The "physical process" version of the first law and the generalized second law for charged and rotating black holes

TL;DR

The paper develops a general framework for the physical process version of the first law in charged and rotating black holes and analyzes the generalized second law under quasi-static lowering of matter toward a black hole. It derives explicit expressions for the first-order variations of mass and angular momentum in Einstein-Maxwell theory and proves the first-law relation that ties horizon area changes to fluxes across the horizon via the Raychaudhuri equation. The authors then model a surrounding thermal atmosphere in local thermodynamic equilibrium, invoking Tolman scaling and a Gibbs–Duhem relation to show that the total generalized entropy is extremized when the atmosphere and horizon temperatures and potentials match the black hole’s, thereby supporting the GSL for lowering and release processes without matter entropy bounds. Overall, the work extends black hole thermodynamics to charged and rotating cases, clarifies the role of the thermal atmosphere, and provides a model-independent pathway to validate the GSL in quasi-static dissipation processes.

Abstract

We investigate both the ``physical process'' version of the first law and the second law of black hole thermodynamics for charged and rotating black holes. We begin by deriving general formulas for the first order variation in ADM mass and angular momentum for linear perturbations off a stationary, electrovac background in terms of the perturbed non-electromagnetic stress-energy, $δT_{ab}$, and the perturbed charge current density, $δj^a$. Using these formulas, we prove the "physical process version" of the first law for charged, stationary black holes. We then investigate the generalized second law of thermodynamics (GSL) for charged, stationary black holes for processes in which a box containing charged matter is lowered toward the black hole and then released (at which point the box and its contents fall into the black hole and/or thermalize with the ``thermal atmosphere'' surrounding the black hole). Assuming that the thermal atmosphere admits a local, thermodynamic description with respect to observers following orbits of the horizon Killing field, and assuming that the combined black hole/thermal atmosphere system is in a state of maximum entropy at fixed mass, angular momentum, and charge, we show that the total generalized entropy cannot decrease during the lowering process or in the ``release process''. Consequently, the GSL always holds in such processes. No entropy bounds on matter are assumed to hold in any of our arguments.

Figures (1)

• Figure 1: In a stationary spacetime, a box is held in place by an agent who holds a massless string connected to the box. The surface, $C$, enclosing the box and string is represented by the dotted line.