_{1}

^{*}

As shown earlier, a linear transformation with the same form for the spatial coordinates as the Lorentz transformation (LT), and that allows for time dilation, but leaves simultaneity invariant instead of the one-way speed of light, predicts the same results as the LT for the usual tests of special relativity. Such a transformation is allowed by general covariance. A complementarity between the invariance of the one-way speed of light and the invariance of simultaneity is discussed. Using this transformation, interpreted as involving external synchronization, it is shown that two frames moving uniformly with equal and opposite velocities, v and –v relative to a third inertial frame, in which clocks are synchronized so that the one-way speed of light is c, can be related by a Galilean-like transformation with a relative velocity 2v/(1-(v/c)^{2}).These transformations do not form a group, hence the term “pseudo-Galilean” is used to distinguish them from the Galilean transformations. An analogy with the Sagnac effect is discussed, and consistency with the LT for stellar aberration, and the Doppler effect is shown. Implication of the above complementarity for the possible unification of quantum theory and gravitation is briefly discussed, as well as the inferred physical significance of general covariance.

It was shown in the author’s Ph.D. dissertation [

The main purpose of this work is to show, with the aid of this transformation, given in Section 2, that two frames moving uniformly with equal and opposite velocities v and -v relative to a third inertial frame in which clocks have been synchronized in accordance with SR, so that the one-way speed of light is c in all directions, are related by a Galilean-like transformation with relative velocity

Because of the utilization of external synchronization, the transformation will be referred to here as the externally synchronized transformation (EST) rather than the absolute Lorentz transformation (ALT) used in [

As given in [

where

the inverse of the EST is not of the same form as the EST, and clocks in the station run faster than clocks on the train, and rods on the station are longer than rods on the train, when measured by clocks and rods on the train. This contrasts with the LT for which there is complete reciprocity, and is due to the fact that the ESTs do not form a group, even for velocity boosts along the same axis, unlike the LTs which do. Likewise, the EST does not keep the one-way speed of light invariant since

Hence, in the forward direction, the speed of light is

Although the transformation given in Equations (1) and (2) is obviously allowed mathematically according to Einstein’s principle of general covariance, one might wonder whether there is a more physical justification for it? A possible answer to this question emerges from the following consideration about the relativity of simultaneity occasioned by the LT for the time coordinate which in the transformed frame will be denoted by

As Einstein first recognized, if two events are spatially separated in the unprimed frame,

It is of course assumed one is working in a patch of space-time so sufficiently small that one can treat it as flat, and that the CMBR is perfectly uniform.

Returning to the case of clocks on a train moving uniformly with speed v through the station, treated as an inertial frame, it is shown in [

There are interesting analogies with the Sagnac effect [

where use has been made of the fact that for the EST, in contrast to the LT, the length D in the unprimed frame appears longer in the primed frame, so that

Assume one has a disk of radius R as measured in an inertial frame S, in which clocks have been synchronized so that the speed of light is c all directions, so that a vacuum is assumed, and also that the disk is rotating in the x-y plane, with angular velocity

Customarily, upon neglect of the higher-order term in the denominator, the above is written as

then because of the time dilation factor

will read is

In comparison with Equation (8), if

The above discussion assumes that light travels through a vacuum, but the analogy with the EST holds when light is constrained to be traveling through an optical fiber, and furthermore, the analogy makes an interesting prediction that is borne out by observation.

Suppose that on the uniformly moving train the two sets of separated clocks are connected by an optical fiber of index of refraction n. The EST clocks are related to the LT clocks by the local time transformation given in (7) so that

One can of course obtain the independence of the time difference on n directly by introducing cylindrical coordinates, and noting that in the plane of the disk, the line element takes the form

element becomes

nifesting a circular relativity principle in which the speed of light both clockwise and counterclockwise is the same when

However, to understand why the train analogy works, one introduces instead of

where

Assume there are two inertial frames

Upon substituting the inverse of this transformation, which is given by Equations (3) and (4) with

This transformation is exactly of the same form as a Galilean transformation for which the relative velocity u of

in which u ranges, analogous to the Galilean transformation,

The expression for u in (15) follows from (11) and (12), since

This Galilean-like transformation, that keeps simultaneity invariant, indicates that when the lengths of similar rods and the rates of similar clocks are compared between the two frames, they are the same, in contrast with the prediction of the Lorentz transformation, which would lead observers in either frame to find the other frame’s rods are shorter in the direction of motion, their clocks keep time more slowly, and simultaneity is relative. This resolves the apparent paradox that is often raised concerning this situation, as discussed e.g., in Bridgman [

However, some caution is necessary in using the transformation, since unlike the Galilean transformation, the relative velocity of

On the other hand, the product rule is not obeyed, since

in which V is the relativistic sum of velocities,

When the velocities are not along the x-axis, the EST is obtained by using the vector expression for the LT for the spatial coordinates, but making the appropriate change for the transformation of the time coordinate. The EST in vector form, is then given by

Similar equations relate

Equation (20) can alternatively be written in tensor form as

Written in this way, there is of course no restriction to three spatial dimensions, so the spatial coordinate dimensionality can have the range

Upon inserting these local time transformations into the corresponding Minkowskian line elements for the two Lorentz frames, one has

After transforming Equation (25) by setting

If one asks for the inverse to the metric in Equation (25), or in Equation (26), one finds a surprising result: it is the metric that would result if one had transformed the Minkowski metric from the

It will now be convenient to set

so that the Galilean metric tensor is the inverse of EST metric tensor, and hence is of same form as the contravariant EST metric tensor

This result was given in [

The matrix

where

Since

and hence

Since the inverse of an LT is an LT, and the LTs leave the Minkowski line element invariant, and consequently the corresponding quadratic form, one has

For the GT,

Then since

and hence the relation given in (28) follows.

The above formalism enables one to obtain the PGT very readily. One goes back to (33), and upon restoring the arguments for both F and G, it becomes,

after using,

It will next be shown that between any two inertial frames in relative motion, there always exists a pseudo- Galilean transformation connecting them. Choose one of the frames

The solution to the resulting quadratic equation for

Upon expanding to first order, one has

It is interesting to show how the PGT may be used to give an alternative treatment of stellar aberration that agrees with the prediction of SR and the LT. Before showing this, it is important to note that the EST should agree with the LT since the spatial coordinates for the two transformations are the same, and since all one measures in aberration is an angle, and since the tangent of the angle is the ratio of the spatial displacements, the result must be the same for the two transformations, even though the light travels with different velocities for the LT and EST observers. In the comparable analysis of Puccini and Selleri [

As indicated in Section 1, considerations about the ether are omitted here for brevity, but given in [

The aberration will be worked out first for the LT, in the notation used previously, so as to facilitate comparison. The light source and the observer will be taken to be in the x ? y plane, with the observer at rest, and the source traveling in the negative x-direction with velocity ?w relative to the observer. Then with frame

This is the standard SR expression for stellar aberration, and note, importantly, that in computing the tangent, the time interval

Since v is the earth’s velocity relative to the center-of?velocity frame, and ?v the velocity of the star relative to that frame, the relativistic relative velocity w of the earth with respect to the star in terms of v is

which is another form of (40). After introducing

This establishes consistency between the PGT and the LT for stellar aberration.

The Doppler effect will be calculated for the PGT for motion along the x-axis and will be shown to reduce to the standard expression based on the LT. The notation will be the same as above. When dealing with the EST or PGT, or non-Lorentz transformations more generally, there can be significant differences between the covariant and contravariant components of tensors, since more than a minus sign is usually involved. Thus, in order to relate the PGT expression for the Doppler effect with that using the LT, equivalent expressions for frequency must be used, albeit in the PGT case, expressed as a function of v, while in the LT case, expressed as a function of w, also there has to be agreement as to the direction in which the light is being observed. To obtain equivalent expressions for frequency, it is helpful to note that the phase

and since

The expressions for

phase in S is given by the standard expression

transforming to

To express

in which

This somewhat lengthy analysis to obtain the relation between

After one sets

which is the standard expression for the relativistic Doppler effect based on the LT. This again establishes consistency between the PGT and the LT.

The fact that the one-way speed of light is not invariant under the EST, while simultaneity between the two inertial frames does remain invariant, and in contrast, the one-way speed of light is invariant under the LT, while simultaneity is no longer invariant, but becomes relative, can be seen as a manifestation of a complementarity between maintaining invariance of the one-way speed of light and maintaining invariance of simultaneity in the same transformation. This situation provides an illustration of Bohr’s [

The preceding analysis shows that, in accordance with the principle of general covariance, one can consistently use a linear transformation between two inertial frames based on an external synchronization that keeps simultaneity invariant, but not the one-way speed of light, while keeping the out-and-back speed invariant. This external synchronization leads to the EST, and contrasts with the LT, based on internal synchronization. Also, as was shown in Section 4, the EST enables one to set up an exact analogy with the Sagnac effect, for a circular path in a rotating frame, both in a vacuum and in a dielectric medium, again in contrast with the LT. However, the main finding of the present study is the pseudo-Galilean transformation (PGT), as discussed in Sections 5 - 7. As discussed in Section 8, there is a complementarity between maintaining invariance of the one-way speed of light, and maintaining invariance of simultaneity in the same linear transformation, that supports the physical significance of general covariance.

Future applications of the PGT might be to the decay of a particle into two particles of equal mass, since the parent particle’s rest frame is the center-of-velocity frame. Also, the PGT might lead to simplifications in analyzing colliding beam experiments between particles of the same mass, since the laboratory, ideally, would be the center-of-velocity frame, and the colliding beams could be related by the PGT, so that observers traveling with either colliding particle, would not see the other particle, say a proton in the LHC, as “pancaked.” But whether these applications will lead to any simplifications in the study of particle decays and collisions will require further study.

A preliminary version of this work was presented at the 2011 Anaheim, California meeting of the American Physical Society [

I would like to acknowledge very helpful and stimulating correspondence with Dr. Gregory B. Malykin of the Russian Academy of Sciencs in Nizhny Novgorod, and also for sending me a copy of Dr. Franz Harress’ dissertation. The author is also indebted to Prof. Eric Sheldon for reference [