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Minimal Elements of the Causal Boundary with Applications to Spacetime Splitting

Leonardo García-Heveling

Abstract

In 1972, Geroch, Kronheimer, and Penrose introduced what is now called the causal boundary of a spacetime. This boundary is constructed out of Terminal Indecomposable Past sets (TIPs) and their future analogues (TIFs), which are the pasts and futures of inextendible causal curves. The causal boundary is a key tool to understand the global structure of a spacetime. In this paper, we show that in a spacetime with compact Cauchy surfaces, there is always at least one minimal TIP and one minimal TIF, minimal meaning that it does not contain another TIP (resp.\ TIF) as a proper subset. We then study the implications of the minimal TIP and TIF meeting each other. This condition generalizes some of the "no observer horizon" conditions that have been used in the literature to obtain partial solutions of the Bartnik splitting conjecture. We also show that such a no observer horizons condition is satisfied when the spacetime has a (possibly discrete) timelike conformal symmetry, generalizing a result of Costa e Silva, Flores, and Herrera about conformal Killing vector fields.

Minimal Elements of the Causal Boundary with Applications to Spacetime Splitting

Abstract

In 1972, Geroch, Kronheimer, and Penrose introduced what is now called the causal boundary of a spacetime. This boundary is constructed out of Terminal Indecomposable Past sets (TIPs) and their future analogues (TIFs), which are the pasts and futures of inextendible causal curves. The causal boundary is a key tool to understand the global structure of a spacetime. In this paper, we show that in a spacetime with compact Cauchy surfaces, there is always at least one minimal TIP and one minimal TIF, minimal meaning that it does not contain another TIP (resp.\ TIF) as a proper subset. We then study the implications of the minimal TIP and TIF meeting each other. This condition generalizes some of the "no observer horizon" conditions that have been used in the literature to obtain partial solutions of the Bartnik splitting conjecture. We also show that such a no observer horizons condition is satisfied when the spacetime has a (possibly discrete) timelike conformal symmetry, generalizing a result of Costa e Silva, Flores, and Herrera about conformal Killing vector fields.
Paper Structure (5 sections, 13 theorems, 18 equations, 2 figures)

This paper contains 5 sections, 13 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries about the causal boundary
  3. Existence of minimal TIPs and TIFs
  4. The min-min condition and spacetime splitting
  5. Timelike conformal symmetries

Key Result

Theorem 1.2

Let $(M,g)$ be a spacetime containing a compact Cauchy surface. Then every TIP contains a minimal TIP, and every TIF contains a minimal TIF. In particular, there exists at least one minimal TIP and one minimal TIF in the spacetime.

Figures (2)

  • Figure 1: The spacetime of Example \ref{['ex:sharp']}, with its intersecting minimal TIP $U$ and minimal TIF $V$.
  • Figure 2: The spacetime of Examples \ref{['ex:notminmin']} and \ref{['ex:flow']}.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1: GKP
  • Lemma 2.2: GKP
  • Definition 2.3: GKP
  • ...and 26 more