Conformal compactification and affine-null metric formulation of the Einstein equations

Abstract

In principle, global properties of solution of Einstein equations need to be addressed using the conformal Einstein equations, because this conformal compactification allows a clean definition of the `infinities' (spacelike, timelike and null infinity) of General Relativity. However, in numerical calculations often compactified coordinates in the physical space are used to reach these infinities. In this note, we discuss the conformal Einstein equations in spherical symmetry coupled to a massless scalar field and compare them with corresponding equations using a compactified coordinate in physical spacetime. The derivation of the field equations is based on metrics, in which the radial coordinate is an affine parameter along outgoing null rays. We show that the conformal equations within an affine-null metric formulation can be cast in a natural hierarchical form after the introduction of suitable auxiliary fields. The system of partial differential equations associated with the resulting (unphysical) conformal field equations proves to be identical to a system that employs a compactified coordinate in physical space along with well-constructed regularized fields. The reason for this equivalence is the introduction of new regularized fields in the physical spacetime after coordinate compactification to obtain a regular system of equations on the complete domain of the compactified coordinate. As part of this work, we also present the solution of the conformal field equations in affine-null coordinates near the conformal boundary, where the Bondi mass loss formula for a massless scalar field is recovered. The validity of the balance law of the mass loss at null infinity is demonstrated numerically.