Properties of dynamical black hole entropy

We study the first law for non-stationary perturbations of a stationary black hole whose event horizon is a Killing horizon, that relates the first-order change in the mass and angular momentum to the change in the entropy of an arbitrary horizon cross-section. Recently, Hollands, Wald and Zhang [1] have shown that the dynamical black hole entropy that satisfies this first law, for general relativity, is Sdyn = (1 - v partial derivative v)SBH, where v is the affine parameter of the null horizon generators and SBH is the Bekenstein-Hawking entropy, and for general diffeomorphism covariant theories of gravity Sdyn = (1 - v partial derivative v)SWall, where SWall is the Wall entropy. They obtained the first law by applying the Noether charge method to non-stationary perturbations and arbitrary cross-sections. In this formalism, the dynamical black hole entropy is defined as an "improved" Noether charge, which is unambiguous to first order in the perturbation. In the present article we provide a pedagogical derivation of the physical process version of the non-stationary first law for general relativity by integrating the linearised Raychaudhuri equation between two arbitrary horizon cross-sections. Moreover, we generalise the derivation of the first law in [1] to non-minimally coupled matter fields that are smooth on the horizon, using boost weight arguments rather than Killing field arguments, and we relax some of the gauge conditions on the perturbations by allowing for non-zero variations of the horizon Killing field and surface gravity. Finally, for f(Riemann) theories of gravity we show explicitly using Gaussian null coordinates that the improved Noether charge is Sdyn = (1 - v partial derivative v)SWall, which is a non-trivial check of [1].