The Entropy of the Complex Scalar Field in a Charged Kerr Black Hole

By using the brick wall method we calculate the thermodynamic potential of the complex scalar field in a charged Kerr black hole. Using it we show that in the Hartle-Hawking state the leading term of the entropy is proportional to AH ǫ , which becomes divergent as the system approaches the black hole horizon. The origin of the divergence is that the density of states diverges at the horizon. PACS numbers: 04.62.+v, 04.70.Dy e-mail : [email protected] 1 By comparing the black hole physics with the thermodynamics Bekenstein showed that the black hole entropy is proportional to the horizon area [1, 2], and Hawking’s discovery of the black hole evaporation confirmed that. In Euclidean path integral approach it was shown that the tree level contribution of the gravitation action gives the black hole entropy [3]. However the exact statistical origin of the Bekenstein-Hawking black hole entropy is unclear. Recently many efforts have been concentrated on understanding the statistical origin of black hole thermodynamics, specially the black hole entropy by various methods [4]: the brick wall method, the conical singularity method, and the entanglement entropy method [5, 6]. The leading term of the entropy obtained by those methods is proportional to the surface area of the horizon. However the proportional coefficient diverges as the cut-off goes to zero. The divergence is because of an infinite number of states near the horizon, which can be explained by the equivalence principle [7]. For the black hole with a rotation the entropy of the neutral scalar field was calculated by authors [9]. It was shown that the leading term of the entropy, if the quantum field is at the Hartle-Hawking state, is proportional to the horizon area. In this paper to understand more deeply the black hole entropy we will investigate the entropy of the complex scalar field interacting with the gauge field Aμ in the charged Kerr black hole background [10]. Let us consider a minimally coupled complex scalar field with mass μ in thermal equilibrium at temperature 1/β in the charged Kerr black hole spacetime. The line element of the charged Kerr black hole spacetime and the electromagnetic vector potential in Boyer-Lindquist coordinates are given by [10, 11] ds = − ( ∆− a2 sin θ Σ ) dt − 2a sin 2 θ (r2 + a2 −∆) Σ dtdφ