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On energy bounds in asymptotically locally AdS spacetimes

Virinchi Rallabhandi

TL;DR

This work develops energy bounds for asymptotically locally AdS spacetimes by leveraging Witten–Nester spinor methods with imaginary Killing spinors and a background-subtracted energy definition. It connects boundary geometry on cross-sections to conserved quantities and derives a general positive energy theorem, then specializes to ALAdS with cross-sections to obtain explicit BPS-type inequalities in 4D and 5D minimal gauged supergravity, including magnetic-field effects. The results recover familiar 4D and 5D bounds in appropriate limits and clarify how spin-structure compatibility and asymptotic symmetries influence the applicability of the bounds. The work also analyzes concrete boundary geometries (torus, sphere, lens spaces) and discusses subtleties from magnetic charges and spin structures, with implications for AdS/CFT-inspired energy constraints and supersymmetric solutions.

Abstract

This work considers positive energy theorems in asymptotically, locally AdS spacetimes. Particular attention is given to spacetimes where conformal infinity has compact, Einstein cross-sections admitting Killing or parallel spinors; a positive energy theorem is derived for such spacetimes in terms of geometric data intrinsic to the cross-section. This is followed by the first complete proofs of the BPS inequalities in (the bosonic sectors of) 4D and 5D minimal, gauged supergravity, including with magnetic fields, provided the Maxwell field is exact. The BPS inequalities are proven for asymptotically AdS spacetimes, but also generalised to the aforementioned class of asymptotically, locally AdS spacetimes.

On energy bounds in asymptotically locally AdS spacetimes

TL;DR

This work develops energy bounds for asymptotically locally AdS spacetimes by leveraging Witten–Nester spinor methods with imaginary Killing spinors and a background-subtracted energy definition. It connects boundary geometry on cross-sections to conserved quantities and derives a general positive energy theorem, then specializes to ALAdS with cross-sections to obtain explicit BPS-type inequalities in 4D and 5D minimal gauged supergravity, including magnetic-field effects. The results recover familiar 4D and 5D bounds in appropriate limits and clarify how spin-structure compatibility and asymptotic symmetries influence the applicability of the bounds. The work also analyzes concrete boundary geometries (torus, sphere, lens spaces) and discusses subtleties from magnetic charges and spin structures, with implications for AdS/CFT-inspired energy constraints and supersymmetric solutions.

Abstract

This work considers positive energy theorems in asymptotically, locally AdS spacetimes. Particular attention is given to spacetimes where conformal infinity has compact, Einstein cross-sections admitting Killing or parallel spinors; a positive energy theorem is derived for such spacetimes in terms of geometric data intrinsic to the cross-section. This is followed by the first complete proofs of the BPS inequalities in (the bosonic sectors of) 4D and 5D minimal, gauged supergravity, including with magnetic fields, provided the Maxwell field is exact. The BPS inequalities are proven for asymptotically AdS spacetimes, but also generalised to the aforementioned class of asymptotically, locally AdS spacetimes.

Paper Structure

This paper contains 11 sections, 23 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conventions
  3. Set-up and general positive energy theorem
  4. Energy bounds and boundary geometry
  5. Example boundaries
  6. Torus
  7. Sphere
  8. Compatibility of spin structures
  9. BPS inequalities
  10. Acknowledgements
  11. Inverting the Dirac operator

Key Result

Theorem 2.6

Figures (1)

  • Figure 1: This is a Penrose diagram for defining asymptotically, locally AdS spacetimes and energy within them. Conformal infinity, $\mathcal{I}$, is $\{r = \infty\}$, where $r$ is the Fefferman-Graham coordinate. $\mathcal{I}$ is topologically $\mathbb{R}\times S$ with metric $f_{(0)mn}$. $t$ is a coordinate along the $\mathbb{R}$ direction, $\Sigma_t$ is a spacelike hypersurface which has constant $t$ near $\mathcal{I}$ and $n^a$ is the future-directed, timelike, unit normal to $\Sigma_t$. Energy is measured at $\Sigma_{t, \infty} = \mathcal{I}\cap \Sigma_t$.

Theorems & Definitions (59)

  • Definition 2.1: Asymptotically, locally AdS
  • Definition 2.2: Background metric, $\bar{g}$
  • Definition 2.3: Energy
  • Definition 2.4: Background Killing spinor
  • Definition 2.5: $\nabla_a$, $\mathcal{A}_a$, $\mathbb{M}$, Witten-Nester 2-form and $Q(\varepsilon)$
  • Theorem 2.6: Lichnerowicz identity
  • proof
  • Lemma 2.7
  • proof
  • Definition 2.8: $p_M$
  • ...and 49 more