Einstein's field equations for metrics with rotation.

A system of field equations for the problem of supermassive equilibrium rotating configurations is developed on the basis of the general theory of relativity. The Einstein field equations in the presence of matter are obtained for a four-dimensional interval that is invariant relative to a simultaneous change in the azimuthal and time variables and independent of the time variable (a metric with rotation). It is found that only six of the ten components of the Ricci tensor are absolutely nonvanishing; these components are presented in explicit form as second-order quasi-linear differential operators that act on the four functionally independent components of the metric tensor. The four de Donder harmonicity conditions for the metric considered are reduced to two quasi-linear differential equations involving the four functionally independent components of the metric tensor.