Entropy in the interior of a Kerr black hole

Abstract

Christodoulou et al have shown that the interior volume of a Schwarzschild black hole grows linearly with time. Subsequently, their conclusion has been extended to the Reissner–Nordström (RN) and Kerr black holes. Meanwhile, the entropy of the scalar field inside a Schwarzschild black hole has also been calculated. In this paper, a general method calculating the number of quantum states of the scalar field inside the black hole is given, which can be used in an arbitrary black hole. After introducing the two important assumptions as the black-body radiation assumption and the quasi-static process assumption, the entropy of the scalar field inside a Kerr black hole is calculated using the differential form, and we find that the variation of the entropy is proportional to the variation of the Bekenstein–Hawking entropy except the ending of the black hole evaporation. Similarly, we recalculate the entropy of the scalar field inside a Schwarzschild black hole and demonstrate that the entropy inside a Kerr black hole can exactly degenerate to the Schwarzschild black hole. As well as, we find that the proportionality coefficient between the entropy of the scalar field and the Bekenstein–Hawking entropy in Schwarzschild case, which is obtained using the differential form, is half of that given in the previous literature. Furthermore, we investigate the total entropy of Kerr and Schwarzschild black holes and find that they all increase with time. It means that the black holes, evolution with Hawking radiation satisfies the second law of thermodynamics. Finally, the black hole information paradox is brought up again and discussed.