Initial data sets with vanishing mass are contained in pp-wave spacetimes

TL;DR

This paper proves that initial data sets with vanishing mass are contained in pp-wave spacetimes, extending the positive mass theorem's rigidity to include these cases across all dimensions.

Contribution

It demonstrates that zero mass initial data sets must be contained in pp-wave spacetimes, broadening the understanding of the positive mass theorem's rigidity in general relativity.

Findings

Initial data sets with zero mass are contained in pp-wave spacetimes.

The proof uses spinorial methods combined with spacetime harmonic functions.

Constructs non-vacuum initial data with zero mass in the borderline decay case.

Abstract

In 1981, Schoen-Yau and Witten showed that in General Relativity both the total energy $E$ and the total mass $m$ of an initial data set modeling an isolated gravitational system are non-negative. Moreover, if $E=0$, the initial data set must be contained in Minkowski space. In this paper, we show that if $m=0$, i.e. if $E$ equals the total momentum $|P|$, the initial data set must be contained in a pp-wave spacetime. Our proof combines spinorial methods with spacetime harmonic functions and works in all dimensions. Additionally, we find the decay rate threshold where the embedding has to be within Minkowski space and construct non-vacuum initial data sets with $m=0$ in the borderline case. As a consequence, this completely settles the rigidity of the spacetime positive mass theorem for spin manifolds.