⚛️ general relativity

Excess inner-horizon entropy and the thermodynamics of Kerr formation

This paper develops a charge-matched thermodynamic framework for regular rotating black holes to quantify the excess inner-horizon entropy generated during Kerr formation, deriving exact evolution equations and establishing bounds that demonstrate how rotation redistributes horizon contributions while preventing regular configurations from attaining the entropy bound without losing their Killing horizon.

Original authors: Francisco S. N. Lobo, Jorge Ovalle, Manuel E. Rodrigues

Published 2026-11-03

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CC BY 4.0

Original authors: Francisco S. N. Lobo, Jorge Ovalle, Manuel E. Rodrigues

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

Black holes are often imagined as simple, one-way traps where gravity is so strong that nothing, not even light, can escape. In the standard view of how these objects form, a massive star collapses inward until it becomes a point of infinite density, surrounded by a boundary known as the event horizon. For decades, physicists have understood that these objects are not just gravitational pits but also thermodynamic systems, possessing a temperature and an amount of disorder, or entropy, that is directly tied to the size of their surface area. This connection between geometry and heat is one of the most profound discoveries in modern physics. However, this picture becomes complicated when the black hole spins. A spinning black hole, known as a Kerr black hole, is more complex than a stationary one; it contains a second, hidden boundary deep inside called the inner horizon. While the outer horizon is the point of no return for the outside world, the inner horizon is a region of intense instability that has long puzzled scientists. The question of what happens to the hidden geometry inside a spinning black hole, and how much "hidden" information it holds, has remained a difficult problem to solve.

A team of researchers has now developed a new way to measure the hidden interior of a spinning black hole as it forms, revealing that the amount of hidden disorder depends on the specific details of how the star collapsed, details that are invisible to an outside observer. The scientists focused on a theoretical model of a "regular" black hole, which is a version of the object that avoids the infinite, crushing singularity at its center by having a smooth, finite core. They studied the process of such a black hole forming and spinning up, tracking how the inner boundary moves and changes. Their central finding is that while the outer surface of the black hole looks exactly like a standard spinning black hole, the inner boundary can sit in different places depending on the internal structure. To make sense of this, the researchers created a new accounting method. They realized that because a standard spinning black hole already has an inner boundary, one cannot simply count the total area of the new inner boundary as "new" information. Instead, they calculated the extra area that exists beyond what a standard spinning black hole would have. This difference, which they call "excess entropy," represents the hidden internal state of the object.

The researchers proved that this excess entropy is always a positive number, meaning the regular black hole always has more hidden area than its standard counterpart. They showed that this extra amount is not infinite; it has a strict upper limit that depends on the mass and spin of the black hole. As the black hole spins faster, approaching the maximum speed allowed by physics, this capacity for hidden information shrinks. Interestingly, if the black hole stops spinning entirely, this new measurement perfectly matches the known hidden entropy of non-spinning black holes, showing that their new method is consistent with previous discoveries. The study also explored what happens as the black hole settles into its final spinning state. They found that the hidden entropy changes as the object evolves, eventually reaching zero once the black hole becomes a standard, smooth spinning hole. However, the researchers emphasize that the mathematical equations alone do not prove that this decrease happens steadily or monotonically; the entropy could fluctuate, storing and releasing information in intermediate phases before reaching its final state. Furthermore, the path to this final state involves a regime where the geometry becomes singular, meaning the object leaves the category of "regular" black holes before it fully settles into the standard Kerr configuration.

A significant part of the work involved understanding the mathematical path the black hole takes as it forms. The researchers discovered that the equations describing this process contain a point where the numbers seem to blow up, a mathematical glitch that usually signals a breakdown in the model. However, they demonstrated that this glitch is only an illusion; by looking more closely at the mass distribution inside the star, they could smooth out the equation and continue the calculation without error. This allowed them to track the inner boundary continuously from the start of the collapse to the final spinning state. Along this path, they identified distinct stages where the nature of the object changes. For instance, at one point, the core of the object stops behaving like empty space, a shift they call "Minkowski breaking," which happens before the inner boundary disappears. This shows that the loss of the inner horizon is not the same thing as the core changing its fundamental nature.

The study also clarifies why the total amount of disorder in a spinning black hole behaves the way it does. In a standard spinning black hole, the sum of the entropy on the outer surface and the inner surface is a fixed number determined only by the mass. When the spin increases, the outer entropy goes down, but the inner entropy goes up by exactly the same amount, keeping the total constant. The new "excess" entropy measured by the researchers sits on top of this balance. It represents the extra geometric contribution from the regular interior that is not part of the standard spinning black hole's baseline. This excess is what distinguishes a regular black hole with a smooth core from a standard one, even if they look identical from the outside. The researchers emphasize that while they have mapped out the geometry and the bookkeeping of this entropy, they have not yet proven that this hidden area corresponds to a physical flow of heat or radiation. The connection to the laws of thermodynamics for the inner region remains a conditional idea, waiting for a full understanding of how matter and energy move during the collapse.

Ultimately, this work provides a precise tool for measuring the hidden interior of rotating black holes. It confirms that the interior of a regular black hole can hold a significant amount of geometric information, far more than a standard black hole, without changing the mass or spin that an outside observer can measure. This hidden information is tied to the specific way the star collapsed, offering a potential window into the microscopic structure of spacetime. The researchers conclude that while the mathematical framework is now in place and the geometric limits are well-defined, the next step is to understand the physical stability of these inner regions and to determine exactly how this hidden entropy is exchanged with the rest of the universe. The study does not claim to have solved the mystery of black hole interiors, but it has successfully isolated the specific contribution of the regular core, separating it from the standard features of a spinning black hole and setting the stage for future investigations into the true nature of these cosmic objects.

Technical Summary: Excess Inner-Horizon Entropy and the Thermodynamics of Kerr Formation

Problem Statement
The paper addresses the thermodynamic interpretation of regular rotating black holes, specifically focusing on the entropy associated with their inner horizons. While the thermodynamics of regular Schwarzschild black holes has been interpreted as carrying a "hidden" internal entropy (the area of the inner horizon, which vanishes in the singular Schwarzschild limit), this interpretation cannot be directly transplanted to rotating (Kerr) geometries. A standard Kerr black hole already possesses an inner Cauchy horizon. Consequently, the total area of a regular rotating black hole's inner horizon cannot be entirely attributed to "hidden" entropy removed during formation; a portion of this area corresponds to the intrinsic inner horizon of the final Kerr state. The authors identify the need for a "charge-matched" framework to isolate the excess geometric contribution of the regular interior relative to the standard Kerr solution with the same mass ( $M$ ) and angular momentum ( $J$ ).

Methodology
The authors utilize a specific family of exact, regular, rotating black-hole solutions based on the Gurses–Gursey Kerr–Schild form. The geometry is defined by a piecewise mass profile $m(r)$ that matches the Kerr exterior exactly at the outer horizon $h$ but deviates in the interior. The study is divided into two main components:

Stationary Analysis: The authors analyze the static limit of the family, defining the inner horizon radius $h_c$ and comparing it to the Kerr inner horizon radius $r_-^K$ . They employ algebraic decomposition of the horizon areas and surface gravities.
Time-Dependent Dynamics: The framework is extended to a time-dependent model describing the final stage of Kerr formation. Here, the rotation parameter $a$ and the interior profile are promoted to time-dependent quantities $a(v)$ and $m(v,r)$ . The authors track the evolution of the roots of the metric function $\Delta(v,r)=0$ , distinguishing between algebraic roots and true dynamical horizons.

Key Contributions and Results

Definition of Excess Entropy ( $S_{exc}$ ): The central contribution is the definition of the excess inner-horizon entropy:
$S_{exc} \equiv S_c - S_-^K = \pi [h_c^2 - (r_-^K)^2]$
where $S_c$ is the geometric entropy of the regular inner horizon and $S_-^K$ is the entropy of the Kerr inner horizon with identical asymptotic charges. This subtraction isolates the internal geometric contribution that is absent in the final Kerr state.
Positivity and Bounds: The authors prove a theorem establishing that for nonextremal stationary configurations with physically motivated interior mass profiles, the regular inner horizon lies strictly outside the Kerr inner horizon ( $r_-^K < h_c < h$ ). This implies $S_{exc} > 0$ . Furthermore, they derive a sharp geometric upper bound:
$S_{exc} < 4\pi M \sqrt{M^2 - a^2}$
This bound is a supremum; it is approached by quasi-extremal configurations where $h_c \to h$ but is not attained by any regular black hole with a smooth Killing horizon (the limit corresponds to a "Kerr mimicker" with a cusp).
Entropy Redistribution: The paper demonstrates that for a fixed mass, the sum of the Kerr outer and inner entropies is constant ( $S_+^K + S_-^K = 4\pi M^2$ ). Therefore, increasing the spin parameter $a$ at fixed mass merely redistributes entropy between the outer and inner Kerr horizons. The quantity $S_{exc}$ uniquely measures the additional contribution from the regular interior structure.
Entropy Product Deformation: The authors relate $S_{exc}$ to the deformation of the Kerr entropy product. They show that $S_+ S_c - 4\pi^2 J^2 = S_+ S_{exc}$ . This deviation from the mass-independent area product theorem is attributed to the non-vacuum, anisotropic matter source supporting the regular interior, distinguishing it from electrovacuum universality results.
Dynamical Evolution and Singularities: In the time-dependent model, the authors derive exact evolution equations for the inner root $h_c(v)$ and the excess entropy $S_{exc}(v)$ . They analyze a minimal trajectory ( $N=1$ ) where the interior profile exponent $n$ evolves from a regular state to the Kerr endpoint ( $n=-1$ ).
- They demonstrate that the apparent pole at $n=2$ in the mass profile is removable, yielding a continuous logarithmic mass function.
- However, they clarify that while the root equation is continuous, the geometry becomes singular at $n=2$ (Kretschmann scalar divergence) and undergoes "Minkowski breaking" at $n=0$ (change in the equatorial core limit of $g_{tt}$ ).
- Crucially, Minkowski breaking ( $n=0$ ) occurs before the inner horizon disappears or reaches the Kerr limit, indicating that the disappearance of the inner horizon is not the sole criterion for the loss of regularity in this trajectory.
Conditional Entropy Balance: The authors formulate a conditional entropy balance law. They define a bookkeeping quantity $S_{book} = S_{env} + S_+ + S_c$ . If one assumes a coarse-grained description where $S_{book}$ is non-decreasing, the decrease in $S_{exc}$ during the collapse to Kerr implies a lower bound on the entropy generated in the environment ( $\Delta S_{env} \ge S_{exc}(v_i)$ ). The authors explicitly state this is a consistency condition, not a derived Generalized Second Law (GSL), as the roots tracked are not yet proven to be true dynamical horizons.

Significance and Claims
The paper claims to provide a precise, charge-matched thermodynamic bookkeeping framework for regular rotating black holes. Its significance lies in:

Resolving the Rotational Ambiguity: It clarifies that the "hidden entropy" of regular black holes in the rotating case is not the total inner horizon area, but the excess relative to the Kerr baseline.
Geometric Diagnostic: $S_{exc}$ serves as a thermodynamic diagnostic for secondary interior structure. It distinguishes different interior geometries that share the same primary asymptotic charges ( $M, J$ ) and the same Kerr exterior, without introducing new conserved charges.
Analytical Control: The framework offers an analytically controlled way to study the approach to the Kerr endpoint, identifying distinct stages of the trajectory (regularity, Minkowski breaking, singularity onset) that are independent of the inner horizon's existence.
Modesty regarding Dynamics: The authors are careful to distinguish between exact algebraic identities for horizon roots and physical thermodynamic laws. They assert that while the root-tracking provides exact geometric bookkeeping, a full dynamical thermodynamic interpretation (involving fluxes and true marginal surfaces) requires further analysis of the stress-energy content and stability, which remains an open problem.

In conclusion, the work establishes that regular rotating black holes possess a well-defined, positive, and bounded internal geometric sector invisible to asymptotic observers, which is isolated via subtraction of the Kerr inner horizon contribution. This sector can support macroscopic entropy even in the absence of near-extremal rotation, but its physical interpretation as a dynamical entropy flux remains conditional on future identification of true dynamical horizons and stability analysis.

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