[Paper Review] High-frequency limits and null dust shell solutions in general relativity

Jonathan Luk, Igor Rodnianski|arXiv (Cornell University)|Sep 18, 2020

Cosmology and Gravitation Theories51 references7 citations

TL;DR

This paper establishes a rigorous connection between high-frequency limits of vacuum Einstein solutions and the Einstein–null dust system, proving that weak limits of vacuum spacetimes with $ L^2 $-bounded null derivatives converge to solutions of the Einstein–null dust equations with measure-valued null dust. It provides the first local existence and uniqueness result for such systems with measure-valued sources, including interacting null dust shells.

ABSTRACT

Consider the characteristic initial value problem for the Einstein vacuum equations without any symmetry assumptions. Impose a sequence of data on two intersecting null hypersurfaces, each of which is foliated by spacelike $2$-spheres. Assume that the sequence of data is such that the derivatives of the metrics along null directions are only uniformly bounded in $L^2$ but the derivatives of the metrics along the directions tangential to the $2$-spheres obey higher regularity bounds uniformly. By the results in [J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, Camb. J. Math. 5(4), 2017], it follows that the sequence of characteristic initial value problems gives rise to a sequence of vacuum spacetimes $(\mathcal M, g_n)$ in a fixed double-null domain $\mathcal M$. Since the existence theorem requires only very low regularity, the sequence of solutions may exhibit both oscillations and concentrations, and the limit need not be vacuum. We prove nonetheless that, after passing to a subsequence, the metrics converge in $C^0$ and weakly in $W^{1,2}$ to a solution of the Einstein-null dust system with two families of (potentially measure-valued) null dust. We show moreover that all sufficiently regular solutions to the Einstein-null dust system (with potentially measure-valued null dust) adapted to a double null coordinate system arise locally as weak limits of solutions to the Einstein vacuum system in the manner described above. As a consequence, we also give the first general local existence and uniqueness result for solutions to the Einstein-null dust system for which the null dusts are only measures. This in particular includes as a special case solutions featuring propagating and interacting shells of null dust.

Motivation & Objective

• To understand the effective matter fields arising in weak limits of high-frequency vacuum solutions to the Einstein equations.
• To characterize the limiting behavior of sequences of vacuum spacetimes with low regularity in the null directions but higher regularity on 2-spheres.
• To establish a local existence and uniqueness theory for the Einstein–null dust system with measure-valued null dust, including null shell solutions.
• To show that all sufficiently regular solutions of the Einstein–null dust system (with measure-valued dust) arise as weak limits of vacuum solutions.

Proposed method

• Uses the double null foliation gauge to analyze the characteristic initial value problem with data on intersecting null hypersurfaces foliated by 2-spheres.
• Imposes uniform $ L^2 $ bounds on null derivatives of the metric and higher regularity on angular derivatives, enabling weak compactness arguments.
• Applies compensated compactness and weak convergence techniques in $ W^{1,2} $ and $ C^0 $ to extract limits of Ricci coefficients and curvature components.
• Derives weak formulations of the Einstein vacuum and Einstein–null dust equations in the double null gauge, including renormalized Bianchi identities.
• Employs frequency localization and Plancherel-type estimates to control weak limits of products of high-frequency components.
• Constructs approximations of measure-valued null dust data by smooth data and then by vacuum data, proving weak approximation theorems.

Experimental results

Research questions

• RQ1 Can high-frequency limits of vacuum Einstein solutions be characterized as solutions to the Einstein–null dust system?
• RQ2 What is the regularity required for the limiting spacetime to satisfy the Einstein–null dust equations with measure-valued dust?
• RQ3 Can all sufficiently regular solutions of the Einstein–null dust system with measure-valued dust be realized as weak limits of vacuum solutions?
• RQ4 What is the role of angular regularity in ensuring compactness and convergence of Ricci coefficients under low regularity assumptions?

Key findings

• After passing to a subsequence, the metrics converge in $ C^0 $ and weakly in $ W^{1,2} $ to a solution of the Einstein–null dust system with two families of measure-valued null dust.
• All sufficiently regular solutions of the Einstein–null dust system with measure-valued null dust arise locally as weak limits of vacuum solutions in the described high-frequency regime.
• The first local existence and uniqueness result is established for the Einstein–null dust system with measure-valued null dust, including null shell solutions.
• The limit spacetime satisfies the Einstein–null dust equations weakly, with non-vanishing Ricci curvature components arising from the null dust.
• Compensated compactness techniques ensure weak convergence of $ abla ext{tr} ilde{ heta} $ and $ abla ext{tr} ilde{ heta} $, even under low regularity.
• The weak approximation theorem shows that measure-valued null dust data can be approximated by vacuum data in the high-frequency limit.