The first law of black hole mechanics in the Einstein-Maxwell theory revisited
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Springer
Received: July 1, 2020 Accepted: August 5, 2020 Published: September 3, 2020
Zachary Elgood,a Patrick Meessenb,c and Tom´ as Ort´ına a
Instituto de F´ısica Te´ orica UAM/CSIC, C/ Nicol´ as Cabrera, 13–15, C.U. Cantoblanco, E-28049 Madrid, Spain b HEP Theory Group, Departamento de F´ısica, Universidad de Oviedo, Avda. Calvo Sotelo s/n, E-33007 Oviedo, Spain c Instituto Universitario de Ciencias y Tecnolog´ıas Espaciales de Asturias (ICTEA), Calle de la Independencia, 13, E-33004 Oviedo, Spain
E-mail: [email protected], [email protected], [email protected] Abstract: We re-derive the first law of black hole mechanics in the context of the EinsteinMaxwell theory in a gauge-invariant way introducing “momentum maps” associated to field strengths and the vectors that generate their symmetries. These objects play the role of generalized thermodynamical potentials in the first law and satisfy generalized zeroth laws, as first observed in the context of principal gauge bundles by Prabhu, but they can be generalized to more complex situations. We test our ideas on the d-dimensional Reissner-Nordstr¨ om-Tangherlini black hole. Keywords: Black Holes, Classical Theories of Gravity ArXiv ePrint: 2006.02792
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)026
JHEP09(2020)026
The first law of black hole mechanics in the Einstein-Maxwell theory revisited
Contents 1 Introduction
1
2 Covariant Lie derivatives and momentum maps 2.1 Lie-Maxwell derivatives 2.2 Lie-Lorentz derivatives
4 4 6 10 10 11
4 Wald-Noether charge for the E-M theory
13
5 The first law of black hole mechanics in the E-M theory
15
6 Discussion
17
1
Introduction
Black-hole thermodynamics1 is probably one of the most active fields of research in Theoretical Physics. It interconnects seemingly disparate areas of Physics such as Gravity, Quantum Field Theory and Information Theory providing deep insights in all of them. Black-hole thermodynamics originates in the analogy between the behaviour of the area of the event horizon A and the second law obeyed by the thermodynamic entropy S noticed by Bekenstein [2, 3] in the results obtained by Christodoulou and Hawking [4–7]. Shortly afterwards, in ref. [8] Bardeen, Carter and Hawking extended this analogy by proving another three laws of black hole mechanics similar to the other three laws of thermodynamics involving the event horizon’s surface gravity κ and angular velocity Ω and the black hole’s mass M . However, the analogy was only taken seriously after Hawking’s discovery that black holes radiate as black bodies with a temperature T = κ/2π [9], which implied the relation S = A/4, both in c = GN = ~ = k = 1 units. Ever since the formulation of these four laws, it has been tried to extend their domain of application and validity with the inclusion of matter fields and terms of higher-order in the curvature, for instance. In refs. [10–12] Wald and collaborators developed a new approach to demonstrate the first law of black hole mechanics
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