NONCIRCULAR AXISYMMETRICAL STATIONARY SPACETIMES

A formalism is presented to treat axisymmetric stationary spacetimes i n the most general case, when the stress-energy tensor is not assumed to be circular, so that one cannot make the usual foliation of spaceti me into two orthogonal families of two-surfaces. Such a study is motiv ated by the consideration of rotating relativistic stars with strong t oroidal 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 sp acetimes. We propose a certain class of coordinates which is interpret able in terms of extremal three- and two-surfaces. This choice leads t o well-behaved elliptic operators in the equations for the metric coef ficients. Consequently, in the case of a starlike object, the proposed coordinates are global ones, i.e., they can be extended to spatial in finity. These coordinates are also appropriate for obtaining initial c onditions for (instability triggered) evolution, since they match natu rally with coordinates proposed for dynamical evolution, especially wi th 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 diff erential equations obtained by specialization of the equations to a ce rtain subclass of the proposed coordinates.