General relativistic axisymmetric rotating systems: Coordinates and equations

The Einstein equations for rotating axisymmetric configurations in asymptotically flat spacetimes are put in a form suitable for numerical calculations of dynamics. The discussion is motivated by the astrophysical problem of gravitational collapse with generation of gravitational radiation and possibly black hole formation. In the context of topologically spherical coordinates there are two spatial gauge conditions which greatly simplify the Einstein equations and are compatible with regularity at the origin. We focus on one, the radial gauge, which generalizes Schwarzschild coordinates and is asymptotically a transverse-traceless gauge for gravitational radiation. The shift vector equation and the Hamiltonian constraint are parabolic equations in the radial gauge, rather than the usual elliptic equations. Two hypersurface conditions are explored in detail, the maximal hypersurface condition and another “polar” hypersurface condition which fits naturally with the radial gauge.