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Asymptotically Flat Initial Data with Prescribed Regularity at Infinity

Sergio Dain, Helmut Friedrich

TL;DR

This work advances the construction of asymptotically flat vacuum initial data with nonzero mass and angular momentum by enforcing near-space-like infinity expansions for both the metric and extrinsic curvature via the conformal method. It develops a robust elliptic framework for solving the Lichnerowicz and momentum constraint equations, including a careful treatment of asymptotics near the point $i$ representing spatial infinity. The authors prove existence and regularity of the conformal factor, derive detailed near-$i$ expansions, and show how logarithmic terms arise when linear momentum is nonzero. The results enable controlled, non-time-symmetric initial data suitable for studying Friedrich’s regular finite initial value problem and for numerical investigations of spacetime evolution with prescribed global charges.

Abstract

We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate.

Asymptotically Flat Initial Data with Prescribed Regularity at Infinity

TL;DR

This work advances the construction of asymptotically flat vacuum initial data with nonzero mass and angular momentum by enforcing near-space-like infinity expansions for both the metric and extrinsic curvature via the conformal method. It develops a robust elliptic framework for solving the Lichnerowicz and momentum constraint equations, including a careful treatment of asymptotics near the point representing spatial infinity. The authors prove existence and regularity of the conformal factor, derive detailed near- expansions, and show how logarithmic terms arise when linear momentum is nonzero. The results enable controlled, non-time-symmetric initial data suitable for studying Friedrich’s regular finite initial value problem and for numerical investigations of spacetime evolution with prescribed global charges.

Abstract

We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate.

Paper Structure

This paper contains 13 sections, 38 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Hamiltonian Constraint
  4. Existence
  5. Asymptotic expansions near $i$ of solutions to the Lichnerowicz equation
  6. Proof of Theorem \ref{['T1']}
  7. Solution of the Hamiltonian Constraint with Logarithmic terms
  8. The Momentum Constraint
  9. The Momentum Constraint on Euclidean Space
  10. The general case: existence
  11. Asymptotic expansions near $i$ of solutions to the momentum constraint
  12. On Hölder functions
  13. An additional result

Key Result

Theorem 1.1

Let $h_{ab}$ be a smooth metric on $S$ with positive Ricci scalar $R$. Assume that $\Psi_{ab}$ is smooth in $\tilde{S}$ and satisfies on $B_a$ Then there exists on $\tilde{S}$ a unique solution $\theta$ of equation (Lich), which is positive, satisfies (thetai), and has in $B_a$ the form

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Sobolev imbedding
  • Theorem 2.2
  • Theorem 2.3: Rellich-Kondrakov
  • Theorem 2.4: Schauder fixed point
  • Theorem 2.5: $L^p$ regularity
  • Theorem 2.6: Schauder elliptic regularity
  • Theorem 2.7: Fredholm alternative
  • Theorem 2.8: Weak Maximum Principle
  • ...and 30 more