# Bowen-York Solution

### From GRwiki

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

which yields

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.