COVARIANT ETHER THEORIES AND SPECIAL RELATIVITY

A.L. Kholmetskii\*\
Department of Physics, Belarus State University\
4, F. Skorina Avenue, 220080 Minsk, BELARUS

**Abstract**

The present paper defines 'covariant ether theories' as all space-time
theories that are alternative to special relativity theory (SRT), but
nevertheless satisfy the general relativity principle. A general
analysis of the properties of admissible space-time transformations
shows that the number of such 'covariant ether theories' is infinite,
and that all these theories are indistinguishable from SRT with all
experiments in space-time physics reported to date. Some new theoretical
and experimental approaches for unambiguous verification of SRT are
discussed. A class of phenomena for which SRT and covariant ether
theories (CET's) give different predictions is identified. It has been
concluded, that under modern development of experimental technique a
crucial choice between SRT and CET's can be made by means of Mössbauer
spectroscopy.

PACS 03.30+p

**\*** e-mail: <kholm@phys.bsu.unibel.by>

**1. Introduction**

Modern physics accepts two relativity principles: the special
Einstein\'s relativity principle (ERP) asserting that fundamental
physical equations do not change (they are form-invariant) under
transformations between inertial reference frames in an empty space, and
the general relativity principle (GRP) stating that fundamental physical
equations do not change their form (they are covariant) under
transformations between any frames of references. (Strictly speaking,
the GRP requires a covariance of physical equations with respect to
\"admissible\" space-time transformations, which keep the requirements
*g*~00~\>0, *g~αβ~dx^α^dx^β^*\<0; **g** is the metric tensor, and *α*,
*β*=1\...3). The GRP is one of the deepest principles of physics and it
means that any phenomenon can be described from any reference frame
\[1\]. Thus, it does not require any experimental test. The GRP
constitutes one of corner stones of modern knowledge, and we strongly
believe in its validity.

During a long time there were a widely distributed opinion that ERP is a
direct consequence of GRP in case of inertial motion in an empty space.
If it would be actually so, that there was no meaning to test ERP
experimentally. Indeed, in such a case an experimental test of ERP would
mean simultaneously a test of GRP, that seems to be senseless. However,
the ERP in not, in general, a consequence of GRP; it represents an
independent physical assumption. Only in case of pseudo-Euclidean
geometry of an empty space-time (that ERP requires) we get a
form-invariance of physical equations with respect to the Lorentz
transforms as a special inference of the covariance principle. At the
same time, from a viewpoint of formal logic we are free to suppose a
curvilinear geometry for an empty space-time in arbitrary inertial
reference frame, with \"admissible\" transformations between different
inertial frames. Omitting a question about compatibility of such a
theory with the available experimental facts, one may nevertheless state
that this theory would be in contradiction with the ERP, but in
agreement with GRP. Moreover, recent decades have witnessed the
development of ether theories of such a kind, and they turned out to be
indistinguishable from SRT in terms of experiments gathered up to date
\[1-15\]. On the other hand, for many years such covariant ether
theories (CETs) were often considered as pure mathematical games, even
as 'ether formulations of relativity' \[8\], which do not predict really
observable non-relativistic phenomena.

For a better analysis of CET's, the present paper develops a new
approach based on the difference between 'physical' and 'measured'
values (and their transformation rules) in a hypothetical curvilinear
geometry of empty space-time. This approach identifies a class of
phenomena for which SRT and covariant ether theories give unambiguously
different predictions, and suggests new experiments for qualitatively
new tests of SRT.

# 2. Geometries of Empty Space-Time

It is known that SRT establishes a pseudo-Euclidean geometry for empty
physical space-time. Thus, any alternative theories of empty space-time
should assume flat curvilinear geometry in arbitrary inertial reference
frames. This Section gives some remarks about pseudo-Euclidean and
alternative geometries of empty space-time.

In the analysis of pseudo-Euclidean and curvilinear geometries of
space-time, there is an essential methodological feature that has to be
taken into account. Although this feature was stressed many years ago by
Reichenbach (*e.g*., \[16\]), present ether theories do not take it into
account explicitly.

It may be natural to believe that in any inertial reference frame we are
able to construct a method for measurement of space and time intervals
such that the result of measurement directly gives the physical
magnitude of the corresponding interval. But strictly speaking, this is
property exclusive to pseudo-Euclidean geometry. Only in this kind of
geometry can we omit a distinction between measured and physical
space-time four-vectors \[11, 13, 14\]. That is, in general we have
measured *x*~ex~ and physical *x*~ph~ four-vectors, and only in
pseudo-Euclidean geometry do we have

$x_{\text{ph}} = x_{\text{ex}} = x_{L}$,

where *x*~L~ is the Minkowskian four-vector, subjected to the Lorentz
transformation **L** between two arbitrary inertial frames K and K':

$x_{Li} = L_{\text{ij}}x'_{L}^{j}$, *i*, *j*=0\...3. (1)

The essential property of any hypothetical curvilinear geometry of empty
space-time is the difference between measured and physical space-time
four-vectors in arbitrary inertial reference frames:
$x_{\text{ph}} \neq x_{\text{ex}}$. The necessity to distinguish the
measured and physical four-vectors can be easily demonstrated with the
Fitzgerald-Lorentz contraction hypothesis, which was first invoked to
explain the null result of the Michelson-Morley experiment. According to
this hypothesis, if a rod initially at rest in the hypothetical
'absolute' frame has the length *l*, then under motion at a constant
'absolute' speed *v* along its axis, the length of the rod becomes
$l\sqrt{1 - v^{2}/c^{2}}$. However, due to proportional contraction of
the unit scale in an attached inertial reference frame, an experimenter
in this frame measures the same length *l* as in the case *v*=0:
Fitzgerald-Lorentz contraction is not observable. Thus, we see that the
length of the rod in physical space-time is
*l*~ph~=$l\sqrt{1 - v^{2}/c^{2}}$, while the measured length is equal to
*l*~ex~=*l*, and *l*~ph~≠*l*~ex~.

One can easy demonstrate that the same conclusion is valid for time
intervals in any curvilinear geometry of an empty space-time:
*t*~ph~≠*t*~ex~. Thus, the four-vectors in physical space-time
(hereinafter 'physical four-vectors') are not in general equal to the
four-vectors, whose components are constituted from corresponding
experimentally measured space and time intervals (hereinafter 'measured
four-vectors'). Hence, in any alternative to SRT theory we have to
separately derive the transformation rules for both physical and
measured four-vectors.

Further, among curvilinear geometries, where we develop a space-time
ether theory, a physical meaning can be prescribed only to geometries
that maintain space-time homogeneity, called 'oblique-angled
geometries'. Some essential properties of these geometries are
considered in the next Section.

# 

# 3. From Pseudo-Euclidean to Oblique-Angled Geometry

Now let us formally construct an oblique-angled geometry by means of a
transformation **B** from pseudo-Euclidean geometry:

$\left( x_{\text{ph}}^{N} \right)_{i} = B_{\text{ij}}\left( x_{\text{ph}}^{E} \right)^{j}$,
(2)

where the coefficients $B_{\text{ij}}$ are some constant values and the
upper superscripts E and N signify the pseudo-Euclidean and new
oblique-angled geometry, respectively.

As shown above, in a non-Euclidean geometry the physical
$x_{\text{ph}}^{N}$ and measured $x_{\text{ex}}^{N}$space-time
four-vectors are not in general equal to each other. Hence, a
corresponding transformation from pseudo-Euclidean to oblique-angled
geometry for *x*~ex~ should be written as

$\left( x_{\text{ex}}^{N} \right)_{i} = C_{\text{ij}}\left( x_{\text{ex}}^{E} \right)^{j}$,
(3)

where, in general, the matrix **C** differs from the matrix **B** in
(2), but obviously depends on **B**.

In this connection one may ask the following question: what should be
the form of matrix **B** in order to provide the equality **C**=**E**,
where **E** is the 4×4 identity matrix? In this particular case we will
have
$\left( x_{\text{ex}}^{N} \right)_{i} = \left( x_{\text{ex}}^{E} \right)_{i}$,
and an experimenter in the oblique-angled geometry will fail to detect a
deflection of his geometry from pseudo-Euclidean.

In order to answer this question, let us introduce the conventional
measuring methods in space-time: the use of a 'unit scale' for
measurement of length, the use of 'standard clocks' for measurement of
time, and the Einstein's method for synchronization of distant clocks.
Then the magnitude of the measured length
$\left( x_{\text{ex}}^{N} \right)^{\alpha}$ in physical space-time
$x_{\text{ph}}^{N}$ is equal to the ratio
$\left( (x_{\text{ph}}^{N})^{\alpha}/(x_{\text{phu}}^{N})^{\alpha} \right)$,
where $\left( x_{\text{phu}}^{N} \right)^{\alpha}$ is the unit scale in
the oblique-angled space-time (hereinafter the Greek subscripts
correspond to three-dimensional space, *α*=1,2,3).

Now let us write a relationship between space components of
$x_{\text{ph}}^{N}$ and four-vector $x_{\text{ph}}^{E}$:

$\left( x_{\text{ph}}^{N} \right)_{\alpha} = B_{\alpha j}\left( x_{\text{ph}}^{E} \right)^{j} = B_{\alpha 0}\left( x_{\text{ph}}^{E} \right)^{0} + B_{\text{αβ}}\left( x_{\text{ph}}^{E} \right)^{\beta}$.
(4)

For unit scale $\left( x_{\text{phu}}^{N} \right)_{\alpha}$ in physical
space-time we can write the similar relation:

$\left( x_{\text{phu}}^{N} \right)_{\alpha} = B_{\alpha 0}\left( x_{\text{ph}}^{E} \right)^{0} + B_{\text{αβ}}\left( x_{\text{phu}}^{E} \right)^{\beta}$,
(5)

where $\left( x_{\text{phu}}^{E} \right)^{\alpha}$ is the corresponding
unit scale in Euclidean space. Dividing (4) by (5), one obtains:

$\frac{\left( x_{\text{ph}}^{N} \right)_{\alpha}}{\left( x_{\text{phu}}^{N} \right)_{\alpha}} = \frac{B_{\alpha 0}\left( x_{\text{ph}}^{E} \right)^{0} + B_{\text{αγ}}\left( x_{\text{ph}}^{E} \right)^{\gamma}}{B_{\alpha 0}\left( x_{\text{ph}}^{E} \right)^{0} + B_{\text{αβ}}\left( x_{\text{phu}}^{E} \right)^{\beta}}\text{.}$.
(6)

This expression allows further transformation in the particular case
*B~α~*~0~=0, taking account of the obvious equality for Euclidean space

$\left( x_{\text{ph}}^{E} \right)^{\beta}/\left( x_{\text{ph}}^{E} \right)^{\alpha} = \left( x_{\text{phu}}^{E} \right)^{\beta}/\left( x_{\text{phu}}^{E} \right)^{\alpha}$.

Hence,

$\begin{array}{r}
\frac{\left( x_{\text{ph}}^{N} \right)_{\alpha}}{\left( x_{\text{phu}}^{N} \right)_{\alpha}} = \frac{B_{\text{αγ}}\left( x_{\text{ph}}^{E} \right)^{\gamma}}{\left( x_{\text{phu}}^{E} \right)^{\alpha}\left\lbrack B_{\text{αα}} + \left( \frac{1}{\left( x_{\text{phu}}^{E} \right)^{\alpha}} \right)\sum_{\beta \neq \alpha}^{}{B_{\text{αβ}}\left( x_{\text{phu}}^{E} \right)^{\beta}} \right\rbrack} = \\
\frac{B_{\text{αγ}}\left( x_{\text{ph}}^{E} \right)^{\gamma}}{\left( x_{\text{phu}}^{E} \right)^{\alpha}\left\lbrack B_{\text{αα}} + \left( \frac{1}{\left( x_{\text{ph}}^{E} \right)^{\alpha}} \right)\sum_{\beta \neq \alpha}^{}{B_{\text{αβ}}\left( x_{\text{ph}}^{E} \right)^{\beta}} \right\rbrack} = \\
\frac{\left( x_{\text{ph}}^{E} \right)_{\alpha}\left( B_{\text{αγ}}\left( x_{\text{ph}}^{E} \right)^{\gamma} \right)}{\left( x_{\text{phu}}^{E} \right)_{\alpha}\left( B_{\text{αβ}}\left( x_{\text{ph}}^{E} \right)^{\beta} \right)} = \frac{\left( x_{\text{ph}}^{E} \right)_{\alpha}}{\left( x_{\text{phu}}^{E} \right)_{\alpha}}\text{.}
\end{array}$ (7)

Equality (7) means that the measured magnitudes
$\left( x_{\text{ex}}^{N} \right)_{\alpha}$ in $x_{\text{ph}}^{N}$
coordinates coincide with the measured magnitudes
$\left( x_{\text{ex}}^{E} \right)_{\alpha}$ in
$x_{\text{ph}}^{E} \equiv x_{L}$ coordinates, *i.e.*,

$\left( x_{\text{ex}}^{N} \right)_{\alpha} = \left( x_{\text{ex}}^{E} \right)_{\alpha} = \left( x_{L} \right)_{\alpha}$
(8)

under the condition *B~α~*~0~=0 adopted for the transformation (2).
Physically, this result signifies that the distortion of length *x^α^*
being induced by the transformation from pseudo-Euclidean to
oblique-angled geometry is not detectable experimentally under
*B~α~*~0~=0 because of proportional distortion of the 'unit scale'
$x_{u}^{\alpha}$.

Further, let us write the relationship between time components of the
four-vectors $x_{\text{ph}}^{N}$ and $x_{\text{ph}}^{E}$:

$\left( x_{\text{ph}}^{N} \right)_{0} = B_{\text{00}}\left( x_{\text{ph}}^{E} \right)^{0} + B_{0\alpha}\left( x_{\text{ph}}^{E} \right)^{\alpha}$.
(9)

For two events at a fixed spatial point
($\left( x_{\text{ph}}^{E} \right)^{\alpha}$=0)

$\left( x_{\text{ph}}^{N} \right)_{0} = B_{\text{00}}\left( x_{\text{ph}}^{E} \right)^{0}$.
(10)

Hence, the coefficient $B_{\text{00}}$ describes the change of clock
rate at a fixed spatial point under the transformation from
pseudo-Euclidean to oblique-angled geometry. Such a change takes place
for both a standard clock and a physical time interval. Therefore, the
measured time interval at a fixed spatial point is

$\left( x_{\text{ex}}^{N} \right)_{0} = \frac{1}{B_{\text{00}}}\left( x_{\text{ph}}^{N} \right)_{0}$.
(11)

For time intervals at two different spatial points separated by the
distance $\left( x_{\text{ph}}^{N} \right)_{\alpha}$, one should write

$\left( x_{\text{ex}}^{N} \right)_{0} = \frac{1}{B_{\text{00}}}\left\lbrack \left( x_{\text{ph}}^{N} \right)_{0} + \Delta\left( x_{\text{ph}}^{N} \right)_{0} \right\rbrack,$,
(12)

where $\Delta\left( x_{\text{ph}}^{N} \right)_{0}$ is the error of
synchronization of clocks separated by the distance
$\left( x_{\text{ph}}^{N} \right)_{\alpha}$ in oblique-angled
space-time. The value of $\Delta\left( x_{\text{ph}}^{N} \right)_{0}$
can be found from the equality

$\left( x_{\text{ph}}^{N} \right)_{\text{02}} = \left( x_{\text{ph}}^{N} \right)_{\text{01}}/2$
(13)

(Einstein's method of clocks synchronization), where
$\left( x_{\text{ph}}^{N} \right)_{\text{01}}$ is the time for light
propagation from the first clock Cl~1~ (at the origin of coordinates) to
the second clock Cl~2~ (at the point
$\left( x_{\text{ph}}^{N} \right)_{\alpha}$) and back according to
Cl~1~, while $\left( x_{\text{ph}}^{N} \right)_{\text{02}}$ is the
indication of Cl~2~ at the moment of arrival of the light pulse. In
oblique-angled geometry the propagation time of light from Cl~1~ to
Cl~2~ $\left( x_{\text{ph}}^{N} \right)_{0 +}$ is not equal, in general,
to the propagation time in the reverse direction
$\left( x_{\text{ph}}^{N} \right)_{0 -}$. Hence, an implementation of
the equality (13) is possible only in the case where the readings of
both clocks at the initial moment of time differ by the value
$\Delta\left( x_{\text{ph}}^{N} \right)_{0}$, and

$\begin{array}{r}
\left( x_{\text{ph}}^{N} \right)_{\text{01}} = \frac{1}{B_{\text{00}}}\left\lbrack \left( x_{\text{ph}}^{N} \right)_{0 +} + \left( x_{\text{ph}}^{N} \right)_{0 -} \right\rbrack, \\
\left( x_{\text{ph}}^{N} \right)_{\text{02}} = \frac{1}{B_{\text{00}}}\left\lbrack \left( x_{\text{ph}}^{N} \right)_{0 +} + \Delta\left( x_{\text{ph}}^{N} \right)_{0} \right\rbrack\text{.}
\end{array}$. (14)

Hence, with account of Eq. (13) we obtain:

$\Delta\left( x_{\text{ph}}^{N} \right)_{0} = \frac{1}{2}\left\lbrack \left( x_{\text{ph}}^{N} \right)_{0 -} - \left( x_{\text{ph}}^{N} \right)_{0 +} \right\rbrack$.
(15)

Expressions for $(x_{\text{ph}}^{N})_{0 +}$ and
$(x_{\text{ph}}^{N})_{0 -}$ can be found from Eq. (2):

$\begin{array}{r}
\left( x_{\text{ph}}^{N} \right)_{0 +} = B_{\text{00}}\left( x_{\text{ph}}^{E} \right)^{0} + B_{0\alpha}\left( x_{\text{ph}}^{E} \right)^{\alpha}, \\
\left( x_{\text{ph}}^{N} \right)_{0 -} = B_{\text{00}}\left( x_{\text{ph}}^{E} \right)^{0} - B_{0\alpha}\left( x_{\text{ph}}^{E} \right)^{\alpha}\text{.}
\end{array}$. (16)

Substituting Eq. (16) into Eq. (15), one gets:

$\Delta\left( x_{\text{ph}}^{N} \right)_{0} = - B_{0\alpha}\left( x_{\text{ph}}^{E} \right)^{\alpha}$.
(17)

Further substitution of Eqs. (17) & (9) into Eq. (12) gives:

$\left( x_{\text{ex}}^{N} \right)_{0} = \left( x_{\text{ph}}^{E} \right)_{0} = \left( x_{\text{ex}}^{E} \right)_{0} = \left( x_{L} \right)_{0}$.
(18)

Therefore, we conclude that for any admissible transformation **B**, an
experimenter will not detect a deflection of his geometry from
pseudo-Euclidean under measurement of the time intervals. Besides, under
the condition *B~α~*~0~=0 the same conclusion is additionally valid for
the measurement of length, so a deflection of geometry of physical
space-time from pseudo-Euclidean is not detectable experimentally. On
the other hand, the transformation **B** under *B~α~*~0~=0 belongs to a
class of transformations
$x'^{0} = x'^{0}(x^{i}),x'^{a} = x'^{\alpha}(x^{\beta})$ acting within
fixed reference frames. Thus, we have proved the following general
theorem:

> Under any admissible transformation **B** from pseudo-Euclidean to
> oblique-angled geometry within a fixed frame of references
> (*B~α~*~0~=0) a deflection of geometry from pseudo-Euclidean is not
> experimentally observable, and the equalities (8) and (18) obtained
> above can be written simultaneously as

$\left( x_{\text{ex}}^{N} \right)^{i} = \left( x_{\text{ex}}^{E} \right)^{i} = \left( x_{L} \right)^{i}$.
(19)

This theorem will play a key role in physical interpretation of
covariant ether theories considered in the next Section.

# 4. Space-time transformations in Covariant Ether Theories

The equality (19) obtained above means that for any hypothetical
oblique-angled geometry of an empty physical space-time, related with
the pseudo-Euclidean one by means of transformation **B** under
*B~α~*~0~=0, the measured space-time four-vectors in properly
constructed inertial reference frames are subject to Lorentz
transformations. In order to derive physical inferences from this
result, it is necessary to find a possible physical meaning for such
**B**-transformations. This can be done in the following way:

Let us consider a class of space-time theories of inertial motion in an
empty space satisfying to the following general principles: space-time
homogeneity; space isotropy; causality principle; general relativity
principle (GRP). In these theories, a general transformation between two
arbitrary inertial reference frames K and K' in physical space-time can
be written as

$x_{\text{ph}i} = A_{\text{ij}}x'_{\text{ph}}^{j}$, (20)

where allowable transformations **A** in (20) are linear due to the
space-time homogeneity and constitute a ten-parametrical Lee group due
to adoption of the GRP \[17\]. For simplicity we further omit trivial
translations and rotations of space, considering three-parametrical
transformations **A**($v$), where $v$ is a relative velocity of two
arbitrary inertial frames. In addition, the GRP also requires the
validity of a reciprocity principle \[18\]: the mutual velocities of two
inertial reference frames should differ only by sign; *i.e.*,

**A**^-1^($v$)=**A**($- v$)**.** (21)

In its turn, the equality (21) and causality principles ensure that
\[18\]

det**A**=1. (22)

Thus, the transformations **A** are 'special orthogonal'.

Further, we do not in general suppose that the geometry of physical
space-time has to be pseudo-Euclidean in any inertial reference frame.
At the same time, due to the isotropy of physical space, there exists an
inertial reference frame K~0~ such that the speed of light is isotropic
and equal to *c*; *i.e.*, its geometry should be pseudo-Euclidean under
any particular choice of admissible **A-**transformation. In order to
specify this requirement, the existence of at least one K~0~ frame with
pseudo-Euclidean geometry for any admissible transformation **A**, we
can formally introduce into consideration the Minkowskian coordinates
*x*~L~ and demand the equality (23) for the frame K~0~:

$x'_{\text{ph}i}\overset{˙}{=}x'_{Li}$ (23)

(hereinafter the primed four-vectors belong to K~0~), and in general
case **A**≠**L** equality (23) is valid only for this ('absolute')
frame. One can additionally show that Eq. (23) simultaneously ensures
that the $x_{\text{ph}}$ coordinates are 'admissible' wherein the known
relationships between components of metric tensor **g** take place
\[11\], and a velocity of light is finite \[13\].

Using Eqs. (1), (20), (23), one can find a relation between
$x_{\text{ph}}$ and $x_{L}$ in an inertial frame K moving at the
constant velocity $v$ in K~0~:

$x_{\text{ph}i} = B_{\text{ij}}(v)x_{L}^{j},$ (24)

where the introduced matrix **B** is determined by the relationship

$B_{\text{ij}}(v) = A_{i}^{k}(v)L_{\text{kj}}^{- 1}(v)\text{.}$ (25)

At the same time, the four-vector $x_{L}$, defined in the frame K, does
not depend on the velocity $v$ of K in K~0~. Therefore, $x_{L}$ should
be considered as a constant in Eq. (24). Its constant magnitude is
determined for the particular case *v*=0, where the matrices **A**,
**L** are equal to the unit matrix, and

$x_{L}^{j} = x_{\text{ph}}^{j}(v = 0)\text{.}$. (26)

Substituting Eq. (26) into Eq. (24), one gets the dependence of the
physical space-time four-vectors $x_{\text{ph}}$ on the 'absolute'
velocity $v$, which will be useful in further consideration:

$x_{\text{ph}i}(v) = B_{\text{ij}}(v)x_{\text{ph}}^{j}(v = 0)\text{.}$.
(27)

Further, using a known form of the matrix **L** (see, for instance
\[19\]) one obtains from Eq. (25) a relationship between the matrices
**B** and **A**:

$B_{\text{00}} = \frac{\gamma}{A_{\text{00}}};$ (28a)

$B_{\alpha 0} = 0;$ (28b)

$B_{0\alpha} = A_{0\alpha} + A_{\text{00}} \times \lbrack\frac{v_{\alpha}}{c^{2}}\gamma + (\frac{1}{A_{\text{00}}^{2}} - 1)\frac{v_{\alpha}}{v^{2}}(\gamma - 1)\rbrack;$
(28c)

$B_{\text{αβ}} = A_{\text{αβ}} + A_{\alpha 0}\frac{v_{\beta}}{v^{2}}(1 - \frac{1}{\gamma}),$
(28d)

where $\gamma = 1/\sqrt{1 - \left( v^{2}/c^{2} \right)}$. (The
derivation of Eqs. (28a) and (28b) takes into account Eq. (21)).

Paying attention to Eq. (28b), we see that the coefficient *B~α~*~0~ is
always equal to zero for any admissible transformation **A**; this is a
necessary and sufficient condition for implementation of the equality
(19). Hence, we conclude that for any space-time theory of inertial
motion satisfying to general symmetries of space-time and the general
relativity principle, the space-time four-vectors measured in
experiments always obey the Lorentz transformation **L** in Galilean
inertial reference frames (corresponding to Cartesian frames in
three-dimensional space). From here follows the fact that Lorentz
transformation for measured space and time intervals formally gives no
information about the geometry of physical space-time, and does not
generally reject ether theories with oblique-angled space-time.

Let us show that a number of such admissible theories is infinite.
Indeed, in the general case the number of coefficients in the matrix
**A** is equal to 16. In the simplest one-dimensional case this number
is reduced to 4. Indeed, in the one-dimensional case the transformation
(20) is written as

$\begin{array}{r}
\left( x_{\text{ph}} \right)_{0} = A_{\text{00}}\left( x'_{\text{ph}} \right)^{0} + A_{\text{01}}\left( x'_{\text{ph}} \right)^{1}, \\
\left( x_{\text{ph}} \right)_{1} = A_{\text{10}}\left( x'_{\text{ph}} \right)^{0} + A_{\text{11}}\left( x'_{\text{ph}} \right)^{1}\text{.}
\end{array}$. (29)

However, in such a case we are able to get only three equations for
coefficients of **A**. One can show that the reciprocity principle (Eq.
(21)) provides the equality:

$A_{\text{11}} = A_{\text{00}}$. (30)

The relationship of *A*~10~ with *A*~00~ can be found from the
requirement:

$\left\lbrack \left( x'_{\text{ph}} \right)^{1}/\left( x'_{\text{ph}} \right)^{0} \right\rbrack_{\left( x_{\text{ph}} \right)_{1} = 0} = v$.

The last requirement and Eq. (29) lead to the equality:

$A_{\text{10}} = - \text{vA}_{\text{00}}\text{.}$ (31)

Further, Eqs. (22), (30) and (31) allow expressing *A*~01~ as:

$A_{\text{01}} = \frac{1 - \left( A_{\text{00}} \right)^{2}}{\text{vA}_{\text{00}}}$.
(32)

Thus, the three obtained Eqs (30)-(32) establish a relationship between
the coefficient *A*~00~ and all other coefficients of the matrix **A**
in one-dimensional case. However, there are no restrictions on the
concrete choice of *A*~00~. This means the number of space-time theories
where the measured space-time four-vectors are subjected to the Lorentz
transforms is infinite. That is why all the experiments for verification
of Lorentz transforms (beginning with Michelson-Morley experiment and
finishing with modern Champeney experiment \[20\]) find an infinite
number of alternative explanations of their results.

We conditionally name the acceptable theories 'covariant ether
theories'. The exclusive place of SRT among all such covariant ether
theories is defined by the fact that it directly asserts the equality of
measured and physical space-time four-vectors, *i.e.*, the equality of
the matrices **A** and **L**. On the other hand, this assertion follows
from nothing: in general, a question about coincidence of 'measured' and
'physical' values even under optimal measurements is one of fundamental
questions to Nature. That is why it is especially interesting to analyze
a negative answer to this question. One can easily see that a negative
answer means that **A**≠**L**. Under **A**≠**L**, the geometry of
physical space-time is oblique-angled, and we must distinguish
space-time transformations for physical and measured four-vectors:

$x_{\text{ph}i} = A_{\text{ij}}x'_{\text{ph}}^{j}$, (33)

$x_{\text{ex}i} = L_{\text{ij}}x'_{\text{ex}}^{j}$, (34)

where the primed four-vectors, as before, belong to the 'absolute' frame
K~0~. This means that these transformations do not yet solve the main
kinematical problem (determination of space-time transformations between
two arbitrary inertial frames): it acts only in the special case defined
by the equality (23). In order to find a transformation between two such
arbitrary inertial frames K and K", we should write

$x_{\text{ex}i} = L_{\text{ij}}(v_{1})x'_{\text{ex}}^{j};x\text{ rSub \{ size 8\{}ex\text{i\} \} =L rSub \{ size 8\{ ital }ij\text{\} \}  \textbackslash( v rSub \{ size 8\{2\} \}  \textbackslash) x' rSub \{ size 8\{}ex\text{\} \}  rSup \{ size 8\{j\} \} \} \{}$,
(35)

$x_{\text{ph}i} = A_{\text{ij}}(v_{1})x'_{\text{ph}}^{j};x\text{ rSub \{ size 8\{}ph\text{i\} \} =A rSub \{ size 8\{ ital }ij\text{\} \}  \textbackslash( v rSub \{ size 8\{2\} \}  \textbackslash) x' rSub \{ size 8\{}ph\text{\} \}  rSup \{ size 8\{j\} \} ,\} \{}$
(36)

where $v_{1},v_{2}$ are the 'absolute' velocities of the frames K and
K", respectively. Excluding four-vector $x'_{\text{ex}}^{j}$ from Eqs.
(35), and $x'_{\text{ph}}^{j}$ from Eq. (36), we obtain general
transformations for measured and physical space-time four-vectors in two
arbitrary inertial frames:

$x_{\text{ex}i} = L_{\text{ij}}(v_{1})\lbrack L^{- 1}(v_{2})\rbrack^{\text{jk}}x\text{ rSub \{ size 8\{}ex\text{k\} \} \} \{}$,
(37)

$x_{\text{ph}i} = A_{\text{ij}}(v_{1})\lbrack A^{- 1}(v_{2})\rbrack^{\text{jk}}x\text{ rSub \{ size 8\{}ph\text{k\} \} \} \{}$,
(38)

where the matrix **A** can be taken in arbitrary admissible form.

Thus in contrast to SRT, under the hypothesis **A**≠**L**, Nature does
not 'know' a direct relative velocity of two arbitrary inertial frames K
and K": it is always composed as a sum $v_{1} \oplus v_{2}$, where
$v_{1}$ and $v_{2}$ are the corresponding velocities of K and K" in the
'absolute' frame K~0~. This means, in particular, that direct
rotation-free Lorentz transformation between measured space-time
four-vectors in K and K" is impossible: according to general group
properties of these transformations, an additional rotation of the
coordinate axes of the frames K and K" appears at so called
Thomas-Wigner angle Ω, depending on $v_{1}$ and $v_{2}$. It is quite
important that such a rotation occurs in measured coordinates, *i.e.*,
it can be really detected. It defines a principal possibility to
experimentally distinguish the hypotheses **A**=**L** and **A**≠**L**.

Among admissible space-time theories that assume **A**≠**L**, the
simplest case corresponds to the choice **A**=**G**, where **G** is the
matrix of Galilean transformation: *G~ii~*=1, *G~α0~* =-*v~α~*, and all
others *G~ij~* =0. Substituting matrix **G** in place of matrix **A** in
Eq. (28), one gets the following coefficients of matrix **B**:

$B_{\text{00}} = \gamma,B_{\alpha 0} = 0,B_{0\alpha} = \frac{v_{\alpha}}{c^{2}}\gamma,B_{\text{αβ}} = \delta_{\text{αβ}}\frac{v_{\alpha}v_{\beta}}{v^{2}}\left( 1 - \frac{1}{\gamma} \right)$,
(39)

where $\delta_{\text{αβ}}$ is the Kronekker symbol. Further substitution
of Eq. (39) into Eq. (27) allows one to determine a dependence of
physical space-time four-vectors on the 'absolute' velocity $v$ of some
arbitrary inertial reference frame K:

$r_{\text{ph}}(v) = r_{\text{ph}}(v = 0) + \frac{v\left( r_{\text{ph}}(v = 0),v \right)}{v^{2}}\left\lbrack \sqrt{1 - (\frac{v^{2}}{c^{2}})} - 1 \right\rbrack,$,
(40)

$t_{\text{ph}}(\overrightarrow{v}) = \frac{t_{\text{ph}}(v = 0)}{\sqrt{1 - (\frac{v^{2}}{c^{2}})}} + \frac{r_{\text{ph}}(v = 0)v}{c^{2}\sqrt{1 - (\frac{v^{2}}{c^{2}})}}$.
(41)

For the time intervals in a fixed spatial point of the frame K
($r_{\text{ph}}$=0), we obtain the dependence of $t_{\text{ph}}$ on $v$
\[see, Eq. (41)\]:

$t_{\text{ph}}(v) = t_{\text{ph}}(v = 0\frac{)}{\sqrt{1 - (\frac{v^{2}}{c^{2}})}},$
(42)

that means an 'absolute' dilation of time by factor
$\sqrt{1 - (\frac{v^{2}}{c^{2}})}$. Furthermore, one obtains from Eq.
(40):

$\begin{array}{r}
\left( r_{\text{ph}}(v),v \right) = \left( r_{\text{ph}}(v = 0),v \right)\sqrt{1 - v^{2}/c^{2}} \\
\left\lbrack r_{\text{ph}}(v) \times v \right\rbrack = \left\lbrack r_{\text{ph}}(v = 0) \times v \right\rbrack,
\end{array}$, (43)

that means an 'absolute' contraction of moving scale along a vector of
'absolute' velocity (Fitzgerald-Lorentz hypothesis). Finally,
transformation (38) (under **A**=**G**)

$$x_{\text{ph}i} = \lbrack G_{\text{ij}}(v_{1} - v_{2})\rbrack x\text{ rSub \{ size 8\{}ph\text{\} \}  rSup \{ size 8\{j\} \} \} \{}$$

leads to the Galilean law of speed addition for the physical light
velocity $c_{\text{ph}}$.

Thus, we have got a full set of the Lorentz ether postulates in case
**A**=**G**.[^1] However, the physical space-time in the Lorentz ether
theory is not observable in an arbitrary inertial reference frame, while
the measured four-vectors *x*~ex~ obey to the Lorentz transformations in
form of (37). (This important circumstance about a difference of
physical and measured four-vectors in non-Euclidean geometry of moving
inertial frame was dropped by Lorentz and his successors). Therefore, we
may consider the Lorentz ether theory as one of the CET's defined above,
and the simplest among them. Due to this fact, the application of the
Lorentz ether postulates for explanation of 'null' results of all
experiments searching for 'ether wind speed' was always successful.

# 5. Experimental Test of CET's

So, we have found that the difference between SRT (**A**=**L)** and
CET's (**A**≠**L)** appears on an experimental level only in successive
space-time transformations. It follows from there that an instrument for
measuring hypothetical 'absolute' velocity must contain moving inertial
parts, in order to deal with such transformations. Then a general idea
of an experiment for a choice between SRT and CET's can be described
with help of the diagram in Fig. 1. It shows the absolute frame K~0~,
laboratory frame K (moving at the constant absolute velocity $v$) and
frame K~i~, attached to some moving inertial part of a measuring
instrument in K.

In our laboratory we specify a velocity $u$ of K~i~ in K. In such a case
for the hypothesis **A**=**L** we apply direct rotation-free Lorentz
transformation K→K~i~ for calculation of the indication of the measuring
device. Hence, according to SRT we get a vanishing value of absolute
velocity. Under the hypothesis **A**≠**L**, Nature does not 'know' a
direct rotation-free Lorentz transformation between K and K~i~, and
'operates' with the absolute velocities of these frames $v$ and
$v \oplus u$. Hence, in order to calculate an indication of the
measuring device, we must apply the successive transformations
K→K~0~→K~i~ according to Eq. (37). (A direct Lorentz transformation from
K to K~i~ is also possible, but it will not be rotation-free). In this
case the axes of the frames K and K~i~ are turned out at Thomas-Wigner
angle Ω, that, in principle, changes the state of the measuring
instrument. Since Ω depends on the absolute velocity $v$ of the
laboratory frame K, the state of the measuring instrument will depend on
$v$, too. There is only one particular case ($v$ is collinear to $u$)
where Ω=0, and the state of the measuring instrument has to be unchanged
for any magnitude of absolute velocity of the laboratory frame. This is
an unambiguous inference from the GRP. All experiments searching for
'ether wind' velocity with experimental instruments containing moving
inertial parts, aiming to measure non-relativistic effects under
collinear $v$ and $u$ (see, *e.g*., \[20-22\]), in fact checked the GRP,
not the Einstein relativity principle.

Thus, an experiment for qualitatively new test of SRT must contain
moving inertial part (parts) with non-collinear velocities $v$ and $u$,
and be aimed to measure a dependence of the angle Ω on the absolute
velocity of the laboratory frame. To the order of magnitude *c*^-2^, and
for orthogonal vectors of $v$ and $u$, this dependence is defined by the
expression (see, e.g. \[19\])

$\Omega = - \text{uv}/2c^{2}$.

A direct measurement of this dependence in a laboratory-scale experiment
is impractical. Indeed, the absolute speed *v* could be taken as about
10^-3^*c* (typical velocities of Galaxy objects). The maximum value of
*u* could be about 10^3^ m/s. Hence, the angle Ω takes on the value
3×10^-9^, *i.e.*, well below any limit of practicability in a laboratory
experiment.

An analysis of possible experimental schemes for indirect measurement of
the angle Ω can be greatly simplified under numerical estimation of
eventual non-relativistic effects, proceeding from their dimension.
Indeed, the experiments, looking for the change of length associated
with $\Omega(v)$ dependence, give an effect in the order of magnitude
*L*Ω. Here *L* is some length, which is equal to about 1 m in the
laboratory scale. In such a case we get *L*Ω≈3×10^-9^ m, which is
impossible to measure in practice. A corresponding change of time has a
dimension $L\Omega/u = \text{Lv}/c^{2}$≈3 ps, a time interval within the
range of present technology, but not for mechanical parts necessarily
involved. Finally, one can rearrange the experiment into a 'speed
experiment' looking at the term $\Omega u$, and the latter arrangement
could be further transformed into frequency measurement via the Doppler
effect ($\Omega u/c$). For the last case one is looking at the term
$u^{2}v/c^{3}$, the latter being about 10^-14^ (for *u*≈10^3^ m/s) - a
value accessible practically conveniently only by the Mössbauer effect,
at least as far as a laboratory-scale experiment is concerned.

Thus, an indirect measurement of the $\Omega(v)$ dependence in a
Mössbauer experiment is a real way to unambiguously check SRT. It can be
accomplished in Champeney-type experiments, using the most recent
methodological developments in the Mössbauer spectroscopy: resonant
method of registration of recoil-free radiation \[13, 23\] or nuclear
resonant scattering of synchrotron radiation \[15\].

# 5. Conclusions

**1)** Consideration of all hypothetical theories of empty space-time
with curvilinear geometry should be based on distinguishing between
physical and measured space-time four-vectors. General analysis of the
properties of admissible space-time transformations shows that in any
theory adopting the general relativity principle and symmetries of
space-time, the measured space and time intervals always obey the
Lorentz transformation, regardless of a concrete choice of physical
space-time transformation. The latter circumstance makes it possible to
explain all known experimental results in space-time physics within an
infinite number of admissible space-time theories, called 'covariant
ether theories'.

**2)** SRT is unique among admissible theories of empty space-time
because it directly asserts an equality between measured and physical
four-vectors under optimal measurements. Adoption of such an equality
defines the possibility of direct rotation-free Lorentz transformation
between two arbitrary inertial frames. This is impossible in all other
admissible space-time theories, and they lead to a dependence of
rotation angle Ω on an 'absolute' velocity $v$ for the measured
space-time co-ordinates, subjected to the transformations (37). Hence,
the hypothetical dependence $\Omega(v)$ is the sole observable physical
phenomenon admitting an unambiguous test of SRT.

**3)** The most convenient method for the measurement of $\Omega(v)$
dependence under modern development of laboratory experimental technique
is the Mössbauer spectroscopy: either a resonant method of registration
of recoil-free radiation, or nuclear resonant scattering of synchrotron
radiation in Champeney-type experiments.

**Acknowledgment**

The author warmly thanks Professor Dr. Cynthia Kolb Whitney (Tufts
University Electro-Optics Technology Center, USA) for helpful
discussions on the subject and linguistic corrections of the paper. The
author also warmly thanks Dr. Walter Potzel (Technical University of
Munich, Germany), Dr. Ralf Röhlsberger (Rostock University, Germany),
Professor Dr. Erich Gerdau (Hamburg University, Germany) and Dr. Uwe van
Buerck (Technical University of Munich, Germany), for the idea and
concrete experimental schemes for test of CET's by nuclear resonant
scattering of synchrotron radiation, that makes topical a presentation
and further development of covariant ether theories.

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![](media/image1.wmf)

FIGURE CAPTIONS

Fig. 1. General idea for an experiment to provide a new test of SRT.

[^1]: Let us recall the postulates of Lorentz ether theory in its modern
    form:

    **1)** There is an "absolute" reference frame K~0~, wherein a light
    velocity is isotropic and equal to *c*. **2)** In arbitrary
    reference frame K, moving at constant velocity ![](media/image2.wmf)
    in K~0~, the velocity of light is equal to $c' = c - v$. **3)** In
    this reference frame K time is dilated by $\sqrt{1 - v^{2}/c^{2}}$
    times. **4)** In this reference frame K a linear scale is contracted
    by $\sqrt{1 - v^{2}/c^{2}}$ times along the vector $v$.