### 2.4 Stationary solutions

When there is sufficient symmetry present, it is possible to construct initial data that are in true
equilibrium. These solutions possess at least two Killing vectors, one that is timelike at large distances and
one that is spatial, representing an azimuthal symmetry. When these symmetries are present, solving for the
initial data produces a global solution of Einstein’s equations and the solution is said to be stationary. The
familiar Kerr–Neumann solution for rotating black holes is an example of a stationary solution in
vacuum. Stationary configurations supported by matter are also possible, but the matter sources
must also satisfy the Killing symmetries, in which case the matter is said to be in hydrostatic
equilibrium [8].
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 as

This form of the metric can describe any stationary spacetime. Notice that the lapse is related to the
conformal factor by
and that the shift vector has only one component
I have used the usual conformal decomposition of the 3-metric (16) and have written the conformal 3-metric
with two parameters as
The four functions , , , and are functions of and only.
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

With the given metric and shift, we find that and the divergence of the shift also vanishes. This
means we can write the tracefree part of the extrinsic curvature as
We find that the Hamiltonian and momentum constraints take on the forms given by the conformal
thin-sandwich decomposition (51) with and . Only one of the momentum
constraint equations is non-trivial, and we find that the constraints yield elliptic equations for and
. What remains unspecified as yet are and (i.e., the conformal 3-metric).
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.