On the Unruh effect in an effective Kleinian geometry

Abstract

Direct experimental verification of the Unruh effect has eluded physicists for a long time. However, this

scenario may be changing. Metamaterial analog systems may provide promising alternatives for the obser- vation of the Unruh effect. Furthermore, photons, in some of these systems, behave as if they were in a

spacetime endowed with a Kleinian signature (+ +−−) — a Kleinian space. In this work, the Unruh effect in an effective geometry with a Kleinian signature arising from an anisotropic medium, such as a hyperbolic metamaterial, is investigated. The path integral approach to the Unruh effect is applied in the context of a background space with Kleinian signature, where problems arise. It is noted that any metamaterial analog system might be, in fact, mimicking a known physical phenomenon but in an exotic space-time where some established physical properties may break down. In the case of this work, an analog accelerated frame in Kleinian space is derived and an induced compactification of a timelike dimension as well as the absence of “wedges”, in contrast to what occurs in Rindler space, are observed. Although an analog Unruh effect in Kleinian space is not shown, some perspectives and alternatives are discussed.