Thermodynamics of Kerr black hole: Tsallis-Cirto composition law and entropy quantization

Abstract

The processes of splitting and merging of black holes obey the composition law generated by the Tsallis-Cirto $\delta=2$ statistics. The same composition law expresses the full entropy of the Reissner-Nordstr\"om black hole via the entropies of its outer and inner horizons. Here we apply this composition law to the thermodynamics of the Kerr black hole. As distinct from Reissner-Nordstr\"om black hole, where the full entropy depends only on mass $M$ and does not depend on its charge $Q$, the entropy of Kerr black hole is the sum of contributions from its mass $M$ and angular momentum $J$, i.e. $S(M,J)=S(M,0) + 4\pi \sqrt{J(J+1)}$. Here $S(M,0)$ is the entropy of the Schwarzschild black hole. This demonstrates that when the Kerr black hole with $J\gg 1$ absorbs or emits a massless particle with spin $s_z=\pm 1/2$, its entropy changes by $|\Delta S| = 2\pi$. We also considered the quantization of entropy suggested by the toy model, in which the black hole thermodynamics is represented by the ensemble of the Planck-scale black holes -- Planckons. The Tsallis-Cirto composition law is also extended to the thermodynamics of Kerr-Newman black hole.