Accelerating relativistic reference frames in Minkowski space-time

TL;DR

This paper develops a framework for describing accelerating relativistic reference frames in Minkowski space-time using harmonic gauge, deriving coordinate transformations and equations of motion that extend the Poincaré group for accelerated observers.

Contribution

It introduces a novel method to construct accelerated relativistic frames via harmonic potentials and dynamical conditions, extending the Poincaré group to include acceleration effects.

Findings

Derived explicit post-Galilean coordinate transformations for accelerated frames

Presented a relativistic proper reference frame balancing external forces and fictitious forces

Extended the Poincaré group to include transformations for accelerating observers

Abstract

We study accelerating relativistic reference frames in Minkowski space-time under the harmonic gauge. It is well-known that the harmonic gauge imposes constraints on the components of the metric tensor and also on the functional form of admissible coordinate transformations. These two sets of constraints are equivalent and represent the dual nature of the harmonic gauge. We explore this duality and show that the harmonic gauge allows presenting an accelerated metric in an elegant form that depends only on two harmonic potentials. It also allows reconstruction of the spatial structure of the post-Galilean coordinate transformation functions relating inertial and accelerating frames. The remaining temporal dependence of these functions together with corresponding equations of motion are determined from dynamical conditions, obtained by constructing the relativistic proper reference frame of an accelerated test particle. In this frame, the effect of external forces acting on the observer is balanced by the fictitious frame-reaction force that is needed to keep the test particle at rest with respect to the frame, conserving its relativistic linear momentum. We find that this approach is sufficient to determine all the terms of the coordinate transformation. The same method is then used to develop the inverse transformations. The resulting post-Galilean coordinate transformations extend the Poincar\'e group on the case of accelerating observers. We present and discuss the resulting coordinate transformations, relativistic equations of motion, and the structure of the metric tensors corresponding to the relativistic reference frames involved.