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Near-Boundary Asymptotics and Unique Continuation for the AdS--Einstein--Maxwell System

Simon Guisset

TL;DR

This work investigates fully nonlinear AdS--Einstein--Maxwell dynamics in spacetimes with a timelike conformal boundary $\mathcal{I}$ and negative cosmological constant. It establishes a Maxwell-augmented Fefferman–Graham expansion (the Maxwell-FG-aAdS segment) near $\mathcal{I}$, identifying free boundary data $\mathfrak{g}^{(0)},\mathfrak{g}^{(n)},\mathfrak{f}^{1,(0)},\mathfrak{f}^{0,((n-4)_+)}$ and deriving the associated holographic constraints. The second pillar proves a local unique continuation result from boundary data under a Generalised Null Convexity Criterion (GNCC), by recasting the Einstein–Maxwell equations as a wave–transport system, introducing renormalised difference fields, and applying Carleman estimates adapted to the AdS geometry. The results extend previous vacuum analyses to the electrovacuum setting, showing that boundary data determine bulk solutions locally (up to diffeomorphisms) when the GNCC holds, and providing a precise link between boundary holographic data and bulk geometric information. Together, the expansions and unique continuation results contribute to the mathematical underpinnings of AdS/CFT-type correspondences and nonlinear rigidity phenomena in asymptotically AdS spacetimes with matter.

Abstract

In this article, we extend the results of both Shao and Holzegel-Shao to the AdS-Einstein-Maxwell system $({M}, g, F)$. We study the asymptotics of the metric $g$ and the Maxwell field $F$ near the conformal boundary ${I}$ for the fully nonlinear coupled system. Furthermore, we characterise the holographic (boundary) data used in the second part of this work. We also prove the local unique continuation property for solutions of the coupled Einstein equations from the conformal boundary. Specifically, the prescription of the coefficients $(\mathfrak{g}^{(0)}, \mathfrak{g}^{(n)})$ in the near-boundary expansion of $g$, along with the boundary data for the Maxwell fields $(\mathfrak{f}^{0}, \mathfrak{f}^{1})$, on a domain ${D} \subset {I}$ uniquely determines $(g, F)$ near ${D}$. The geometric conditions required for unique continuation are identical to those in the vacuum case, regardless of the presence of the Maxwell fields. This work is part of the author's thesis.

Near-Boundary Asymptotics and Unique Continuation for the AdS--Einstein--Maxwell System

TL;DR

This work investigates fully nonlinear AdS--Einstein--Maxwell dynamics in spacetimes with a timelike conformal boundary and negative cosmological constant. It establishes a Maxwell-augmented Fefferman–Graham expansion (the Maxwell-FG-aAdS segment) near , identifying free boundary data and deriving the associated holographic constraints. The second pillar proves a local unique continuation result from boundary data under a Generalised Null Convexity Criterion (GNCC), by recasting the Einstein–Maxwell equations as a wave–transport system, introducing renormalised difference fields, and applying Carleman estimates adapted to the AdS geometry. The results extend previous vacuum analyses to the electrovacuum setting, showing that boundary data determine bulk solutions locally (up to diffeomorphisms) when the GNCC holds, and providing a precise link between boundary holographic data and bulk geometric information. Together, the expansions and unique continuation results contribute to the mathematical underpinnings of AdS/CFT-type correspondences and nonlinear rigidity phenomena in asymptotically AdS spacetimes with matter.

Abstract

In this article, we extend the results of both Shao and Holzegel-Shao to the AdS-Einstein-Maxwell system . We study the asymptotics of the metric and the Maxwell field near the conformal boundary for the fully nonlinear coupled system. Furthermore, we characterise the holographic (boundary) data used in the second part of this work. We also prove the local unique continuation property for solutions of the coupled Einstein equations from the conformal boundary. Specifically, the prescription of the coefficients in the near-boundary expansion of , along with the boundary data for the Maxwell fields , on a domain uniquely determines near . The geometric conditions required for unique continuation are identical to those in the vacuum case, regardless of the presence of the Maxwell fields. This work is part of the author's thesis.
Paper Structure (52 sections, 61 theorems, 555 equations)

This paper contains 52 sections, 61 theorems, 555 equations.

Table of Contents

  1. Introduction
  2. Fefferman-Graham expansion
  3. Unique continuation
  4. Setting-up the problem
  5. Wave--transport system and renormalisation
  6. Carleman estimates and unique continuation
  7. Gauge freedom
  8. Future directions of study
  9. Generalisations to nontrivial matter models
  10. Optimality of the GNCC
  11. Global rigidity result
  12. Preliminaries
  13. Basic definitions
  14. Mixed covariant formalism
  15. General formulas and notations
  16. ...and 37 more sections

Key Result

Theorem 1.1

Let $(\mathcal{M}, g)$ and $({\mathcal{M}}, \check{g})$ be two near--boundary regions of aAdS solutions, with $g,\, \check{g}$ solving Einstein_intro with $T=0$, and let $\mathscr{D}\subset \mathcal{I}$. Assuming that $\mathscr{D}$ satisfies a certain null-pseudoconvexity condition, and if the bound up to a diffeomorphism preserving $\mathcal{I}$.

Theorems & Definitions (179)

  • Theorem 1.1: Informal version of Theorem 6.7 of Holzegel22
  • Definition 1.2: FG-aAdS segment, informal
  • Example 1
  • Theorem 1.3: Theorem \ref{['theorem_main_fg']}, informal
  • Theorem 1.4: Theorem \ref{['thm_UC_main']}, informal version
  • Definition 2.1: aAdS Region
  • Definition 2.2: Vertical Bundle/Vertical Tensor Fields
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 169 more