Abstract
We construct a proof of the second law of thermodynamics in an arbitrary diffeomorphism invariant theory of gravity working within the approximation of linearized dynamical fluctuations around stationary black holes. We achieve this by establishing the existence of an entropy current defined on the horizon of the dynamically perturbed black hole in such theories. By construction, this entropy current has non-negative divergence, suggestive of a mechanism for the dynamical black hole to approach a final equilibrium configuration via entropy production as well as the spatial flow of it on the null horizon. This enables us to argue for the second law in its strongest possible form, which has a manifest locality at each space-time point. We explicitly check that the form of the entropy current that we construct in this paper exactly matches with previously reported expressions computed considering specific four derivative theories of higher curvature gravity. Using the same set up we also provide an alternative proof of the physical process version of the first law applicable to arbitrary higher derivative theories of gravity.
Highlights
1 Introduction and summary In Einstein’s classical theory of gravity, black holes are interesting solutions that behave like thermodynamic objects [1,2,3,4] and one can associate notions like temperature and entropy to them
They should maintain the laws of thermodynamics provided we could correctly identify the thermodynamic properties with the geometric properties of black holes
The originality of our results, lies in the fact that we have been able to justify the ultra-local version of the second law via the entropy current on the horizon with an analysis that is based on general principles like Noether charge for diffeomorphism invariance and most importantly it makes a statement applicable to any higher derivative theory of gravity
Summary
In Einstein’s classical theory of gravity, black holes are interesting solutions that behave like thermodynamic objects [1,2,3,4] and one can associate notions like temperature and entropy to them. The analysis of [22] explicitly showed that, only for specific four derivative theories of gravity, introducing the notion of an entropy current, the second law for black holes could be formulated in its most potent possible form, i.e., in an ultra-local version which is local in both time and space This statement has been recast even in a more robust sense as follows: in a general four derivative theory of gravity, working up to linear order in the amplitude of the dynamical fluctuations, the spatial components of the entropy current has to be accounted for if one aims to prove an ultra-local version of the second law (as described above). The other appendices provide useful technical details for our computations
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