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

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 theimmediate-" 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.