⚛️ high-energy theory

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

By analyzing the generalized entropy of an evaporating rotating regular black hole, the study demonstrates that the mass cannot drop below a finite limit ( $m_{\rm ext}+\alpha_1$ ) where the correction term vanishes, thereby predicting the formation of a stable remnant in the final stage of evaporation.

Original authors: Shingo Takeuchi

Published 2026-07-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Shingo Takeuchi

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

Imagine a black hole not as a cosmic vacuum cleaner that swallows everything forever, but as a glowing, spinning ember that slowly cools down. For decades, physicists have been worried about what happens when this ember burns out completely. The scary thought was that all the information about the stuff that fell in would just vanish into thin air, breaking the fundamental rules of how the universe works. This is the famous "information paradox."

But this paper suggests a different ending to the story. Instead of disappearing entirely, the spinning black hole might stop shrinking at a specific, tiny size and stay there forever. Think of it like a car that runs out of gas but doesn't just dissolve; it rolls to a halt and becomes a permanent, tiny statue. The authors call this leftover piece a "remnant."

The Cosmic Balloon and the Magic Number

To figure this out, the authors looked at a special kind of black hole: one that spins and has a "regular" center (meaning it doesn't have a weird, infinite point of destruction in the middle). They imagined the black hole losing mass over time, like a balloon slowly leaking air.

They used a clever trick to track this process. They said the black hole's mass is made of two parts: a "floor" mass (the smallest it can possibly be while still spinning) and a little bit extra, which they called $\alpha$ (alpha). As the black hole evaporates, $\alpha$ gets smaller and smaller.

The team calculated the "entropy" (a measure of disorder or information) of the radiation coming off the black hole. They found something interesting: as $\alpha$ shrinks, a correction term in their math—which accounts for the weird quantum effects near the center—starts to vanish.

Here is the kicker: Entropy is like a score in a game; it can't be negative. The authors argue that the rules of the game shouldn't suddenly change just because the black hole is getting tiny. So, they assumed the "signs" of the math terms stay the same throughout the whole process.

Because of this rule, they found that $\alpha$ cannot just go down to zero. It hits a hard stop at a specific, finite value they call $\alpha_1$ . Once the black hole reaches this point, it can't shrink any further. It's stuck. This means the black hole doesn't vanish; it leaves behind a tiny, stable remnant with a mass of $m_{ext} + \alpha_1$ .

What They Rejected and Why

The paper is very careful about what it doesn't say.

No "Vanishing Act": The authors explicitly argue against the idea that the black hole evaporates completely until nothing is left. Their math suggests that if it did, the entropy would behave in a way that breaks the rules of physics (becoming negative or changing signs abruptly).
No "Firewalls": The paper doesn't support the idea that a wall of fire destroys everything at the edge. Instead, it focuses on the structure of the center and the math of the radiation.
No "Pair Creation" in the End: While some theories say black holes lose mass by constantly creating pairs of particles (one falling in, one escaping), the authors suggest a different picture for the final stage. They propose that the particles inside the black hole are simply emitted out, which helps explain how the information gets out without breaking the rules.

How Sure Are They?

It is important to note that this isn't a lab experiment where they built a black hole in a jar. This is a theoretical study based on complex equations and mathematical models. The authors have calculated that a remnant is likely to form based on their specific assumptions about how entropy works and how the black hole's center behaves.

They found that the value $\alpha_1$ is finite, which is a strong mathematical hint that a remnant exists. However, because this is a theoretical calculation involving approximations (like expanding the math in small steps), they are suggesting this is the most logical outcome of their model, rather than proving it with a physical measurement.

The Big Picture

So, if you imagine the universe as a giant library, and a black hole as a book that is being read and then burned, the old worry was that the story would be lost forever. This paper suggests that the book doesn't burn to ash. Instead, it shrinks down to a tiny, indestructible bookmark that stays on the shelf forever, keeping the story safe. The spinning black hole doesn't disappear; it just settles into a permanent, tiny state, solving the mystery of where the information went.

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

Problem Statement
This study addresses the black hole information paradox and the fate of evaporating black holes in their final stages. While the "island formula" derived from quantum extremal surfaces (QES) has successfully reproduced the Page curve for eternal black holes, its application to shrinking black holes remains less explored. Specifically, there is a tension between the assumption that Hawking radiation (HR) originates solely from pair-creation near the horizon and the requirement for unitary evolution, which suggests that information must be preserved. Furthermore, standard models often predict that black holes evaporate completely, leading to a violation of unitarity if distinct initial states evolve into the same final state of pure radiation. This paper investigates whether a "remnant"—a stable or quasi-stable object remaining after evaporation—emerges naturally when considering a rotating regular black hole (RBH) where the central singularity is resolved by quantum gravity effects (regularization).

Methodology
The authors employ a semi-classical approach combining the island formula with the geometry of a specific rotating regular black hole spacetime. The methodology proceeds as follows:

Spacetime Model: The study utilizes a rotating regular black hole metric where the central singularity is removed by introducing a Planck-scale parameter $\ell_p$ . The mass $m_0$ is parametrized as $m_0 = m_{ext} + \alpha$ , where $m_{ext}$ is the mass at the extremal limit and $\alpha$ is a small parameter representing the deviation from extremality. The analysis focuses on the near-extremal regime where $\alpha \to 0$ .
Horizon Analysis: The authors solve the horizon equation $\tilde{\Delta}=0$ perturbatively up to order $\ell_p^3$ to determine the horizon radii ( $r_+$ , $r_-$ , and others) in both non-extremal and extremal cases. They demonstrate that these solutions connect continuously as $\alpha \to 0$ .
Near-Horizon Geometry: The effective action of a scalar field in the near-horizon region is analyzed to derive the two-dimensional metric and the surface gravity ( $\tilde{\kappa}$ ). Unlike non-rotating black holes where surface gravity diverges in the final stage, the rotating RBH exhibits surface gravity that vanishes in the extremal limit.
Generalized Entropy Calculation: Using the island formula, the generalized entropy ( $S_{gen}$ ) of the Hawking radiation is calculated. This includes the area term ( $A/4G_N$ ) and the matter field correction term ( $S_{field}$ ). The calculation is performed in the final stage of evaporation ( $t_a, t_b \gg 1$ ), utilizing the Kruskal coordinates of the maximally extended spacetime.
Extremization and Bounds: The authors extremize the generalized entropy with respect to the positions of the island boundaries ( $r_a$ ) and the cutoff surface ( $r_b$ ). Crucially, they impose the physical constraint that entropy must be non-negative. They assume the signs of the area and correction terms remain consistent throughout evaporation.

Key Contributions and Results

Finite Lower Bound on Mass: By analyzing the generalized entropy in the final stage, the authors find that the correction term vanishes at a specific finite value of the parameter $\alpha$ , denoted as $\alpha_1$ . Since entropy is non-negative, $\alpha$ cannot decrease below this value without violating the assumed sign structure of the entropy contributions.
Emergence of a Remnant: The existence of a finite lower bound $\alpha_1$ implies that the black hole mass cannot shrink below $m_{ext} + \alpha_1$ . This result is interpreted as the emergence of a remnant at the final stage of evaporation, preventing the black hole from disappearing completely.
Regularization Necessity: The study highlights that the fine structure of the central region (regularization via $\ell_p$ ) becomes relevant in the final stage. The specific form of the regularized metric allows for a continuous connection between non-extremal and extremal horizons, which is essential for the derivation.
Surface Gravity Behavior: The paper confirms that for rotating regular black holes, the surface gravity approaches zero in the extremal limit, contrasting with the behavior of non-rotating black holes. This behavior is critical for the specific approximations used in the entropy calculation.

Significance and Claims
The paper claims to provide a concrete analytical framework for the emergence of black hole remnants based on the generalized entropy and the island formula, specifically within the context of rotating regular black holes. The authors argue that their findings offer a resolution to the unitarity problem by suggesting that black holes do not evaporate completely but leave behind a remnant with a finite mass determined by the regularization scale.

The study emphasizes that this result is derived without assuming pair-creation in the late stage, instead allowing for the possibility that particles inside the black hole are emitted directly, facilitating the purification of the radiation. The authors note that while concrete analyses of remnants have been limited due to the extreme spacetime curvature involved, their perturbative approach using the island formula provides a viable path to investigating this phenomenon. The work is presented as a step toward understanding the final state of black hole evaporation and the potential existence of remnants as a solution to the information paradox.

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