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On the ADM mass of critical area-normalized capacitors

Simon Raulot

TL;DR

This work introduces the notion of critical area-normalized capacitors on asymptotically flat manifolds, tying the boundary capacity potential to geometric data through an overdetermined boundary condition. It establishes a mass-capacity inequality linking the ADM mass and the boundary capacity, with equality signaling the exterior Schwarzschild geometry, and provides dimensional relaxations in the 3D case. The authors then leverage these inequalities to obtain rigidity results for static AF vacua, including uniqueness theorems for spacetimes with photon surfaces and for static manifolds with equipotential boundaries, under spin assumptions and curvature bounds. The proofs combine mass-capacity techniques (Hirsch–Miao), positive mass theorems on manifolds with boundary, and conformal changes to extend the reach of the rigidity results to the spin and three-dimensional settings. Collectively, the results yield Schwarzschild rigidity in a broad AF static context and sharpen uniqueness theorems for related geometries in general relativity.

Abstract

In this note, we prove mass-capacity inequalities for asymptotically flat manifolds whose boundary capacity potential satisfies an overdetermined problem, referred to as critical area-normalized capacitors. As a consequence, we obtain uniqueness results for the Schwarzschild metric, from which improvements in the uniqueness theorems for spin asymptotically flat spacetimes containing a connected photon surface, as well as for spin asymptotically flat static manifolds with boundary are obtained.

On the ADM mass of critical area-normalized capacitors

TL;DR

This work introduces the notion of critical area-normalized capacitors on asymptotically flat manifolds, tying the boundary capacity potential to geometric data through an overdetermined boundary condition. It establishes a mass-capacity inequality linking the ADM mass and the boundary capacity, with equality signaling the exterior Schwarzschild geometry, and provides dimensional relaxations in the 3D case. The authors then leverage these inequalities to obtain rigidity results for static AF vacua, including uniqueness theorems for spacetimes with photon surfaces and for static manifolds with equipotential boundaries, under spin assumptions and curvature bounds. The proofs combine mass-capacity techniques (Hirsch–Miao), positive mass theorems on manifolds with boundary, and conformal changes to extend the reach of the rigidity results to the spin and three-dimensional settings. Collectively, the results yield Schwarzschild rigidity in a broad AF static context and sharpen uniqueness theorems for related geometries in general relativity.

Abstract

In this note, we prove mass-capacity inequalities for asymptotically flat manifolds whose boundary capacity potential satisfies an overdetermined problem, referred to as critical area-normalized capacitors. As a consequence, we obtain uniqueness results for the Schwarzschild metric, from which improvements in the uniqueness theorems for spin asymptotically flat spacetimes containing a connected photon surface, as well as for spin asymptotically flat static manifolds with boundary are obtained.
Paper Structure (6 sections, 11 theorems, 59 equations)

This paper contains 6 sections, 11 theorems, 59 equations.

Table of Contents

  1. Statements of the main results
  2. Rigidity results for static asymptotically flat vacuums with equipotential boundaries
  3. An uniqueness theorem for asymptotically flat static vacuum spacetimes with a connected photon surface as inner boundary
  4. An uniqueness result for asymptotically flat static manifolds with boundary
  5. Proof of Theorem \ref{['SpinCase']} and Theorem \ref{['ThreeDimensionalCase']}
  6. Two positive mass theorems on manifold with boundary

Key Result

Theorem 1

Let $(M^n,g)$ be a spin critical area-normalized capacitor with a connected boundary $\Sigma^{n-1}:=\partial M^n$. Assume that $R\geq 0$ as well as $H>0$ and that there exists $c>1$ such that Then and equality occurs if and only if $(M^n,g)$ is isometric to the region outside a rotationally symmetric sphere in an $n$-dimensional Schwarzschild manifold.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 1 more