“Physical process version” of the first law and the generalized second law for charged and rotating black holes

We investigate both the “physical process” version of the first law and the generalized second law of black hole thermodynamics for charged and rotating black holes. We begin by deriving general formulas for the first order variation in the Arnowitt-Deser-Misner mass and angular momentum for linear perturbations off a stationary, electrovac background in terms of the perturbed nonelectromagnetic stress-energy $\delta T_{\mathrm{ab}}$ and the perturbed charge current density $\delta j^a.$ 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.