Black hole thermodynamic geodesics

Abstract

Put forward in recent years, the formalism of geometrothermodynamics (GTD) makes it possible to describe the geometry of thermodynamics via Legendre-invariant metrics. It has been applied, with success, to a number of diverse thermodynamic systems, including black holes. In this dissertation, starting from the appropriate GTD metric and working in the mass representation, a pair of differential equations is obtained to describe the geodesics in the space of thermodynamic equilibrium states of a Reissner-Nordstrom black hole. In this case the said geodesics are expected to be curves that extremise changes in the black hole mass. The differential equations are solved numerically by considering the processes of Hawking radiation and the Schwinger mechanism to derive a set of constraints. If it is instead assumed that the equations governing these two processes hold at every point of a black hole's trajectory, it turns out that the resulting curves share certain similarities with the geodesics, implying that the combined Hawking and Schwinger mechanisms might be nature's way of 'selecting' thermodynamic equilibrium states that lie along a geodesic. The analysis is then extended to the Kerr black hole. In this case, Hawking radiation is the only process considered here that is responsible for the evolution of the black hole. The geodesics obtained numerically for both types of black hole have a common feature: in general, the closer the constraining values are to extremality, the lower the entropy along a trajectory turns out to be. The geodesics pertaining to the Reissner-Nordstrom black hole all reach a maximum close to the point specified by the constraints, with the decrease in entropy being larger if a quantity of charge is gained than if the same quantity is lost. On the other hand, the geodesics followed by a Kerr black hole only have stationary points if the constraints are very close to extremality. This mirrors the fact that, unlike the pair-production of electrons and positrons by the electric field surrounding a charged black hole, the slowing down/speeding up of a rotating black hole affects the entropy directly. The Kerr geodesics are instead characterised by a point of inflection that gets more pronounced as the constraining value for the black hole's speed increases.