⚛️ general relativity

First-Law Entropy and a Degenerate Extremal Remnant in a Minimal-Length Simpson--Visser-Type Regular Black Hole: Geometrothermodynamics, Phase Structure, and Observational Discriminants

This paper constructs a first-law-consistent entropy for a minimal-length deformed Schwarzschild black hole that predicts evaporation terminates in a degenerate extremal remnant with vanishing temperature and entropy, while revealing unique phase transitions in its geometrothermodynamic curvature and proposing observational discriminants via shadow degeneracy and photon-ring flux enhancements.

Original authors: T. Toghrai, N. Mansour, A. Daassou, R. Benbrik

Published 2026-09-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: T. Toghrai, N. Mansour, A. Daassou, R. Benbrik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, physicists have been haunted by a single, stubborn problem: the center of a black hole. According to our best theories of gravity, when a massive star collapses, it crushes everything down to a point of infinite density and zero size, a place where the laws of physics simply break down. This "singularity" is a sign that our understanding is incomplete. Many researchers believe that at the tiniest scales, space itself is not smooth and continuous but has a fundamental graininess, a smallest possible length below which the concept of distance loses its meaning. This idea, often called a minimal length, suggests that nature has a built-in limit to how much it can compress matter. If this is true, the crushing point of a black hole should never actually form; instead, the collapse should stop at a tiny, finite size, leaving behind a smooth, regular core.

The question then becomes: what happens to the black hole as it slowly evaporates and loses energy? Standard theory predicts that a black hole shrinks until it vanishes completely, potentially taking any information it swallowed with it, which creates a deep paradox. However, if a minimal length exists, the black hole might stop shrinking before it disappears, leaving behind a stable, tiny remnant. Understanding the nature of this final state is crucial, not just for solving the mystery of the singularity, but for figuring out whether the universe preserves the information of everything that falls into a black hole.

In a new study, researchers have constructed a specific model of such a black hole to test these ideas. They started with the standard description of a black hole but replaced the idea of a collapsing point with a smooth, curved geometry that respects a minimum size limit. Instead of a point where space ends, their model features a tiny, finite sphere at the center that acts as a hard stop for the collapse. This modification changes the behavior of the black hole as it cools down. The researchers calculated how the black hole's temperature and energy change as it shrinks, finding that it does not simply vanish. Instead, the evaporation process slows down and eventually halts completely.

The endpoint of this process is a surprising object: a degenerate extremal regular black hole. In this final state, the black hole's temperature drops to absolute zero, and its entropy, a measure of its internal disorder, also drops to zero. Unlike other models where a black hole might leave behind a "naked" core with no event horizon, this object retains a horizon, but one that is perfectly degenerate, meaning the inner and outer boundaries of the horizon merge into a single surface. The center of the black hole and this horizon coincide at a specific, tiny radius. The researchers showed that this state is stable against small disturbances, meaning it would not spontaneously collapse or explode. It represents a quiet, frozen end to the life of a black hole, a stable relic that persists indefinitely.

To understand how this object would look to an observer, the team examined how light behaves around it. They found that the shadow cast by the black hole—the dark region where light is swallowed—is identical to the shadow of a standard black hole, regardless of the size of the minimum length. This means that current telescopes looking at the shadows of black holes, such as those observed by the Event Horizon Telescope, cannot yet distinguish this model from the classical version. However, the model does leave a subtle fingerprint in the rings of light that surround the shadow. These rings, formed by light that orbits the black hole multiple times before escaping, would be slightly brighter and spaced differently than in a standard black hole. The researchers calculated that the brightness of these rings could be enhanced by a measurable amount, offering a potential way to detect the existence of this minimal length in the future with more sensitive instruments.

The study also explored the thermodynamic stability of this object, analyzing how it responds to changes in heat and energy. They discovered that the black hole undergoes a distinct phase transition as it cools, shifting from an unstable state to a stable one before finally reaching its zero-temperature remnant. This transition is marked by a sharp change in the black hole's ability to store heat, a feature that the researchers identified using a geometric approach to thermodynamics. This geometric analysis revealed a unique signature at the very end of the evaporation process, a mathematical divergence that signals the presence of the degenerate horizon. This finding suggests that the final state is not just a random leftover but a highly specific, physically distinct configuration.

One of the most significant aspects of this work is how it handles the information paradox. Because the final remnant has zero entropy and zero temperature, it carries no residual micro-state degeneracy, unlike scenarios where information is stored in a large number of hidden states. This implies that if the black hole evaporates down to this state, the information it contained must have been released during the evaporation process, rather than being trapped inside the remnant. However, the researchers emphasize that this does not definitively prove the universe preserves information; distinguishing between unitary evolution (where information is released) and genuine information loss requires tracking fine-grained radiation correlations through the near-endpoint regime, a calculation beyond the semi-classical scope of this paper. The regularity and the smooth limit to zero temperature established here are necessary, but not by themselves sufficient, conditions for a unitary evaporation history.

The researchers also compared their findings with other theories that attempt to resolve the singularity problem. Some previous models suggested that black holes leave behind a stable core with a large amount of hidden information, or that they simply disappear. This new model differs by showing that the black hole can end in a state with a genuine horizon that is degenerate, rather than a horizonless object. This distinction is important because it changes the mathematical properties of the final state and how it interacts with the surrounding space. The study confirms that the geometry of the black hole remains smooth and free of infinite curvature everywhere, even at the very center, validating the idea that a minimal length can successfully remove the singularity.

Looking ahead, the authors suggest several ways to test these ideas further. They propose that future observations of the rings of light around black holes could provide the first evidence for this minimal length, as the predicted brightness enhancement is within the reach of next-generation telescopes. They also note that understanding the quantum stability of this final remnant—whether it could decay through quantum tunneling—remains an open question that requires deeper investigation. For now, the study offers a clear, mathematically consistent picture of a black hole that does not end in a singularity or a total disappearance, but in a quiet, frozen state that respects the fundamental limits of space itself. This provides a new perspective on the ultimate fate of black holes and the information they hold, turning a theoretical puzzle into a potentially observable reality.

Technical Summary: First-Law Entropy and a Degenerate Extremal Remnant in a Minimal-Length Simpson–Visser-Type Regular Black Hole

Problem Statement
The paper addresses the thermodynamic and geometric properties of a regular black hole model constructed by applying a minimal-length deformation to the Schwarzschild spacetime. While the Simpson–Visser (SV) metric is a well-studied "black-bounce" solution that interpolates between a wormhole and a regular black hole by allowing the radial coordinate $r \in (-\infty, +\infty)$ , this work restricts the domain to $r \in [0, +\infty)$ . This restriction transforms the $r=0$ surface from a wormhole throat into a terminating regular center, creating a simply connected spacetime. The authors aim to determine the consistent thermodynamic entropy, phase structure, and evaporation endpoint of this specific topology, distinguishing their results from both the standard SV branch and other regular black hole models (e.g., Bardeen, Hayward, GUP-based). A central challenge is defining the entropy boundary condition at the degenerate extremal limit where the horizon coincides with the regular center, a regime where standard area-law entropy and third-law expectations may diverge.

Methodology
The authors employ a geometric deformation of the areal radius, $R(r) = \sqrt{r^2 + \ell^2}$ , where $\ell$ is a fundamental minimal length parameter, while retaining the Schwarzschild-like metric function $f(R) = 1 - 2M/R$ . The metric is defined on $r \ge 0$ with $g_{rr} = -f(R)^{-1}$ .

Geometric Analysis: The authors analyze the Einstein tensor to characterize the effective geometric source, compute the Kretschmann scalar to verify curvature regularity, and construct Penrose–Carter diagrams to map the global causal structure. They utilize a $\mathbb{Z}_2$ reflection quotient of the maximal two-sided extension to justify the boundary at $r=0$ as a totally geodesic, tensionless surface.
Thermodynamics: Entropy is derived by integrating the first law of black hole thermodynamics, $dM = T_H dS$ , using the Hawking temperature $T_H$ . The integration constant is fixed by the physical boundary condition $S(r_h=0)=0$ , justified via the third law of thermodynamics and the properties of degenerate extremal horizons.
Stability and Phase Structure: The authors analyze the heat capacity $C$ and Helmholtz free energy $F$ to identify phase transitions. They apply Geometrothermodynamics (GTD), specifically Quevedo's Legendre-invariant formalism, to construct a thermodynamic metric and calculate its curvature scalar $R_{GTD}$ to identify singularities corresponding to phase transitions and endpoints.
Observational Signatures: The photon sphere location, black hole shadow size, and gravitational lensing coefficients (weak and strong field) are calculated. The strong-field deflection is analyzed using Bozza's formalism to determine the photon-ring flux ratio.
Stability Analysis: Linear mode stability against massless scalar perturbations is established using the Regge–Wheeler potential and an energy argument.

Key Contributions and Results

Entropy and Boundary Conditions: The authors derive a closed-form entropy $S(r_h) = \pi[r_h R_h + \ell^2 \ln((r_h + R_h)/\ell)]$ . While the functional form matches the semiclassical term found by Joshi and Joshi [28] for the SV branch, this work adopts it as the complete physical entropy (rather than a piece awaiting tunneling corrections) based on the boundary condition $S(0)=0$ . This choice is motivated by the third law ( $T_H \to 0 \implies S \to 0$ ) and the statistical mechanics of extremal horizons, distinguishing it from the Bekenstein–Hawking area law which would yield non-zero entropy at the endpoint.
Degenerate Extremal Remnant: The evaporation process terminates naturally at a minimum mass $M_{min} = \ell/2$ . At this point, the coordinate horizon radius vanishes ( $r_h=0$ ), but the areal radius remains finite ( $R_h = \ell$ ). The surface $r=0$ becomes a degenerate Killing horizon of quadratic order ( $f \approx r^2/2\ell^2$ ). The resulting object is a degenerate extremal regular black hole with vanishing temperature, vanishing entropy, and finite curvature everywhere. This is distinct from horizonless remnant scenarios; the horizon persists but degenerates.
Thermodynamic Phase Structure: The system exhibits a two-phase structure separated by a Davies-type transition at $M^* = \ell/\sqrt{2}$ $M^{*} = ℓ / 2$ .
- Phase I ( $M > M^*$ ): Unstable ( $C < 0$ ).
- Phase II ( $\ell/2 < M < M^*$ ): Locally stable ( $C > 0$ ).
- The transition is second-order, characterized by a divergence in heat capacity.
Geometrothermodynamics (GTD): The thermodynamic curvature scalar $R_{GTD}$ $R_{GT D}$ diverges at two distinct points:
1. At the Davies point $M^*$ , corresponding to the heat capacity divergence.
2. At the degenerate extremal endpoint $M_{min}$ . The authors note this second divergence is novel and specific to the presence of a genuine (degenerate) horizon at $S=0$ , a feature absent in other minimal-length models where remnants are typically horizonless.
Observational Discriminants:
- Shadow: The angular radius of the black hole shadow is exactly $3\sqrt{3}M/D$ , identical to the Schwarzschild value and independent of $\ell$ . This implies exact shadow degeneracy with the classical black hole.
- Photon Ring: The strong-field lensing coefficient $a = 3/\sqrt{9 - \ell^2/M^2}$ depends on $\ell$ . Consequently, the flux ratio between consecutive photon-ring images, $r_n = e^{-2\pi/a}$ , is enhanced relative to the Schwarzschild value. This provides a measurable discriminant for next-generation very-long-baseline interferometry (VLBI).
Stability: The degenerate extremal remnant is shown to be linearly mode-stable against massless scalar perturbations, as the Regge–Wheeler potential is non-negative throughout the static exterior.

Significance and Claims
The paper claims to provide the first explicit Penrose–Carter structure for a degenerate extremal regular black hole arising from a minimal-length deformation on a simply connected topology. The primary significance lies in the identification of a specific evaporation endpoint that is neither a horizonless core nor a singularity, but a degenerate horizon with zero entropy.

The authors emphasize that while the metric-level quantities (temperature, photon sphere, shadow size) coincide with the Simpson–Visser results on the $r \ge 0$ branch, the thermodynamic interpretation and the resulting phase structure are unique to this work. Specifically:

The adoption of the first-law-consistent entropy with $S(0)=0$ as the complete physical entropy, rather than a semiclassical approximation.
The identification of the $M_{min}$ endpoint as a degenerate extremal black hole with a genuine horizon, rather than a horizonless object.
The discovery of a second, independent divergence in the GTD curvature scalar at the remnant endpoint, which the authors argue is a geometric signature of the degenerate horizon at zero entropy.
The proposal that while the shadow is indistinguishable from Schwarzschild, the photon-ring flux ratio offers a potential observational test for the minimal length parameter $\ell$ .

The paper remains modest regarding the quantum stability of the remnant, noting that while linear classical stability is proven, non-perturbative quantum stability (e.g., tunneling or decay) remains an open question requiring a microscopic derivation. Similarly, the resolution of the information paradox is framed as sharpened by the regularity and zero-entropy endpoint, but not solved, as tracking fine-grained correlations through the near-endpoint regime requires a full quantum-gravitational calculation beyond the paper's semi-classical scope.

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