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%TCIDATA{Created=Tue Oct 05 13:28:08 1999}
%TCIDATA{LastRevised=Fri Feb 02 14:18:25 2001}

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\begin{center}
\textbf{RELATIVELY MOVING SYSTEMS IN ``TRUE TRANSFORMATIONS RELATIVITY''}%
\bigskip

Tomislav Ivezi\'{c}\bigskip

Ru\textit{{%
\mbox
 {\it{d}\hspace{-.15em}\rule[1.25ex]{.2em}{.04ex}\hspace{-.05em}}}er Bo\v
{s}kovi\'{c} Institute, P.O.B. 180, 10002 Zagreb, Croatia}

ivezic@rudjer.irb.hr\bigskip
\end{center}

%\begin{abstract}
%\noindent 
In this paper the physical systems consisting of relatively moving
subsystems are considered in the ''true transformations relativity''. It is
found in a manifestly covariant way that there is a second-order electric
field outside stationary current-carrying conductor. It is also found that
there are opposite charges on opposite sides of a square loop with current
and these charges are invariant charges.\bigskip
%\begin{abstract}

\noindent Key words: covariant length, current, electric field and
charge.\bigskip

\noindent \textit{Henceforth space by itself, and time by itself, are doomed 
\newline
to fade away into mere shadows and only a kind of union of \newline
the two will preserve an independent reality.} H. Minkowski\newline
\newline
\newline
\textbf{1. INTRODUCTION}\newline
\newline

In the recent paper \cite{ivezic} I have shown that due to the fundamental
difference between the true transformations (TT) and the apparent
transformations (AT) (see \cite{ivezic} and \cite{rohrl1}) one can speak
about two forms of relativity: the ''TT relativity'' and the ''AT
relativity.'' The ''TT relativity,'' which is a covariant formulation of
relativity, is based on the TT of physical quantities as 4-dimensional (4D)
spacetime tensors, i.e., on the covariant definition of the spacetime
length, and the covariant electrodynamics with 4-vectors $E^{\alpha }$ and $%
B^{\alpha },$ see \cite{ivezic} and \cite{ivsc98}. This formulation of
electrodynamics is equivalent to the usual covariant electrodynamics with
the electromagnetic field tensor $F^{\alpha \beta }$, as shown in \cite
{ivezic}. \textit{The TT are the transformations of 4D spacetime tensors
referring to the same quantity (in 4D spacetime) considered in different
inertial frames of reference (IFRs), or in different coordinatizations of
some IFR.} The TT do conform with the special relativity as the theory of 4D
spacetime with pseudo-Euclidean geometry, i.e., they leave the interval $ds$
and thus the geometry of spacetime unchanged. An example of the TT are the
Lorentz transformations (LT) of 4D tensor quantities. The ''AT relativity''
is the conventional special relativity based on Einstein's relativity of
simultaneity and on the synchronous definition of the spatial length, i.e.,
on the AT of the spatial length (the Lorentz contraction, see \cite
{ivezic,rohrl1,ivsc98,gamba}) and the time distance (the conventional
dilatation of time), and, as shown in \cite{ivezic} (see also \cite{ivsc98}%
), on the AT of the electric and magnetic three-vectors (3-vectors) $\mathbf{%
E}$ and $\mathbf{B}$\/\ (the conventional transformations of $\mathbf{E}$
and $\mathbf{B}$). \textit{The AT are not the transformations of 4D
spacetime tensors and they do not refer to the same quantity (in 4D
spacetime), but, e.g., they refer to the same measurement in different IFRs.}

In this paper we investigate physical systems consisting of relatively
moving subsystems, as it is a current-carrying conductor (CCC), using a
covariant formulation of physical quantities and physical phenomena, i.e.,
the ''TT relativity''. First we examine the covariant definition of length
when defined in geometrical terms and in different coordinatizations of an
IFR. We also report an expression for the Lorentz transformations, which is
independent of the chosen synchronization, i.e., coordinatization of an IFR.
Further, the AT of the spatial length - the Lorentz contraction - is
examined in detail. Then the covariant definition of length in Einstein's
coordinatization is applied to the consideration of the well-known
''relativistic'' paradox ''Car and garage paradox.'' It is found that in the
''TT relativity'' and, \textit{if one wants to retain the connection with
the prerelativistic physics in which one deals with the ''spatial length''},
then only the rest length (volume) of the object is well defined quantity.

>From this result and the covariant definition of charge we also find that in
the ''TT relativity'' the charge density as the three-dimensional (3D)
quantity has definite physical meaning only for charges at rest. In order to
avoid from the beginning the misunderstanding of the ''TT relativity'' and
of our choice of the rest frame of the object, as the starting frame for the
definitions of 4D quantities, we emphasize that the ''TT relativity'' is
covariant in the usual sense. \textit{In the ''TT relativity'' one can
define 4D physical quantities and investigate physical laws connecting such
4D quantities in any IFR, not only in the rest frame of the object. The LT
will correctly connect the results of measurements of}\emph{\ }\textit{the
same 4D quantity in two, arbitrary, relatively moving IFRs.} Thus, the ''TT
relativity'' does not use a preferred reference frame. Our choice of the
rest frame of the object does not mean in any way that this frame is a
preferred IFR. The rest frame is, in fact, the most convenient for the
purpose of comparison with the prerelativistic physics, in which one does
not deal with 4D quantities but with ''3+1'' quantities (the quantities
defined in ''3+1'' space and time), and with the ''AT relativity'', in which
one works in 4D spacetime but with quantities, e.g., the spatial length, the
time distance, the 3-vectors \textbf{E }and\textbf{\ B, }etc\textbf{.,} that
are not 4D tensor quantities. Taking this into account we show that the
current density 4-vector $j^{\mu }$ for a CCC in an arbitrary IFR is
determined as the sum $j_{+}^{\mu }+j_{-}^{\mu }$, where the current density
4-vectors $j_{+}^{\mu }$ and $j_{-}^{\mu }$ for positive and negative
charges, respectively, have to be found in their own rest frames, and then
transformed by the Lorentz transformation to the considered IFR. Then in
Sec.3.2 we quote the covariant Maxwell equations when written by the
electromagnetic field tensor $F^{\alpha \beta }$ and by the 4-vectors $%
E^{\alpha }$ and $B^{\alpha },$ (both forms were already found in \cite
{ivezic}), and also we report a new form - the covariant Majorana form of
Maxwell's equations. Then the 4-vectors $E^{\alpha }$ and $B^{\alpha }$ are
determined for a CCC (instead of the usual 3-vectors $\mathbf{E}$ and $%
\mathbf{B)}$ and it is obtained in such a covariant way, i.e., in the ''TT
relativity'', that, for the observers at rest in the rest frame of that CCC,
there is a second-order electric field outside stationary conductor with
steady current. Such fields are already theoretically predicted on different
grounds in \cite{ivezpla}, see also \cite{ivescle}. In contrast to previous
works we also find in such a covariant manner that there are opposite
charges on opposite sides of a square loop with current and these charges
are Lorentz invariant charges. In the usual approach, i.e., in the ''AT
relativity'', it is found that there is an electric moment $\mathbf{P}$%
\textbf{\ }for a moving loop with a current. However we find that such loop,
regarding the electric effects, always behaves at long distances as an
electric quadrupole.\newline
\newline
\newline
\textbf{2. COVARIANT AND SYNCHRONOUS DEFINITIONS OF \newline
LENGTH}\newline
\newline

As discussed in \cite{ivezic}, (and \cite{ivsc98}) according to the
''modern'' point of view the special relativity is the theory of 4D
spacetime with pseudo-Euclidean geometry. Quantities of physical interest,
both local and nonlocal, are represented in the special relativity by
spacetime tensors, i.e., as covariant quantities, and the laws of physics
are written in a manifestly covariant way as tensorial equations. The
geometry of the spacetime is generally defined by the invariant
infinitesimal spacetime distance $ds$ of two neighboring points, $%
ds^{2}=dx^{a}g_{ab}dx^{b}.$ I adopt the following convention with regard to
indices. Repeated indices imply summation. Latin indices $a,b,c,d,...$ are
to be read according to the abstract index notation, see \cite{Wald},
Sec.2.4.. They designate geometric objects and they run from 0 to 3. Thus $%
dx^{a,b}$ and $g_{ab},$ and of course $ds,$ are defined independently of any
coordinate system, e.g., $g_{ab}$ is a second-rank covariant tensor (whose
Riemann curvature tensor $R_{bcd}^{a}$ is everywhere vanishing; the
spacetime of special relativity is a flat spacetime, and this definition
includes not only the IFRs but also the accelerated frames of reference).
Greek indices run from 0 to 3, while latin indices $i,j,k,l,...$ run from 1
to 3, and they both designate the components of some geometric object in
some coordinate chart, e.g., $x^{\mu }(x^{0},x^{i})$ and $x^{\prime \mu
}(x^{\prime 0},x^{\prime i})$ are two coordinate representations of the
position 4-vector $x^{a}$ in two different inertial coordinate systems $S$
and $S^{\prime },$ and $g_{\mu \nu }$ is the $4\times 4$ matrix of
components of $g_{ab}$ in some coordinate chart.\newline
\newline
\newline
\textbf{2.1. The spacetime or the TT length}\newline
\newline

In general, in 4D spacetime of special relativity it is not possible to
separate the spatial and temporal parts of $ds$, or according to Minkowski's
words, quoted here as a motto, \textit{the spatial and temporal parts taken
separately loose their physical meaning}. Therefore, only the invariant
spacetime length (the Lorentz scalar) between two points (events) in 4D
spacetime does have definite physical meaning in the ''TT relativity'' and
it is defined as

\begin{equation}
l=(l^{a}g_{ab}l^{b})^{1/2},  \label{covlen}
\end{equation}
where $l^{a}(l^{b})$ is the distance 4-vector between two events $A$ and $B$%
, $l^{a}=x_{B}^{a}-x_{A}^{a}$, $x_{A,B}^{a}$ are the position 4-vectors, and 
$g_{ab}$ is the metric tensor.

Using different coordinatizations of a given reference frame, which can be
realized, for example, by means of different synchronizations, we find
different expressions, i.e., different representations of the spacetime
length $l,$ Eq.(\ref{covlen}). Obviously the coordinates $x^{\mu }$ of an
event, when written in some coordinatization of an IFR, do not have an
intrinsic meaning in 4D spacetime. However the spacetime length $l$ (\ref
{covlen}) does have the same value for all relatively moving inertial
coordinate systems and it represents an intrinsic feature of the spacetime.

Different synchronizations are determined by the parameter $\varepsilon $ in
the relation $t_{2}=t_{1}+\varepsilon (t_{3}-t_{1})$, where $t_{1}$ and $%
t_{3}$ are the times of departure and arrival, respectively, of the light
signal, read by the clock at $A$, and $t_{2}$ is the time of reflection at $%
B $, read by the clock at $B$, that has to be synchronized with the clock at 
$A $. Usually physicists prefer Einstein's synchronization convention with $%
\varepsilon =1/2$ in which the measured coordinate velocity of light (the
one-way speed of light) is constant and isotropic. A nice example of a
non-standard synchronization is ''everyday'' clock synchronization \cite
{Leub} in which $\varepsilon =0$ and there is an absolute simultaneity; see
also \cite{Caval} for an absolute simultaneity in the special relativity,
and for the review on synchronisation and test theories see the recent
article \cite{anders}. As explained in \cite{Leub}: ''For if we turn on the
radio and set our clock by the standard announcement ''...at the sound of
the last tone, it will be 12 o'clock'', then we have synchronized our clock
with the studio clock in a manner that corresponds to taking $\varepsilon =0$
in $t_{2}=t_{1}+\varepsilon (t_{3}-t_{1})."$

When Einstein's synchronization of distant clocks and cartesian space
coordinates $x_{e}^{i}$ are used in an IFR $S$ (this coordinatization will
be named Einstein's or ''e'' coordinatization) then, e.g., the geometric
object $g_{ab}$ is represented by the $4\times 4$ matrix of components of $%
g_{ab}$ in that coordinate chart, i.e., it is the Minkowski metric tensor $%
g_{\mu \nu e}=diag(-1,1,1,1),$ where ''e'' stands for Einstein's
coordinatization. With such $g_{\mu \nu e}$ the space $x_{e}^{i}$ and time $%
t_{e}$ $(x_{e}^{0}\equiv ct_{e})$ components of $x_{e}^{\mu }$ do have their
usual meaning. Then $ds^{2}$ can be written with the separated spatial and
temporal parts, $ds^{2}=(dx_{e}^{i}dx_{ie})-(dx_{e}^{0})^{2}$, and the same
happens with the spacetime length $l$ (\ref{covlen})$,$ $%
l^{2}=(l_{e}^{i}l_{ie})-(l_{e}^{0})^{2}$. Such separation remains valid in
other inertial coordinate systems with the Minkowski metric tensor, and in $%
S^{\prime }$ one finds $l^{\prime 2}=(l_{e}^{\prime i}l_{ie}^{\prime
})-(l_{e}^{\prime 0})^{2},$ where $l_{e}^{\prime \mu }$ in $S^{\prime }$ is
connected with $l_{e}^{\mu }$ in $S$ by the LT.

In the usual form the LT connect two coordinate representations (in the
''e'' coordinatization) $x_{e}^{\mu },x_{e}^{\prime \mu }$ of a given event. 
$x_{e}^{\mu },x_{e}^{\prime \mu }$ refer to two relatively moving IFRs (with
the Minkowski metric tensor) $S$ and $S^{\prime },$ 
\[
x_{e}^{\prime \mu }=L^{\mu }{}_{\nu ,e}x_{e}^{\nu
},\,\,\,L^{0}{}_{0,e}=\gamma _{e},L^{0}{}_{i,e}=L^{i}{}_{0,e}=-\gamma
_{e}V_{e}^{i}/c,L^{i}{}_{j,e}=\delta _{j}^{i}+(\gamma
_{e}-1)V_{e}^{i}V_{je}/V_{e}^{2},
\]
where $V_{e}^{i}=dx_{e}^{i}/dt_{e}$ are the components of the ordinary
velocity 3-vector, and $\gamma _{e}\equiv (1-V_{e}^{2}/c^{2})^{1/2}$. As
explained in \cite{Fahn}, when such usual representations of pure Lorentz
transformations are applied to covariant expressions they destroy the
covariant form : ''because they employ three-vector notation, because they
treat the spatial and temporal components separately, and because they are
parametrized by the ordinary velocity three-vector $\mathbf{V.}$'' In order
to obtain a covariant expression for $L^{\mu }{}_{\nu ,e}$ the ordinary
velocity is replaced in \cite{Fahn} by the proper velocity 4-vector $%
v_{e}^{\mu }\equiv dx_{e}^{\mu }/d\tau =(\gamma _{e}c,\gamma _{e}v_{e}^{i}),$
$d\tau \equiv dt_{e}/\gamma _{e}$ is the scalar proper-time, the unit vector 
$n_{e}^{\mu }\equiv (1,0,0,0)$ along the temporal axis is introduced, and $%
\delta _{j}^{i}$ is replaced with the Minkowski metric tensor $g^{\mu
}{}_{\nu e}.$ This shows that the cartesian space coordinates $x_{e}^{i}$
and Einstein's synchronization of distant clocks are explicitly chosen in 
\cite{Fahn}. In such a way the covariant expression for $L^{\mu }$ $_{\nu ,e}
$ in the ''e'' coordinatization is found in \cite{Fahn}, Eq.(5), 
\[
L^{\mu }{}_{\nu ,e}\equiv L^{\mu }{}_{\nu ,e}(v)=g^{\mu }{}_{\nu e}-\frac{%
2n_{e}^{\mu }v_{\nu e}}{c}+\frac{(n_{e}^{\mu }+v_{e}^{\mu }/c)(n_{\nu
e}+v_{\nu e}/c)}{1-n_{e}\cdot v_{e}/c}.
\]
Since we want to use the LT in different coordinatizations we generalize the
expression for $L^{\mu }{}_{\nu ,e}$ from \cite{Fahn} and find 
\begin{equation}
L^{a}{}_{b}\equiv L^{a}{}_{b}(v)=g^{a}{}_{b}-\frac{2n^{a}v_{b}}{c}+\frac{%
(n^{a}+v^{a}/c)(n_{b}+v_{b}/c)}{1-n\cdot v/c}.  \label{fah}
\end{equation}
Such form (\ref{fah}) of the LT can be applied to an arbitrary inertial
coordinate system in which the metric tensor can be different than the
Minkowski metric tensor, and thus the form of the covariant 4D Lorentz
transformations (\ref{fah}) is independent of the chosen synchronization,
i.e., coordinatization of reference frames. But we have to note that $n^{a}$
in (\ref{fah}) is a specific quantity. Namely it always has to be taken as
the unit vector along the temporal axis in the chosen IFR and the chosen
coordinatization. Nevertheless $L^{a}{}_{b}$ correctly transforms some 4D
tensor quantity from an IFR to another relatively moving IFR. For example,
when $L^{a}{}_{b}$ is applied to the position 4-vector $x^{a}$ one finds (in
the abstract index notation) 
\begin{equation}
x^{\prime a}=x^{a}+\frac{\left[ n\cdot x-(2\gamma +1)v\cdot x/c\right]
n^{a}+(n\cdot x+v\cdot x/c)v^{a}/c}{1-n\cdot v/c}.  \label{fahn2}
\end{equation}
\newline

Let us examine the relations (\ref{fah}) and (\ref{fahn2}) in two different
coordinatizations. First in the ''e'' coordinatization, in which the
Minkowski metric tensor is used, $n^{a}$ becomes $n_{e}^{\mu }\equiv
(1,0,0,0),$ $v_{e}^{\mu }=(\gamma _{e}c,\gamma _{e}v_{e}^{i}),$ and $\gamma
_{e}=-n_{e}^{\mu }v_{\mu e}/c,$ as in \cite{Fahn}. From the general relation 
$v^{a}v_{a}=-c^{2}$ one finds, in the ''e'' coordinatization, that $%
v_{e}^{0}=(c^{2}+v_{e}^{i}v_{ie})^{1/2},$ which shows that the expression
for $L^{\mu }{}_{\nu ,e}$ is parametrized essentially by the three spatial
components $v_{e}^{i}$ of the proper velocity 4-vector $v_{e}^{\mu }.$ Then,
using the above expressions for $n_{e}^{\mu },$ $v_{e}^{\mu },$ and $\gamma
_{e}$ one finds from (\ref{fah}) and (\ref{fahn2}) the usual expressions for
pure LT, as in \cite{Fahn}, i.e., the above mentioned $L^{\mu }{}_{\nu ,e}$
and $x_{e}^{\prime \mu }$, but with $v_{e}^{i}$ replacing the components of
the ordinary velocity 3-vector $\mathbf{V.}$ Also, we find the above
mentioned usual expressions in the ''e'' coordinatization for $%
ds^{2}=ds_{e}^{2}=(dx_{e}^{i}dx_{ie})-(dx_{e}^{0})^{2}$ and $%
l^{2}=l_{e}^{2}=(l_{e}^{i}l_{ie})-(l_{e}^{0})^{2},$with the separated
spatial and temporal parts.

In the similar way we use the relations (\ref{fah}) and (\ref{fahn2}) to
write the corresponding expressions in another coordinatization, ''r''
coordinatization, of an IFR, which is found in \cite{Leub}, where
''everyday'' or ''radio'' synchronization of distant clocks is used. For
simplicity we consider 2D spacetime as in \cite{Leub}. Then the metric
tensor $g_{ab}$ becomes $g_{\mu \nu r}=\left( 
\begin{array}{cc}
-1 & -1 \\ 
-1 & 0
\end{array}
\right) ,$ where ''r'' stands for ''radio'' (it differs from that one in 
\cite{Leub} since the Minkowski tensors are different). The LT $L^{\mu
}\,_{\nu ,r}$ in the ''r'' coordinatization can be easily found from (\ref
{fah}), from the known $g_{\mu \nu r},$ with $n_{r}^{\mu }=(1,0)$ and $%
\gamma _{r}=-n_{r}^{\mu }v_{\mu r}/c=\gamma _{e}.$ These relations can be
found as in \cite{Leub}, or by means of the matrix $T^{\mu }\,_{\nu },$
which is given below. Thus the pure Lorentz transformation matrix $%
L^{a}{}_{b}$ (\ref{fah}) becomes in the ''r'' coordinatization 
\[
L^{\mu }\,_{\nu ,r}=\left( 
\begin{array}{ll}
K & 0 \\ 
-\beta _{r}/K & 1/K
\end{array}
\right) . 
\]
Also we find the ''r'' representation $x_{r}^{\prime \mu }$ of $x^{\prime a}$
(\ref{fahn2}), 
\[
x_{r}^{\prime 0}=Kx_{r}^{0},\,\,x_{r}^{\prime 1}=(1/K)(-\beta
_{r}x_{r}^{0}+x_{r}^{1}), 
\]
where $K=(1+2\beta _{r})^{1/2}$ and $\beta _{r}=dx_{r}^{1}/dx_{r}^{0}$ is
the velocity of the frame $S^{\prime }$ as measured by the frame $S.$
Further $ds^{2}=dx^{a}g_{ab}dx^{b}$ becomes in the ''r'' coordinatization $%
ds^{2}=ds_{r}^{2}=-\left[ (dx_{r}^{0})^{2}+2dx_{r}^{0}dx_{r}^{1}\right] .$
We see that in the ''r'' coordinatization the spatial and temporal parts of $%
ds^{2}$ are not separated, that is different than in the coordinatization
with the Minkowski metric tensor. The same holds for the spacetime length $%
l, $ which is in the ''r'' coordinatization determined as $%
l^{2}=l_{r}^{2}=-\left[ (l_{r}^{0})^{2}+2l_{r}^{0}l_{r}^{1}\right] .$
Expressing $dx_{r}^{\mu },$ or $l_{r}^{\mu }$, in terms of $dx_{e}^{\mu },$
or $l_{e}^{\mu }$ (the transformation matrix between ''r'' and ''e''
coordinatizations is 
\[
T^{\mu }\,_{\nu }=\left( 
\begin{array}{cc}
1 & -1 \\ 
0 & 1
\end{array}
\right) , 
\]
whence, e.g., $x_{r}^{0}=x_{e}^{0}-x_{e}^{1},$ $x_{r}^{1}=x_{e}^{1},$ and $%
\beta _{r}=\beta _{e}/(1-\beta _{e}),$ see \cite{Leub}) one finds that $%
ds_{r}^{2}=ds_{e}^{2},$ and also, $l_{r}^{2}=l_{e}^{2},$ as it must be.

The whole preceding discussion about the geometric quantities $%
x^{a},l^{a},ds,l,..$ and their different representations can be illustrated
in a way which better clarifies the difference between two sorts of
quantities. Again we consider the TT length (\ref{covlen}) (we use the words
- the TT length, the spacetime length, and the covariantly defined length as
synonyms) in two relatively moving IFRs $S$ and $S^{\prime }$ and in two
coordinatizations ''e'' and ''r'' in these IFRs. Now, let the spacetime be
endowed with base vectors, the temporal and the spatial base vectors. The
bases $\left\{ e_{\mu }\right\} $, with the base vectors $\left\{
e_{0},e_{1}\right\} $, and $\left\{ r_{\mu }\right\} ,$ with the base
vectors $\left\{ r_{0},r_{1}\right\} ,$ are associated with ''e'' and ''r''
coordinatizations, respectively, of a given IFR. The temporal base vector $%
e_{0}$ is the unit vector directed along the world line of the clock at the
origin. The spatial base vector by definition connects \textit{simultaneous}
events, the event ''clock at rest at the origin reads 0 time'' with the
event ''clock at rest at unit distance from the origin reads 0 time'', and
thus it is synchronization-dependent. The spatial base vector $e_{1}$
connects two above mentioned simultaneous events when Einstein's
synchronization ($\varepsilon =1/2$) of distant clocks is used. The temporal
base vector $r_{0}$ is the same as $e_{0}.$ The spatial base vector $r_{1}$
connects two above mentioned simultaneous events when ''everyday'' clock
synchronization ($\varepsilon =0$) of distant clocks is used. All the
spatial base vectors $r_{1},r_{1}^{\prime },..$ are parallel and directed
along an (observer-independent) light line. Hence, two events that are
everyday (''r'') simultaneous in $S$ are also ''r'' simultaneous for all
other IFRs. The connection between the bases $\left\{ e_{\mu }\right\} $ and 
$\left\{ r_{\mu }\right\} $ is\quad $r_{0}=e_{0},\;r_{1}=e_{0}+e_{1},$ see 
\cite{Leub}. Then the geometrical quantity, e.g., the distance 4-vector $%
l_{AB}^{a}$ between two events $A$ and $B,$ will be represented by the
vector in 2D spacetime, which have different decompositions,
representations, with respect to $\left\{ e_{\mu }\right\} ,\left\{ e_{\mu
}^{\prime }\right\} $ and $\left\{ r_{\mu }\right\} ,\left\{ r_{\mu
}^{\prime }\right\} $ bases. \textit{Note that in the ''TT relativity'' the
same distance 4-vector }$l_{AB}^{a}$ \textit{is considered (measured) in
different relatively moving IFRs and in different coordinatizations of these
IFRs.}

In order to retain the connection with the prerelativistic physics and to
facilitate the comparison with the ''AT relativity'' we consider a
particular choice for the 4-vector $l_{AB}^{a}$ (in the usual ''3+1''
picture it corresponds to an object, a rod, that is at rest in an IFR $S$
and situated along the common $x_{e}^{1},x_{e}^{\prime 1}-$ axes). In the
''e'' coordinatization the position 4-vectors of the events $A,$ $x_{A}^{a},$
and $B,$ $x_{B}^{a},$ in $S$ are decomposed with respect to $\left\{ e_{\mu
}\right\} $ base as $%
x_{A}^{a}=x_{Ae}^{0}e_{0}+x_{Ae}^{1}e_{1}=0e_{0}+0e_{1}, $ and $%
x_{B}^{a}=x_{Be}^{0}e_{0}+x_{Be}^{1}e_{1}=0e_{0}+l_{0}e_{1},$ and the
distance 4-vector $l_{AB}^{a}=x_{B}^{a}-x_{A}^{a}$ is decomposed as 
\[
l_{AB}^{a}=l_{e}^{0}e_{0}+l_{e}^{1}e_{1}=0e_{0}+l_{0}e_{1}. 
\]
Thus in $S$ the position 4-vectors $x_{A,B}^{a}$ are determined
simultaneously, $x_{Be}^{0}-x_{Ae}^{0}=l_{e}^{0}=0,$ i.e., the temporal part
of $l_{AB}^{a}$ is zero. The spacetime length $l$ is written in the $\left\{
e_{\mu }\right\} $ base as 
\[
l=l_{e}=(l_{e}^{\mu }l_{\mu e})^{1/2}=(l_{e}^{i}l_{ie})^{1/2}=l_{0}, 
\]
as in the prerelativistic physics; it is in that case a measure of the
spatial distance, i.e., of the rest spatial length of the rod. The observers
in all other IFRs will look at the same events but associating with them
different coordinates; it is the essence of the covariant description. They
all obtain the same value $l$ for the spacetime length. It has to be pointed
out that \textit{in the ''TT relativity'' it is not necessary to start in
this example with the rest frame of the object and to choose the events }$A$%
\textit{\ and }$B$\textit{\ to be simultaneous in that frame. The whole
consideration can be done in the same covariant manner for other choices of
IFRs and of the events }$A$\textit{\ and }$B$\textit{\ in the chosen IFR}.
For any starting choice the covariant LT (\ref{fah}) will correctly connect
the results of measurements of \textit{the same 4D quantity} in two
relatively moving IFRs. The rest frame of the object and the simultaneity of
the events $A$ and $B$ in it are chosen only to have the connection with the
prerelativistic physics, which deals with ''3+1'' quantities and not with 4D
quantities.

Let us then consider the same 4-vector $l_{AB}^{a}$ in $S^{\prime },$ (where
in ''3+1'' picture the rod is moving). The position 4-vectors $x_{A}^{a}$
and $x_{B}^{a}$ of the events $A$ and $B$ respectively are decomposed with
respect to $\left\{ e_{\mu }^{\prime }\right\} $ base as $%
x_{A}^{a}=x_{Ae}^{\prime 0}e_{0}^{\prime }+x_{Ae}^{\prime 1}e_{1}^{\prime
}=0e_{0}^{\prime }+0e_{1}^{\prime },$ and $x_{B}^{a}=x_{Be}^{\prime
0}e_{0}^{\prime }+x_{Be}^{\prime 1}e_{1}^{\prime }=-\beta _{e}\gamma
_{e}l_{0}e_{0}^{\prime }+\gamma _{e}l_{0}e_{1}^{\prime },$ and the distance
4-vector is decomposed as 
\[
l_{AB}^{a}=x_{B}^{a}-x_{A}^{a}=l_{e}^{\prime 0}e_{0}^{\prime }+l_{e}^{\prime
1}e_{1}^{\prime }=-\beta _{e}\gamma _{e}l_{0}e_{0}^{\prime }+\gamma
_{e}l_{0}e_{1}^{\prime }. 
\]
Note that in the ''e'' coordinatization, commonly used in the ''AT
relativity,'' there is a dilatation of the spatial part $l_{e}^{\prime
1}=\gamma _{e}l_{0}$ with respect to $l_{e}^{1}=l_{0}$ and not the Lorentz
contraction as predicted in the ''AT relativity.'' Hovewer it is clear from
the above discussion that comparison of only spatial parts of the components
of the distance 4-vector $l_{AB}^{a}$ in $S$ and $S^{\prime }$ is physically
meaningless in the ''TT relativity.'' All components of the distance
4-vector $l_{AB}^{a}$ are transformed by the LT from $S$ to $S^{\prime }.$ $%
l_{e}^{\mu }$ and $l_{e}^{\prime \mu }$ are different representations of the
same physical quantity $l_{AB}^{a}$ measured in two relatively moving IFRs $%
S $ and $S^{\prime }$. The invariant spacetime length of that object in $%
S^{\prime }$ is 
\[
l=l_{e}^{\prime }=(l_{e}^{^{\prime }\mu }l_{\mu e}^{\prime })^{1/2}=l_{0}. 
\]
Note that if $l_{e}^{0}=0$ then $l_{e}^{\prime \mu }$ in any other IFR $%
S^{\prime }$ will contain the time component $l_{e}^{\prime 0}\neq 0.$ We
conclude from the above discussion that if one wants in the ''TT
relativity'' to compare in a physically meaningful sense the ''lengths'' of
two different objects than it is possible only by comparing their invariant
spacetime lengths.

In the ''r'' coordinatization the position 4-vectors of the events $A$ and $%
B,$ $x_{A}^{a}$ and $x_{B}^{a},$ in $S$ are decomposed with respect to $%
\left\{ r_{\mu }\right\} $ base as $%
x_{A}^{a}=x_{Ar}^{0}r_{0}+x_{Ar}^{1}r_{1}=0r_{0}+0r_{1},$ and $%
x_{B}^{a}=x_{Br}^{0}r_{0}+x_{Br}^{1}r_{1}=-l_{0}r_{0}+l_{0}r_{1},$ and the
distance 4-vector $l_{AB}^{a}=x_{B}^{a}-x_{A}^{a}$ is decomposed as 
\[
l_{AB}^{a}=l_{r}^{0}r_{0}+l_{r}^{1}r_{1}=-l_{0}r_{0}+l_{0}r_{1}, 
\]
and the TT length $l$ is 
\[
l=l_{r}=(l_{r}^{\mu }l_{\mu r})^{1/2}=l_{e}=l_{0} 
\]
as it must be.

In $S^{\prime }$ and in the $\left\{ r_{\mu }^{\prime }\right\} $ base the
position 4-vectors of the events $A$ and $B$ are $x_{A}^{a}=0r_{0}^{\prime
}+0r_{1}^{\prime }$ and $x_{B}^{a}=x_{Br}^{\prime 0}r_{0}^{\prime
}+x_{Br}^{\prime 1}r_{1}^{\prime }=-Kl_{0}r_{0}^{\prime }+(1+\beta
_{r})(1/K)l_{0}r_{1}^{\prime }$, and the components $l_{r}^{\prime \mu }$ of
the distance 4-vector $l_{AB}^{a}$ are equal to the components $%
x_{Br}^{\prime \mu }$, i.e., $l_{r}^{\prime \mu }=x_{Br}^{\prime \mu }.$
Thus $l_{AB}^{a}$ is decomposed as 
\[
l_{AB}^{a}=l_{r}^{\prime 0}r_{0}^{\prime }+l_{r}^{\prime 1}r_{1}^{\prime
}=-Kl_{0}r_{0}^{\prime }+(1+\beta _{r})(1/K)l_{0}r_{1}^{\prime }. 
\]
If only spatial parts of $l_{r}^{\mu }$ and $l_{r}^{\prime \mu }$ are
compared than one finds that $\infty \succ l_{r}^{\prime 1}\geq l_{0}$ for $%
-1/2\prec \beta _{r}\leq 0$ and $l_{0}\leq l_{r}^{\prime 1}\prec \infty $
for $0\leq \beta _{r}\prec \infty $ , which once again shows that such
comparison is physically meaningless in the ''TT relativity.'' Hovewer the
invariant spacetime length always takes the same value 
\[
l=l_{r}^{\prime }=(l_{r}^{^{\prime }\mu }l_{\mu r}^{\prime
})^{1/2}=l_{r}=l_{0}, 
\]
and as already said, it can be compared in a physically meaningful sense in
the ''TT relativity.'' One concludes from this discussion that, e.g., our
particular 4-vector $l_{AB}^{a}$ (a geometrical quantity) is represented in
different bases $\left\{ e_{\mu }\right\} ,\left\{ e_{\mu }^{\prime
}\right\} ,\left\{ r_{\mu }\right\} $ and $\left\{ r_{\mu }^{\prime
}\right\} $ by its coordinate representations $l_{e}^{\mu },l_{e}^{\prime
\mu },l_{r}^{\mu }$ and $l_{r}^{\prime \mu }$, respectively$.$

We see that in the ''TT relativity'' the geometrical quantities, e.g., the
4-vectors $x^{a},l^{a},$..., have different representations depending on the
chosen IFR and the chosen coordinatization in that IFR, e.g., $x_{e,r}^{\mu
},l_{e,r}^{\prime \mu },..$. Although the Einstein coordinatization is
preferred by physicists due to its simplicity and symmetry it is nothing
more ''physical'' than others, e.g., the ''r'' coordinatization. The
coordinate dependent quantities have not an intrinsic physical meaning. The
spacetime length $l$ is an example of a well defined quantity that is
independent of the chosen IFR and also of the coordinatization taken in that
IFR; it is an intrinsic property of the spacetime. From this consideration
an important conclusion emerges; the usual 3D length of a moving object
cannot be defined in the 4D spacetime of the TT relativity in an adequate
way, since it is only the spatial length and not a 4D tensor quantity.%
\newline
\newline
\newline
{2.2. \textbf{The AT of length}}\newline
\newline

In contrast to the covariant definition of the spacetime length and the TT
of the spacetime tensors considered in the ''TT relativity'' \textit{the
synchronous definition of spatial length, introduced by Einstein} \cite
{Einst} \textit{defines length as the spatial distance between two spatial
points on the (moving) object measured by simultaneity in the rest frame of
the observer.} To see the difference with respect to the ''TT relativity''
we determine the spatial length of the rod considered in the previous
section. As shown above in the ''TT relativity,'' in contrast to the ''AT
relativity,'' one cannot speak about the spatial distance, e.g., the rest
spatial length of a rod, as a correctly defined physical quantity,.but only
about 4D tensor quantities, the geometrical quantities - the position
4-vectors $x_{A,B}^{a}$, the distance 4-vector $l_{AB}^{a}$, the spacetime
length $l$, etc., and their 4D representations, $x_{A,B,..e,r,..}^{\mu
},l_{ABe,r,..}^{\mu },l_{e,r,..}.$

Instead of to work with geometrical quantities $x_{A,B}^{a},l_{AB}^{a}$ and $%
l$ one deals, in the ''AT relativity,'' only with the spatial, or temporal,
parts of their coordinate representations $x_{Ae,r}^{\mu },x_{Be,r}^{\mu }$
and $l_{e,r}^{\mu }.$ First the ''e'' coordinatization, which is almost
always used in the ''AT relativity,'' is considered. According to Einstein's
definition \cite{Einst} of the spatial length the spatial ends of the rod
must be taken simultaneously in the chosen coordinatization. In 4D (at us
2D) spacetime and in the ''e'' coordinatization the simultaneous events $A$
and $B$ (whose spatial parts correspond to the spatial ends of the rod) are
the intersections of $x_{e}^{1}$ axis (that is along the spatial base vector 
$e_{1}$) and the world lines of the spatial ends of the rod that is at rest
in $S$ and situated along the $x_{e}^{1}$ axis.The position 4-vectors (in
the ''e'' base) $x_{Ae}^{\mu }$ and $x_{Be}^{\mu }$ of the simultaneous (at $%
t_{e}=a=0)$ events $A$ and $B$ in $S$ are $x_{Ae}^{\mu }=0e_{0}+0e_{1},$ or,
in short, $x_{Ae}^{\mu }=(0,0),$ and $x_{Be}^{\mu }=(0,l_{0}),$ and the
distance 4-vector (in the ''e'' base) $l_{ABe}^{\mu }=x_{Be}^{\mu
}-x_{Ae}^{\mu }=(0,l_{0}).$ We emphasize that it is necessary in the ''AT
relativity'' to take the end points of the spatial length of the rod to be
simultaneous, whereas in the ''TT relativity'' the events $A$ and $B$ can
be, in principle, taken at arbitrary $x_{Ae}^{0}\neq x_{Be}^{0}.$ Then in $S$%
, the rest frame of the object, the spatial part $l_{ABe}^{1}=l_{0}$ of $%
l_{ABe}^{\mu }$ is considered to define the rest spatial length (the
temporal part of $l_{ABe}^{\mu }$ is taken to be zero). Further one uses the
inverse Lorentz transformations to express $x_{Ae}^{\mu },$ $x_{Be}^{\mu },$
and $l_{ABe}^{\mu }$ in $S$ in terms of the corresponding quantities in $%
S^{\prime }$, in which the rod is moving. This procedure yields $%
x_{A,Be}^{0}=ct_{A,Be}=\gamma _{e}(ct_{A,Be}^{\prime }+\beta
_{e}x_{A,Be}^{\prime 1}),$ and $x_{A,Be}^{1}=\gamma _{e}(\beta
_{e}ct_{A,Be}^{\prime }+x_{A,Be}^{\prime 1}),$ whence 
\begin{equation}
l_{ABe}^{0}=ct_{Be}-ct_{Ae}=\gamma _{e}(ct_{Be}^{\prime }-ct_{Ae}^{\prime
})+\gamma _{e}\beta _{e}(x_{Be}^{\prime 1}-x_{Ae}^{\prime 1})=\gamma
_{e}l_{ABe}^{\prime 0}+\gamma _{e}\beta _{e}l_{ABe}^{\prime 1}
\label{elnula}
\end{equation}
and 
\begin{equation}
l_{ABe}^{1}=x_{Be}^{1}-x_{Ae}^{1}=\gamma _{e}(x_{Be}^{\prime
1}-x_{Ae}^{\prime 1})+\gamma _{e}\beta _{e}(ct_{Be}^{\prime
}-ct_{Ae}^{\prime })=\gamma _{e}l_{ABe}^{\prime 1}+\gamma _{e}\beta
_{e}l_{ABe}^{\prime 0}.  \label{eljedan}
\end{equation}
Now comes the main difference between the two forms of relativity. Instead
of to work with 4D tensor quantities and their LT (as in the ''TT
relativity'') in the ''AT relativity'' one forgets about the transformation
of the temporal part $l_{ABe}^{0},$ Eq.(\ref{elnula}), and considers only
the transformation of the spatial part $l_{ABe}^{1},$ Eq.(\ref{eljedan})$.$
Further, in that relation for $l_{ABe}^{1}$ one assumes that $t_{Be}^{\prime
}=t_{Ae}^{\prime }=t_{e}^{\prime }=b,$ i.e., that $x_{Be}^{\prime 1}$ and $%
x_{Ae}^{\prime 1}$ are simultaneously determined at some arbitrary $%
t_{e}^{\prime }=b$ in $S^{\prime }.$ However, in 4D (at us 2D) spacetime
such an assumption means that in $S^{\prime }$ one actually does not
consider the same events $A$ and $B$ as in $S$ but some other two events $C$
and $D,$ whence $t_{Be}^{\prime }=t_{Ae}^{\prime }$ has to be replaced with $%
t_{De}^{\prime }=t_{Ce}^{\prime }=b.$ The events $C$ and $D$ are the
intersections of the line (the hypersurface $t_{e}^{\prime }=b$ with
arbitrary $b$) parallel to the spatial axis $x_{e}^{\prime 1}$ (which is
along the spatial base vector $e_{1}^{\prime }$) and of the above mentioned
world lines of the spatial end points of the rod. Then in the above
transformation for $l_{ABe}^{1}$ (\ref{eljedan}) one has to write $%
x_{De}^{\prime 1}-x_{Ce}^{\prime 1}=l_{CDe}^{\prime 1}$ instead of $%
x_{Be}^{\prime 1}-x_{Ae}^{\prime 1}=l_{ABe}^{\prime 1}.$ The spatial parts $%
l_{ABe}^{1}$ and $l_{CDe}^{\prime 1}$ are the \textit{spatial distances}
between the events $A,$ $B$ and $C$, $D,$ respectively. \textit{The spatial
distance }$l_{ABe}^{1}=x_{Be}^{1}-x_{Ae}^{1}$ \textit{defines in the ''AT
relativity,''} \textit{and in the ''e'' base}, \textit{the spatial length of
the rod} \textit{at rest in} $S,$ while $l_{CDe}^{\prime 1}=x_{De}^{\prime
1}-x_{Ce}^{\prime 1}$ \textit{is considered in the ''AT relativity,'' and in
the ''e'' base, to define} \textit{the spatial length of the moving rod} 
\textit{in }$S^{\prime }.$ With these definitions we find from the equation
for $l_{ABe}^{1}$ (\ref{eljedan}) the relation between $l_{e}^{\prime
1}=l_{CDe}^{\prime 1}$ and $l_{e}^{1}=l_{ABe}^{1}=l_{0}$ as the famous
formulae for the Lorentz contraction of the moving rod 
\begin{equation}
l_{e}^{\prime 1}=x_{De}^{\prime 1}-x_{Ce}^{\prime 1}=l_{0}/\gamma
_{e}=(x_{Be}^{1}-x_{Ae}^{1})(1-\beta _{e}^{2})^{1/2},  \label{contr}
\end{equation}
with $t_{Ce}^{\prime }=t_{De}^{\prime },$ $\,$and $t_{Be}=t_{Ae},$ where $%
\beta _{e}=V_{e}/c$, $V_{e}$ is the relative velocity of $S$ and $S^{\prime
}.$ Note that the spatial lengths $l_{0}$ and $l_{e}^{\prime 1}$ refer not
to the same 4D tensor quantity, as in the ''TT relativity,'' but to two
different quantities in 4D spacetime. These quantities are obtained by the
same measurements in $S$ and $S^{\prime };$ the spatial ends of the rod are
measured simultaneously at some $t_{e}=a$ in $S$ and also at some $%
t_{e}^{\prime }=b$ in $S^{\prime },$ and $a$ in $S$ and $b$ in $S^{\prime }$
are not related by the LT or any other coordinate transformation. While in
the ''TT relativity'' one deals with events as correctly defined quantities
in 4D spacetime in Einstein's approach \cite{Einst} the spatial and temporal
parts of events are treated separately, and moreover the time component is
not transformed in the Lorentz contraction.

The LT (\ref{fah}) is the transformation in 4D spacetime and it transforms
some 4D tensor quantity $Q_{b..}^{a..}(x^{c},x^{d},..)$ from $S$ to $%
Q_{b..}^{\prime a..}(x^{\prime c},x^{\prime d},..)$ in $S^{\prime },$ (all
parts of the quantity are transformed), which means that in 4D spacetime is
not possible to neglect the transformation of $l^{0}$ as a part of $l^{\mu
}, $ as done in the derivation of the Lorentz contraction (\ref{contr}).
However, if one does not forget the transformation of the temporal part $%
l_{ABe}^{0},$ Eq.(\ref{elnula}), and takes in it that $t_{Be}^{\prime
}=t_{Ae}^{\prime },$ $t_{Be}=t_{Ae}$ (as in the derivation of the Lorentz
contraction), then one finds from (\ref{elnula}) that $x_{Be}^{\prime
1}=x_{Ae}^{\prime 1},$ which is in the obvious contrast with the formulae
for the Lorentz contraction.

Let us also see does the Lorentz contraction, as the coordinate
transformation, change the interval $ds,$ which defines the geometry of the
spacetime. In $S$ and in the ''e'' base the interval $ds$ is $%
ds^{2}=ds_{e}^{2}=(dx_{e}^{1})^{2}-(c^{2}dt_{e})^{2},$ and with $dt_{e}=0,$
as assumed in the derivation of the Lorentz contraction, it becomes, in $%
S\quad ds^{2}=(dx_{e}^{1})^{2}.$ In $S^{\prime },$ where it is assumed that $%
dt_{e}^{\prime }=0,$ and with the relation for the Lorentz contraction (\ref
{contr}), $dx_{e}^{\prime 1}=dx_{e}^{1}/\gamma _{e},$ the infinitesimal
spacetime distance $ds^{\prime }$ becomes, in $S^{\prime }$\quad $ds^{\prime
2}=(dx_{e}^{1})^{2}/\gamma _{e}^{2},$ and thus $ds^{\prime }\neq ds.$

Let us now consider the Lorentz ''contraction'' in the ''r''
coordinatization. According to Einstein's definition \cite{Einst} of the
spatial length the spatial ends of the rod must be taken simultaneously in
the chosen coordinatization. In 4D (at us 2D) spacetime and in the ''r''
base the spatial ends of the considered rod, that is at rest in $S,$ must
lie on the light line, i.e., on the $x_{r}^{1}$ axis (that is along the
spatial base vector $r_{1}$). Hence the simultaneous events $E$ and $F$
(whose spatial parts correspond to the spatial ends of the rod) are the
intersections of $x_{r}^{1}$ axis and the world lines of the spatial ends of
the rod. Note that in our 2D spacetime the events $E$ and $F$ are not the
same events as the events $A$ and $B,$ considered in the ''e'' base for the
same rod at rest in $S,$ since the simultaneity of the events is defined in
different ways. The $\left\{ r_{\mu }\right\} $ representations of the
position 4-vectors $x_{E}^{a}$ and $x_{F}^{a}$ of the events $E$ and $F$ in $%
S$ are $x_{Er}^{\mu }=(0,0)$ and $x_{Fr}^{\mu }=(0,l_{0}),$ and of the
distance 4-vector $l_{EF}^{a}$ is $l_{EFr}^{\mu }=x_{Fr}^{\mu }-x_{Er}^{\mu
}=(0,l_{r}^{1})=(0,l_{0}).$ However, as noticed above, in 4D spacetime the
spatial length in the ''r'' base $l_{r}^{1}=l_{0}$ (with $%
x_{Fr}^{0}=x_{Er}^{0}$) is not the same 4D quantity as the spatial length in
the ''e'' base $l_{e}^{1}=l_{0}$ (with $x_{Be}^{0}=x_{Ae}^{0}),$ since the
simultaneity is defined in a different way. Applying the same procedure as
in the case of the derivation of the Lorentz contraction in the ''e'' base
we find the relations for $l_{r}^{0}$ and $l_{r}^{1}$ corresponding to (\ref
{elnula}) and (\ref{eljedan}), respectively, 
\begin{equation}
l_{EFr}^{0}=x_{Fr}^{0}-x_{Er}^{0}=(1/K)(x_{Fr}^{\prime 0}-x_{Er}^{\prime
0})=(1/K)l_{EFr}^{\prime 0},  \label{elnul1}
\end{equation}
\begin{equation}
l_{EFr}^{1}=x_{Fr}^{1}-x_{Er}^{1}=(\beta _{r}/K)(x_{Fr}^{\prime
0}-x_{Er}^{\prime 0})+K(x_{Fr}^{\prime 1}-x_{Er}^{\prime 1})=(\beta
_{r}/K)l_{EFr}^{\prime 0}+Kl_{EFr}^{\prime 1},  \label{eljed1}
\end{equation}
$K=(1+2\beta _{r})^{1/2}.$ Further, in the ''r'' base, one again forgets the
transformation of the temporal part $l_{EFr}^{0}$ (\ref{elnul1}) of $%
l_{EFr}^{\mu }$ and assumes that in the relation for $l_{EFr}^{1}$ (\ref
{eljed1}) $x_{Fr}^{\prime 1}$ and $x_{Er}^{\prime 1}$ are simultaneously
determined at some $x_{Fr}^{\prime 0}=x_{Er}^{\prime 0}=b$ in $S^{\prime }.$
However, in the same way as in the ''e'' base, in 4D (at us 2D) spacetime
such an assumption means that in $S^{\prime }$ one actually does not
consider the same events $E$ and $F$ as in $S$ but some other two events $G$
and $H,$ and that the equality $x_{Fr}^{\prime 0}=x_{Er}^{\prime 0}=b$ has
to be replaced by $x_{Hr}^{\prime 0}=x_{Gr}^{\prime 0}=b.$ The events $G$
and $H$ are the intersections of the line (the hypersurface $x_{Hr}^{\prime
0}=x_{Gr}^{\prime 0}=b$ with arbitrary $b$) parallel to the spatial axis $%
x_{r}^{\prime 1}$ (which is along the spatial base vector $r_{1}^{\prime }$)
and of the above mentioned world lines of the spatial end points of the rod.
Then in the above transformation for $l_{EFr}^{1}$ (\ref{eljed1}) $%
l_{GHr}^{\prime 0}=0$, and $l_{EFr}^{\prime 1}=x_{Fr}^{\prime
1}-x_{Er}^{\prime 1}$ is replaced by $l_{GHr}^{\prime 1}=x_{Hr}^{\prime
1}-x_{Gr}^{\prime 1}.$ Now, \textit{in the ''r'' base}, \textit{the spatial
distance }$l_{EFr}^{1}=x_{Fr}^{1}-x_{Er}^{1}$ \textit{defines in the ''AT
relativity''} \textit{the spatial length of the rod} \textit{at rest in} $S,$
while $l_{GHr}^{\prime 1}=x_{Hr}^{\prime 1}-x_{Gr}^{\prime 1}$ \textit{%
defines} \textit{the spatial length of the moving rod} \textit{in }$%
S^{\prime }.$ Then, from the equation for $l_{EFr}^{1}$ (\ref{eljed1}), and
with these definitions, we find the relation between $l_{r}^{\prime
1}=l_{GHr}^{\prime 1}$ and $l_{r}^{1}=l_{EFr}^{1}=l_{0}$ as the Lorentz
''contraction'' of the moving rod in the ''r'' base, 
\begin{equation}
l_{r}^{\prime 1}=x_{Hr}^{\prime 1}-x_{Gr}^{\prime
1}=l_{0}/K=(1/K)(x_{Fr}^{1}-x_{Er}^{1}),  \label{contra1}
\end{equation}
with $\,x_{Hr}^{\prime 0}=x_{Gr}^{\prime 0}$ $\,$and $x_{Fr}^{0}=x_{Er}^{0}.$
In contrast to the ''e'' coordinatization we find that in the ''r'' base
there is a length dilatation $\infty \succ l_{r}^{\prime 1}\succ l_{0}$ for $%
-1/2\prec \beta _{r}\prec 0$ and the standard ''length contraction'' $%
l_{0}\succ l_{r}^{\prime 1}\succ 0$ for positive $\beta _{r},$ which clearly
shows that the ''Lorentz contraction'' is not physically correctly defined
transformation.

We see from the preceding discussion that - \textit{the Lorentz contraction
is the transformation connecting different quantities (in 4D spacetime) in
different IFRs and different coordinatizations, and also it changes the
infinitesimal spacetime distance }$ds$\textit{\ and consequently the
pseudo-Euclidean geometry of the 4D spacetime. Such characteristics of the
Lorentz contraction as the coordinate transformation clearly show that the
Lorentz contraction belongs to - the AT. }In the same way\textit{\ }one can
show that the usual ''time dilatation'' does have the same characteristics
as the Lorentz contraction, i.e., that it is also - an AT, but this will not
be done here.

Although the Lorentz contraction is an AT it is still widely used in
numerous textbooks and papers as an ''important relativistic effect.'' Thus,
for example, it is almost generally accepted in ultra-relativistic nuclear
collisions, see, e.g., \cite{geiger}: ''that in the center-of-mass frame two
highly Lorentz contracted nuclei pass through each other .... .'' In the
recent paper \cite{pardy} it is supposed that: ''... \v {C}erenkov radiation
of the charged two-particle system involves the Lorentz contraction of their
rest distance.'' An experiment, based on this idea, is suggested in \cite
{pardy} for the verification of the Lorentz contraction. Moreover it is
argued in \cite{doder} that the authors have experimentally succeeded to
observe the Lorentz contraction of magnetic flux quanta (vortices) in
Josephson tunnel junction. In all these examples it is understood that a
Lorentz boost transforms the rest length to the contracted length. But, as
it is already explained above, a Lorentz boost is a TT transforming from an
IFR $S$ to another IFR $S^{\prime }$, e.g., all four coordinates as a
4-vector; the same events are considered in $S$ and $S^{\prime }$. Also, a
Lorentz boost transforms a physical quantity represented by a 4D spacetime
tensor, e.g., $Q(x)$ in $S$, to $Q^{\prime }(x^{\prime })$ in $S^{\prime }$,
thus again considering the same quantity in $S$ and $S^{\prime }$. On the
contrary, as already said, in the Lorentz contraction, Eqs.(\ref{contr}) and
(\ref{contra1}), the time component is not transformed and the Lorentz
contraction is an AT from the relativity viewpoint, which has nothing in
common with a Lorentz boost as a TT. In the ''TT relativity'' one cannot say
that the nucleus must contract (as argued in literature on
ultra-relativistic nuclear collisions), or that the rest distance between
two charged particles undergoes the Lorentz contraction (as considered in 
\cite{pardy}), or that the vortices contract when moving (as argued to be
proved in experiments \cite{doder}), since the Lorentz contraction is
certainly not a relativistic relation, i.e., the relation belonging to the
''TT relativity,'' and cannot be used either to illustrate or to test any
part of the ''TT relativity.''

The above discussion reveals the main differences between \textit{the
spacetime length} considered in the ''TT relativity'' and \textit{the
spatial length} considered in the ''AT relativity.'' The same example (a rod
at rest in $S$) is investigated in the ''TT relativity,'' Sec.2.1, and in
the ''AT relativity,'' this section. The ''TT relativity'' deals with 4D
quantities in 4D spacetime. We associate with the mentioned rod a 4D
quantity - a distance 4-vector $l_{AB}^{a},$ and consider this quantity $%
l_{AB}^{a}$ in two IFRs $S$ and $S^{\prime }$ (in which the rod is moving)
and in two coordinatizations, ''e'' and ''r''. Four different
decompositions, representations, are found for the same $l_{AB}^{a}.$
Different representations of $l_{AB}^{a}$ in $S$ and $S^{\prime }$ are
connected by the TT - the LT. In terms of $l_{AB}^{a}$ \textit{the spacetime
length} $l$ (\ref{covlen}) is constructed and it \textit{does have the same
value for all four representations of} $l_{AB}^{a}.$ An essentially
different treatment of that rod is performed in the ''AT relativity.'' This
form of relativity does not deal with 4D quantities in 4D spacetime. In the
''AT relativity'' we associate with that rod \textit{four different spatial
lengths, i.e., four different 3D quantities in 4D spacetime}; in the ''e''
base they are $l_{ABe}^{1}$ in $S$ and $l_{CDe}^{\prime 1}$ in $S^{\prime },$
and in the ''r'' base they are $l_{EFr}^{1}$ in $S$ and $l_{GHr}^{\prime 1}$
in $S^{\prime }.$ The quantities in the same base but in different IFRs are
connected by the AT - the Lorentz ''contraction''; $l_{ABe}^{1}$ and $%
l_{CDe}^{\prime 1}$ with (\ref{contr}), and $l_{EFr}^{1}$ and $%
l_{GHr}^{\prime 1}$ with (\ref{contra1}). None of these quantities is well
defined in 4D spacetime. We conclude from the whole previous consideration
that \textit{when the 4D structure of our spacetime is correctly taken into
account then there is no place for the Lorentz contraction formulae, and
only the spacetime length and the spacetime quantities are well defined
quantities. }\newline
\newline
\newline
\textbf{2.3. ''Car and garage paradox''}\newline
\newline

In the previous sections we have examined the main characteristics of both
forms of relativity. Now we want to show the difference between the
treatments of relatively moving systems in the ''AT relativity'' and the
''TT relativity.'' Usually such systems are treated in a noncovariant
manner, i.e., in the ''AT relativity,'' but here we shall present the
treatment of such systems in a manifestly covariant manner, i.e., in the
''TT relativity.'' In order to see the differences between both treatments
we do not need to work completely in geometrical terms, but we can choose
some specific coordinatization, e.g., the simplest one, the ''e''
coordinatization. Therefore, in the following, we restrict ourselves to the
''e'' coordinatization and, for simplicity in notation, we omit the
subscript -e- in all quantities. However, we shall often write the important
relations in geometrical terms, and also we shall explain which results and
conclusions are independent of the chosen coordinatization.

As already discussed, if in an IFR $S^{\prime }$ in which the time component
of the distance 4-vector $l_{AB}^{\prime \mu }$, i.e., $l_{AB}^{\prime 0},$
is zero (simultaneously determined events $A$ and $B$), then $l_{AB}^{\prime
\mu }$ comprises only spatial components. Hence, in $S^{\prime }$ the
invariant spacetime length $l^{\prime }$ is given as the usual 3D distance
between $A$ and $B$. But, in such a case, $l_{AB}^{\mu }$ in the rest frame $%
S$ of the object does have $l_{AB}^{0}\neq 0$, and the spacetime length $l$
in $S$ (it is $=l^{\prime }$) could take different values depending on the
chosen IFR $S^{\prime }$. Such an arbitrariness in $l$, although not
forbidden by any physical law, would complicate both theory and experiment.
Furhermore, one wants to retain the connection with the prerelativistic
concept of the spatial length. Therefore, the most convenient choice for the
frame in which the time component $l_{AB}^{\prime 0}$ of $l_{AB}^{\prime \mu
}$ is zero is the $S$ frame. Thus in $S$ the position 4-vectors $%
x_{A,B}^{\mu }$ are determined simultaneously, $%
x_{B}^{0}-x_{A}^{0}=l_{AB}^{0}=0,$ and the spacetime length $l$ becomes in $%
S,$ the rest frame of the object, the rest spatial length $l_{0},$ i.e., $%
l=(l_{AB}^{i}l_{ABi})^{1/2}=l_{0}$, as in the prerelativistic physics. The
observers in all other IFRs will look at the same events but associating
with them different coordinates; they all find (measure) the same value $%
l=l_{0}$ for the spacetime length. Note, as we have mentioned, that if $%
l_{AB}^{0}=0$ then $l_{AB}^{\prime \mu }$ in any other IFR $S^{\prime }$
will contain the time component $l_{AB}^{\prime 0}\neq 0$. We once again
emphasize that the choice of the rest frame of the object as the starting
frame for the consideration is not dictated by physical requirements. This
choice is, in fact, determined only by our desire to have a quantity that
corresponds in the 4D spacetime to the prerelativistic spatial length. As
already said at the end of Sec.2.1 the usual 3D length of a moving object
cannot be defined in the 4D spacetime, i.e., in the ''TT relativity'', in an
adequate way. Only the spacetime length $l$ does have a definite theoretical
and experimental meaning and it is an invariant quantity. This holds for all
possible synchronizations. With our choice of the rest frame of the object
as the starting frame, i.e., with $l=l_{0},$ the spatial rest length
determined simultaneously in $S$ obtains the properties of the spacetime
length $l.$ Then, the coordinate measurements of $x_{A,B}^{\prime \mu }$ in
an IFR $S^{\prime }$ in which an object is moving are not of interest in
their own right but they have to enable one to find the rest spatial length $%
l_{0}$. In the prerelativistic ''3+1'' picture, and in the ''AT
relativity,'' one can compare \textit{the spatial lengths} of two relatively
moving objects. But in the ''TT relativity'' \textit{the spacetime lengths}
(that contain both spatial and temporal parts) are well defined quantities
in 4D spacetime and they, or the rest spatial lengths, can be compared in a
physically meaningful way.

Let us illustrate the preceding discussion considering the well-known ''Car
and garage paradox'' (see, e.g., \cite{Wald}, p.9). The common assertion
about this ''paradox'' is that it comes out due to, \cite{Wald}: ''The lack
of a notion of absolute simultaneity in special relativity ....,'' and
consequently due to the relativity of the Lorentz contraction. However, as
discussed above, the relativity of simultaneity is a coordinate dependent
effect and, for example, for $\varepsilon =0$, \cite{Leub}, the absolute
simultaneity is preserved. Also, the Lorentz contraction is an AT and it has
nothing to do with the 4D pseudo-Euclidean geometry of the special
relativity. Therefore, we discuss this ''paradox'' using covariant 4D
quantities, i.e., in the ''TT relativity''. But for our purposes, as it is
already explained, there is no need to discuss the ''paradox'' in
geometrical terms, than it can be considered in the inertial coordinate
systems with the Minkowski metric tensors, that is in the ''e''
coordinatization. The frame in which a garage is at rest is denoted by $S$
while that one in which a car is at rest by $S^{\prime }$. The unprimed
quantities are in $S$ and the primed ones in $S^{\prime }$. Instead of 4D
spacetime we work here with 2D spacetime. The garage and the car are along
the common $x^{1},x^{\prime 1}-$ axes, and they are of equal proper lengths $%
l_{g}=l_{c}^{\prime }=l$. First we consider the ''paradox'' in the $S$
frame. The distance 4-vector $l_{g}^{\mu }$ for the garage is determined
directly in $S$. However the distance 4-vector $l_{c}^{\mu }$ for the moving
car in $S$ has to be determined in such a way that one first finds it in the
car own rest frame $S^{\prime }$, and then transforms it by the LT to the
rest frame of the garage $S$. This procedure follows from the preceding
general discussion where it is shown that only the rest length, i.e., the
invariant spacetime length, of a moving object is well defined quantity in
the ''TT relativity''. The position 4-vectors of the spatial end points $A$
and $B$ of the garage in $S$ are taken to be $x_{Ag}^{\mu }=(0,0)$ and $%
x_{Bg}^{\mu }=(0,l),$ and the distance 4-vector is $l_{g}^{\mu }=x_{Bg}^{\mu
}-x_{Ag}^{\mu }=(0,l).$ (In the remaining part of this paper we shall use $%
A,B,C,D,...$ and $A^{\prime },B^{\prime },...$ to denote the spatial points
on the (moving) object.) The invariant spacetime length of the garage is $%
l_{g}=(l_{g}^{\mu }l_{\mu g})^{1/2}=l.$ The frame $S^{\prime }$, together
with the car, moves relative to $S$ with 4-velocity $v^{\mu }=(\gamma
c,\gamma V),$ $\gamma =(1-\beta ^{2})^{1/2},\beta =V/c.$ Let the origin of $%
S^{\prime }$ (with the left end of the car $A^{\prime }$ attached to it)
coincides with the origin of $S$ (with the left end of the garage $A$
attached to it) at the moment $t=t^{\prime }=0$. The position 4-vectors of
the end points $A^{\prime }$ and $B^{\prime }$ of the car in $S^{\prime }$
are $x_{A^{\prime }c}^{\prime \mu }=(0,0)$ and $x_{B^{\prime }c}^{\prime \mu
}=(0,l)$, and the distance 4-vector is $l_{c}^{\prime \mu }=x_{B^{\prime
}c}^{\prime \mu }-x_{A^{\prime }c}^{\prime \mu }=(0,l)$. The invariant
spacetime length of the car is $l_{c}^{\prime }=(l_{c}^{^{\prime }\mu
}l_{\mu c}^{\prime })^{1/2}=l$. To find the position 4-vector of the right
end $B^{\prime }$ of the car and the distance 4-vector of the car in $S$ one
applies the LT to $x_{B^{\prime }c}^{\prime \mu }$ and $l_{c}^{\prime \mu }$%
. Then one finds $x_{B^{\prime }c}^{\mu }=l_{c}^{\mu }=(\gamma \beta
l,\gamma l)$. The spacetime length of the car is now determined in $S$, and
is as before $l_{c}=l$, equal to its rest length. Obviously, in the ''TT
relativity'' nothing could be said whether the \textit{moving }car fit into
the garage or not, as the comparison of the spatial parts of the distance
4-vectors for the garage and the car has no physical meaning. (Note that
there is a \textit{dilatation }of the spatial part of the moving car, and
not the Lorentz contraction as in the ''AT relativity''.) Only if both
objects are relatively at rest their spatial parts can be compared in
physically meaningful sense. The same conclusions hold from the point of
view of the observer in $S^{\prime }$, or any other IFR. These conclusions
hold not only for Einstein's coordinatization (which is used here) but for
any other possible coordinatization of IFRs.\newline
\newline
\newline
\textbf{3. CURRENT-CARRYING CONDUCTOR AND EXTERNAL ELECTRIC FIELDS IN THE
''TT RELATIVITY''}\newline
\newline

Let us now apply these ideas to the consideration of a CCC in the ''TT
relativity''. An infinite straight wire with a steady current is situated
along the $x^{1}$ axis. A current is flowing in $-x^{1}$ direction and
accordingly electrons move in $+x^{1}$ direction. We suppose that positive
and negative charge densities are of equal magnitude when both subsystems
are relatively at rest, i.e., before a current is established in the wire.
In a CCC the wire (i.e., the ions) is supposed to be at rest in $S$, while
the electrons are at rest in $S^{\prime }$.

Before determining the current density 4-vectors $j^{\mu }$ in $S$ and $%
S^{\prime }$ we give the manifestly covariant definition of a charge within
a boundary $\delta H$ of an arbitrary hypersurface $H$ (see, e.g., \cite
{Bilic}, \cite{Baroni}), 
\begin{equation}
Q_{\delta H}=\int_{H}j^{\mu }d\sigma _{\mu ,}  \label{charge}
\end{equation}
where $d\sigma ^{\mu }$ is the 4-vector of an element of the hypersurface $H$%
. (This expression can be written in a more general form, i.e., in
geometrical terms, replacing Greek index $\mu $ by the abstract index $a.$)
The invariance of charge defined by (\ref{charge}) is proved in \cite{Bilic}
for a linear CCC, and also for the general case of 4D spacetime in \cite
{Baroni}. If the hypersurface $H$ is chosen in the rest frame of charges in
such a way that it is the space-like plane $t=const.,$ then the charge $dQ$
is given as in the prerelativistic physics $dQ=\rho dV$; the 4-vector $%
j^{\mu }$ has only the time component $j^{0}=c\rho $, since we are in the
rest frame of charges, and $d\sigma _{\mu }$ also has only time component
which is the synchronously defined rest volume $dV$. Consequently, the
charge density $\rho $ is defined as the ratio of $dQ$ given by (\ref{charge}%
), but taken simultaneously in the rest frame of the charges, and the
synchronously defined rest volume $dV,$ $\rho =dQ/dV,$ and it is well
defined quantity from the ''TT relativity'' viewpoint. Obviously the charge
density has the common prerelativistic meaning only in the rest frame of the
charges. The charge density of moving charges is not a well defined quantity
from the ''TT relativity'' point of view in the same way as the spatial
length or the volume of a moving object are not correctly defined quantities
in the ''TT relativity''. This is in contrast with the ''AT relativity'' and
the Lorentz contraction, where the charge density of the moving charges is
defined; it is enhanced by $\gamma =(1-\beta ^{2})^{1/2}$ relative to the
proper charge density due to the Lorentz contraction of the moving volume.
Thereby, when determining the current density 4-vector $j^{\mu }$ in some
IFR in which the charges are moving one first has to find that vector in the
rest frame of the charges, where the space component $\mathbf{j}=0$ and $%
\gamma =1,$ and then to transform by the LT so determined $j^{\mu }$ to the
considered IFR. According to this consideration the simplest and the correct
way, from the ''TT relativity'' viewpoint, to determine the current density
4-vector $j^{\mu }$ in some IFR for a CCC is the following: The current
density 4-vectors $j_{+}^{\mu }$ and $j_{-}^{\mu }$ for positive and
negative charges, respectively, have to be determined in their rest frames
and then transformed by the LT to the given IFR. It has to be noted that in
the ''TT relativity'' it is not necessary to determine $j^{\mu }$ for a CCC,
in an arbitrary IFR, in the mentioned way. In that frame we could start in (%
\ref{charge}) with an arbitrary space-like hypersurface $H$ and determine $%
j^{a}$ and $d\sigma ^{a}$ in some coordinatization that is different than
the ''e'' coordinatization. But then we loose the connection with the
prerelativistic notions, the charge density $\rho ,$ the current density $%
\mathbf{j}$ (3-vector), the spatial length and volume, etc., and with the
prerelativistic relation $dQ=\rho dV.$\newline
\newline
\newline
\textbf{3.1. The current density $j^{\mu }$ in the ions' rest frame S}%
\newline

Hence, the current density 4-vector in $S$, for the considered wire with
current, is $j^{\mu }=j_{+}^{\mu }+j_{-}^{\mu },$ where $j_{+}^{\mu }=(c\rho
_{0},0).$ The positive charge density $\rho _{+}$ is $=\rho _{0},$ where $%
\rho _{0}$ is the positive charge density for the wire at rest but without a
current. To find $j_{-}^{\mu }$ in $S$ one has, as already said, to find the
electrons' charge density $\rho _{-}^{\prime }$, and the current density
4-vector of the electrons $j_{-}^{\prime \mu }$ in their rest frame $%
S^{\prime }$, where $\rho _{-}^{\prime }$ is well defined quantity, and then
to transform them to the ions' rest frame $S$. In the rest wire, but without
a current, the charge density of the electrons, which are at rest there, is $%
-\rho _{0}$. Then, it follows from the previous consideration that in $%
S^{\prime }$, where the electrons in that wire, but with a current, are at
rest, the proper charge density $\rho _{-}^{\prime }$ of the electrons must
again be equal to $-\rho _{0},$ i.e., 
\begin{equation}
\rho _{-}^{\prime }=-\rho _{0},\quad j_{-}^{\prime \mu }=(-c\rho _{0},0).
\label{ronula}
\end{equation}
By means of (\ref{ronula}) and the LT we find the current densities in $S$
as 
\begin{equation}
j_{-}^{\mu }=(-c\gamma \rho _{0},-c\gamma \beta \rho _{0}),\quad j^{\mu
}=(c(1-\gamma )\rho _{0},-c\gamma \beta \rho _{0}).  \label{jotmi}
\end{equation}
Eqs. (\ref{ronula}) and (\ref{jotmi}) are in contrast to all previous works
from the time of Clausius, Clausius hypothesis, see \cite{Edwards}, until
today. In the Clausius hypothesis \textit{it is simply supposed} that in the
ions' rest frame $S$ the charge density of the moving electrons $\rho
_{-}=-\rho _{0}.$ However the same equations were already obtained in \cite
{ivezpla}, where the ''AT relativity'' with the Lorentz contraction is used.
This may seem surprising that the same equations exist in \cite{ivezpla}
(with the ''AT relativity'') and here, where the ''TT relativity'' is
considered and thus only the covariant quantities are used. But, we must
note that the results obtained in \cite{ivezpla} are not actually based on
the AT, i.e., on the Lorentz contraction, than on the \textit{assumption}
that in the electrons' rest frame $S^{\prime }$ the electrons' charge
density $\rho _{-}^{\prime }$ is $=-\rho _{0}.$ In a covariant approach,
i.e., in the ''TT relativity,'' Eq. (\ref{ronula}) is neither \textit{%
hypothesis} (as in the traditional approach) nor the \textit{assumption} (as
in \cite{ivezpla}), but it is a consequence of the covariant definition of
an invariant charge (\ref{charge}) and of the invariance of the rest length,
i.e., it resulted from the use of correctly defined covariant quantities.%
\newline
\newline
\newline
\textbf{3.2. The $F^{\alpha \beta }$ and the $E^{\alpha },B^{\alpha }$
formulations of electrodynamics}\newline
\newline

Having determined the sources $j^{\mu }$ we find the electric and magnetic
fields for that infinite wire with current. One way is to start with the
covariant Maxwell equations with $F^{\alpha \beta }$ and its dual $%
^{*}F^{\alpha \beta }$ 
\begin{equation}
\partial _{\alpha }F^{a\beta }=-j^{\beta }/\varepsilon _{0}c,\quad \partial
_{\alpha }\ ^{*}F^{\alpha \beta }=0  \label{covef}
\end{equation}
where $^{*}F^{\alpha \beta }=-(1/2)\varepsilon ^{\alpha \beta \gamma \delta
}F_{\gamma \delta }$ and $\varepsilon ^{\alpha \beta \gamma \delta }$ is the
totally skew-symmetric Levi-Civita pseudotensor. In such a covariant
formulation F$^{\alpha \beta }$ is the primary quantity; it is the solution
of (\ref{covef}), or the corresponding wave equation 
\begin{equation}
\partial ^{\sigma }\partial _{\sigma }F_{\alpha \beta }-(1/\varepsilon
_{0}c)(\partial _{\beta }j_{\alpha }-\partial _{\alpha }j_{\beta })=0,
\label{wave}
\end{equation}
and it conveys all the information about the electromagnetic field. There is
no need to introduce either the intermediate electromagnetic 4-potential $%
A^{\mu }$ or the connection of the components of $F^{\alpha \beta }$ with
the usual 3-vectors $\mathbf{E}$ and $\mathbf{B.}$ The general solution in
the retarded representation of (\ref{wave}) or (\ref{covef}) is 
\begin{equation}
F^{\alpha \beta }(x^{\mu })=(2k/i\pi c)\int \left\{ \frac{\left[ j^{\alpha
}(x^{\prime \mu })(x-x^{\prime })^{\beta }-j^{\beta }(x^{\prime \mu
})(x-x^{\prime })^{\alpha }\right] }{\left[ (x-x^{\prime })^{\sigma
}(x-x^{\prime })_{\sigma }\right] ^{2}}\right\} d^{4}x^{\prime },
\label{elten}
\end{equation}
where $x^{\alpha },x^{\prime \alpha }$ are the position 4-vectors of the
field point and the source point respectively, $k=1/4\pi \varepsilon _{0}.$
After transforming by the LT (\ref{covef}) to the $S^{\prime }$ frame one
finds the same equations with primed quantities replacing the unprimed ones,
since the transformations of all quantities in (\ref{covef}) are -- the TT.

Instead of to work with $F^{\alpha \beta }$- formulation one can
equivalently use the $E^{\alpha },B^{\alpha }$- formulation, which is
presented in \cite{ivezic} and \cite{ivsc98}. It is shown there that in the
''TT relativity'' one has to use the 4-vectors $E^{\alpha }$ and $B^{\alpha
} $ instead of the usual 3-vectors $\mathbf{E}$ and $\mathbf{B}.$ The usual
transformations of $\mathbf{E}$ and $\mathbf{B,}$ obtained by the
identification of the components of $\mathbf{E}$ and $\mathbf{B}$ with the
components of $F^{\alpha \beta }$ ($E_{i}=F^{0i}$ and $B_{i}=^{*}F^{0i}$),
are shown to be the AT referring to the same measurements in different IFRs
and not to the same quantity, see \cite{ivezic} and \cite{ivsc98}. In that
way it is found in \cite{ivezic} (and \cite{ivsc98}) that contrary to the
common belief the usual noncovariant formulation with the 3-vectors $\mathbf{%
E}$ and $\mathbf{B}$ is not equivalent to the covariant formulations. $%
E^{\alpha }$ and $B^{\alpha }$ are determined by the covariant Maxwell
equations derived in \cite{ivezic} (and \cite{ivsc98}), 
\begin{eqnarray}
\partial _{\alpha }(\delta _{\mu \nu }^{\alpha \beta }v^{\mu }E^{\nu
})+c\partial _{\alpha }(\varepsilon ^{\alpha \beta \mu \nu }B_{\mu }v_{\nu
}) &=&-j^{\beta }/\varepsilon _{0},  \nonumber \\
\partial _{\alpha }(\delta _{\mu \nu }^{\alpha \beta }v^{\mu }B^{\nu
})+(1/c)\partial _{\alpha }(\varepsilon ^{\alpha \beta \mu \nu }v_{\mu
}E_{\nu }) &=&0,  \label{maeb}
\end{eqnarray}
where $E^{\alpha }$ and $B^{\alpha }$ are the electric and magnetic field
4-vectors measured by a family of observers moving with 4-velocity $v^{\mu }$%
, and $\delta _{\mu \nu }^{\alpha \beta }=\delta _{\mu }^{\alpha }\delta
_{\nu }^{\beta }-\delta _{\nu }^{\alpha }\delta _{\mu }^{\beta }.$ For the
given sources $j^{\mu }$ one could solve these equations and find the
general solutions for $E^{\alpha }$ and $B^{\alpha }.$

We note that it is possible to write the equations (\ref{maeb}) in a
somewhat simpler form, the covariant Majorana form, introducing $\Psi
^{\alpha }=E^{\alpha }-icB^{\alpha }.$ Then the covariant Majorana form of
Maxwell's equations becomes 
\begin{equation}
(\gamma ^{\mu })^{\beta }\ _{\alpha }\partial _{\mu }\Psi ^{\alpha
}=-j^{\beta }/\varepsilon _{0},  \label{Major}
\end{equation}
where the $\gamma $-matrices are 
\begin{equation}
(\gamma ^{\mu })^{\beta }\ _{\alpha }=\delta _{\rho \gamma }^{\mu \beta
}v^{\rho }g_{\alpha }^{\gamma }+i\varepsilon ^{\mu \beta }\ _{\alpha \gamma
}v^{\gamma }.  \label{gama}
\end{equation}
In the case that $j^{\mu }=0$ Eq. (\ref{Major}) becomes Dirac-like
relativistic wave equation for free photons 
\begin{equation}
(\gamma ^{\mu })^{\beta }\ _{\alpha }\partial _{\mu }\Psi ^{\alpha }=0.
\label{gam1}
\end{equation}
We shall not further discuss the covariant Majorana formulation since it
will be reported elsewhere.\newline
\newline
\newline
\textbf{3.3. $E^{\alpha }$ for a CCC in the ''TT relativity''}\newline
\newline

Instead of to solve (\ref{maeb}) or (\ref{Major}) to find $E^{\alpha }$ and $%
B^{\alpha }$ for a CCC we first find $F^{\alpha \beta }$ from (\ref{elten})
inserting into it $j^{\mu }$ from (\ref{jotmi}) and performing the
integration. Then we use the relations given in \cite{ivezic} (and \cite
{ivsc98}), which connect $F^{\alpha \beta }$ and $E^{\alpha },B^{\alpha }$-
covariant formulations, 
\begin{equation}
E^{\alpha }=(1/c)F^{\alpha \beta }v_{\beta ,}\quad B^{\alpha
}=(1/c^{2})^{*}F^{\alpha \beta }v_{\beta }.  \label{vezaEF}
\end{equation}
The inverse relations connecting the $E^{\alpha },B^{\alpha }$ and $%
F^{\alpha \beta }$- covariant formulations are also given in \cite{ivezic}
(and \cite{ivsc98}) and they are 
\begin{equation}
F^{\alpha \beta }=(1/c)\delta _{\mu \nu }^{\alpha \beta }v^{\mu }E^{\nu
}+\varepsilon ^{\alpha \beta \mu \nu }B_{\mu }v_{\nu },\ ^{*}F^{\alpha \beta
}=\delta _{\mu \nu }^{\alpha \beta }v^{\mu }B^{\nu }+(1/c)\varepsilon
^{\alpha \beta \mu \nu }v_{\mu }E_{\nu }.  \label{vezFE}
\end{equation}
Taking that the family of observers who measures $E^{\alpha }$ is at rest in
the S frame, i.e., that $v_{\mu }=(-c,0)$ one finds from (\ref{vezaEF}) that 
$E^{0}=0,\ E^{i}=F^{0i}$ whence 
\begin{equation}
E^{1}=0,\ E^{2}=2k(1-\gamma )\rho _{0}y(y^{2}+z^{2})^{-1},\
E^{3}=2k(1-\gamma )\rho _{0}z(y^{2}+z^{2})^{-1}.  \label{eovi}
\end{equation}
\textit{The equation} (\ref{eovi}) \textit{shows that the observer who is at
rest relative to a wire with steady current will see, i.e., measure, the
second order electric field outside such a CCC}. Thus the result which is
for such fields predicted on different grounds in \cite{ivezpla} is proved
to be correct in the ''TT relativity'' too, i.e., when all quantities are
treated in a covariant manner, see the discussion at the end of Sec.3.1. We
thus find that these electric fields naturally come out in the ''TT
relativity'' treatment of physical systems consisting of relatively moving
subsystems. Such fields were not searched for and, it seems, were not
observed earlier due to their extreme smallness. (To be more precise, the
similar second-order electric fields ($\propto v^{2}/c^{2}$) have been
detected in \cite{Edwards} and \cite{Edwar2}, but it is not sure that they
are caused by the effect predicted in \cite{ivezpla} and here.) However, I
suppose that such fields must play an important role in many physical
phenomena with steady currents, particularly in tokamaks and astrophysics,
where high currents exist, and in superconductors, where the electric fields
of zeroth order outside CCCs are absent, (see \cite{ivepra}).

Similarly, the magnetic field 4-vector $B^{\mu }$ can be also obtained from (%
\ref{vezaEF}) and the expression for $F^{\alpha \beta }$ (\ref{elten}). For
the observers with $v_{\mu }=(-c,0)$ one finds $B^{0}=0$ and $%
B^{i}=(-1/c)^{*}F^{i0}.$ In terms of the known $^{*}F^{i0}$ we find for $%
B^{i}$ the usual expression for the magnetic field of an infinite straight
wire with current, (only the current density is $\gamma $ times bigger).%
\newline
\newline
\newline
\textbf{4. CHARGES ON A CURRENT LOOP IN THE ''TT RELATIVITY''}\newline
\newline

In this section we discuss the macroscopic charge of a square loop with a
steady current in the ''TT relativity''. This is already discussed in
numerous previous works but from the ''AT relativity'' viewpoint by using
the synchronous definition of length and the Lorentz contraction. Let a
square loop is at rest in the $x,y-$plane in an IFR $S$. The spatial $x,y-$%
coordinates of the corners are: $A(0,0),B(1,0),E(1,1),F(0,1).$ The electrons
move from $A$ to $B$ in the $\overline{AB}$ side. We first consider the
charge in the $\overline{AB}$ side in two frames; in $S$, the ions' rest
frame, and $S_{AB}^{\prime }$, the electrons' rest frame. The evaluation of
the charge in the $\overline{AB}$ side in $S$ and $S_{AB}^{\prime }$ using (%
\ref{charge}) is already correctly performed in \cite{Bilic}. However there
is an important difference between the calculation of $Q_{AB}$ in \cite
{Bilic} and in this paper. In \cite{Bilic} the charge density of the
electrons in $S_{AB}^{\prime }$ is not specified but simply taken to have
some undetermined value $-\overline{\lambda }$ and its value in $S$ is found
by means of the LT. The covariance of the definition (\ref{charge}) (the
Lorentz scalar) will yield that $Q_{AB}$ in $S$ is equal to $Q_{AB}^{\prime
} $ in $S_{AB}^{\prime }$ for any choice of $-\overline{\lambda }$. Our
discussion of an infinite wire with a current reveals that $\rho
_{-}^{\prime }$ and $j_{-}^{\prime \mu }$ in $S_{AB}^{\prime }$ are
determined by (\ref{ronula}). Then $j_{-}^{\mu }$ in $S$ is obtained by the
LT, and it is given by (\ref{jotmi}). Hence, we find from (\ref{charge})
that 
\begin{equation}
Q_{AB}=(1/c)\int_{0}^{l}(j_{+}^{0}+j_{-}^{0})dx=(1-\gamma )\rho _{0}l
\label{charAB}
\end{equation}
This result is already found in \cite{ivezpla}. Similarly, to find $%
Q_{AB}^{\prime }$ in $S_{AB}^{\prime }$ one first determines $j_{+}^{\mu }$
in $S,$ $j_{+}^{\mu }=(c\rho _{0},0),$ and then by the LT one obtains $%
j_{+}^{\prime \mu }=(c\gamma \rho _{0},-c\gamma \beta \rho _{0})$ in $%
S_{AB}^{\prime }$. $Q_{AB}^{\prime }$, for the moving loop in $%
S_{AB}^{\prime }$, is found from (\ref{charge}) in the same way as in \cite
{Bilic} and it is, of course, equal to $Q_{AB}$. The evaluation of $Q_{EF}$
in the $\overline{EF}$ side, which is parallel to the $\overline{AB}$ side,
proceeds in the same way as for $Q_{AB}$. However the relative velocity of
the $S_{EF}^{\prime }$ frame, the rest frame for the electrons in the $%
\overline{EF}$ side, and the $S$ frame is now $v^{\mu }=(\gamma c,-\gamma v)$%
. Hence, $j_{-}^{\mu }$ is now $j_{-}^{\mu }=(-c\gamma \rho _{0},c\gamma
\beta \rho _{0})$ and 
\begin{equation}
Q_{EF}=(1/c)\int_{l}^{0}(j_{+}^{0}+j_{-}^{0})dx=-Q_{AB}.  \label{charEF}
\end{equation}
Thus we find that there are charges $Q_{AB}$ and $-Q_{AB}$ on the sides $%
\overline{AB}$ and $\overline{EF}$, respectively, in the $S$ frame in which
the loop is at rest. According to (\ref{charge}) $Q_{EF}$ is an invariant
charge, hence $Q_{EF}^{\prime }$ for the moving loop is $=Q_{EF}=-Q_{AB}$.
Moreover, it can be immediately concluded that the charge $Q_{BE}$, in $S$,
in the vertical side $\overline{BE}$must be the same as $Q_{AB}$; it is the
simple change of the spatial axes x and y. Thus $Q_{BE}=Q_{AB}$. Similarly
the charge $Q_{FA}$, in $S$, in the vertical side $\overline{FA}$ is $%
Q_{FA}=-Q_{AB}$. The total charge in $S$, the rest frame of the loop with
current, is zero $Q=Q_{AB}+Q_{BE}+Q_{EF}+Q_{FA}=0$, as it must be. It
remains zero in every IFR since the charges in all sides are invariant
charges according to (\ref{charge}). Thus, we find the same behaviour for a
moving loop with current and for the same loop but at rest in a given IFR.

In the traditional ''AT relativity'' approach with the synchronous
definition of length and the Lorentz contraction (see, e.g., \cite{Panof})
the charges on all sides are supposed to be zero in the rest frame of a loop
with current. In the IFR in which the loop with current is moving it is
found that $Q_{AB}^{\prime }\neq Q_{AB},$ due to the Lorentz contraction,
and that the charge on the $\overline{EF}$ side is ''-'' of that one on the $%
\overline{AB}$ side. Furthermore, it is obtained that the charges on the
vertical sides of a loop with current are zero for both the loop at rest and
moving loop, since for vertical sides there is no Lorentz contraction. Such
results obtained in the common approach led the physics community (I am not
aware of any exception) to conclude that there is an electric moment $%
\mathbf{P}$ for a moving loop with current, (see \cite{Panof}, Eq.(18-58)).
The appearance of this ''relativistic'' effect and its consequences are
discussed in numerous papers and books. Contrary to all these works we find
in the ''TT relativity'' that at long distances a loop with current always
(in any IFR) behaves as a quadrupole. The existence of a quadrupole electric
moment instead of dipole electric moment will influence many previous
results from the classical electrodynamics up to the intepretation of an
electric moment for a moving neutron.\newline
\newline
\newline
\textbf{5. CONCLUSIONS}\newline
\newline

In this paper the ''TT relativity'' is consistently applied to the
investigation of relatively moving systems. We presented the expressions for
the covariantly defined spacetime length (\ref{covlen}) and for covariant 4D
Lorentz transformations (\ref{fah}) when both are written in geometrical
terms, and in the ''e'' and ''r'' coordinatizations. It is also shown that
it is not the spatial length but the spacetime length (\ref{covlen}) which
is well defined quantity from the ''TT relativity'' viewpoint; in the rest
frame of the object and when the temporal part of $l_{AB}^{a}$ is zero the
spacetime length $l$ becomes the rest spatial length of the object. Using
this result and the covariant definition of charge (\ref{charge}) the
expressions (\ref{jotmi}) for the current density 4-vectors of a CCC were
found in the ions' rest frame. Then the 4-vectors $E^{\alpha }$ and $%
B^{\alpha }$ for a CCC are determined by means of the known $F^{\alpha \beta
}$ (\ref{elten}) and the relations (\ref{vezaEF}) and (\ref{vezFE}), which
connect the $F^{\alpha b}$ and $E^{\alpha },B^{\alpha }$ formulations of
electrodynamics. This yields Eq. (\ref{eovi}), which is one of the main
results found in this paper. It shows that for the obsevers at rest in the
ions' rest frame the spatial components $E^{i}$ of $E^{\alpha }$ are
different from zero outside a CCC. The second important result, which is
also found in a completely covariant manner, i.e., in the ''TT relativity''
treatment, is the existence of invariant charges on a loop with current.
There are opposite charges on opposite sides of a square loop with current,
but the total charge of that loop is zero. These results are different from
those found in all previous works in which mainly the ''AT relativity'' is
used.\newline
\newline
\newline
\textbf{ACKNOWLEDGEMENTS}\newline
\newline
I am indebted to Prof. F. Rohrlich for reading the first version of this
paper and for useful suggestions and encouragement, and also to an anonymous
referee for useful comments.

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\bibitem{Caval}  G. Cavalleri and C. Bernasconi, \textit{Nuovo Cimento B} 
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\bibitem{anders}  R. Anderson, I. Vetharaniam, G.E. Stedman, \textit{Phys.
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\bibitem{Bilic}  N. Bili\'{c}, \textit{Phys. Lett. A} \textbf{162,} 87
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\bibitem{Baroni}  L. Baroni, E. Montanari and A.D. Pesci, \textit{Nuovo
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\bibitem{Edwards}  W.F. Edwards, C.S. Kenyon and D.K. Lemon, \textit{Phys.
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\bibitem{Edwar2}  D.K. Lemon, W.F. Edwards and C.S. Kenyon, \textit{Phys.
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\bibitem{ivepra}  T. Ivezi\'{c}, \textit{Phys. Rev. A} \textbf{44,} 2682
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\bibitem{Panof}  W.K.H. Panofsky and M. Phillips, \textit{Classical
electricity and magnetism,} 2nd edn. (Addison-Wesley, Reading, Mass., 1962).
\end{thebibliography}

\end{document}

physics0102014.extracted_bib
\begin{thebibliography}{99}
\bibitem{ivezic}  T.Ivezi\'{c}, \textit{Found. Phys. Lett}. \textbf{12}, 105
(1999).

\bibitem{rohrl1}  F. Rohrlich, \textit{Nuovo Cimento B} \textbf{45}, 76
(1966).

\bibitem{ivsc98}  T Ivezi\'{c}, preprint SCAN-9802018 (on the CERN server).

\bibitem{gamba}  A. Gamba, \textit{Am. J. Phys}. \textbf{35}, 83 (1967).

\bibitem{ivezpla}  T.Ivezi\'{c}, \textit{Phys. Lett. A} \textbf{144}, 427
(1990).

\bibitem{ivescle}  T.Ivezi\'{c}, preprint SCAN-9802017 (on the CERN server).

\bibitem{Wald}  R.M. Wald, \textit{General relativity} (The University of
Chicago Press, Chicago, 1984).

\bibitem{Leub}  C. Leubner, K. Aufinger and P. Krumm, \textit{Eur. J. Phys.} 
\textbf{13,} 170 (1992).

\bibitem{Caval}  G. Cavalleri and C. Bernasconi, \textit{Nuovo Cimento B} 
\textbf{104}, 545 (1989).

\bibitem{anders}  R. Anderson, I. Vetharaniam, G.E. Stedman, \textit{Phys.
Rep}. \textbf{295}, 93 (1998).

\bibitem{Fahn}  D.E. Fahnline, \textit{Am. J. Phys}. \textbf{50}, 818 (1982)

\bibitem{Einst}  A. Einstein, \textit{Ann. Physik} \textbf{17, 891 }(1905),
tr. by W. Perrett and G.B. Jeffery, in \textit{The principle of relativity}
(Dover, New York).

\bibitem{geiger}  K. Geiger, \textit{Phys. Rep.} \textbf{258,} 240 (1995).

\bibitem{pardy}  M. Pardy, \textit{Phys. Rev. A} \textbf{55,} 1647 (1997).

\bibitem{doder}  A. Laub, T. Doderer, S.G. Lachenmann and R.P. Huebener, 
\textit{Phys. Rev. Lett}. \textbf{75,} 1372 (1995).

\bibitem{Bilic}  N. Bili\'{c}, \textit{Phys. Lett. A} \textbf{162,} 87
(1992).

\bibitem{Baroni}  L. Baroni, E. Montanari and A.D. Pesci, \textit{Nuovo
Cimento B} \textbf{109}, 1275 (1994).

\bibitem{Edwards}  W.F. Edwards, C.S. Kenyon and D.K. Lemon, \textit{Phys.
Rev. D} \textbf{14,} 922 (1976).

\bibitem{Edwar2}  D.K. Lemon, W.F. Edwards and C.S. Kenyon, \textit{Phys.
Lett. A }\textbf{62}, 105 (1992).

\bibitem{ivepra}  T. Ivezi\'{c}, \textit{Phys. Rev. A} \textbf{44,} 2682
(1991).

\bibitem{Panof}  W.K.H. Panofsky and M. Phillips, \textit{Classical
electricity and magnetism,} 2nd edn. (Addison-Wesley, Reading, Mass., 1962).
\end{thebibliography}