On the Newtonian Limit of General Relativity

TL;DR

This paper introduces a new variable choice in the 3+1 formulation of general relativity that results in a symmetric-hyperbolic system with a smooth Newtonian limit, facilitating numerical computations.

Contribution

It presents a novel variable redefinition and formulation that achieves a symmetric-hyperbolic system with a well-defined Newtonian limit for arbitrary lapse and shift.

Findings

System has a smooth Newtonian limit when variables are scaled appropriately.

Provides a method for initial data with controlled radiation content.

Formulation is local and suitable for practical numerical applications.

Abstract

We find a choice of variables for the 3+1 formulation of general relativity which casts the evolution equations into (flux-conservative) symmetric-hyperbolic first order form for arbitrary lapse and shift, for the first time. We redefine the lapse function in terms of the determinant of the 3-metric and a free function U which embodies the lapse freedom. By rescaling the variables with appropriate factors of 1/c, the system is shown to have a smooth Newtonian limit when the redefined lapse U and the shift are fixed by means of elliptic equations to be satisfied on each time slice. We give a prescription for the choice of appropriate initial data with controlled extra-radiation content, based on the theory of problems with different time-scales. Our results are local, in the sense that we are not concerned with the treatment of asymptotic regions. On the other hand, this local theory is all what is needed for most problems of practical numerical computation.