Group Analysis of Differential Equations and Integrable Systems − 2018

Georgy I. Burde (Ben-Gurion University of the Negev, Israel)

Groups of transformations between inertial frames with the existence of a preferred frame

A framework, which may be named 'special relativity with a preferred frame', is developed. It, like the standard relativity theory, is based on the symmetry properties of the special relativity theory, the relativity principle and the universality of the speed of light but, nevertheless, includes the existence of a preferred reference frame as an essential element. It is shown that the reconciliation and synthesis of such seemingly incompatible concepts as the relativity principle and existence of a preferred frame is possible in the analysis, based on invariance of the equation of anisotropic light propagation with respect to the space-time transformations between inertial frames. Since any one-way speeds of light, consistent with the two-way speed equal to c, are acceptable, a preferred frame is defined as the only frame, in which the one-way speed of light is isotropic, while it is anisotropic in any other frame moving with respect to a preferred frame. Thus, a degree of anisotropy of the one-way speed acquires meaning of a characteristic of the really existing anisotropy caused by motion of an inertial frame relative to the preferred frame. Correspondingly, the anisotropy parameter in the equation of light propagation is considered as a variable that takes part in the group transformations, varying from frame to frame. The Lie group theory apparatus is applied to define groups of transformations leaving the equation of anisotropic light propagation invariant. In such a framework, the principle of relativity is preserved since the preferred frame, in which the anisotropy parameter is zero, enters the analysis on equal footing with other frames -- the transformations from/to that frame are not distinguished from other members of the group of transformations. Nevertheless, the existence of a preferred frame becomes an essential element of the analysis when an argument, that a size of the anisotropy in a specific frame is determined by its velocity with respect to the preferred frame, is used. It allows to define the form of the group generator for transformations of the anisotropy parameter, which remains undefined in the course of the symmetry analysis of the equation of light propagation, and to specify the form of the transformations that way. It is an unusual feature in the Lie group applications to differential equations.