Inertial-to-Rindler Coordinates, with applications to the Twin Paradox, Radar Time and the Unruh Temperature
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915757444562944 |
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| author | Alsing, Paul M. |
| author_facet | Alsing, Paul M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18956 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inertial-to-Rindler Coordinates, with applications to the Twin Paradox, Radar Time and the Unruh Temperature Alsing, Paul M. General Relativity and Quantum Cosmology 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. |
| title | Inertial-to-Rindler Coordinates, with applications to the Twin Paradox, Radar Time and the Unruh Temperature |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2601.18956 |