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Transformation equation for frames undergoing non-uniform acceleration such as SHM and rotational motion

Ranchhaigiri Brahma, A. K. Sen

TL;DR

The paper extends Lorentz/Rindler transformations to non-uniformly accelerated frames by deriving explicit, approximate coordinate transformations for frames undergoing simple harmonic motion (SHM) along one axis and for frames moving in uniform circular motion (UCM). By treating the frame's proper acceleration as a time-dependent function and employing a small-velocity expansion with the parameter $r_0\omega/c$, the authors obtain SHM- and circle-based transformations that reduce to the Rindler/Lorentz form in the appropriate limits. The SHM cases yield time and space corrections up to $\mathcal{O}((r_0\omega/c)^3)$, while the circular case introduces higher-order relativistic corrections that reflect anharmonic effects in SR. The results are framed as locally Lorentz-type (Fermi-Walker transported) and have potential applications in relativistic astrophysics, such as pulsating stars and binary systems, as well as offering a basis for extending Unruh-type analyses to non-uniform acceleration in flat spacetime.

Abstract

Lorentz transformation equations provide us a set of relations between the spacetime coordinates as observed from two different inertial frames. In case, one of the frames is moving with a uniform rectilinear acceleration we have Rindler's transformation equations under such a situation. In the present work, we extend the Rindler's equations to a situation where we have in general non-uniform acceleration. After that we consider the non-inertial frame to undergo simple harmonic motion (SHM) and as a second case we consider the non-inertial frame to move uniformly along a circle. This set of transformation equations will have applications in various branches of Physics and in general in Astrophysics.

Transformation equation for frames undergoing non-uniform acceleration such as SHM and rotational motion

TL;DR

The paper extends Lorentz/Rindler transformations to non-uniformly accelerated frames by deriving explicit, approximate coordinate transformations for frames undergoing simple harmonic motion (SHM) along one axis and for frames moving in uniform circular motion (UCM). By treating the frame's proper acceleration as a time-dependent function and employing a small-velocity expansion with the parameter , the authors obtain SHM- and circle-based transformations that reduce to the Rindler/Lorentz form in the appropriate limits. The SHM cases yield time and space corrections up to , while the circular case introduces higher-order relativistic corrections that reflect anharmonic effects in SR. The results are framed as locally Lorentz-type (Fermi-Walker transported) and have potential applications in relativistic astrophysics, such as pulsating stars and binary systems, as well as offering a basis for extending Unruh-type analyses to non-uniform acceleration in flat spacetime.

Abstract

Lorentz transformation equations provide us a set of relations between the spacetime coordinates as observed from two different inertial frames. In case, one of the frames is moving with a uniform rectilinear acceleration we have Rindler's transformation equations under such a situation. In the present work, we extend the Rindler's equations to a situation where we have in general non-uniform acceleration. After that we consider the non-inertial frame to undergo simple harmonic motion (SHM) and as a second case we consider the non-inertial frame to move uniformly along a circle. This set of transformation equations will have applications in various branches of Physics and in general in Astrophysics.
Paper Structure (8 sections, 65 equations, 1 figure)

This paper contains 8 sections, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Transformation relations for a frame of reference executing SHM
  3. The frame $S^{\prime}$ undergoing SHM in $X-$direction:
  4. The frame $S^{\prime}$ undergoing SHM in $Y-$direction with initial phase of $\pi/2$ radians:
  5. Transformation relations for the frame moving in a circular path
  6. Comparison with some existing relativistic coordinate transformations
  7. Discussion of results and conclusions
  8. Non-uniform acceleration & Fermi-Walker transported frame

Figures (1)

  • Figure 1: Schematic of $S'$ frame moving in a circular path on the $XY$ plane.