Relativistic Symmetry of The Space-time of The Frames with a Constant Proper Acceleration

Abstract

This chapter reveals that, besides the relativistic symmetry expressed by the Lorentz group of coordinate transformations which leave invariant the Minkowski metric of space-time of inertial frames, there exists one more relativistic symmetry expressed by a group of coordinate transformations leaving invariant the space-time metric of the frames with a constant proper-acceleration. It is noteworthy that there are only those two relativistic symmetries that exist in space-time, which correspond to groups of continuous transformations that leave the metric of space-time of extended rigid reference frames unaltered. As such, the Lorentz symmetry and the new relativistic symmetry ought to be seen equally. The Lie group analysis is used to determine the groups of transformations that leave invariant the metric of the space-time of constant proper-acceleration. This process is enhanced by the need for the group to contain transformations from an accelerated frame to an inertial one. Two-parameter groups of two-dimensional (1 + 1), three-dimensional (2 + 1), and four-dimensional (3 + 1) transformations, with the group parameters related to the ratio of accelerations of the frames and the relative velocity of the frame space origins at the initial moment, can be considered as counterparts of the Lorentz group of corresponding dimensions. Defining the form of the interval and the groups of coordinate transformations satisfying the relativity principle paves the way to defining the invariant forms of the laws of dynamics and electrodynamics in accelerated frames. It is appropriate to define the class of non-inertial frames of reference as the special theory of relativity’s equivalent of the class of inertial frames. It implies that one should take into account the frames that are stretched throughout space, signifying a continuity of observers.