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

TL;DR

The paper introduces Inertial-to-Rindler (I2R) coordinates, a two-parameter framework with $(a_0, \beta_0)$ that interpolates between inertial motion and uniform acceleration in flat spacetime. It derives the I2R metric $ds^2_{I2R}=(1+a(t)x)^2 dt^2 - dx^2$ with time-dependent proper acceleration $a(t)$ and analyzes the resulting radar-time surfaces and twin-paradox scenarios, showing how turnaround can be realized as either an immediate or gradual process by adjusting $\beta_0$. The work extends the Unruh-temperature analysis to nonuniform acceleration, deriving perturbative corrections in two limits ($\beta_0\ll1$ and $\beta_0\approx1$) and proposing a velocity-dependent temperature $T_{I2R}=\frac{\hbar a(t)}{2\pi c}$ that connects smoothly to zero at zero acceleration and to the standard Unruh temperature at constant acceleration. The framework provides a coherent, continuous description of noninertial observers in Minkowski space, with implications for particle content, radar-time semantics, and potential links to analogs of Hawking radiation and cosmological particle production. Overall, it offers a versatile tool for analyzing noninertial motion and its quantum-field-theoretic consequences beyond idealized, purely inertial or purely uniform-accelerating cases.

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.

Figures (8)

• Figure 1: Schematic definition of radar time$\tau(x)$.
• Figure 2: Orbits and lines of simultaneity ${\tau}_0$ for $a_0 = 0.1$ and ${\beta}_0 = \{0.0001, 0.1, 0.25\}$. (solid black) $I2R$ orbit (Irwin), (dashed red) lightcone $T=\pm X$, (dashed blue) Irwin's asymptotic velocity $T=\pm {\beta}_0 X$, (dashed gray) lines of simultaneity ${\tau}_0$ (labeled).
• Figure 3: Orbits and lines of simultaneity ${\tau}_0$ for $a_0 = 0.1$ and ${\beta}_0 = \{0.5, 0.75, 0.9999\}$. (solid black) $I2R$ orbit (Irwin), (dashed red) lightcone $T=\pm X$, (dashed blue) Irwin's asymptotic velocity $T=\pm {\beta}_0 X$, (dashed gray) lines of simultaneity ${\tau}_0$ (labeled).
• Figure 4: Acceleration profiles for $a_0 = 0.1$ and ${\beta}_0 = \{0.0001, 0.1, 0.25\}$.
• Figure 5: Acceleration profiles for $a_0 = 0.1$ and ${\beta}_0 = \{0.5, 0.75, 0.9999\}$.