The freely specifiable data in the conformal transverse-traceless decomposition of the constraints is
The physical metric is . Let denote the covariant derivative built from the conformal metric. (Indices are raised with the inverse conformal metric.) Let denote the covariant derivative build from the physical metric.
Choose maximal slicing (), conformal flatness ( is a flat metric) and . With
the momentum constraint can be written as
This is solved by
if the vector satisfies
The Bowen-York solution is
In these expressions the components of and are constants in Cartesian coordinates. Also, is the coordinate distance from the origin, is the radial normal vector, and is the alternating symbol. Indices on these quantities are lowered with the conformal metric.