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We propose an entropy current for dynamical black holes in a theory witharbitrary four derivative corrections to Einstein's gravity, linearized arounda stationary black hole. The Einstein-Gauss-Bonnet theory is a special case ofthe class of theories that we consider. Within our approximation, ourconstruction allows us to write down a completely local version of the secondlaw of black hole thermodynamics, in the presence of the higher derivativecorrections considered here. This ultra-local, stronger form of the second lawis a generalization of a weaker form, applicable to the total entropy,integrated over a compact `time-slice' of the horizon, a proof of which hasbeen recently presented in arXiv:1504.08040. We also provide a generalalgorithm to construct the entropy current for the four derivative theories,which may be straightforwardly generalized to arbitrary higher derivativecorrections to Einstein's gravity. This algorithm highlights the possibleambiguities in defining the entropy current.

An entropy current for dynamical black holes in four-derivative theories of gravity