Initial data sets with dominant energy condition admitting no smooth dec spacetime extension

TL;DR

This paper investigates the relationship between initial data sets satisfying the dominant energy condition and their realization as slices of Lorentzian manifolds, providing a counterexample in the smooth category.

Contribution

It demonstrates that not all smooth initial data sets with the dominant energy condition can be embedded into a smooth Lorentzian manifold satisfying the same condition.

Findings

Counterexample showing no smooth embedding exists

Highlights limitations of the dominant energy condition in smooth settings

Clarifies the distinction between initial data and spacetime extensions

Abstract

There are two versions of the dominant energy condition (=dec): The original one for Lorentzian manifolds and an associated one for initial data sets. If a Lorentzian manifold satisfies dec, then so does the induced initial set on any embedded spacelike hypersurface. In this article, we discuss the question of a potential converse of this: Is every dec initial data set the induced one on a spacelike hypersurface within a suitably chosen dec Lorentzian manifold? We provide an example showing that in general the answer is no if we require all structures to be smooth.