Effect of the inner horizon on the black hole thermodynamics: Reissner-Nordström black hole and Kerr black hole

Abstract

For the Schwarzschild black hole the Bekenstein-Hawking entropy is proportional to the area of the event horizon. For the black holes with two horizons the thermodynamics is not very clear, since the role of the inner horizons is not well established. Here we calculate the entropy of the Reissner-Nordström black hole and of the Kerr black hole, which have two horizons. For the spherically symmetric Reissner-Nordström black hole we used several different approaches. All of them give the same result for the entropy and for the corresponding temperature of the thermal Hawking radiation. The entropy is not determined by the area of the outer horizon, and it is not equal to the sum of the entropies of two horizons. It is determined by the correlations between the two horizons, due to which the total entropy of the black hole and the temperature of Hawking radiation depend only on mass M of the black hole and do not depend on the black hole charge Q. For the Kerr and Kerr-Newman black holes it is shown that their entropy has the similar property: it depends only on mass M of the black hole and does not depend on the angular momentum J and charge Q.

Thermodynamics of multi-horizon spacetimes

“…It is not fully understood whether there exists a global temperature for the multi-horizon spacetimes or not. If more than one horizons for a spacetime are present, it may contribute to the temperature of the spacetime [33][34][35][36][37]. Also, the total entropy of the spacetime can not be determined only by the outer horizon.…”

“…Also, the total entropy of the spacetime can not be determined only by the outer horizon. The correlations between the multi horizons determine it [33,34,37].…”

“…Recently, It has been shown that a global temperature can exist for a Reissner-Nordstrom black hole [33,34]. This temperature depends only on the mass of the black hole and does not depend on the black hole charge.…”

“…This temperature depends only on the mass of the black hole and does not depend on the black hole charge. Also, the total entropy of a Reissner-Nordstrom black hole, a Kerr black hole, and a Kerr-Newman black hole can be determined by the correlations between inner and outer horizons [33,34].…”

“…We obtain a global temperature of radiation that does not coincide with the conventional Hawking temperature related to the outer cosmological horizon. We also compute the entropy by using the approach based on the method of the singular coordinate transformations [33,34] for this spacetime. We show that the total entropy for this spacetime can * chiranjeeb.singha@saha.ac.in not be determined only by the outer cosmological horizon.…”