Conformal deformations of initial data sets to the strict dominant energy condition using a spacetime Poisson equation

TL;DR

This paper presents a PDE-based method to conformally deform initial data sets in general relativity to satisfy the strict dominant energy condition, while controlling boundary properties and energy variations.

Contribution

It provides an explicit PDE construction for conformal deformation to the strict dominant energy condition, improving upon previous implicit methods.

Findings

Explicit PDE solution for conformal deformation

Preserves boundary character and minimally alters energy

Applicable to 3-manifolds, extendable to higher dimensions

Abstract

We give an alternate proof of one of the results given in [16] showing that initial data sets with boundary for the Einstein equations $(M, g, k)$ satisfying the dominant energy condition can be conformally deformed to the strict dominant energy condition, while preserving the character of the boundary (minimal, future trapped, or past trapped) while changing the area of the boundary and ADM energy of the initial data set by an arbitrarily small amount. The proof relies on solving an equation that looks like the equation for spacetime harmonic functions studied in [7], but with a Neumann boundary condition and non-zero right hand side, which we refer to as a spacetime Poisson equation. One advantage of this method of proof is that the conformal deformation is explicitly constructed as a solution to a PDE, as opposed to only knowing the solution exists via an application of the implicit function theorem as in [16]. We restrict ourselves to the physically relevant case of a $3$-manifold $M$, though the proof can be generalized to higher dimensions.