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

TL;DR

The paper addresses how the Tsallis-Cirto non-extensive entropy with δ=2, which yields a square-root composition law, applies to Kerr black holes and whether entropy exhibits quantization. By expressing the Kerr entropy via horizon entropies using the δ=2 composition, the authors derive $S_{\rm Kerr}(M,J)=S_{\rm Schwarzschild}(M) + 4\pi \sqrt{J(J+1)}$ and show that in the large-$J$ limit $\partial S/\partial J|_{M} \to 4\pi$, while absorption or emission of a massless spin-1/2 particle changes the entropy by $|\Delta S|=2\pi$. The analysis highlights a contrast with Reissner-Nordström black holes, where the full entropy depends only on mass and is charge-independent under the same framework, and explores a Planckon toy model where entropy becomes a function of two quantum numbers $N$ and $J$. These results provide a concrete non-extensive-thermodynamics perspective on black-hole entropy and offer a route to entropy quantization and connections to Planck-scale physics, suggesting extensions to Kerr-Newman and other multi-horizon spacetimes.

Abstract

The processes of splitting and merging of black holes obey the composition law generated by the Tsallis-Cirto $δ=2$ statistics. The same composition law expresses the full entropy of the Reissner-Nordström black hole via the entropies of its outer and inner horizons. He we apply this composition law to the entropy of the Kerr black hole. As distinct from Reissner-Nordström 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. $S(M,J)=S(M,0) + 4π\sqrt{J(J+1)}$. 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 $|ΔS| = 2π$. 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.