A strongly hyperbolic and regular reduction of Einstein's equations for axisymmetric spacetimes

TL;DR

This paper develops a strongly hyperbolic, regular reduction of Einstein's equations tailored for axisymmetric spacetimes, suitable for stable numerical evolution with well-defined boundary conditions.

Contribution

It introduces a novel (2+1)+1 formalism with a generalized harmonic gauge that ensures regularity and hyperbolicity for axisymmetric Einstein equations.

Findings

Constructed a strongly hyperbolic first-order evolution system.

Derived constraint-preserving stable outer boundary conditions.

Ensured regularity of all terms on the symmetry axis.

Abstract

This paper is concerned exclusively with axisymmetric spacetimes. We want to develop reductions of Einstein's equations which are suitable for numerical evolutions. We first make a Kaluza-Klein type dimensional reduction followed by an ADM reduction on the Lorentzian 3-space, the (2+1)+1 formalism. We include also the Z4 extension of Einstein's equations adapted to this formalism. Our gauge choice is based on a generalized harmonic gauge condition. We consider vacuum and perfect fluid sources. We use these ingredients to construct a strongly hyperbolic first-order evolution system and exhibit its characteristic structure. This enables us to construct constraint-preserving stable outer boundary conditions. We use cylindrical polar coordinates and so we provide a careful discussion of the coordinate singularity on axis. By choosing our dependent variables appropriately we are able to produce an evolution system in which each and every term is manifestly regular on axis.