The basic approach for finding stationary solutions begins by simplifying the metric to take into account the symmetries. Many different forms have been used for the metric (cf. Refs. [8, 9, 22, 34, 63, 21]). I will use a decomposition that makes comparison with the previous decompositions straightforward. First, define the interval asany stationary spacetime. Notice that the lapse is related to the conformal factor by
The equations necessary to solve for these four functions are derived from the constraint equations (14) and (15), and the evolution equations (12) and (13). For the evolution equations, we use the fact that and . The metric evolution equation (13) defines the extrinsic curvature in terms of derivatives of the shift
The conformal 3-metric is determined by the evolution equations for the traceless part of the extrinsic curvature. Of these five equations, one can be written as an elliptic equation for , and two yield complementary equations that can each be solved by quadrature for . The remaining equations are redundant as a result of the Bianchi identities.
Of course, the clean separation of the equations I have suggested above is an illusion. All four equations must be solved simultaneously, and clever combinations of the four metric quantities can greatly simplify the task of solving the system of equations. This accounts for the numerous different systems used for solving for stationary solutions.
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