Abstract
We propose an entropy current for dynamical black holes in a theory with arbitrary four derivative corrections to Einstein’s gravity, linearized around a stationary black hole. The Einstein-Gauss-Bonnet theory is a special case of the class of theories that we consider. Within our approximation, our construction allows us to write down a completely local version of the second law of black hole thermodynamics, in the presence of the higher derivative corrections considered here. This ultra-local, stronger form of the second law is a generalization of a weaker form, applicable to the total entropy, integrated over a compact ‘time-slice’ of the horizon, a proof of which has been recently presented in [1]. We also provide a general algorithm to construct the entropy current for the four derivative theories, which may be straightforwardly generalized to arbitrary higher deriva- tive corrections to Einstein’s gravity. This algorithm highlights the possible ambiguities in defining the entropy current.
Highlights
Introduction and summaryIt is widely believed that Einstein’s theory of gravity must admit an adequate UV completion when we approach length scales comparable to Planck length
We propose an entropy current for dynamical black holes in a theory with arbitrary four derivative corrections to Einstein’s gravity, linearized around a stationary black hole
The limit that we shall consider here, is the one, where the length scale associated with the curvatures of the space-time and those associated with the variations of the matter fields, are much larger compared to ls
Summary
It is widely believed that Einstein’s theory of gravity must admit an adequate UV completion when we approach length scales comparable to Planck length. This is the central result of our note. We would like to mention that, the notion of an entropy current for black hole dynamics in higher derivative theories of gravity, is not completely new in this note. This idea has been previously introduced in [33,34,35]. Given the form of the metric (2.1), let us outline the broad strategy of the proof of second law provided in [1]
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