Ground and thermal states for the Klein-Gordon field on a massless hyperbolic black hole with applications to the anti-Hawking effect

TL;DR

This paper constructs ground and thermal states for a scalar field on a hyperbolic black hole, analyzes the detector response, and finds that the anti-Hawking effect observed in lower dimensions does not occur in four dimensions.

Contribution

It provides a detailed construction of quantum states for scalar fields on hyperbolic black holes and investigates the anti-Hawking effect across different dimensions.

Findings

Anti-Hawking effect occurs in 3D but not in 4D hyperbolic black holes.

Constructed explicit two-point functions for ground and thermal states.

Analyzed detector responses to understand quantum effects near hyperbolic black holes.

Abstract

On an $n$-dimensional, massless, topological black hole with hyperbolic sections, we construct the two-point function both of a ground state and of a thermal state for a real, massive, free scalar field arbitrarily coupled to scalar curvature and endowed with Robin boundary conditions at conformal infinity. These states are used to compute the response of an Unruh-DeWitt detector coupled to them for an infinite proper time interval along static trajectories. As an application, we focus on the massless conformally coupled case and we show, numerically, that the anti-Hawking effect, which is manifest on the three-dimensional case, does not occur if we consider a four-dimensional massless hyperbolic black hole. On the one hand, we argue that this result is compatible with what happens in the three- and four dimensional Minkowski spacetime, while, on the other hand, we stress that it generalizes existing results concerning the anti-Hawking effect on black hole spacetimes.