Microlocal analysis near null infinity in asymptotically flat spacetimes

TL;DR

This work develops a fully microlocal edge-b framework for studying linear waves on asymptotically flat spacetimes near null infinity. By building edge-b vector fields, Sobolev spaces, and a robust edge-b pseudodifferential calculus, the authors control regularity and decay through radial sets and along the null-infinity boundary, even in the presence of nontrivial geometry. They prove propagation of edge-b regularity and decay estimates for forward solutions, with sharp results on Minkowski space and applications to nonlinear stability problems in general relativity. The framework also provides a foundation for global analysis in spacetimes with asymptotically stationary regions, via inversion of edge normal operators and leading-order decay control at null infinity.

Abstract

We present a novel approach to the analysis of regularity and decay for solutions of wave equations in a neighborhood of null infinity in asymptotically flat spacetimes of any dimension. The classes of metrics and wave type operators we consider near null infinity include those arising in nonlinear stability problems for Einstein's field equations in $1+3$ dimensions. In a neighborhood of null infinity, in an appropriate compactification of the spacetime to a manifold with corners, the wave operators are of edge type at null infinity and totally characteristic at spacelike and future timelike infinity. On a corresponding scale of Sobolev spaces, we demonstrate how microlocal regularity propagates across or into null infinity via a sequence of radial sets. As an application, inspired by work of the second author with Baskin and Wunsch, we prove regularity and decay estimates for forward solutions of wave type equations on asymptotically flat spacetimes which are asymptotically homogeneous with respect to scaling in the forward timelike cone and have an appropriate structure at null infinity. These estimates are new even for the wave operator on Minkowski space. The results obtained here are also used as black boxes in a global theory of wave type equations on asymptotically flat and asymptotically stationary spacetimes developed by the first author.

Figures (10)

• Figure 1.1: The radial compactification of $\mathbb{R}^{n+1}$, and the light cone at infinity $Y\cong\mathbb{S}^{n-1}$, in the case $n=1$.
• Figure 1.2: The blow-up $\tilde{M}$ of $\overline{\mathbb{R}^{n+1}}$ at $Y$, and labels for the boundary hypersurfaces.
• Figure 1.3: Structure of the null-bicharacteristic flow near null infinity in $2+1$ spacetime dimensions. The cross sections of the cylinder are cross sections of the future light cones inside of each fiber of the eb-phase space over $\mathscr I^+$. The thick black sets are the radial sets (the two antipodal points over $\mathscr I^+\cap I^0$ forming a connected radial set in higher dimensions). See §\ref{['SsMF']} and Figure \ref{['FigMF']} for details.
• Figure 2.1: A neighborhood of $H_1\cap H_2$ inside $M$, and the local coordinates $x,y,z$ used in \ref{['EqEBSe2']}. The fibration $\phi$ of $H_2$ and the base $Y$ are indicated in red, and the fibers of $H_2$ in blue.
• Figure 3.1: The resolution $\tilde{M}$ of the radial compactification $\overline{\mathbb{R}^{n+1}}$ at the light cone $Y$ at future infinity, and the boundary hypersurfaces of $\tilde{M}$. The manifold $M$ is the square root blow-up of $\tilde{M}$ at $\tilde{\mathscr I}^+$, and $\upbeta$ is the total blow-down map $M\to\overline{\mathbb{R}^{n+1}}$.

Theorems & Definitions (112)

• Theorem 1.1: Edge-b-regularity and decay of waves in the exterior domain
• Theorem 1.2: Global edge-b-regularity of waves
• Remark 1.3: Geometric singular analysis on $\tilde{M}$
• Remark 1.4: Parabolic scaling
• Remark 1.5: Conormal coefficients at null infinity
• Remark 1.6: Klein--Gordon equation
• Remark 1.7: b-perspective
• Definition 2.1: Lie algebras of vector fields
• Lemma 2.2: Hamiltonian vector field on ${}^{{\mathrm{e,b}}}T^*M$
• proof