The $C^3$-null gluing problem: linear and nonlinear analysis

TL;DR

This paper studies the $C^3$-null gluing problem for Einstein vacuum equations, identifying 20 obstructions related to third-order derivatives, with 10 being new, in the near-Minkowski regime.

Contribution

It extends the null gluing analysis to third-order derivatives, revealing new obstructions and conserved quantities specific to the $C^3$ setting.

Findings

Solvability up to 20 obstructions near Minkowski data

Identification of 10 new linearly conserved charges

Extension of previous $C^2$-null gluing results

Abstract

In this paper, we investigate the $C^3$-null gluing problem for the Einstein vacuum equations, that is, we consider the null gluing of up to and including third-order derivatives of the metric. In the regime where the characteristic data is close to Minkowski data, we show that this $C^3$-null gluing problem is solvable up to a $20$-dimensional space of obstructions. The obstructions correspond to $20$ linearly conserved quantities: $10$ of which are already present in the $C^2$-null gluing problem analysed by Aretakis, Czimek and Rodnianski, and $10$ are novel obstructions inherent to the $C^3$-null gluing problem. The $10$ novel obstructions are linearly conserved charges calculated from third-order derivatives of the metric.