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On the Asymptotics for the Vacuum Einstein Constraint Equations

Justin Corvino, Richard M. Schoen

TL;DR

The paper develops a rigorous gluing framework for vacuum Einstein initial data, showing that any asymptotically flat data can be deformed so that outside a large compact set it matches a Kerr slice (or another admissible family) while preserving the interior data. It introduces a robust use of the constraint map and its L^2-adjoint, identifies a ten-dimensional obstruction tied to energy-momentum and angular momentum, and combines local deformation with degree-theoretic arguments to achieve exact solutions with controlled asymptotics. The results establish that AC-satisfying data are dense and provide a mechanism to approximate arbitrary AF data by data that evolve to spacetimes with well-behaved null infinity. This work generalizes prior time-symmetric gluing by incorporating the full momentum and angular momentum structure of the constraints. The techniques have broad implications for the global behavior of solutions to the Einstein equations and for the construction of spacetimes with prescribed asymptotic properties.

Abstract

Given asymptotically flat initial data on M^3 for the vacuum Einstein field equation, and given a bounded domain in M, we construct solutions of the vacuum constraint equations which agree with the original data inside the given domain, and are identical to that of a suitable Kerr slice (or identical to a member of some other admissible family of solutions) outside a large ball in a given end. The data for which this construction works is shown to be dense in an appropriate topology on the space of asymptotically flat solutions of the vacuum constraints. This construction generalizes work of the first author, where the time-symmetric case was studied.

On the Asymptotics for the Vacuum Einstein Constraint Equations

TL;DR

The paper develops a rigorous gluing framework for vacuum Einstein initial data, showing that any asymptotically flat data can be deformed so that outside a large compact set it matches a Kerr slice (or another admissible family) while preserving the interior data. It introduces a robust use of the constraint map and its L^2-adjoint, identifies a ten-dimensional obstruction tied to energy-momentum and angular momentum, and combines local deformation with degree-theoretic arguments to achieve exact solutions with controlled asymptotics. The results establish that AC-satisfying data are dense and provide a mechanism to approximate arbitrary AF data by data that evolve to spacetimes with well-behaved null infinity. This work generalizes prior time-symmetric gluing by incorporating the full momentum and angular momentum structure of the constraints. The techniques have broad implications for the global behavior of solutions to the Einstein equations and for the construction of spacetimes with prescribed asymptotic properties.

Abstract

Given asymptotically flat initial data on M^3 for the vacuum Einstein field equation, and given a bounded domain in M, we construct solutions of the vacuum constraint equations which agree with the original data inside the given domain, and are identical to that of a suitable Kerr slice (or identical to a member of some other admissible family of solutions) outside a large ball in a given end. The data for which this construction works is shown to be dense in an appropriate topology on the space of asymptotically flat solutions of the vacuum constraints. This construction generalizes work of the first author, where the time-symmetric case was studied.

Paper Structure

This paper contains 14 sections, 13 theorems, 91 equations.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Basic Notation.
  4. Linearization of the Constraint Map.
  5. Constructing Solutions with Good Asymptotics.
  6. Solving the Deformation Problem.
  7. The Basic Estimate.
  8. Variational Method.
  9. Pointwise estimates and the nonlinear problem.
  10. Handling the cokernel.
  11. Computing the parameter map.
  12. Preservation of the AC Condition
  13. Controlling the resulting constraint data.
  14. Solving the constraints.

Key Result

Lemma 2.1

Assume that $\rho$ is bounded from above. For $k\geq 1$, the subspace $H^k(\Omega)$ (and hence $C^{\infty}(\overline{\Omega})$) is dense in $H^k_\rho(\Omega)$.

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 1
  • Proposition 3.1
  • Theorem 2
  • Theorem 3
  • Lemma 4.1
  • Lemma 4.2
  • ...and 3 more