Extremal black hole entropy satisfying the Nernst theorem
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O R I G I N A L A RT I C L E
Extremal black hole entropy satisfying the Nernst theorem Li-Qin Mi
Received: 4 September 2012 / Accepted: 15 October 2012 / Published online: 26 October 2012 © Springer Science+Business Media Dordrecht 2012
Abstract Quantum entropy of extremal black holes has been discussed by many authors. However the contribution of inner event horizon has not been considered. In this paper, we consider the Reissner-Nordström-anti-de Sitter black hole, and show that the contributions lead extremal black hole entropy to obey the Nernst theorem, naturally. Therefore the contributions are important and cannot be neglected in near-extremal and extremal black hole cases. Keywords Extremal black hole · Quantum entropy · Event horizon
It is well known that entropy of an ordinary thermodynamic system vanishes at the absolute zero of temperature, which is called Nernst theorem. Although is commonly considered to be a fundamental law of thermodynamics, many open questions remain and a proper entropy is still lacking for extremal black hole. In fact, one can easily find that the Bekenstein-Hawking entropy at absolute zero temperature is nonvanishing and violates the Nernst theorem (Wald 1997). Hawking et al. (1995) proposed that the entropy of an extremal Reissner-Nordström black hole is zero, despite the fact that its horizon has nonzero area. Ghosh and Mitra (1997), Mitra (1998) showed that the entropy depends very significantly on whether quantization is carried out first or extremalization: in the former case, the answer is a quarter of the area, and in the latL.-Q. Mi () College of Science, Zhejiang University of Technology, Hangzhou, 310023, China e-mail: [email protected]
ter, zero. On the other hand, ’t Hooft (1985) introduced the brick wall model. In this model, the black hole entropy includes the entropy of the quantized fields in its neighborhood. Using the brick-wall model, Winstanley (2001) found that, for a quantum scalar field, extremalization after quantization, the entropy diverges as in the nonextremal case, and may be absorbed in a suitable renormalization of the coupling constants. The main purpose of this paper is to attempt to obtain a black hole entropy, which naturally obeys the Nernst theorem when black hole becomes extremal. In 2000 year, Li (2000) starting from Teukolsky-type master equation, investigated the quantum entropy of the nonextremal Reissner-Nordström black hole due to arbitrary spin field by using the brick-wall model, and discovered first the spin-dependent correction to the entropy. Subsequently, the spin effect of quantum entropy of various nonextremal black hole has been studied (Li 2006; Jing and Yan 2001a, 2001b; López-Ortega 2003; Mi and Li 2006). However, the entropy is negative when one assumes that the black holes are extremal. In this paper we consider the Reissner-Nordström-anti-de Sitter(RNAdS) black hole, we demonstrate that, using the brick-wall model, one can obtain a black hole entropy which satisfies the Nernst theorem when the inner event horizon is t
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