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Cauchy problems for Einstein equations in three-dimensional spacetimes

Piotr T. Chruściel, Wan Cong, Théophile Quéau, Raphaela Wutte

TL;DR

This work analyzes the Cauchy problem for Einstein equations in 2+1 dimensions with negative cosmological constant, focusing on asymptotically locally hyperbolic initial data and their boundary behavior. It develops a robust conformal approach to both vacuum and non-vacuum data, introducing a novel class of vacuum spacelike data with poles at the conformal boundary that still possess finite energy. The paper establishes mass and angular momentum notions, proves positivity-type bounds via a Witten-type spinor argument, and identifies circumstances under which these charges can be minimized or become rigid, with extremal BTZ cases achieving equality. It further extends Maskit-type gluing results to two dimensions, studies the characteristic (null) Cauchy problem, and analyzes matter models such as Maxwell fields and scalar fields within the ALH framework, culminating in explicit BTZ-related metrics and detailed asymptotic symmetry structures.

Abstract

We analyze existence and properties of solutions of two-dimensional general relativistic initial data sets with a negative cosmological constant, both on spacelike and characteristic surfaces. A new family of such vacuum, spacelike data parameterised by poles at the conformal boundary at infinity is constructed. We review the notions of global Hamiltonian charges, emphasising the difficulties arising in this dimension, both in a spacelike and characteristic setting. One or two, depending upon the topology, lower bounds for energy in terms of angular momentum, linear momentum, and center of mass are established.

Cauchy problems for Einstein equations in three-dimensional spacetimes

TL;DR

This work analyzes the Cauchy problem for Einstein equations in 2+1 dimensions with negative cosmological constant, focusing on asymptotically locally hyperbolic initial data and their boundary behavior. It develops a robust conformal approach to both vacuum and non-vacuum data, introducing a novel class of vacuum spacelike data with poles at the conformal boundary that still possess finite energy. The paper establishes mass and angular momentum notions, proves positivity-type bounds via a Witten-type spinor argument, and identifies circumstances under which these charges can be minimized or become rigid, with extremal BTZ cases achieving equality. It further extends Maskit-type gluing results to two dimensions, studies the characteristic (null) Cauchy problem, and analyzes matter models such as Maxwell fields and scalar fields within the ALH framework, culminating in explicit BTZ-related metrics and detailed asymptotic symmetry structures.

Abstract

We analyze existence and properties of solutions of two-dimensional general relativistic initial data sets with a negative cosmological constant, both on spacelike and characteristic surfaces. A new family of such vacuum, spacelike data parameterised by poles at the conformal boundary at infinity is constructed. We review the notions of global Hamiltonian charges, emphasising the difficulties arising in this dimension, both in a spacelike and characteristic setting. One or two, depending upon the topology, lower bounds for energy in terms of angular momentum, linear momentum, and center of mass are established.

Paper Structure

This paper contains 35 sections, 378 equations, 1 figure.

Table of Contents

  1. Introduction
  2. All conformally smooth spacetimes which are vacuum near the conformal boundary
  3. Vacuum time-symmetric geometrically finite initial data sets
  4. Spacelike general relativistic initial data
  5. Vacuum time symmetric data
  6. The vector constraint equation and conformal transformations
  7. The vacuum vector constraint equation and Airy functions
  8. The vacuum vector constraint equation and holomorphic functions
  9. Poles at the conformal boundary
  10. Matter currents and the conformal vector Laplacian
  11. A "finite-mass" condition?
  12. The Lichnerowicz equation
  13. Boundary regularity
  14. Explicit solutions of the vacuum Cauchy problem
  15. Angular momentum and mass
  16. ...and 20 more sections

Figures (1)

  • Figure 6.1: The set $\Omega$ and its boundaries. In this section the boundary $\partial M$ is given by $\{z=0\}$. The boundary of $\Omega$ has three components, one is a subset of $\partial M$. The function $x$ vanishes precisely on both the blue and red components of the boundary of $\Omega$.

Theorems & Definitions (1)

  • proof