}}=
{\frac{1}{8{\pi}}}{\tilde E}_{(t)r}{\hat{\tilde E}}_{(t)}^r=
{\frac{t_0^2}{t^2}}{\frac{{\bar N}_t^2Q^2}{8{\pi}r^4}}{\equiv}
{\frac{t^2_0}{t^2}}{\bar N}_t^{-2}{\bar{\rho}}_{\text m}^{(EM)}c^2,
\ena
\eqa
T_{(t)r}^{(EM)r}=-T_{(t){\bar {\perp}}{\bar {\perp}}}^{(EM)}{\equiv}
-{\frac{t^2_0}{t^2}}{\bar N}_t^{-2}{\bar{p}}^{(EM)}=
-{\frac{t^2_0}{t^2}}{\bar N}_t^{-2}{\bar{\rho}}_{\text m}^{(EM)}c^2, 
\ena
\eqa
T_{(t){\theta}}^{(EM){\theta}}=T_{(t){\phi}}^{(EM){\phi}}=
-T_{(t)r}^{(EM)r}, \qquad
T_{(t){\bar {\perp}}r}^{(EM)}=T_{(t){\bar {\perp}}{\theta}}^{(EM)}=
T_{(t){\bar {\perp}}{\phi}}^{(EM)}=0,
\ena
where ${\bar{\rho}}_{\text m}^{(EM)}$ and ${\bar p}^{(EM)}$ are the so-called 
coordinate density and coordinate pressure of active electromagnetic 
mass-energy. 

We may now insert equations (35), (36) and (37) into the field equation (6)
(equation (7) becomes vacuous). The result is similar to the case of a perfect
fluid [3], the only difference being that the pressure is not isotropic.
Using the equation derived in [3] we find (with 
$r_{\text Q0}{\equiv}{\sqrt{2G}}Q/c^2$)
\eqa
\hspace*{-3mm}&&{\Big (}1-{\frac{r^2}{{\bar B}(r){\Xi}_0^2}}{\Big )}
{\frac{{\bar B}''(r)}{{\bar B}(r)}}-
{\Big (}1+{\frac{3r^2_{\text Q0}{\bar B}(r)}{4r^2}}-
{\frac{2r^2}{{\bar B}(r){\Xi}_0^2}}{\Big )}
{\frac{{\bar B}'^2(r)}{{\bar B}^2(r)}} \nonumber \\
\hspace*{-3mm}&&+{\frac{2}{r}}{\Big (}1+
{\frac{3r^2_{\text Q0}{\bar B}(r)}{4r^2}}-
{\frac{3r^2}{2{\bar B}(r){\Xi}_0^2}}{\Big )}{\frac{{\bar B}'(r)}{{\bar B}(r)}} 
-{\frac{r}{4}}{\Big (}1-{\frac{r^2_{\text Q0}{\bar B}(r)}{2r^2}}{\Big )}
{\frac{{\bar B}'^3(r)}{{\bar B}^3(r)}}={\frac{r^2_{\rm Q0}{\bar B}(r)}{r^4}}.
\ena
We now seek a solution of this equation in the region ${\cal R}<r<{\Xi}_0$. 
However, equation (38) is too complicated to have any hope of finding an 
{\em exact} solution. But it can be simplified if rewritten in terms of an
isotropic radial coordinate ${\rho}$ defined by $r{\equiv}
{\sqrt {\bar B}}{\rho}$, so that the family of line elements takes the form
given in equation (5). Equation (38) then transforms to
\eqa
(1-{\frac{{\rho}^2}{{\Xi}_0^2}}){\frac{{\bar B},_{{\rho}{\rho}}}{\bar B}}+
{\frac{2}{\rho}}(1-{\frac{3{\rho}^2}{2{\Xi}_0^2}})
{\frac{{\bar B},_{\rho}}{\bar B}}={\frac{r_{\rm Q0}^2}{{\rho}^4}}.
\ena
An (unique) exact solution of equation (39), given the correspondence with the
uncharged case [3], yields
\eqa
{\bar B}({\rho})={\cosh}{\Big [}{\frac{r_{\rm Q0}}{\rho}}{\sqrt{1-
{\frac{{\rho}^2}{{\Xi}_0^2}}}}{\Big ]}-{\frac{r_{\rm s0}}{r_{\rm Q0}}}
{\sinh}{\Big [}{\frac{r_{\rm Q0}}{\rho}}
{\sqrt{1-{\frac{{\rho}^2}{{\Xi}_0^2}}}}{\Big ]},
\ena
where the constant $r_{\rm s0}$ is found by integrating equation (6) once and 
then comparing with ${\bar B},_{\rho}$ obtained from equation (40). The result 
is
\eqa
r_{\text s0}{\equiv}{\frac{2M_{t_0}G}{c^2}}{\rm sech}
{\Big [}{\frac{r_{\rm Q0}}{{\rho}_{\rm s}}}
{\sqrt{1-{\frac{{\rho}_{\rm s}^2}{{\Xi}_0^2}}}}{\Big ]}+
r_{\rm Q0}{\tanh}{\Big [}{\frac{r_{\rm Q0}}
{{\rho}_{\rm s}}}{\sqrt{1-{\frac{{\rho}_{\rm s}^2}{{\Xi}_0^2}}}}{\Big ]},
\ena
\eqa
{\rho}_{\rm s}{\equiv}{\frac{\cal R}{\sqrt{\bar B({\cal R})}}}, \qquad
M_{t_0}{\equiv}c^{-2}{\int}{\int}{\int}{\bar N}_{t_0}{\Big [}
T_{(t_0){\bar {\perp}}{\bar {\perp}}}+
{\hat T}^i_{(t_0)i}{\Big ]}d{\bar V}_{t_0}.
\ena
Here the integration is taken over the central source (i.e., $r{\leq}{\cal R}$)
and ${\bf T}_t$ is the total active stress-energy tensor of the source. The 
quantity $r_{\text s0}$ is thus the Schwarzschild radius of the source at 
epoch $t_0$ modified with terms representing the electrostatic field energy.

Notice that, since isolated systems do not exist except as an approximation in 
quasi-metric gravity [3], the solution (40) is expected to be realistic only 
for ${\rho}{\ll}{\Xi}_0$. We may also ask how ${\bar B}({\rho})$ behaves for 
small distances in the case of a point source, i.e., in the limit where 
${\cal R}{\rightarrow}0$. Requiring correspondence with the uncharged case 
[3], we impose the condition ${\bar B}({\rho}){\rightarrow}0$ in the limit 
${\rho}{\rightarrow}C$, where $C$ is found from equation (40). The result is
\eqa
C={\frac{r_{\rm Q0}}{{\sqrt{{\rm artanh}^2
[{\frac{r_{\rm Q0}}{r_{\rm s0}}}]+{\frac{r_{\rm Q0}^2}{{\Xi}_0^2}}}}}}.
\ena
By taking the limit $r_{\rm Q0}{\rightarrow}0$ in equation (43) one finds the
right correspondence with the uncharged case. This means that the qualitative 
behaviour of the gravitational field does not depend on whether or not the 
source has a net charge.

One may find an implicit expression for ${\bar B}(r)$ by substituting 
${\rho}={\frac{r}{\sqrt {\bar B}}}$ into equation (40). From this we see that 
${\bar B}(r)$ cannot be written in closed form. However, the implicit
expression may be used to find a series solution for weak fields, in terms of 
the small quantities ${\frac{r_{\rm s0}}{r}}$, ${\frac{r_{\rm Q0}}{r}}$ and
${\frac{r}{{\Xi}_0}}$. After some straightforward work one may show that an 
approximative expression for ${\bar B}(r)$ is given by
\eqa
{\bar B}(r)&=&1-{\frac{r_{\text s0}}{r}}+{\Big (}1+
{\frac{r_{\text Q0}^2}{r_{\text s0}^2}}{\Big )}{\frac{r_{\text s0}^2}{2r^2}}+
{\frac{r_{\text s0}r}{2{\Xi}_0^2}}-{\Big (}1+
{\frac{22r_{\text Q0}^2}{3r_{\text s0}^2}}{\Big )}
{\frac{r_{\text s0}^3}{8r^3}}+{\cdots}.
\ena
Moreover, using equation (29) we may derive an expression of ${\bar A}(r)$ in 
terms of ${\bar B}(r)$. We find
\eqa
{\bar A}(r)={\bar B}^2(r){\Bigg \{}
{\Big (}{\sqrt{1-{\frac{r^2}{{\bar B}(r){\Xi}_0^2}}}}
+{\frac{{\sqrt{{\bar B}(r)}}r_{\rm s0}}{2r}}{\Big )}{\cosh}{\Big [}
{\frac{r_{\rm Q0}{\sqrt{{\bar B}(r)-{\frac{r^2}{{\Xi}_0^2}}}}}{r}}{\Big ]}
\nonumber \\ -{\Big (}{\frac{r_{\rm s0}}{r_{\rm Q0}}}
{\sqrt{1-{\frac{r^2}{{\bar B}(r){\Xi}_0^2}}}}+
{\frac{{\sqrt{{\bar B}(r)}}r_{\rm Q0}}{2r}}{\Big )}
{\sinh}{\Big [}{\frac{r_{\rm Q0}{\sqrt{{\bar B}(r)-
{\frac{r^2}{{\Xi}_0^2}}}}}{r}}{\Big ]}{\Bigg \}}^{-2},
\ena
and using equation (44) this yields the weak field expansion
\eqa
{\bar A}(r)&=&1-{\frac{r_{\text s0}}{r}}+{\Big (}1+
4{\frac{r_{\text Q0}^2}{r_{\text s0}^2}}{\Big )}
{\frac{r_{\text s0}^2}{4r^2}}+{\frac{r^2}{{\Xi}_0^2}}
%+{\Big (}1+10{\frac{r_{Q0}^2}{r_{s0}^2}}{\Big )}{\frac{r_{s0}^3}{8r^3}}
+{\cdots}.
\ena
To construct the metric family ${\bf g}_t$ represented in the form
\eqa
ds^2_t=-B(r)(dx^0)^2+({\frac{t}{t_0}})^2{\Big (}
A(r)dr^2+r^2d{\Omega}^2{\Big )},
\ena
from the family ${\bf {\bar g}}_t$ given in equation (28), one uses the method
described in [1, 2]. (See also [3] for the case when $r_{\text Q0}$ 
vanishes.) To construct ${\bf g}_t$ we need the quantity
\eqa
v(r)={\frac{cr}{2}}{\frac{{\bar B}'(r)}{{\bar B}(r)}}=
c[{\frac{r_{\text s0}}{2r}}-{\frac{r_{\text Q0}^2}{2r^2}}+O(3)],
\ena
and $A(r)$ and $B(r)$ may then be found from equations (2) and (4) (since
${\bf b}_{\cal F}=r{\frac{\partial}{{\partial}r}}$ for all spherically 
symmetric cases [1, 2])
\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
Inserting equation (48) into equation (49) and equation (49) into equation 
(47) we find
\eqa
ds^2_t&=&-{\Bigg (}1-{\frac{r_{\text s0}}{r}}+{\frac{r_{\text Q0}^2}{2r^2}}+
{\frac{r_{\text s0}r}{2{\Xi}_0^2}}+{\Big (}1+
{\frac{2r_{\text Q0}^2}{9r_{\text s0}^2}}{\Big )}
{\frac{3r_{\text s0}^3}{8r^3}}+{\cdots}{\Bigg )}(dx^0)^2 \nonumber \\
&&+({\frac{t}{t_0}})^2{\Bigg (}{\Big \{}1+{\frac{r_{\text s0}}{r}}
+{\Big (}1-4{\frac{r_{\text Q0}^2}{r_{\text s0}^2}}{\Big )}
{\frac{r_{\text s0}^2}{4r^2}}+{\frac{r^2}{{\Xi}_0^2}}+
{\cdots}{\Big \}}dr^2+r^2d{\Omega}^2{\Bigg )}.
\ena
This expression represents the gravitational field (to the relevant accuracy)
outside a spherically symmetric, metrically static and charged source in 
quasi-metric gravity. Its correspondence with the Reissner-Nordstr\"{o}m
solution in GR may be found by setting ${\frac{t}{t_0}}$ equal to unity in
equation (50) and then taking the limit ${\Xi}_0{\rightarrow}{\infty}$.
Notice that an extension of the Reissner-Nordstr\"{o}m solution, adapted to a 
Friedmann-Robertson-Walker background, was found in [4]. But of course 
equation (50) does not have any further correspondence with this extended 
solution, since the cosmic expansion as described in the QMF is fundamentally 
different from its counterpart in the MF. That is, the way the cosmic expansion
is fit into the extended Reissner-Nordstr\"{o}m solution has no counterpart in
equation (50).

It is also possible construct an exact representation for ${\bf g}_t$ by first
calculating the quantity $v({\rho})={\frac{{\bar B},_{\rho}}{\bar B}}
b_{\cal F}^{\rho}c$, the result being
\eqa
v({\rho})={\frac{{\Big (}{\frac{r_{\rm s0}}{2{\rho}}}{\cosh}{\Big [}
{\frac{r_{\rm Q0}{\sqrt{1-{\frac{{\rho}^2}{{\Xi}_0^2}}}}}{\rho}}{\Big ]}
-{\frac{r_{\rm Q0}}{2{\rho}}}{\sinh}{\Big [}{\frac{r_{\rm Q0}
{\sqrt{1-{\frac{{\rho}^2}{{\Xi}_0^2}}}}}{\rho}}{\Big ]}{\Big )}c}
{{\Big (}{\sqrt{1-{\frac{{\rho}^2}{{\Xi}_0^2}}}}
+{\frac{r_{\rm s0}}{2{\rho}}}{\Big )}{\cosh}{\Big [}
{\frac{r_{\rm Q0}{\sqrt{1-{\frac{{\rho}^2}{{\Xi}_0^2}}}}}{{\rho}}}{\Big ]}
-{\Big (}{\frac{r_{\rm s0}}{r_{\rm Q0}}}
{\sqrt{1-{\frac{{\rho}^2}{{\Xi}_0^2}}}}+
{\frac{r_{\rm Q0}}{2{\rho}}}{\Big )}
{\sinh}{\Big [}{\frac{r_{\rm Q0}{\sqrt{1-
{\frac{{\rho}^2}{{\Xi}_0^2}}}}}{\rho}}{\Big ]}}},
\ena
and then using equations (2), (4) and (5) to obtain
\eqa
ds^2_t={\bar B}({\rho}){\Big \{}-{\Big (}1-{\frac{v^2({\rho})}{c^2}}{\Big )}^2
(dx^0)^2+({\frac{t}{t_0}})^2{\Big [}
{\Big (}{\frac{1+{\frac{v({\rho})}{c}}}{1-{\frac{v({\rho})}{c}}}}{\Big )}^2
{\frac{d{\rho}^2}{1-{\frac{{\rho}^2}{{\Xi}_0^2}}}}+{\rho}^2d{\Omega}^2{\Big ]}
{\Big \}}.
\ena
Substituting ${\bar B}({\rho})$ and $v({\rho})$ into equation (52) now yields
an explicit, exact expression for the metric family ${\bf g}_t$. Moreover, a 
change of radial coordinates to $r{\equiv}{\sqrt{\bar B}}{\rho}$ yields another
exact, but implicit expression. Note that the qualitative behaviour of the 
exact solution (52) is quite similar to that found for the uncharged case 
presented in [3].

To calculate the paths of charged test particles one uses the quasi-metric
equations of motion (10), (11). But to apply these equations one needs to know 
the Lorentz force from equation (15). This means that we must first calculate 
${\bf {\bar F}}_t$ from equation (16) and then find ${\bf F}_t$ from $v$ and
${\bf {\bar F}}_t$. That is, using equation (16) we first find the passive 
electric field ${\bf {\bar E}}_t$:
\eqa
{\bar E}_{(t)r}{\equiv}{\bar F}_{(t){\bar {\perp}}r}{\equiv}-
{\bar n}_{(t)}^{\mu}{\bar F}_{(t){\mu}r}=
{\frac{t_0}{t}}{\bar N}_t^{-1}{\tilde F}_{(t){\bar {\perp}}r}
={\frac{t_0}{t}}{\frac{{\sqrt{{\bar A}(r)}}Q}{r^2}}.
\ena
Note that since ${\bar A}(r)$ and ${\bar B}(r)$ are not inverse functions,
${\bar F}_{(t_0)r0}={\sqrt{{\bar B}(r)}}{\bar E}_{(t_0)r}$ does not take the 
Euclidean form. Second, we are able to find $F_{(t)0r}=
(1+{\frac{v(r)}{c}})^2{\bar F}_{(t)0r}$ using equation (3), and we then get 
(using equation (49))
\eqa
E_{(t)r}{\equiv}F_{(t){\perp}r}{\equiv}-n_{(t)}^{\mu}F_{(t){\mu}r}
={\Big (}{\frac{1+{\frac{v(r)}{c}}}{1-{\frac{v(r)}{c}}}}{\Big )}
{\bar F}_{(t){\bar {\perp}}r}={\frac{t_0}{t}}{\frac{{\sqrt{A(r)}}Q}{r^2}}.
\ena
We may now calculate the passive mass-energy density ${\rho}_{\text m}^{(EM)}$ 
and pressure $p^{(EM)}$ of the electromagnetic field as measured in the
metrically static frame. This yields
\eqa
{\bar {\cal T}}^{(EM)}_{(t){\bar {\perp}}{\bar {\perp}}}=
{\rho}_{\text m}^{(EM)}c^2=
{\frac{1}{8{\pi}}}{\bar E}_{(t)r}{\hat {\bar E}}_{(t)}^r=
{\frac{t_0^4}{t^4}}{\frac{Q^2}{8{\pi}r^4}},
\ena
\eqa
{\bar {\cal T}}_{(t){\theta}}^{(EM){\theta}}=
{\bar {\cal T}}_{(t){\phi}}^{(EM){\phi}}=-{\bar {\cal T}}_{(t)r}^{(EM)r}=
p^{(EM)}={\bar {\cal T}}^{(EM)}_{(t){\bar {\perp}}{\bar {\perp}}}.
\ena
On the other hand we can define the passive stress-energy tensor 
${\cal T}^{(EM)}_t$ of the electromagnetic field in $({\cal N},{\bf g}_t)$
from ${\bf F}_t$ by
\eqa
{\cal T}_{(t){\alpha}{\beta}}^{(EM)}={\frac{1}{4{\pi}}}{\Big (}
F_{(t){\alpha}}^{{\ }{\ }{\ }{\,}{\,}{\nu}}
F_{(t){\beta}{\nu}}-{\frac{1}{4}}F_{(t){\sigma}{\rho}}
F_{(t)}^{{\sigma}{\rho}}g_{(t){\alpha}{\beta}}{\Big )}
{\sqrt{{\frac{{\bar h}_t}{h_t}}}},
\ena
where the square root factor is necessary to ensure that the total amount of 
passive electromagnetic mass-energy is identical in 
$({\cal N},{\bf {\bar g}}_t)$ and $({\cal N},{\bf g}_t)$. Using equation (57)
we may then calculate
\eqa
{\cal T}^{(EM)}_{(t){\perp}{\perp}}=
{\frac{1}{8{\pi}}}E_{(t)r}{\hat E}_{(t)}^r{\sqrt{\frac{{\bar h}_t}{h_t}}}
={\rho}_{\text m}^{(EM)}c^2{\sqrt{\frac{{\bar h}_t}{h_t}}}=
{\frac{t_0^4}{t^4}}{\frac{Q^2}{8{\pi}r^4}}
{\Big (}{\frac{1-{\frac{v(r)}{c}}}{1+{\frac{v(r)}{c}}}}{\Big )},
\ena
and similar formulae for the other components of ${\cal T}_t^{(EM)}$.
\subsection{The effects of cosmic expansion on electromagnetism}
To see to what extent a classical, electromagnetically bound system is 
affected by the global cosmic expansion we may calculate the path of a charged
test particle with mass $m$ and charge $q$ in the electric field of an 
isolated spherical charge Q. That is, we may use the metric family (47) where 
it is assumed that the gravitational field of the source is negligible, i.e., 
$B(r)={\bar B}(r)=1$. We also set $A(r)={\bar A}(r)=1$ since isolated systems 
are realistic in the QMF only if $r{\ll}{\Xi}_0$ [3]. Using equation (54) and 
the coordinate expression for ${\bf u}_t$, i.e., $u_{(t)}^{\alpha}{\equiv}
{\frac{dx^{\alpha}}{d{\tau}_t}}$ where ${\tau}_t$ is the proper time as 
measured along the path of the test particle, equation (15) yields the 
components of the Lorentz force (neglecting radiative effects and taking care
of the fact that $qQ<0$ since we have a bound system)
\eqa
ma_{(t)}^r=-{\frac{t_0^3}{t^3}}
{\frac{{\mid}qQ{\mid}}{r^2}}{\frac{dx^0}{d{\tau}_t}}, \qquad
ma_{(t)}^0=-{\frac{t_0}{t}}{\frac{{\mid}qQ{\mid}}{r^2}}{\frac{dr}{d{\tau}_t}}.
\ena
Furthermore, we may insert this result into the equations of motion (10). 
Confining the motion of the test particle to the equatorial plane 
${\theta}={\pi}/2$ the result is 
\eqa
{\frac{d^2r}{d{\tau}_t^2}}-r{\Big (}{\frac{d{\phi}}{d{\tau}_t}}{\Big )}^2
+{\frac{1}{ct}}{\frac{dr}{d{\tau}_t}}{\frac{dx^0}{d{\tau}_t}}=
-{\frac{t_0^3}{t^3}}{\frac{{\mid}qQ{\mid}}{mr^2}}{\frac{dx^0}{d{\tau}_t}},
\ena
\eqa
{\frac{d^2x^0}{d{\tau}_t^2}}=-
{\frac{t_0}{t}}{\frac{{\mid}qQ{\mid}}{mr^2}}{\frac{dr}{d{\tau}_t}},
\ena
\eqa
{\frac{d^2{\phi}}{d{\tau}_t^2}}+{\frac{2}{r}}{\frac{d{\phi}}{d{\tau}_t}}
{\frac{dr}{d{\tau}_t}}+
{\frac{1}{ct}}{\frac{d{\phi}}{d{\tau}_t}}{\frac{dx^0}{d{\tau}_t}}=0.
\ena
Equation (62) yields a constant of motion $J$, namely [3]
\eqa
J{\equiv}{\frac{t}{t_0}}r^2{\frac{d{\phi}}{cd{\tau}_t}}.
\ena
Introducing the dimensionless variable ${\xi}{\equiv}{\frac{t}{t_0}}$ and 
neglecting terms of order ${\frac{{\xi}^2}{{\Xi}_0^2}}$, equations (59)-(63)
may be combined and written in the form (since $dx^0=cdt$ along the path of 
the charged particle)
\eqa
(1+{\frac{J^2}{r^2}}){\Big [}{\frac{d^2r}{d{\xi}^2}}+{\frac{1}{\xi}}
{\frac{dr}{d{\xi}}}{\Big ]}+{\frac{{\mid}qQ{\mid}}{{\xi}mcr^2}}
{\sqrt{1+{\frac{J^2}{r^2}}}}{\Big [}{\frac{{\Xi}_0^2}{{\xi}^2}}-
({\frac{dr}{d{\xi}}})^2{\Big ]}-{\frac{{\Xi}_0^2J^2}{{\xi}^2r^3}}=0,
\ena
\eqa
{\frac{dt}{d{\tau}_t}}={\sqrt{1+{\frac{J^2}{r^2}}}},
\ena
where we have used equation (47) (with $B(r)=A(r)=1$). Equation (64) is too
complicated to have any hope of finding an exact solution. However, in the
non-relativistic limit one may neglect the term quadratic in the radial 
velocity. Then, taking as an initial condition that ${\frac{dr}{d{\xi}}}$
should vanish, numeric calculations show that the solution lies close to 
(oscillates around) a straight line $r({\xi}){\propto}{\xi}$. This seems 
reasonable since one expects that the Lorentz force term and the centrifugal 
term in equation (64) should on average scale similarly with respect to the 
factor ${\frac{t}{t_0}}$. 

Since the solution of equation (64) yields that $r(t){\propto}{\frac{t}{t_0}}$ 
on average, we have that according to the QMF, a classical system bound 
solely by electromagnetic forces will experience an even larger secular
expansion than a gravitationally bound system of the same size. However, an 
important difference is that the norm of the (passive) electromagnetic field
is constant at a given distance from the source (as seen from equation (54)), 
whereas the gravitational field at a given distance from the source gets 
stronger due to the secular increase of active mass-energy [3]. (This
last point can also easily be seen from equation (50).)

This means that the effect of the cosmic expansion on the electromagnetic 
field is fundamentally different from its effect on the gravitational field. 
In particular it means that, except for a global cosmic attenuation (and the
associated implicit effects) not noticeable locally, the electromagnetic field 
is unaffected by the global cosmic expansion, unlike the gravitational field. 
And even if it is predicted that a classical, electromagnetically bound system 
should experience cosmic expansion, this expansion implies that the potential 
energy of the system increases and that its kinetic energy decreases. But 
suitable interactions with other systems should then allow the system to 
return to its initial state. On the other hand it is impossible to null out 
the effects of the cosmic expansion on a gravitational system via interactions 
with other systems [3].

For ``small'' classical systems radiative effects will dominate over the 
expansion effect by many orders of magnitude so the expansion would not be 
noticeable. Furthermore, the classical calculation is hardly relevant for
quantum-mechanical systems since quantum-mechanical states should not be 
affected by the cosmic expansion. This is readily seen for the Bohr atom, 
where the radii of the ``permitted'' electron orbits are determined from the 
requirement that angular momenta take the discrete values $L_{\rm n}=n{\hbar}$;
i.e., a set of static values. But this is inconsistent with the above solution 
of the classical system yielding $L{\propto}{\frac{t}{t_0}}$, which is
non-static. This means that there is no reason to expect that 
quantum-mechanical systems such as atoms should behave as purely classic 
systems and increase in size due to the global cosmic expansion. (Rather 
the expansion could perhaps induce spontaneous excitations. To explore this 
possibility a quantum-mechanical calculation should be carried out.) For a 
comparison with GR, see references [5] and [6] for the estimated effects of 
the cosmic expansion on a classical ``atom'' according to GR. Note that the 
results found in these papers differ considerably from the results found here.
\section{Geometric optics in quasi-metric space-time}
In this section we sketch how the fundamental laws of geometric optics are
derived within the quasi-metric framework. Except from small changes these
derivations may be done exactly as for the metric framework, see, e.g., 
reference [7]. 

The main difference from the metric framework is that we are forced to include
the effects of the cosmic expansion on the wavelength, the amplitude and the
polarization of electromagnetic waves propagating through a source-free
region of space-time. But the electromagnetic waves may still be taken to
be locally monochromatic and plane-fronted. 

Similarly to the metric case we start with a vector potential family
${\bf {\bar A}}_t$ of the form
\eqa
{\bar A}_{(t)}^{\mu}=
{\Re}[{\mid}{\bar {\cal A}}_t{\mid}{\exp}(i{\bar {\vartheta}}_t)
{\bar f}_{(t)}^{\mu}], \qquad {\bar f}_{(t)}^{\mu}{\equiv}
{\mid}{\bar {\cal A}}_t{\mid}^{-1}{\bar {\cal A}}_{(t)}^{\mu},
\ena 
where ${\mid}{\bf {\bar {\cal A}}}_t{\mid}$ denotes the norm of the (possibly
complex) vector amplitude ${\bar {\cal A}}_{(t)}^{\mu}
{\frac{\partial}{{\partial}{x^{\mu}}}}$ of ${\bf {\bar A}}_t$ and where 
${\bf {\bar f}}_t$ is the polarization vector family. Moreover the phase
factor ${\bar {\vartheta}}_t$ is defined by (using a GTCS)
\eqa
{\bar {\vartheta}}_t{\equiv}{\bar k}_{(t)0}(x^0-x^0_1)+{\bar k}_{(t)i}x^i, 
\qquad {\bar k}_{(t){\mu}}={\bar {\vartheta}}_{t},_{\mu},
\ena
where ${\bf {\bar k}}_t$ is the wave vector family and where $x^0_1=ct_1$ is
an arbitrary reference epoch. Note that equations (66) and (67) are identical 
to those valid for the metric case except for the dependence on $t$.
The geometric optics approximation holds if ${\mid}{\bar {\cal A}}_t{\mid},_t
=-({\frac{1}{t}}+{\frac{{\bar N},_t}{{\bar N}_t}}){\mid}{\bar {\cal A}}_t
{\mid}$, i.e., if the only implicit $t$-dependence of ${\bf {\bar A}}_t$ is 
via the phase factor ${\bar {\vartheta}}_t$. The $t$-dependence of 
${\bf {\bar k}}_t$ is determined by how it is affected by the global cosmic 
expansion. That is, in the QMF there is a general cosmological attenuation of 
the electromagnetic field, and for electromagnetic radiation this attenuation 
takes the form of a general redshift due to the global cosmic expansion. This 
means that we must have (using a GTCS)
\eqa
{\bar k}_{(t)0},_t=-{\frac{1}{t}}{\bar k}_{(t)0}, \quad 
{\bar k}_{(t)j},_t=0, \quad
{\bar k}_{(t)}^0,_t=-({\frac{1}{t}}+
2{\frac{{\bar N}_{t,t}}{{\bar N}_t}}){\bar k}_{(t)}^0, \quad 
{\bar k}_{(t)}^j,_t=-2({\frac{1}{t}}+{\frac{{\bar N}_{t,t}}{{\bar N}_t}})
{\bar k}_{(t)}^j.
\ena
The given $t$-dependency of ${\mid}{\bar {\cal A}}_t{\mid}$, together with the 
$t$-dependencies given in equation (68), then imply that the covariant 
derivatives of ${\bf {\bar k}}_t$ and of ${\bf {\bar f}}_t$ in the 
$t$-direction are given by (see [1, 2] for the connection coefficients)
\eqa
{\stackrel{\star}{\bf {\bar{\nabla}}}}_{\frac{\partial}{{\partial}t}}
{\bf {\bar k}}_t=-{\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}
{\Big )}{\bf {\bar k}}_t, \qquad
{\stackrel{\star}{\bf {\bar{\nabla}}}}_{\frac{\partial}{{\partial}t}}
{\bf {\bar f}}_t=0.
\ena
We now put equation (66) into the Lorentz gauge condition 
${\bar A}_{(t);{\mu}}^{\mu}=0$ and then into Maxwell's equations (21) without 
sources. These calculations are done explicitly for the metric case in 
reference [7]. Since the derivations for the quasi-metric case are very 
similar we will not repeat them here. Rather we list the results, also very 
similar to those found for the metric case. Firstly, we find that
\eqa
{\bar k}_{(t){\mu}}{\bar f}_{(t)}^{\mu}=0, \qquad 
{\bar k}_{(t){\mu}}{\bar k}_{(t)}^{\mu}=0, \qquad
{\bar k}_{(t){\mu};{\nu}}{\bar k}_{(t)}^{\nu}=0.
\ena
Selecting a suitable affine parameter ${\lambda}$ along a light ray we have
\eqa
{\bf {\stackrel{\star}{\bar {\nabla}}}}_{\frac{\partial}{{\partial}{\lambda}}}
{\bf {\bar k}}_t{\equiv}{\frac{dt}{d{\lambda}}}
{\bf {\stackrel{\star}{\bar {\nabla}}}}_{\frac{\partial}{{\partial}t}}
{\bf {\bar k}}_t+{\frac{dx^{\alpha}}{d{\lambda}}}
{\bf {\stackrel{\star}{\bar {\nabla}}}}_
{\frac{\partial}{{\partial}x^{\alpha}}}{\bf {\bar k}}_t.
\ena
Provided that we make the identification
\eqa
{\frac{dx^{\mu}}{d{\lambda}}}={\frac{t}{t_0}}{\bar k}_{(t)}^{\mu},
\ena
between the tangent vector along the light ray and ${\bf {\bar k}}_t$, 
equations (69), (70) and (71) yield (in component notation, using a GTCS)
\eqa
{\frac{d{\bar k}_{(t)}^{\mu}}{d{\lambda}}}+{\Big (}
{\topstar{\bar {\Gamma}}}^{\mu}_{t{\epsilon}}{\frac{dt}{d{\lambda}}}+
{\topstar{\bar {\Gamma}}}^{\mu}_{{\nu}{\epsilon}}{\frac{dx^{\nu}}{d{\lambda}}}
{\Big )}{\bar k}_{(t)}^{\epsilon}=
-{\Big (}{\frac{1}{t}}+{\frac{{\bar N}_t,_t}{{\bar N}_t}}{\Big )}
{\frac{dt}{d{\lambda}}}{\bar k}_{(t)}^{\mu}.
\ena
Given equation (72), and if one makes a suitable change of parameter along the 
path for the case when ${\bar N}_t$ depends on $t$, equation (73) is identical
to the geodesic equation in $({\cal N},{\bf {\bar g}}_t)$. That is, we have 
derived from Maxwell's equations that light rays are null geodesics in 
$({\cal N},{\bf {\bar g}}_t)$. But since any ``physical'' null vector is
required to remain a null vector under the transformation 
${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$, we may write 
${\bf {\bar k}}_t{\rightarrow}{\bf k}_t$, where ${\bf {\bar k}}_t$ transforms
as shown in equation (1). Similarly, since the norm of ${\bf {\bar f}}_t$ is 
required to be constant under the transformation 
${\bf {\bar g}}_t{\rightarrow}{\bf g}_t$, we may write ${\bf {\bar f}}_t
{\rightarrow}{\bf f}_t$, where ${\bf {\bar f}}_t$ transforms in the same way
as ${\bf {\bar k}}_t$. Furthermore we may define a new phase factor
${\vartheta}_t$ by
\eqa
{\vartheta}_t{\equiv}k_{(t)0}(x^0-x^0_1)+k_{(t)i}x^i, 
\qquad k_{(t){\mu}}={\vartheta}_{t},_{\mu},
\ena
where the $t$-dependence of ${\bf k}_t$ is given by (in a GTCS)
\eqa
k_{(t)0},_t=-{\frac{1}{t}}k_{(t)0}, \qquad k_{(t)j},_t=0, \qquad
k_{(t)}^0,_t=-{\frac{1}{t}}k_{(t)}^0, \qquad 
k_{(t)}^j,_t=-{\frac{2}{t}}k_{(t)}^j.
\ena
From equations (74), (75) and the definitions of ${\bf k}_t$ and ${\bf f}_t$
we get the counterparts to equation (70) in $({\cal N},{\bf g}_t)$, namely
\eqa
k_{(t){\mu}}f_{(t)}^{\mu}=0, \qquad k_{(t){\mu}}k_{(t)}^{\mu}=0, \qquad
k_{(t){\mu};{\nu}}k_{(t)}^{\nu}=0.
\ena
Using the counterparts in $({\cal N},{\bf g}_t)$ to equations (69), (71) and
(72) together with equations (75) and (76) it is now straightforward to derive
the counterpart in $({\cal N},{\bf g}_t)$ to equation (73). Thus we have shown
that light rays must be null geodesics in $({\cal N},{\bf g}_t)$ as well. 

Secondly, we find that just as for the metric case we have
\eqa
{\bar {\cal A}}_{(t);{\nu}}^{\mu}{\bar k}_{(t)}^{\nu}=-{\frac{1}{2}}
{\bar k}_{(t);{\alpha}}^{\alpha}{\bar {\cal A}}_{(t)}^{\mu}, \qquad 
\Rightarrow \qquad
{\bf {\stackrel{\star}{\bar {\nabla}}}}_{{\bf {\bar k}}_t}{\bf {\bar f}}_t
{\equiv}c^{-1}{\bar k}_{(t)}^0
{\bf {\stackrel{\star}{\bar {\nabla}}}}_{\frac{\partial}{{\partial}t}}
{\bf {\bar f}}_t+{\bar k}_{(t)}^{\nu}
{\bf {\stackrel{\star}{\bar {\nabla}}}}_{\frac{\partial}{{\partial}x^{\nu}}}
{\bf {\bar f}}_t=0,
\ena
where we have used equation (69) in the last step. That is, the polarization 
vector family is perpendicular to the light rays and parallel-transported 
along them. From equation (76) we see that the polarization vector family is
perpendicular to the light rays in $({\cal N},{\bf g}_t)$ as well. Besides, 
since light rays are null geodesics in $({\cal N},{\bf g}_t)$, ${\bf f}_t$ is 
parallel-transported along them and 
${\bf {\stackrel{\star}{\nabla}}}_{{\bf k}_t}{\bf f}_t$ thus vanishes.

Thirdly, also just as for the metric case we have that
\eqa
({\mid}{\bar {\cal A}}_t{\mid}^2{\bar k}_{(t)}^{\alpha})_{;{\alpha}}=0, \qquad
{\bf {\stackrel{\star}{\bar {\nabla}}}}{\bf {\cdot}}({\frac{t^3}{t_0^3}}
{\mid}{\bar {\cal A}}_t{\mid}^2{\bf {\bar k}}_t){\equiv}c^{-1}
{\bf {\stackrel{\star}{\bar {\nabla}}}}_{\frac{\partial}{{\partial}t}}
({\frac{t^3}{t_0^3}}{\mid}{\bar {\cal A}}_t{\mid}^2{\bar k}_{(t)}^0)+
{\bf {\stackrel{\star}{\bar {\nabla}}}}_{\frac{\partial}{{\partial}x^{\alpha}}}
({\frac{t^3}{t_0^3}}{\mid}{\bar {\cal A}}_t{\mid}^2{\bar k}_{(t)}^{\alpha})=0,
\ena
or equivalently, that the volume integral ${\int}{\int}{\int}{\mid}
{\bar {\cal A}}_t{\mid}^2{\bar k}_{(t){\bar {\perp}}}{\sqrt{{\bar h}_t}}d^3x$ 
has a constant value when integrating over the 3-volume cut out of the FHSs by
a tube formed of light rays (such that no rays cross the tube walls). This is
the law of conservation of photon number in geometric optics. And since
${\bar k}_{(t){\bar {\perp}}}=k_{(t){\perp}}$ the above integral may be 
written as ${\int}{\int}{\int}{\mid}{\cal A}_t{\mid}^2k_{(t){\perp}}
{\sqrt{h}_t}d^3x$ where ${\mid}{\cal A}_t{\mid}^2{\sqrt{h_t}}{\equiv}
{\mid}{\bar {\cal A}}_t{\mid}^2{\sqrt{{\bar h}_t}}$. So we see that the law of
conservation of photon number holds for $({\cal N},{\bf g}_t)$ also but with a
different ``density of light rays'' ${\mid}{\cal A}_t{\mid}^2k_{(t){\perp}}$ 
on each FHS. However, the total number of light rays in the tube is the same.

The passive electromagnetic field tensor ${\bf {\bar F}}_t$ has the same form 
as for the metric case, i.e.
\eqa
{\bar F}_{(t){\mu}{\nu}}={\Re}[i{\mid}{\bar {\cal A}}_t{\mid}
{\exp}(i{\bar {\vartheta}}_t)({\bar k}_{(t){\mu}}{\bar f}_{(t){\nu}}-
{\bar f}_{(t){\mu}}{\bar k}_{(t){\nu}})].
\ena
We may find ${\bf F}_t$ from ${\bf {\bar F}}_t$ just as ${\bf g}_t$ is
found from ${\bf {\bar g}}_t$ (see equations (2)-(4)). Moreover, we may also
include the active aspects of light rays as long as we are not going beyond 
the geometric optics approximation. That is, we can construct 
${\bf {\tilde F}}_t$ from ${\bf {\bar F}}_t$ using equation (16). The
passive electromagnetic field stress-energy tensor 
${\bf {\bar {\cal T}}}_t^{(EM)}$ then yields the active electromagnetic field
stress-energy tensor ${\bf T}_t^{(EM)}$ via equation (19). Thus, putting 
equation (79) into the standard definition of ${\bf {\bar {\cal T}}}_t^{(EM)}$,
we get the expressions (averaged over one wavelength)
\eqa
{\bar {\cal T}}_{(t){\mu}{\nu}}^{(EM)}={\frac{1}{8{\pi}}}{\mid}
{\bar {\cal A}}_t{\mid}^2{\bar k}_{(t){\mu}}{\bar k}_{(t){\nu}}, \qquad 
\Rightarrow \qquad  T_{(t){\mu}{\nu}}^{(EM)}=
{\frac{t^2}{t_0^2}}{\bar N}_t^2{\frac{1}{8{\pi}}}
{\mid}{\bar {\cal A}}_{t}{\mid}^2{\bar k}_{(t){\mu}}{\bar k}_{(t){\nu}}.
\ena
This expression for ${\bf T}_t^{(EM)}$ can then be used to find the necessary
projections to be inserted into the gravitational field equations (6) and (7). 
So as long as the geometric optics approximation holds, it is thus possible to 
set up a well-defined initial value problem for light rays coupled to gravity 
within the QMF.
\section{Conclusion}
In metric theory the nature of the cosmic expansion is kinematical. That is,
to which degree a given system is influenced by the cosmic expansion is
determined by dynamical laws subject to cosmological initial conditions. This
means that in metric theory, bound systems may in principle be influenced by
the cosmic expansion regardless of the nature of the force holding the system 
together. However, calculations show that the effect of the cosmic expansion 
on realistic local systems should be totally negligible; see e.g. [5] and 
references listed therein. On the other hand, in the QMF the nature of the 
cosmic expansion is non-kinematical, meaning that the expansion is described 
as a secular global change of scale between gravitational and non-gravitational
systems. That is, in the QMF the effects of the global cosmic expansion on
gravitationally bound objects should be fundamentally different from its
effects on objects solely bound by non-gravitational forces, e.g. 
electromagnetism.

In this paper we have shown how to formulate classical electrodynamics coupled 
to gravity in a way consistent with the QMF. This is possible provided that 
the explicit effect of the cosmic expansion on the electromagnetic field takes 
the form of a global cosmic attenuation not noticeable locally. Moreover, as 
an illustrative example, we have calculated the exact solutions for the 
electric and gravitational fields in the electro-vacuum outside a spherically 
symmetric, metrically static charged source. This example shows that the 
effect of the cosmic expansion on the electric field is fundamentally 
different from its effect on the gravitational field. Also calculated is the 
path of a charged test particle moving in the electric field of (and 
electromagnetically bound to) an isolated spherical charge with negligible 
gravity. This calculation shows that the cosmic expansion affects the 
electromagnetically bound system even more than a similar, but gravitationally 
bound system. However, since the cosmic expansion should not affect quantum 
states, there is no reason to think that the calculated expansion of classical 
electromagnetically bound systems should apply to quantum-mechanical systems.

Furthermore we have shown that just as for metric theory, within the QMF it 
is possible to derive the fact that light rays move along geodesics from 
Maxwell's equations in curved space-time. Also valid within the QMF are the 
other two main results of geometric optics in curved space-time. It thus 
seems that the QMF represents a self-consistent framework within to do 
electromagnetism as well as relativistic gravity.
\\ [4mm]
{\bf Acknowledgment} \\ [1mm]
I wish to thank Dr. K{\aa}re Olaussen for making a review of the manuscript.
\\ [4mm]
{\bf References} \\ [1mm]
{\bf [1]} D. {\O}stvang, {\em Gravit. Cosmol.} {\bf 11}, 205 (2005)
(gr-qc/0112025). \\
{\bf [2]} D. {\O}stvang, {\em Doctoral Thesis} (2001) (gr-qc/0111110). \\
{\bf [3]} D. {\O}stvang, gr-qc/0201097 (2002). \\
{\bf [4]} C.J. Gao, S.N. Zhang, {\em Phys. Lett.} {\bf B 595}, 28 (2004). \\
{\bf [5]} W.B. Bonnor, {\em Class. Quantum Grav.} {\bf 16}, 1313 (1999). \\
{\bf [6]} R.H. Price, gr-qc/0508052 (2005). \\
{\bf [7]} C.W. Misner, K.S. Thorne, J.A. Wheeler,
{\em Gravitation}, W.H. Freeman ${\&}$ Co. (1973). 
\end{document}
