Tunnelling processes for Hadamard states through a 2+1 dimensional black hole and Hawking radiation

TL;DR

This paper investigates the behavior of quantum scalar fields near a 2+1 dimensional black hole horizon, demonstrating thermal properties consistent with Hawking radiation and verifying the existence of Hadamard states in this context.

Contribution

It extends the analysis of Hadamard states and tunnelling processes to 2+1 dimensions, including a concrete example with the BTZ black hole.

Findings

Two-point correlation functions show thermal behavior at Hawking temperature near the horizon.

Hadamard states exist for scalar fields with Robin boundary conditions in BTZ spacetime.

The analysis confirms the universality of Hawking radiation features in lower-dimensional black holes.

Abstract

We analyse the local behaviour of the two-point correlation function of a quantum state for a scalar field in a neighbourhood of a Killing horizon in a $2+1$-dimensional spacetime, extending the work of Moretti and Pinamonti in a $3+1$-dimensional scenario. In particular we show that, if the state is of Hadamard form in such neighbourhood, similarly to the $3+1$-dimensional case, under a suitable scaling limit towards the horizon, the two-point correlation function exhibits a thermal behaviour at the Hawking temperature. Since the whole analysis rests on the assumption that a Hadamard state exists in a neighbourhood of the Killing horizon, we show that this is not an empty condition by verifying it for a massive, real scalar field subject to Robin boundary conditions in the prototypic example of a three dimensional black hole background: the non-extremal, rotating BTZ spacetime.