An entropy current for dynamical black holes in four-derivative theories of gravity
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Springer
Received: February 25, 2020 Accepted: May 11, 2020 Published: June 1, 2020
Jyotirmoy Bhattacharya,a Sayantani Bhattacharyya,b Anirban Dindab and Nilay Kunduc a
Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India b School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, Khurda 752050, Odisha, India c Department of Physics, Indian Institute of Technology Kanpur, Kalyanpur, Kanpur 208016, India
E-mail: [email protected], [email protected], [email protected], [email protected] 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 derivative corrections to Einstein’s gravity. This algorithm highlights the possible ambiguities in defining the entropy current. Keywords: Black Holes, Black Holes in String Theory ArXiv ePrint: 1912.11030
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)017
JHEP06(2020)017
An entropy current for dynamical black holes in four-derivative theories of gravity
Contents 1 Introduction and summary
2 7 8 12 12 13 16
3 An entropy current for four derivative theories of gravity HD and the entropy current for theories with four 3.1 Explicit calculation of Evv derivative corrections to Einstein gravity 3.1.1 Ricci scalar square theory 3.1.2 Ricci tensor square theory 3.1.3 Riemann tensor square theory HD 3.2 The most general structure of the ‘zero boost term’ in Evv HD 3.3 Constraints on the ‘zero boost terms’ in Evv 3.4 The general strategy for constructing the entropy current maintaining the boost symmetry 3.5 Einstein-Gauss-Bonnet gravity in d ≥ (4 + 1) 3.6 The Einstein-Gauss-Bonnet theory in d = 3 + 1 HD 3.7 Comments on entropy current for higher boost terms in Evv
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4 Discussions and future directions
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A A general stationary metric can have v dependent components
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B Arguments leading to vanishing of Tvv on any Killing horizon
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C Conventions, notations and useful formulae
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D Detailed expressions D.1 Expressions of Riemann tensors and Ricci tensors off the horizon HD for different theories D.2 Relevant terms on the horizon H, to compute Evv D.3 Ricci scalar square theory D.4 Ricci tensor squared theory D
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