⚛️ general relativity

On the Classical, Penrose, and Reverse Isoperimetric Inequalities in Black Holes: Insights From AdS to Flat Riemannian Backgrounds

This paper argues that violations of the reverse isoperimetric inequality in black holes invariably lead to thermodynamic instability or unphysical solutions, thereby supporting the conjecture that superentropic black holes are unstable and proposing new, stronger isoperimetric inequalities for asymptotically flat spacetimes.

Original authors: Robert B. Mann, Behnam Pourhassan, Ali Dehghani

Published 2026-09-15

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CC BY 4.0

Original authors: Robert B. Mann, Behnam Pourhassan, Ali Dehghani

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 among the most extreme objects in the universe, regions where gravity is so intense that nothing, not even light, can escape. For decades, physicists have studied these cosmic traps not just as gravitational anomalies, but as thermodynamic systems, much like a pot of boiling water or a compressed gas. Just as a pot of water has a temperature, pressure, and volume, a black hole possesses a temperature related to its surface gravity and an entropy linked to the size of its event horizon, the point of no return. In recent years, scientists have expanded this view by treating the cosmological constant—a value describing the energy density of empty space—as a form of pressure. This shift allows them to define a specific "thermodynamic volume" for the black hole, a measure of the space it effectively occupies within the laws of physics, which is distinct from the simple geometric volume one might calculate by measuring its radius. This framework has opened a new door for understanding how the shape of a black hole relates to its energy and stability.

In a new study, researchers Robert Mann, Behnam Pourhassan, and Ali Dehghani have investigated a proposed rule connecting a black hole's volume to the area of its event horizon. This rule, known as the reverse isoperimetric inequality, suggests that for a given volume, a black hole's horizon area cannot exceed a certain maximum limit. In the familiar world of geometry, a sphere is the most efficient shape, enclosing the maximum area for a given volume. The reverse isoperimetric inequality proposes that black holes behave similarly, but with a twist: they are conjectured to always be at least as efficient as a sphere, meaning their area is never larger than what a sphere of the same volume would allow. The researchers set out to test what happens when this rule is broken. They examined a class of theoretical black holes, dubbed "superentropic," which appear to violate this rule by having an entropy, or disorder, that is higher than the maximum allowed by their volume.

The team's investigation reveals that these superentropic black holes are not just geometric curiosities; they are fundamentally unstable. By analyzing the mechanical and thermal properties of these objects, the researchers found that whenever the reverse isoperimetric inequality is violated, the black hole fails to meet the basic requirements for stability. Specifically, these objects exhibit negative compressibility, a condition where the system would expand if squeezed and shrink if pressure were released, a behavior that is physically impossible for a stable object in nature. Furthermore, they found that these unstable configurations often lead to other physical impossibilities, such as negative temperatures or the exposure of a naked singularity, a point of infinite density that is not hidden behind an event horizon. The study concludes that the reverse isoperimetric inequality is likely a fundamental constraint of nature; the authors conjecture that if a black hole violates it, it cannot exist as a stable, physical object, though they note that certain exotic objects like black rings may present exceptions to the broader hierarchy of inequalities.

The researchers also explored whether these unstable, superentropic black holes could exist in a universe without the cosmological constant, effectively transitioning from a universe with a specific type of expanding or contracting background to one that is flat and static, like our own. They found that such a transition is not possible in a smooth manner. If one attempts to force a superentropic black hole into a flat background, the mathematical description either fails to yield a smooth limit, leads to pathological behavior, or results in the loss of the superentropic character itself. This suggests that the existence of these unstable black holes is tied strictly to the specific conditions of the universe they inhabit, and they cannot simply be "turned off" to exist in a flat space. This finding reinforces the idea that the reverse isoperimetric inequality acts as a guardian of cosmic order, preventing the formation of pathological objects that would violate the cosmic censorship hypothesis, which states that singularities must always be hidden.

Finally, the study connects these findings to a famous, older rule known as the Penrose isoperimetric inequality. This older rule relates the mass of a black hole to the area of its horizon, setting a limit on how much mass can be packed into a given surface area without creating a naked singularity. The researchers demonstrated that the reverse isoperimetric inequality is a stronger, more restrictive condition. In the flat space of our universe, satisfying the reverse inequality automatically ensures that the older Penrose inequality is also satisfied. However, the reverse is not true; a black hole could theoretically satisfy the Penrose rule while still violating the newer, stricter reverse rule, which would render it unstable. The authors propose a new relationship, which they call the thermo-volumetric inequality, linking the mass of the black hole directly to its thermodynamic volume. This new bound appears to be a more precise indicator of stability than the older mass-area relationship in many cases, though the authors note that it can fail for certain higher-dimensional rotating solutions with multiple spins, offering a deeper insight into the internal mechanics of these cosmic giants.

Through detailed calculations across various types of black holes, including those with electric charge and rapid rotation, the team confirmed that no stable counterexamples to their findings exist. While some previous studies had suggested that certain exotic black holes might be stable despite violating the reverse inequality, the researchers showed that those conclusions relied on incomplete analyses that ignored mechanical stability or used incorrect definitions of volume. When the full set of physical requirements is applied, the instability of superentropic black holes becomes undeniable. The work suggests that the universe has a built-in mechanism to prevent the formation of these unstable states, ensuring that black holes remain consistent with the laws of thermodynamics and the structure of spacetime. By establishing that the reverse isoperimetric inequality is a fundamental constraint, the study provides a clearer picture of the limits of black hole physics and the delicate balance required for these objects to exist.

Technical Summary: Classical, Penrose, and Reverse Isoperimetric Inequalities in Black Holes

Problem Statement
The paper addresses the physical significance and thermodynamic implications of the Reverse Isoperimetric Inequality (RII) conjecture in black hole thermodynamics. The RII posits that for asymptotically Anti-de Sitter (AdS) and de Sitter (dS) black holes, the isoperimetric ratio $R_{\text{RII}} \geq 1$ , where the ratio compares the thermodynamic volume to the horizon area. While this inequality holds for many black hole families, it is violated by "superentropic" black holes ( $R_{\text{RII}} < 1$ ). Previous literature has presented conflicting views: some suggest superentropic black holes are thermodynamically stable, while others propose a link between superentropicity and instability. Furthermore, the relationship between the RII (formulated in AdS with a cosmological constant) and the classical Penrose Isoperimetric Inequality (PII, formulated in asymptotically flat spacetimes) remains unclear. The authors aim to resolve these contradictions, determine the thermodynamic stability of superentropic black holes, and establish whether these inequalities extend to asymptotically flat backgrounds.

Methodology
The authors employ the framework of extended black hole thermodynamics (often termed "black hole chemistry"), where the cosmological constant $\Lambda$ is treated as a thermodynamic pressure $P = -\Lambda/(8\pi G_N)$ , and its conjugate variable is the thermodynamic volume $V$ .

Thermodynamic Stability Analysis: The authors analyze the mechanical and thermal stability conditions for various black hole families. They explicitly calculate the adiabatic ( $\kappa_S$ ) and isothermal ( $\kappa_T$ ) compressibilities, as well as specific heats ( $C_P, C_V$ ). Stability requires $\kappa_T \geq \kappa_S \geq 0$ and $C_P \geq C_V \geq 0$ .
Case Studies: They examine a wide range of black hole solutions, including:
- Charged and rotating BTZ black holes in 2+1 dimensions.
- Ultraspinning Kerr(-Newman)-AdS black holes in arbitrary dimensions.
- Black holes with nonlinear electrodynamics (Born-Infeld, Euler-Heisenberg limits).
- Generalized exotic BTZ black holes and those with Immirzi hair.
- Myers-Perry black holes in higher dimensions.
Flat Limit Analysis: The authors investigate the asymptotically flat limit ( $\Lambda \to 0$ or AdS radius $\ell \to \infty$ ) of these solutions. They define the thermodynamic volume in flat space via the smooth limit of the AdS enthalpy: $V_{\text{flat}} \equiv (\partial M / \partial P)|_{P=0}$ .
Inequality Verification: They test the validity of the RII, the PII, and a newly proposed "thermo-volumetric inequality" (relating ADM mass and thermodynamic volume) across these families in both AdS and flat Riemannian backgrounds.

Key Contributions and Results

Thermodynamic Instability of Superentropic Black Holes:
The authors demonstrate that black holes violating the RII ( $R_{\text{RII}} < 1$ ) invariably fail the requirements for thermodynamic stability. Specifically, they find that superentropic black holes possess negative compressibility (either $\kappa_S < 0$ or $\kappa_T < 0$ ).
- They refute previous claims of stability in certain superentropic models (e.g., specific Born-Infeld BTZ or ultraspinning Kerr-Newman configurations) by showing that those claims relied on analyzing specific heats ( $C_P, C_V$ ) in isolation while ignoring compressibility constraints.
- Conjecture 1: Violation of the RII necessarily implies thermodynamic instability. Superentropic black holes never satisfy the combined mechanical ( $\kappa_T \geq \kappa_S \geq 0$ ) and thermal ( $C_P \geq C_V \geq 0$ ) stability requirements.
Non-Existence of Flat Limits for Superentropic Black Holes:
The authors argue that superentropic black holes do not possess a smooth asymptotically flat limit.
- Conjecture 2: There is no superentropic black hole spacetime that has a smooth limit to an asymptotically flat background.
- The reasoning is that if such a limit existed, the superentropic character ( $R_{\text{RII}} < 1$ ) would persist in flat space, leading to a violation of the Penrose Isoperimetric Inequality (PII) and the formation of naked singularities, thereby violating cosmic censorship.
Hierarchy of Inequalities in Flat Space:
By taking the $\Lambda \to 0$ limit of extended thermodynamics, the authors derive a hierarchy of isoperimetric inequalities for asymptotically flat black holes. They define a new thermo-volumetric inequality relating the ADM mass ( $M$ ) and the thermodynamic volume ( $V$ ).
- For $D=4$ and $D=5$ , and for singly-rotating Myers-Perry black holes in any dimension, they establish the chain:
  $\left( \frac{16\pi E}{(D-2)A_{D-2}} \right)^{\frac{1}{D-3}} \geq \left( \frac{(D-1)V}{A_{D-2}} \right)^{\frac{1}{D-1}} \geq \left( \frac{A}{A_{D-2}} \right)^{\frac{1}{D-2}}$
- This implies that the Penrose Isoperimetric Inequality (PII) is strictly weaker than the Reverse Isoperimetric Inequality (RII) in the asymptotically flat limit. The RII provides a stronger geometric bound on the mass and volume than the PII.
- The authors note that for higher-dimensional rotating solutions with multiple independent rotation parameters (e.g., Myers-Perry with $D \geq 6$ ), the thermo-volumetric inequality (the leftmost part of the chain) can fail, even though both the PII and RII remain valid individually. The validity of the full hierarchy in these higher-dimensional, multi-spin cases remains an open problem.
Role of Thermodynamic Volume:
The paper emphasizes that the thermodynamic volume (derived from the generalized Komar integral/Killing potential) is the physically preferred definition for isoperimetric inequalities. Unlike the geometric volume, the thermodynamic volume yields consistent inequalities across AdS, dS, and flat backgrounds and correctly encodes gravitational binding energy effects.

Significance
The paper claims that the Reverse Isoperimetric Inequality is of fundamental importance, acting as a geometric constraint that ensures thermodynamic stability and cosmic censorship.

Unification: It unifies geometric inequalities (PII) with thermodynamic stability criteria, suggesting that the RII is a more fundamental constraint than the PII.
Stability Diagnostic: It provides a robust diagnostic tool: any black hole solution violating the RII is physically unstable or pathological (e.g., naked singularities, negative temperature).
Cosmic Censorship: The absence of flat limits for superentropic black holes suggests that the RII acts as a "censor" preventing the formation of naked singularities in flat space limits.
Framework Validation: The results validate the extended thermodynamics framework (treating $\Lambda$ as pressure) as essential for uncovering universal constraints on gravitational systems that are not visible in the standard $\Lambda=0$ formulation.

The authors conclude that while general analytic proofs for arbitrary stationary spacetimes remain an open problem, their explicit calculations across diverse black hole families in $D=4$ and $D=5$ strongly support the conjecture that superentropicity is synonymous with thermodynamic instability and that the RII is a stronger, more fundamental bound than the PII in asymptotically flat spacetimes.

Technical Summary: Classical, Penrose, and Reverse Isoperimetric Inequalities in Black Holes

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