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Maximal Hypersurfaces in Asymptotically Stationary Space-Times

P. T. Chrusciel, R. Wald

TL;DR

This work establishes the existence and foliation by maximal hypersurfaces in asymptotically stationary, asymptotically flat spacetimes with a Killing field $X$ timelike at infinity. It proves maximal slices exist in two settings: (a) spacetimes without a black hole or white hole (ergoregions allowed) and (b) spacetimes with intersecting black/white hole horizons at a compact surface $S$, with slices asymptotic to the ends and orthogonal to $X$. Crucially, the results do not assume Einstein’s equations or energy conditions and employ Killing coordinates and time functions to produce maximal foliations under mild hypotheses. The findings extend Sudarsky–Wald analyses and provide a robust framework for constructing slices through exterior and interior regions in stationary-like spacetimes, with a precise equivalence between no-holes and compact domains of dependence.

Abstract

Existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike ``near infinity''. The first class consists of strongly causal asymptotically flat spacetimes which contain no ``black hole or white hole" (but may contain ``ergoregions" where the Killing orbits fail to be timelike). The second class of spacetimes possess a black hole and a white hole, with the black and white hole horizons intersecting in a compact 2-surface $S$.

Maximal Hypersurfaces in Asymptotically Stationary Space-Times

TL;DR

This work establishes the existence and foliation by maximal hypersurfaces in asymptotically stationary, asymptotically flat spacetimes with a Killing field timelike at infinity. It proves maximal slices exist in two settings: (a) spacetimes without a black hole or white hole (ergoregions allowed) and (b) spacetimes with intersecting black/white hole horizons at a compact surface , with slices asymptotic to the ends and orthogonal to . Crucially, the results do not assume Einstein’s equations or energy conditions and employ Killing coordinates and time functions to produce maximal foliations under mild hypotheses. The findings extend Sudarsky–Wald analyses and provide a robust framework for constructing slices through exterior and interior regions in stationary-like spacetimes, with a precise equivalence between no-holes and compact domains of dependence.

Abstract

Existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike ``near infinity''. The first class consists of strongly causal asymptotically flat spacetimes which contain no ``black hole or white hole" (but may contain ``ergoregions" where the Killing orbits fail to be timelike). The second class of spacetimes possess a black hole and a white hole, with the black and white hole horizons intersecting in a compact 2-surface .

Paper Structure

This paper contains 6 sections, 31 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary Definitions and Results on Asymptotically Stationary Spacetimes
  3. Causal Properties of Spacetimes of Classes (a) and (b)
  4. Maximal Slicings
  5. A generalization --- stationary--rotating space--times.
  6. An Alternate Proof of Existence of a Maximal Foliation for Spacetimes of Class (b) Satisfying the Strong Energy Condition

Key Result

Proposition 2.1

Let $(M, g_{ab})$ with $\Sigma = \Sigma_1 \cup \Sigma'$ be asymptotically stationary with respect to $\Sigma_1$. Then there exists a global coordinate system on $M_1$ such that $M_1 \approx I\!\! R \times (I\!\! R^3 \setminus B(R_1))$ with $\Sigma_1=\{x^0=0\}$, and for some constant $C$; moreover, $\partial_{\sigma_1} \ldots \partial_{\sigma_k} g_{\mu\nu}$ satisfy an obvious weighted Hölder condi

Figures (2)

  • Figure 1.1: The Penrose diagram of a spacetime which illustrates some applications of our theorems. Each point on this Figure represents a two sphere. The arrows represent the direction and character (spacelike or timelike) of Killing orbits. A spherically symmetric, asymptotically stationary spacetime, not necessarily satisfying any reasonable field equations, with the global structure displayed in this Figure can be easily constructed using the methods of Walker.
  • Figure 2.1: A Penrose diagram of a spacetime in which $\cal H$ is not the entire Cauchy horizon of $\Sigma$. In this spacetime $\partial {\cal D}(\Sigma)$ has two connected components, ${\cal H}={\cal H}_+\cup{\cal H}_-$ and ${\cal H}^\prime$.

Theorems & Definitions (38)

  • Definition 2.1
  • Proposition 2.1: Killing time based on $\Sigma\cap M_1$
  • Definition 2.2: Spacetimes of class (a)
  • Definition 2.3: Spacetimes of class (b)
  • Definition 2.4
  • Definition 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.1
  • Lemma 3.1
  • ...and 28 more