The HeatKernel in a Schwarzschild Geometry and the Casimir Energy
Abstract
We obtain an hybrid expression for the heatkernel, 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 zeropoint energy density as a function of the distance from the massive object generating the gravitational field. The contribution to the zeropoint 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.
 Publication:

arXiv eprints
 Pub Date:
 October 1997
 DOI:
 10.48550/arXiv.grqc/9710100
 arXiv:
 arXiv:grqc/9710100
 Bibcode:
 1997gr.qc....10100A
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 LaTeX2e (minor errors corrected, discussion extended and a few new results added)