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Positive energy-momentum theorems for asymptotically AdS spin initial data sets with charge

Simon Raulot

Abstract

For complete spin initial data sets with an asymptotically anti--de Sitter end, we introduce a charged energy--momentum defined as a linear functional arising from the Einstein--Maxwell constraints. Under a dominant energy condition adapted to the presence of a negative cosmological constant, we establish positive energy--momentum theorems, showing in particular that this functional is non--negative on a natural real cone. We place particular emphasis on the case where the manifold carries a compact inner boundary. In the time--symmetric setting, this yields a mass--charge inequality for asymptotically hyperbolic manifolds with charge.

Positive energy-momentum theorems for asymptotically AdS spin initial data sets with charge

Abstract

For complete spin initial data sets with an asymptotically anti--de Sitter end, we introduce a charged energy--momentum defined as a linear functional arising from the Einstein--Maxwell constraints. Under a dominant energy condition adapted to the presence of a negative cosmological constant, we establish positive energy--momentum theorems, showing in particular that this functional is non--negative on a natural real cone. We place particular emphasis on the case where the manifold carries a compact inner boundary. In the time--symmetric setting, this yields a mass--charge inequality for asymptotically hyperbolic manifolds with charge.
Paper Structure (14 sections, 14 theorems, 136 equations)

This paper contains 14 sections, 14 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Definitions and statement of the results
  3. Asymptotically AdS initial data sets with charge
  4. The charged energy-momentum
  5. The dominant energy condition
  6. Positive charged energy-momentum theorems
  7. The Schrödinger-Lichnerowicz formula for initial data sets with charge
  8. Preliminaries on spinors
  9. The modified connection
  10. Proof of the main results
  11. The boundaryless case: proof of Theorem \ref{['PET-Spherical']}
  12. The non-empty boundary case: proof of Theorem \ref{['PositiveETS-Boundary']}
  13. The non-empty boundary case: proof of Theorem \ref{['PositiveETS-Boundary-APS']}
  14. The time-symmetric case: proof of Theorems \ref{['PMTC-Hyp']} and \ref{['PMTCB-Hyp-TS']}

Key Result

Theorem 1

Let $(M^n,g,K,E)$, $n\geq 3$, be a complete spin initial data set with charge containing at least one asymptotically AdS end. Assume that the integrability condition (IntegrabilityCondition) holds, as well as the dominant energy condition (DEC-General). Then the linear form $\Xi\circ\mathcal{K}$ is

Theorems & Definitions (18)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • Theorem 5
  • Remark 2
  • Theorem 6
  • Remark 3
  • ...and 8 more