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Four components of the axisymmetric Einstein equations in 2+1 dimensions withnegative 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 angularmomentum parameters of the BTZ solution of the same name. The integrabilityconditions for the Einstein equations give rise to two conserved stress-energycurrents $\nabla_aj^a_{(M)}=0$ and $\nabla_aj^a_{(J)}=0$. The angular momentumcurrent is just the Noether current due to axisymmetry, but the mass current isunexpected in the presence of rotation. The conserved quantity $M$ exists inall dimensions in spherical symmetry, known as the Misner-Sharp, Hawking orKodama mass, but in 2+1 dimensions $M$ exists also in axisymmetry, even withrotation. We use $M$ and $J$ to give a fully constrained formulation of theaxisymmetric Einstein equations in 2+1 dimensions, where the Einstein equationsare solved by explicit integration from the center along time slices. We usethe two conserved matter currents in the construction of a high-resolutionshock-capturing formulation of the Einstein-perfect fluid system, in which $M$and $J$ momentum are then exactly conserved by construction. We demonstrateconvergence of the code in the test cases of generic dispersion and collapseand stable and unstable rotating stars.

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