Hadamard Renormalisation of the Stress Energy Tensor on the Horizons of a Spherically Symmetric Black Hole Space-Time

TL;DR

This paper develops a method to compute the renormalized stress-energy tensor of a quantum field near black hole horizons using Hadamard renormalization, with applications to lukewarm Reissner-Nordstrom-de Sitter black holes.

Contribution

It introduces a uniform approximation technique for the radial equation to evaluate the stress tensor components on black hole horizons in a general spherically symmetric spacetime.

Findings

Derived conditions for the regularity of the Hartle-Hawking state on black hole horizons.

Applied the method specifically to lukewarm Reissner-Nordstrom-de Sitter black holes.

Provided explicit expressions for the renormalized stress tensor components.

Abstract

We consider a quantum field which is in a Hartle-Hawking state propagating in a general spherically symmetric black hole space-time. We make use of uniform approximations to the radial equation to calculate the components of the stress tensor, renormalized using the Hadamard form of the Green's function, on the horizons of this space-time. We then specialize these results to the case of the `lukewarm' Reissner-Nordstrom-de Sitter black hole and derive some conditions on the stress tensor for the regularity of the Hartle-Hawking state.