The Remnant of an Evaporating Rotating Regular Black Hole from the Generalized Entropy in the Final Stage of Evaporation

Abstract: We express the generalized entropy (GE) of the Hawking radiation in the final stage of an evaporating rotating regular black hole (BH) by writing the mass of the BH as $m_{\rm ext}+α$, where $m_{\rm ext}$ represents the mass at the extremal limit and $α$ is a parameter. Generally, entropy is non-negative. Based on this fact, we assume that, in the GE considered in this study, the signs of the contributions from the area term and the correction term remain unchanged throughout the entire evaporation process of the BH. Therefore, we regard $α$ at which the correction term vanishes as its lower bound and determine it. As a result, we find that such a value of $α$ is finite. Denoting this value by $α<em>1$, this result indicates that the mass of the BH cannot become smaller than $m</em>{\rm ext}+α_1$, which can be interpreted as the emergence of a remnant at the final stage of BH evaporation. The BH considered in this study is a rotating regular BH. The regularization is motivated by the fact that the fine structure of the central region becomes relevant in the final stage of evaporation. A rotating BH is considered from the viewpoint of generality.

• The paper identifies a finite remnant mass in evaporating rotating regular black holes by analyzing generalized entropy in the near-extremal regime.
• It employs perturbative methods with the island formula and quantum extremal surfaces to detail quantum corrections affecting the evaporation process.
• The findings suggest that a nonzero final mass may resolve the information loss paradox by preventing complete black hole evaporation.

Remnants in Evaporating Rotating Regular Black Holes from Generalized Entropy

Introduction and Motivation

The problem of black hole evaporation, specifically the fate of information and the final state of an evaporating black hole (BH), has been a subject of intense interest since Hawking’s discovery of black hole radiation. The standard semiclassical expectation is the complete evaporation of the BH, possibly leaving behind pure Hawking radiation, giving rise to the information loss paradox.

This paper addresses these questions within the context of rotating regular black holes (RBHs). These are nonsingular models where the central singularity is replaced by a region of high curvature regulated by a minimal length scale (taken to be of Planck order). The analysis leverages developments in quantum gravitational entropy—specifically the generalized entropy (GE) arising from the island formula and quantum extremal surfaces (QESs).

The principal focus is the entanglement entropy (EE) of Hawking radiation (HR) in the near-extremal stage of a shrinking, evaporating, rotating RBH. The problem is formalized by expressing the BH mass as $m_0 = m_{\mathrm{ext}} + \alpha$, where $m_{\mathrm{ext}}$ is the extremal mass and $\alpha$ parametrizes the residual mass above extremality. Through a detailed analysis of the GE, the author identifies a finite lower bound $\alpha_2$ for $\alpha$. This imposes a nonzero minimal final mass, interpreted as a remnant at the endpoint of BH evaporation.

Island Formula, Generalized Entropy, and Regular Black Holes

Utilizing the recent advances in the understanding of quantum corrections to gravitational entropy, the analysis builds heavily on the island prescription for gravitational entanglement entropy. The key framework is:

$$S_\text{rad} = \min_X \left\{ \mathrm{ext}_X \left[ S_\text{gen}(\Sigma_X) \right] \right\},$$

where the generalized entropy is

$$S_\text{gen}(\Sigma_X) = \frac{\mathcal{A}(X)}{4G_N} + S_\text{field}(\Sigma_X),$$

with $\mathcal{A}(X)$ the area of the (quantum) extremal surface, and $S_\text{field}$ the bulk matter entropy outside the QES ("island").

The problem is set in the background of a rotating regular black hole modeled by a modified Kerr-type solution, regularized by a Planck-scale parameter $\ell_p$ that smooths the central singularity. Both extremal and non-extremal cases are handled, with particular emphasis on establishing the smoothness of the transition and the analytic control of the horizon structure through a systematic $m_{\mathrm{ext}}$0 expansion.

Computation of the Hawking Radiation Entropy in the Final Evaporation Stage

A substantial technical component is devoted to:

• Construction of the maximally extended spacetime for the rotating RBH, required for a correct implementation of the QES and the island formula.
• Calculation of the surface gravity and the Hawking temperature in the near-extremal regime, essential to analyzing evaporation rates and entropy flows.
• Formulation of the generalized entropy as a function of $m_{\mathrm{ext}}$1 (i.e., as the regular BH evolves toward extremality), including all relevant semiclassical and quantum corrections up to the necessary order in $m_{\mathrm{ext}}$2.

A crucial insight is that in the near-extremal regime, the generalized entropy admits a simplification. The leading terms are separated into:

• $m_{\mathrm{ext}}$3: the area (Bekenstein-Hawking) term, always positive and decreasing with $m_{\mathrm{ext}}$4, approaching a finite value as $m_{\mathrm{ext}}$5.
• $m_{\mathrm{ext}}$6: quantum corrections from the bulk fields, dominated in this limit by a term behaving as $m_{\mathrm{ext}}$7, which diverges negatively as $m_{\mathrm{ext}}$8.

The paper imposes the physical requirement of nonnegativity of the entropy. Since $m_{\mathrm{ext}}$9 becomes negative and unbounded in the extremal limit, the total entropy becomes negative if $\alpha$0 is taken below a critical value $\alpha$1. This leads to the identification of a minimal allowed value of $\alpha$2, corresponding to a nonzero final mass remnant.

Quantitative Results and Main Claims

1. Smooth Connection of Horizon Structure: The horizon equation for the rotating RBH is a quintic in $\alpha$3. A detailed perturbative solution up to $\alpha$4 is provided, explicitly demonstrating smooth continuity between non-extremal and extremal geometries in the presence of the regulating parameter.
2. Generalized Entropy as a Function of Residual Mass: The GE is analytically expanded in terms of $\alpha$5 in the near-extremal regime. The area and quantum field contributions are separated for clarity.
3. Identification of Remnant Mass—Strong Claim: The analysis yields a finite lower bound $\alpha$6 for the residual mass parameter, below which the generalized entropy becomes negative, violating physical principles. The corresponding lower bound for the BH mass is then $\alpha$7. This is interpreted as an unavoidable remnant in the evaporation process.

• Explicitly, the bound is $\alpha$8, with additional Planck-scale corrections.
• In the limits considered, Planckian effects provide quantitative corrections but do not qualitatively alter the mechanism for remnant formation.

1. Comparison and Contrast with Previous Works: Whereas earlier studies of BH entropy and Page curves often assume eternal or static BHs, the present analysis allows for explicit tracking of evaporation and horizon evolution via the parameter $\alpha$9. Additionally, the technical treatment of the cutoff ($\alpha_2$0) in the entanglement entropy integral and Planck-scale regularization are handled in a manner more closely tailored to the physics of a truly evaporating rotating RBH.

Theoretical Implications

The emergence of a finite-mass remnant in rotating RBHs from quantum corrected entropy has several profound implications:

• Resolution to Unitarity and Information Loss: The presence of a remnant can potentially resolve the issue of unitarity in quantum gravity by enabling final states that can encode information that would otherwise be lost in traditional semiclassical evaporation scenarios.
• Constraints from Generalized Entropy: The analysis demonstrates that demanding the generalized entropy remain non-negative throughout evaporation places stringent constraints on the late-time endpoint of black hole physics.
• Evaporation Endpoints and Remnants: The result suggests that, within quantum gravitational corrections handled through the island formula, complete evaporation to zero mass may be dynamically excluded, mandating the presence of Planckian or near-extremal mass remnants which could harbor information.

Future Directions

Several avenues are suggested by this work:

• Mechanism of Remnant Formation: Beyond the proof of existence, the detailed dynamics and microphysical origin of remnants in a unitary theory of gravity require further elucidation. Connections with proposals involving quantum gravitational degrees of freedom or stringy corrections may yield further insights.
• Extensions to General Spacetimes and Non-Rotating Regular Black Holes: The methods employed here can be generalized to non-rotating cases, more complicated matter content, or other regularization schemes.
• Observational Consequences: Possible phenomenological or cosmological implications of remnant formation, particularly if such objects are stable or long-lived, may provide constraints or clues for beyond-Standard Model physics.
• Relation to Holography and AdS/CFT: The compatibility of these findings with holographic dualities and quantum information-theoretic analyses of black hole entropy deserves further investigation.

Conclusion

By explicit analysis of the generalized entropy within the island framework, the study demonstrates that the endpoint of evaporation for a rotating regular black hole is not complete disappearance, but rather a non-zero remnant mass. The derivation is robust with respect to the specifics of the Planck-scale regularization and reflects the power of the island formula and QES methods in constraining the possible final states of evaporating quantum black holes. This work provides a significant contribution toward a unitary quantum theory of gravity in the context of black hole evaporation and the information paradox.

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Reference: "The Remnant of an Evaporating Rotating Regular Black Hole from the Generalized Entropy in the Final Stage of Evaporation" (2607.08661).