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

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-Nordstrom black hole and of the Kerr black hole, which have two horizons. For the spherically symmetric Reissner-Nordstrom 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.

Automated summary

The entropy calculations for black holes with two horizons, such as the Reissner-Nordstrom black hole and the Kerr black hole, show that the entropy is not determined by the area of the outer horizon but by the correlations between the two horizons. The entropy depends only on the mass M of the black hole and is independent of the black hole charge Q.