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

Abstract

Four components of the axisymmetric Einstein equations in 2+1 dimensions with negative cosmological constant can be written as ${\nabla }_{a}M=\dots$ and ${\nabla }_{a}J=\dots$, 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 ${\nabla }_a{j}_{(M)}^a=0$ and ${\nabla }_a{j}_{(J)}^a=0$. 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.

arXiv comment23 pages, 17 figures, Typos corrected. This version has been accepted by PRD

References in corpus (8)

• The Black Hole in Three Dimensional Space Time
• Geometry of the 2+1 Black Hole
• On weakly turbulent instability of anti-de Sitter space
• Generalized Misner-Sharp quasi-local mass in Einstein-Gauss-Bonnet gravity
• General relativistic hydrodynamics in curvilinear coordinates
• Ultrarelativistic fluid dynamics
• Extension of Kodama vector and quasilocal quantities in three-dimensional axisymmetric spacetimes
• Rigidly rotating perfect fluid stars in $2+1$ dimensions

Cited by in corpus (5)

• Extension of Kodama vector and quasilocal quantities in three-dimensional axisymmetric spacetimes
• Geometrical origin of the Kodama vector
• Charged rotating BTZ solution revisited: New coordinates and algebraic classifications
• Critical collapse of an axisymmetric ultrarelativistic fluid in $2+1$ dimensions
• Extremal rotating BTZ black holes cannot be dressed in (anti-)self-dual Maxwell field