The Heat-Kernel in a Schwarzschild Geometry and the Casimir Energy

TL;DR

This paper derives a renormalized expression for the Casimir energy density of a scalar field in Schwarzschild spacetime, revealing effects like entropy generation and bound states due to curvature.

Contribution

It introduces a hybrid heat-kernel expression in Schwarzschild geometry and computes the Casimir energy density, including curvature effects, at finite temperature.

Findings

Entropy is positive outside the horizon for small beta.

Internal energy can be negative outside the horizon.

Total energy inside the horizon is finite but complex, indicating particle creation.

Abstract

We obtain an hybrid expression for the heat-kernel, and from that the density of the free energy, for a minimally coupled scalar field in a Schwarzschild geometry at finite temperature. This gives us the zero-point energy density as a function of the distance from the massive object generating the gravitational field. The contribution to the zero-point energy due to the curvature is extracted too, in this way arriving at a renormalised expression for the energy density (the Casimir energy density). We use this to find an expression for other physical quantities: internal energy, pressure and entropy. It turns out that the disturbance of the surrounding vacuum generates entropy. For $\beta$ small the entropy is positive for $r>2M$. We also find that the internal energy can be negative outside the horizon pointing to the existence of bound states. The total energy inside the horizon turns out to be finite but complex, the imaginary part being interpreted as responsible for particle creation.