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A proof of the reverse isoperimetric inequality using a geometric-analytic approach

Naman Kumar

TL;DR

The paper addresses whether the reverse isoperimetric inequality (RII) holds for AdS black holes in $D\ge4$ Einstein gravity by combining geometric rigidity with variational analysis. It employs a 1+1+2 decomposition and a curvature-preserving conformal deformation of horizon slices, together with Sherif–Dunsby rigidity, to show that the round horizon $S^{D-2}$ is the unique stable extremum at fixed pressure and volume, yielding maximal entropy for AdS--Schwarzschild. A second-variation calculation confirms the round horizon is a strict local maximum of area (entropy) under volume-preserving deformations (with $\ell\ge2$ modes yielding negative second variation), and Kerr–AdS is shown to decrease entropy at fixed $V$, establishing RII for the considered class. The results highlight the role of spacetime curvature and gravitational focusing in entropic extremization and offer a foundation for exploring extensions to other gravity theories, quantum corrections, and holographic interpretations.

Abstract

We present a proof of the reverse isoperimetric inequality -- a central conjecture in extended black hole thermodynamics -- for black holes in Einstein gravity with $D \geq 4$, employing a two-pronged geometric-analytic method. Our analysis shows that the reversal of the usual isoperimetric inequality originates from the structure of curved backgrounds governed by Einstein's equations, thereby underscoring the fundamental role of gravity in the reverse isoperimetric property of AdS black hole horizons.

A proof of the reverse isoperimetric inequality using a geometric-analytic approach

TL;DR

The paper addresses whether the reverse isoperimetric inequality (RII) holds for AdS black holes in Einstein gravity by combining geometric rigidity with variational analysis. It employs a 1+1+2 decomposition and a curvature-preserving conformal deformation of horizon slices, together with Sherif–Dunsby rigidity, to show that the round horizon is the unique stable extremum at fixed pressure and volume, yielding maximal entropy for AdS--Schwarzschild. A second-variation calculation confirms the round horizon is a strict local maximum of area (entropy) under volume-preserving deformations (with modes yielding negative second variation), and Kerr–AdS is shown to decrease entropy at fixed , establishing RII for the considered class. The results highlight the role of spacetime curvature and gravitational focusing in entropic extremization and offer a foundation for exploring extensions to other gravity theories, quantum corrections, and holographic interpretations.

Abstract

We present a proof of the reverse isoperimetric inequality -- a central conjecture in extended black hole thermodynamics -- for black holes in Einstein gravity with , employing a two-pronged geometric-analytic method. Our analysis shows that the reversal of the usual isoperimetric inequality originates from the structure of curved backgrounds governed by Einstein's equations, thereby underscoring the fundamental role of gravity in the reverse isoperimetric property of AdS black hole horizons.

Paper Structure

This paper contains 8 sections, 1 theorem, 75 equations.

Table of Contents

  1. Introduction
  2. AdS--Schwarzschild black holes maximize entropy
  3. Sherif--Dunsby rigidity and maximal entropy
  4. Effective Functional and Its Variation
  5. Conclusion and Discussion
  6. Conformal Rigidity and Sphericity of Compact Hypersurfaces
  7. Boundary-‐term variation for a general York boundary
  8. Extrinsic and mean curvature of a 2--sphere embedded in Euclidean AdS$_4$

Key Result

Theorem A.1

Let $(M^{4},g)$ be a spacetime admitting a $1{+}1{+}2$ covariant decomposition, and let $T\hookrightarrow M$ be a compact, smoothly embedded spacelike hypersurface whose induced metric $h_{ab}$ has Ricci tensor where $\mathbf e$ is the unit "radial" direction and $N$ is the projector onto the orthogonal $2$--sheets. The condition $\alpha\neq\beta$ means that $(T,h)$ is non--Einstein, i.e. $R_{ab}

Theorems & Definitions (1)

  • Theorem A.1: Sherif--Dunsby Sherif:2021zzb, Theorem VII.4