Thermodynamic Phase Transitions and Quantum Entropy Corrections in the Simpson-Visser Regular Black Hole

TL;DR

This work investigates the Simpson-Visser regular black hole as a concrete setting to study how singularity resolution alters black hole thermodynamics and quantum entropy. By combining semiclassical thermodynamics with a Davies-type stability analysis, it identifies a critical point $a_{\text{crit}}=\sqrt{2}\,m$ that separates unstable and stable phases, showing that regularization can qualitatively change evaporation dynamics. Using the Hamilton-Jacobi tunneling approach, the authors derive leading quantum corrections to entropy, revealing that the leading term depends on the regularization scale $a$ and that logarithmic and higher-order corrections emerge, with a final extremal remnant carrying a purely logarithmic entropy $S_{\text{remnant}}\propto \beta_1\ln(a)$. Collectively, these results illustrate that singularity resolution is a thermodynamic and quantum-gravitational phenomenon with implications for black hole remnants and information preservation, and they offer a framework for exploring more general regular spacetimes.

Abstract

Regular black holes offer a compelling framework to explore the consequences of resolving the central singularity of standard black holes. Using the Simpson-Visser "black-bounce" geometry as an elegant, analytically tractable framework, we explore the intricate thermodynamic behavior in such models. We demonstrate that this regular spacetime exhibits a critical instability, marked by a phase transition where the heat capacity is discontinuous. This transition signals a fundamental change in the black hole's evaporation state, which depends on the regularization parameter. Pushing beyond the semiclassical limit, we then derive the leading-order quantum corrections to the entropy via the Hamilton-Jacobi tunneling formalism. Our analysis provides a refined statistical basis for the entropy of non-singular spacetimes and offers a quantitative analysis of the nature of the black hole end-state. These results reveal that singularity resolution is not merely a geometric modification but a profound thermodynamic event, with direct implications for the stability and ultimate fate of evaporating black holes.

Figures (5)

• Figure 1: Metric structure and energy density for the Simpson--Visser spacetime with mass $m = 1$. (a) The metric function $f(r)$ illustrates the transition from a regular black hole ($a=1.0$) to an extremal case ($a=2.0$) and finally to a traversable wormhole ($a=2.5$). (b) The corresponding energy density $\rho(r)$ shows regular, localized matter supporting the geometry. Horizon structure and energy profiles are directly controlled by the parameter $a$. In both panels, the lines correspond to different values of the regularization parameter $a$: the solid blue line represents the regular black hole case ($a=1.0$), the dashed red line represents the extremal black-bounce ($a=2.0$), and the dash-dot green line represents the traversable wormhole ($a=2.5$).
• Figure 2: Normalized heat capacity as a function of the dimensionless regularization parameter $a/(2m)$. The plot reveals the existence of two distinct thermodynamic phases. For $a < \sqrt{2}m$, the heat capacity is negative (blue shaded region), corresponding to an unstable phase analogous to the Schwarzschild black hole. For $\sqrt{2}m < a < 2m$, the heat capacity is positive (red shaded region), indicating a thermodynamically stable phase. The vertical dashed line marks the critical point where the heat capacity diverges, signifying a phase transition between the stable and unstable phases.
• Figure 3: The free energy indicates the thermodynamically favored state of the system. The plot reveals that the free energy is an increasing function of $a$, achieving its global minimum of $F=m/2$ at $a=0$. This implies that the singular Schwarzschild configuration is the thermodynamically preferred state among all regular black hole solutions of the same mass. The physical thermodynamic evolution is instead governed by mass loss due to Hawking evaporation at fixed $a$. As the mass $m$ decreases, the system crosses a critical point at $m=a/\sqrt{2}$, beyond which the heat capacity becomes positive. This signals a change in the thermal response. The vertical dashed line marks the critical point $a_c = \sqrt{2}m$ of the phase transition for reference.
• Figure 4: The quantum-corrected entropy $S$ of the Simpson--Visser black hole as a function of the regularization parameter $a$, calculated for a fixed mass $m=1$. The different colored lines represent distinct phenomenological scenarios for the quantum correction coefficients $(\beta_1, \beta_2)$, as specified. The dashed black line indicates the purely semiclassical entropy for comparison, which is independent of $a$. The plot demonstrates that quantum corrections can significantly alter the entropy, particularly for small $a$ where the geometry is nearly singular. For certain choices of parameters (e.g., strong coupling), the total entropy can become negative, signaling the need for non-perturbative effects in that regime. All corrected entropies converge towards the semiclassical result for large $a$ near the extremal limit. Standard coupling is for the coupling constant $\beta =\frac{-1}{8\pi}$. The enhanced, intermediate and strong coupling coupling are taken arbitrarily to demonstrate the effects of coupling constant on the entropy.
• Figure 5: A detailed decomposition and analysis of the quantum-corrected entropy for various phenomenological scenarios. (a) The deviation of the corrected entropy from the semiclassical value, $\Delta S = S - S_{BH}$, isolating the net effect of the quantum corrections. (b) A decomposition of the total entropy into its constituent parts: the modified semiclassical geometric term ($S_0$), the logarithmic correction term ($S_1 \propto \beta_1$), and the higher-order inverse-mass term ($S_2 \propto \beta_2$). This panel reveals the dominant contribution of the higher-order term at small $a$. (c) The thermodynamic stability derivative, $\partial S / \partial a$, indicating how the total entropy changes with the regularization parameter. (d) The relative entropy enhancement ratio, showing the factor by which quantum corrections enhance or suppress the total entropy relative to the semiclassical value.