Unruh-like effects: Effective temperatures along stationary worldlines

Abstract

We study the detailed balance temperatures recorded along all classes of stationary, uniformly accelerated worldlines in four-dimensional Minkowski spacetime, namely along (i) linear uniform acceleration, (ii) cusped, (iii) circular, (iv) catenary, and (v) helix worldlines, among which the Unruh temperature is the particular case for linear uniform acceleration. As a measuring device, we employ an Unruh-DeWitt detector, modeled as a qubit that interacts for a long time with a massless Klein-Gordon field in the vacuum state. The temperatures in each case (i) - (v) are functions of up to three invariant quantities: curvature or proper acceleration, $\kappa$, torsion, $b$, and hypertorsion, $\nu$, and except for the case (i), they depend on the transition frequency difference of the detector, $\omega$. We investigate numerically the behavior of the frequency-dependent temperatures for different values of $\kappa$, $b$, and $\nu$ along the stationary worldlines (ii) - (v) and evaluate analytically the regimes where the temperatures recorded along the different worldlines coincide with each other in terms of relevant asymptotic limits for $\kappa$, $b$, or $\nu$, and discuss their physical meaning. We demonstrate that the temperatures in cases (ii) - (v) dip under the Unruh temperature at low frequencies and go above the Unruh temperature for large $|\omega|$. It is our hope that this study will be relevant to the design of experiments seeking to verify the Unruh effect or generalizations thereof.