Extended Tsallis-Cirto entropy for black and white holes

Abstract

In reference [1] we considered the black hole thermodynamics with the non-extensive entropy. This entropy obeys the composition rule which coincides with the composition rule in the non-extensive Tsallis-Cirto $δ=2$ statistics. Here we extend this approach to the thermodynamics of white holes. The entropy of the white hole is negative as follows from the rate of macroscopic quantum tunneling from black hole to white hole. The white hole entropy is with the minus sign the entropy of the black hole with the same mass, $S_{\rm WH}(M)=-S_{\rm BH}(M)$. This reflects the anti-symmetry with respect to time reversal, at which the shift vector in the Arnowitt-Deser-Misner formalism changes sign. This symmetry allows one to extend the Tsallis-Chirto entropy by adding a minus sign to the Tsallis-Chirto formula applied to white hole. As a result, the composition rule remains the same, with the only difference being that instead of entropy it contains the entropy modulus. The same non-extensive composition rule is obtained for the entropy of the Reissner-Nordström black hole. This entropy is formed by the positive entropy of the outer horizon and the negative entropy of the inner horizon. The model of the black hole formed by "black hole atoms" with Planck-scale mass is also extended to include the negative entropy of white holes.