Noncircular axisymmetric stationary spacetimes

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Abstract
A formalism is presented to treat axisymmetric stationary spacetimes in the most general case, when the stress-energy tensor is not assumed to be circular, so that one cannot make the usual foliation of spacetime into two orthogonal families of two-surfaces. Such a study is motivated by the consideration of rotating relativistic stars with strong toroidal magnetic field or meridional circulation of matter (convection). The formulation is based on a "(2+1)+1" slicing of spacetime and the corresponding projections of the Einstein equation. It offers a suitable frame to discuss the choice of coordinates appropriate for the description of asymptotically flat and noncircular axisymmetric spacetimes. We propose a certain class of coordinates which is interpretable in terms of extremal three- and two-surfaces. This choice leads to well-behaved elliptic operators in the equations for the metric coefficients. Consequently, in the case of a starlike object, the proposed corrdinates are global ones, i.e., they can be extended to spatial infinity. These coordinates are also appropriate for obtaining initial conditions for (instability triggered) evolution, since they match naturally with coordinates proposed for dynamical evolution, especially with the "maximal time slicing" condition. The formulation is written in an entirely two-dimensional covariant form, but, in order to obtain numerical solutions, we also give the complete system of partial differential equations obtained by specialization of the equations to a certain subclass of the proposed coordinates.