Inertial-to-Rindler Coordinates, with applications to the Twin Paradox, Radar Time and the Unruh Temperature

TL;DR

This paper introduces a two-parameter family of transformations called Inertial-to-Rindler coordinates, bridging inertial and accelerated frames, and explores their implications for the Twin Paradox, radar time, and the Unruh effect.

Contribution

It formulates a new coordinate system interpolating between inertial and Rindler frames, providing insights into relativistic paradoxes and quantum field effects.

Findings

Revealed how I2R coordinates justify different Twin Paradox scenarios.

Analyzed radar time hypersurfaces during acceleration transition.

Derived corrections to the Unruh temperature based on initial/final velocities.

Abstract

In this work we formulate a two-parameter family of transformations in flat Minkowksi spacetime that smoothly interpolates between motion with constant initial/final velocity (inertial coordinates), and with constant acceleration (Rindler coordinates \cite{Rindler:1956}), which we term Inertial-to-Rindler (I2R) coordinates. We revisit the Twin ``Paradox" and show how the new I2R coordinates justify the ``immediate-" and ``gradual-turnaround" scenarios discussed in many texbooks and articles. We also examine the radar time formulation of hypersurfaces of simultaneity by Dolby and Gull \cite{Dolby_Gull:2001} for these new coordinates as we transition from zero to uniform acceleration. Finaly we re-examine the negative frequency content of a purely positive frequency Minkowski plane wave as observed by the I2R observer, and derive perturbative corrections to the Unruh \cite{Unruh:1976} temperature for the two cases of initial/final velocities slightly greater than zero, and slightly less than the speed of light - the latter of which characterizes constant acceleration motion. We argue for a proposed velocity-dependent generalization of the Unruh temperature that smoothly varies from zero at zero-acceleration, to the standard form at constant acceleration.