Exploring the Conformal Constraint Equations

TL;DR

This paper investigates the properties of conformal constraint equations in general relativity, introducing a perturbative method to generate solutions near flat spacetime by addressing degeneracies at null infinity.

Contribution

It develops a perturbative approach to solve the conformal constraint equations, providing insights into their properties and regularity at the boundary.

Findings

Established properties of conformal constraint equations.

Developed a perturbative solution method near flat space.

Addressed degeneracy issues at null infinity.

Abstract

One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where the conformal factor vanishes, namely at the boundary representing null infinity. This problem can be avoided by means of a technique of H. Friedrich, which replaces the Einstein equations in the unphysical spacetime by an equivalent system of equations which is regular at the boundary. The initial value problem for these equations produces a system of constraint equations known as the conformal constraint equations. This work describes some of the properties of the conformal constraint equations and develops a perturbative method of generating solutions near flat space under certain simplifying assumptions.