Stresses and Strains in the First Law for Kaluza-Klein Black Holes

TL;DR

This paper derives a generalized first law for Kaluza-Klein black holes, incorporating moduli variations of the compactification manifold, using Hamiltonian methods and concepts from elasticity theory.

Contribution

It provides a new Hamiltonian-based proof for the first law with moduli variations and extends the framework to 2-torus compactifications involving stress and strain tensors.

Findings

First law includes moduli variation terms for Kaluza-Klein black holes.

Stress and strain tensors describe shape and size changes of the compactification manifold.

Result applies to arbitrary perturbations around static black hole backgrounds.

Abstract

We consider how variations in the moduli of the compactification manifold contribute pdV type work terms to the first law for Kaluza-Klein black holes. We give a new proof for the circle case, based on Hamiltonian methods, which demonstrates that the result holds for arbitrary perturbations around a static black hole background. We further apply these methods to derive the first law for black holes in 2-torus compactifications, where there are three real moduli. We find that the result can be simply stated in terms of constructs familiar from the physics of elastic materials, the stress and strain tensors. The strain tensor encodes the change in size and shape of the 2-torus as the moduli are varied. The role of the stress tensor is played by a tension tensor, which generalizes the spacetime tension that enters the first law in the circle case.