Entropy release from Minkowski breaking in regular Schwarzschild black holes

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Original abstract

The classical formation of a Schwarzschild black hole from a regular, non-singular configuration has recently been shown to be impossible within general relativity: the geometry inevitably develops a discontinuity at the origin, a phenomenon termed Minkowski breaking by Ovalle, Casadio, and Kamenshchik [PRD 113 (2026), 064042]. This obstruction signals that the transition to the Schwarzschild point mass must be a discrete, quantum event. We uncover the thermodynamic footprint of this transition. Using the explicit family of regular Schwarzschild black holes with a de Sitter core, we show that the inner Killing horizon carries a formal Bekenstein-Hawking entropy $S_{\rm inner} = A_{\rm inner}/4$ that is absent in the singular Schwarzschild state. This entropy is hidden from external observers in equilibrium but, assuming the generalized second law, must be released when the inner horizon disappears. As the collapse parameter $n$ decreases, the inner horizon shrinks and its entropy is gradually released during classical evolution, until the horizon finally vanishes at $n=0$ with the Minkowski breaking. The surface gravity diverges as $n\to0^+$, with the semiclassical description breaking down at $n \sim 1/\ln(h/\ell_P)$; the final disappearance is therefore a deep quantum process. For the $n=3$ regular black hole, the stored entropy is approximately $59\%$ of $A/4$; in the semiclassical limit $n\gg1$, it approaches the full $A/4$. The integer nature of $n$ implies a quantized entropy spectrum, with the Schwarzschild black hole as the ground state within the OCK family. We discuss how the classical mass-inflation instability may be circumvented by the quantum disappearance of the Cauchy horizon, and clarify the continuous vs. discrete nature of the collapse.

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