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

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