Special Relativity Kinematics with Anisotropic Propagation of Light and Correspondence Principle

The purpose of the present paper is to develop kinematics of the special relativity with an anisotropy of the one-way speed of light. As distinct from a common approach, when the issue of anisotropy of the light propagation is placed into the context of conventionality of distant simultaneity, it is supposed that an anisotropy of the one-way speed of light is due to a real space anisotropy. In that situation, some assumptions used in developing the standard special relativity kinematics are not valid so that the “anisotropic special relativity” kinematics should be developed based on the first principles, without refereeing to the relations of the standard relativity theory. In particular, using condition of invariance of the interval between two events becomes unfounded in the presence of anisotropy of space since the standard proofs drawing the interval invariance from the invariance of equation of light propagation are not valid in that situation. Instead, the invariance of the equation of light propagation (with an anisotropy of the one-way speed of light incorporated), which is a physical law, should be taken as a first principle. A number of other physical requirements, associativity, reciprocity and so on are satisfied by the requirement that the transformations between the frames form a group. Finally, the correspondence principle is to be satisfied which implies that the coordinate transformations should turn into the Galilean transformations in the limit of small velocities. The above formulation based on the invariance and group property suggests applying the Lie group theory apparatus which includes the following steps: constructing determining equations for the infinitesimal group generators using the invariance condition; solving the determining equations; specifying the solutions using the correspondence principle; defining the finite transformations by solving the Lie equations; relating the group parameter to physical parameters. The transformations derived in such a way, as distinct from the transformations derived in the context of conventionality of distant simultaneity, cannot be converted into the standard Lorentz transformations by a coordinate (synchrony) change. The anisotropic nature of the presented transformations manifests itself in that they do not leave the interval invariant but only provide the conformal invariance of the interval. The relations that represent measurable effects include the conformal factor which depends on the relative velocity of the frames and the anisotropy degree. It is important to note the use of the correspondence principle as a heuristic principle which allows to relate the conformal factor to the anisotropy degree and thus completely specify the transformations and observable quantities.

1. It is claimed sometimes in the literature that Edwards’ [6] and Winnie’s [7] transformations do not form a group while those due to Ungar [8] do. This statement is not completely correct. Winnie’s [7] \(\epsilon \)-Lorentz transformations do not possess the group properties if \(\epsilon _1\ne \epsilon _2\) but, in the case of \(\epsilon _1=\epsilon _2\) (and it is the case considered in Ungar’s paper [8]), they do form a group. It can be checked in a straightforward way but it is evident that the \(\epsilon \)-Lorentz transformations should form a group since they may be obtained by a coordinate change (1.3) from the Lorentz transformations which form a group (see Sect. 5.2).

2. The correspondence principle was taken by Niels Bohr as the guiding principle to discoveries in the old quantum theory. In the context of special relativity, the correspondence principle is traditionally mentioned as a statement that Einstein’s theory of special relativity reduces to classical mechanics in the limit of small velocities. Nevertheless, the correspondence principle has not been properly used as a heuristic principle in developing the special relativity kinematics, there are many other ways to build the theory. In Sect. 2, which is of methodical value, it is shown, in particular, how the correspondence principle may be used, instead of some other common assumptions, in derivation of the Lorentz transformations.

3. It is worth noting that the “\(\epsilon \)-Lorentz transformations” do not satisfy the correspondence principle unless the standard (Einstein) synchrony is used. In the limit of small velocities, the coordinate transformation contains additional terms including the synchronization parameter and light speed which are alien to the framework of the Galilean kinematics. So the correspondence principle applied to the \(\epsilon \)-Lorentz transformations singles out the Einstein synchrony as a “natural” one in a sense.

4. The 2D-analysis of [18] cannot be generalized to this case. The generalization to 4D-case in [16, 17] is based on the 4D-Finslerian metric derived using assumptions that have no direct connections with the basic principles of the relativity theory (see discussion in Sect. 5.1).

5. The use of the group property in derivation of transformations between inertial frames was made in several studies initiated by the works of Ignatowski [21] and Frank and Rothe [20]. In those studies, the group property was complemented by some other requirements, for example, linearity of the transformations and relativity arguments as in [20] where it led to the transformations with an undefined invariant velocity.

6. The form (3.1) of the anisotropic equation of light propagation is usually attributed to the one-dimensional formulation. Nevertheless, it can be shown that, in the three-dimensional case, the equation has the same form if the x-axis is chosen to be along the anisotropy vector \(\mathbf {k}\) (see Appendix A).

7. The following relations are usually derived by considering particular thought experiments (see, e.g., [26]).

8. It has been done for the transformations (5.3) by Winnie [7], Ungar [8] and others, it is repeated here for the sake of comparison with similar calculations for the transformations (3.10) and (3.13).

9. It is evident that the transformations (3.10) and (3.13) are applied to clocks synchronized according to the assumed anisotropy degree k (\(k_{\epsilon }=k\)) as it is seen from comparison of (1.4) and (1.6).

Authors and Affiliations

1. Swiss Institute for Dryland Environmental and Energy Research, Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede-Boker Campus, 84990, Midreshet Ben-Gurion, Israel

   Georgy I. Burde

Corresponding author

Correspondence to Georgy I. Burde.

Appendix: Equation of Light Propagation

We will define the form of the equation of light propagation based on the law (1.6) for the light speed variation. If we use the spherical coordinate system

with the axis x directed along the anisotropy vector \(\mathbf {k}\), then the angle \(\theta _k\) in (1.6) coincides with the polar angle \(\theta \) so that the law of the variation of speed of light in space becomes

To derive the equation for light propagation corresponding to the law (7.2) we start from

with i and k running from 0 to 3 (\(g_{00}>0\)) and \(x^0=c t\), \(x^1=x\), \(x^2=y\), \(x^3=z\). To define \(g_{ik}\) such that (7.3) corresponded to the law (7.2) we will use the expression for the light velocity (see, e.g., [11]):

where Greek letters run from 1 to 3 as distinct from Latin letters that run from 0 to 3. We will also use the relation

Based on the symmetry of the problem we have

Then it follows from (7.4), (7.2) and (7.5) that

With \((n^{1}=\cos \theta ,\; n^{2}=\sin \theta \sin \phi ,\; n^{3}=\sin \theta \cos \phi )\) we obtain \(g_{10}=-k\) and \(g_{11}=k^2-1\) so that the equation for light propagation becomes

Thus, although equation (7.8) (with \(dy=dz=0\)) commonly arises in the traditional one-dimensional arguments it corresponds to the three-dimensional law (7.2). It can be also demonstrated by rewriting (7.8) in the spherical coordinates (7.1) for light rays propagating in radial direction, as follows

Solving (7.9) for \(V=\frac{dr}{dt}\) yields two roots

corresponding to two different directions of the light propagation according to the law (7.2).

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