Black hole entropy from non-proper gauge degrees of freedom: II. The charged vacuum capacitor

TL;DR

This paper investigates the classical degrees of freedom responsible for black hole entropy using a charged vacuum capacitor as a toy model, revealing boundary-induced contributions in gauge theories that relate to thermodynamics.

Contribution

It demonstrates how boundary conditions in gauge theories lead to additional classical contributions to entropy, providing a microscopic derivation in a charged capacitor model.

Findings

Boundary conditions induce non-trivial gauge degrees of freedom.

Classical contributions scale with the area of the capacitor plates.

Microscopic derivation of thermodynamics for the charged capacitor.

Abstract

The question which degrees of freedom are responsible for the classical part of the Gibbons-Hawking entropy is addressed. A physical toy model sharing the same properties from the viewpoint of the linearized theory is a charged vacuum capacitor. In Maxwell's theory, the gauge sector including ghosts is a topological field theory. When computing the grand canonical partition function with a chemical potential for electric charge in the indefinite metric Hilbert space of the BRST quantized theory, the classical contribution originates from the part of the gauge sector that is no longer trivial due to the boundary conditions required by the physical set-up. More concretely, in the benchmark problem of a planar charged vacuum capacitor, we identify the degrees of freedom that, in the quantum theory, give rise to an additional contribution to the standard black body result proportional to the area of the plates, and that allow for a microscopic derivation of the thermodynamics of the charged capacitor.