Conformal regularization of Einstein's field equations

TL;DR

This paper introduces a conformal regularization method for Einstein's field equations, enabling better analysis of asymptotic structures and singularities through dimensionless variables and hyperbolic equations.

Contribution

It develops a new conformal regularization framework for Einstein's equations using conformal transformations and orthonormal frames, enhancing the study of asymptotic behaviors.

Findings

Provides explicit conformal factors and coordinate choices for singularity analysis

Introduces a dimensionless hyperbolic system related to the Hubble-normalized approach

Discusses the relation to existing conformal methods in the literature

Abstract

To study asymptotic structures, we regularize Einstein's field equations by means of conformal transformations. The conformal factor is chosen so that it carries a dimensional scale that captures crucial asymptotic features. By choosing a conformal orthonormal frame we obtain a coupled system of differential equations for a set of dimensionless variables, associated with the conformal dimensionless metric, where the variables describe ratios with respect to the chosen asymptotic scale structure. As examples, we describe some explicit choices of conformal factors and coordinates appropriate for the situation of a timelike congruence approaching a singularity. One choice is shown to just slightly modify the so-called Hubble-normalized approach, and one leads to dimensionless first order symmetric hyperbolic equations. We also discuss differences and similarities with other conformal approaches in the literature, as regards, e.g., isotropic singularities.