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The Penrose inequality in spherical symmetry with charge and in Gauss-Bonnet gravity

Hari K. Kunduri, Juan Margalef-Bentabol, Sarah Muth

TL;DR

This work delivers a unified initial-data proof of Penrose-type inequalities in spherical symmetry across asymptotically flat and AdS settings, extending to Einstein–Maxwell theory with charge and to Gauss–Bonnet gravity. The authors introduce a charged Hawking mass $m_{\mathrm{CH}}$ and a generalized Gauss–Bonnet Hawking mass $m_{\mathrm{GB}}$, prove their monotonicity under the respective constraint equations, and show these masses converge to the total energy (ADM or AdS energy) at infinity. By comparing the limiting energy with the quasi-local horizon mass at the outermost MOTS, they establish lower bounds on the total energy in terms of horizon area and charge (and GB corrections), and prove rigidity statements that equality implies isometric embedding into RN–AdS or Schwarzschild–AdS/GB spacetimes. The results rely on an initial-data perspective, offering a coherent framework that encapsulates both EM and GB extensions and has implications for holography and the stability of gravitational collapse in higher-curvature theories.

Abstract

We establish the spacetime Penrose inequality in spherical symmetry in spacetime dimensions $n+1\geq3$ with charge and cosmological constant from the initial data perspective. We also show that this result extends to the Gauss-Bonnet theory of gravity.

The Penrose inequality in spherical symmetry with charge and in Gauss-Bonnet gravity

TL;DR

This work delivers a unified initial-data proof of Penrose-type inequalities in spherical symmetry across asymptotically flat and AdS settings, extending to Einstein–Maxwell theory with charge and to Gauss–Bonnet gravity. The authors introduce a charged Hawking mass and a generalized Gauss–Bonnet Hawking mass , prove their monotonicity under the respective constraint equations, and show these masses converge to the total energy (ADM or AdS energy) at infinity. By comparing the limiting energy with the quasi-local horizon mass at the outermost MOTS, they establish lower bounds on the total energy in terms of horizon area and charge (and GB corrections), and prove rigidity statements that equality implies isometric embedding into RN–AdS or Schwarzschild–AdS/GB spacetimes. The results rely on an initial-data perspective, offering a coherent framework that encapsulates both EM and GB extensions and has implications for holography and the stability of gravitational collapse in higher-curvature theories.

Abstract

We establish the spacetime Penrose inequality in spherical symmetry in spacetime dimensions with charge and cosmological constant from the initial data perspective. We also show that this result extends to the Gauss-Bonnet theory of gravity.
Paper Structure (21 sections, 7 theorems, 67 equations)

This paper contains 21 sections, 7 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Asymptotically flat case
  3. Asymptotically AdS case
  4. Einstein Maxwell initial data
  5. Gauss-Bonnet gravity
  6. The Einstein-Maxwell theory
  7. Equations of motion
  8. Initial Value Problem
  9. Spherically symmetric initial data set
  10. Definition of charged mass and its properties
  11. Monotonicity
  12. Asymptotic limit
  13. Penrose inequality
  14. The Gauss-Bonnet gravity theory
  15. Equations of motion
  16. ...and 6 more sections

Key Result

Theorem 1.1

Consider a spherically symmetric charged asymptotically flat/hyperbolic initial data set $(M,g,k,E,B)$ of the Einstein-Maxwell equations with uncharged matter sources (with a negative cosmological constant in the latter case) satisfying the constraint equations eq: constraint EM and the dominant ene where $\mathcal{E}$ is the ADM mass (asymptotically flat case, $\ell \to \infty$) or the Chrusciel-

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2: Charged Hawking (Misner-Sharp) mass
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof
  • ...and 5 more