Unruh and analogue Unruh temperatures for circular motion in 3+1 and 2+1 dimensions

TL;DR

This paper analytically and numerically investigates the temperature experienced by circularly accelerating observers in different spacetime dimensions, revealing dimension-dependent differences and potential experimental implications for analogue gravity systems.

Contribution

It provides the first detailed analytic and numerical analysis of circular Unruh temperatures in 3+1 and 2+1 dimensions, including their dependence on parameters and potential for experimental observation.

Findings

In 3+1 dimensions, $T_{circ}$ is comparable to linear Unruh temperature at high energies.

In 2+1 dimensions, $T_{circ}$ is significantly lower at low energies.

Circular Unruh temperature can grow arbitrarily large near the sonic limit in analogue models.

Abstract

The Unruh effect states that a uniformly linearly accelerated observer with proper acceleration $a$ experiences Minkowski vacuum as a thermal state in the temperature $T_{\text{lin}} = a/(2\pi)$, operationally measurable via the detailed balance condition between excitation and de-excitation probabilities. An observer in uniform circular motion experiences a similar Unruh-type temperature $T_{\text{circ}}$, operationally measurable via the detailed balance condition, but $T_{\text{circ}}$ depends not just on the proper acceleration but also on the orbital radius and on the excitation energy. We establish analytic results for $T_{\text{circ}}$ for a massless scalar field in $3+1$ and $2+1$ spacetime dimensions in several asymptotic regions of the parameter space, and we give numerical results in the interpolating regions. In the ultrarelativistic limit, we verify that in $3+1$ dimensions $T_{\text{circ}}$ is of the order of $T_{\text{lin}}$ uniformly in the energy, as previously found by Unruh, but in $2+1$ dimensions $T_{\text{circ}}$ is significantly lower at low energies. We translate these results to an analogue spacetime nonrelativistic field theory in which the circular acceleration effects may become experimentally testable in the near future. We establish in particular that the circular motion analogue Unruh temperature grows arbitrarily large in the near-sonic limit, encouragingly for the experimental prospects, but the growth is weaker in effective spacetime dimension $2+1$ than in $3+1$.