Regularity of the Einstein Equations at Future Null Infinity

TL;DR

This paper demonstrates that Einstein's equations, when expressed in a conformal framework at future null infinity, are actually regular and explicitly evaluable there, despite initial appearances of singularities.

Contribution

It shows that the apparently singular terms in the conformal Einstein equations at null infinity are regular and can be explicitly computed using geometric data, under certain constraints.

Findings

Singular terms are regular at null infinity when constraints are enforced.

Explicit evaluation of conformally regular geometric data at null infinity.

Results may be applicable to other formulations with minor modifications.

Abstract

When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal factor and thus appear to be ill-behaved at this (exterior) boundary. In this article however we show, through an enforcement of the Hamiltonian and momentum constraints to the needed order in a Taylor expansion, that these apparently singular terms are not only regular at the boundary but can in fact be explicitly evaluated there in terms of conformally regular geometric data. Though we employ a rather rigidly constrained and gauge fixed formulation of the field equations, we discuss the extent to which we expect our results to have a more 'universal' significance and, in particular, to be applicable, after minor modifications, to alternative formulations.