Fully constrained, high-resolution shock-capturing, formulation of the Einstein-fluid equations in $2+1$ dimensions

TL;DR

This paper develops a high-resolution shock-capturing numerical formulation for axisymmetric Einstein-fluid equations in 2+1 dimensions, utilizing conserved mass and angular momentum currents to improve accuracy and stability in simulations of gravitational collapse and rotating stars.

Contribution

It introduces a fully constrained, high-resolution shock-capturing formulation of Einstein-fluid equations in 2+1 dimensions using conserved quantities M and J, with demonstrated convergence.

Findings

Code converges in test cases of dispersion and collapse.

Stable and unstable rotating star simulations are successful.

Conservation of M and J is exact in the numerical scheme.

Abstract

Four components of the axisymmetric Einstein equations in 2+1 dimensions with negative cosmological constant can be written as $\nabla_aM=\dots$ and $\nabla_aJ=\dots$, where the dots stand for stress-energy terms, and $M$ and $J$ are scalars. In vacuum, they reduce to the constant mass and angular momentum parameters of the BTZ solution of the same name. The integrability conditions for the Einstein equations give rise to two conserved stress-energy currents $\nabla_aj^a_{(M)}=0$ and $\nabla_aj^a_{(J)}=0$. The angular momentum current is just the Noether current due to axisymmetry, but the mass current is unexpected in the presence of rotation. The conserved quantity $M$ exists in all dimensions in spherical symmetry, known as the Misner-Sharp, Hawking or Kodama mass, but in 2+1 dimensions $M$ exists also in axisymmetry, even with rotation. We use $M$ and $J$ to give a fully constrained formulation of the axisymmetric Einstein equations in 2+1 dimensions, where the Einstein equations are solved by explicit integration from the center along time slices. We use the two conserved matter currents in the construction of a high-resolution shock-capturing formulation of the Einstein-perfect fluid system, in which $M$ and $J$ momentum are then exactly conserved by construction. We demonstrate convergence of the code in the test cases of generic dispersion and collapse and stable and unstable rotating stars.