Bowen-York Solution

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The freely specifiable data in the conformal transverse-traceless decomposition of the constraints is

   {\tilde g}_{ab} &  & {\rm conformal\ metric} \\
    {\tilde M}_{ab} & & {\rm symmetric\  trace\  free\ tensor} \\
   K & & {\rm trace\ of\ the\ extrinsic\ curvature}

The physical metric is g_{ab} = \psi^4{\tilde g}_{ab}. Let {\tilde D}_a denote the covariant derivative built from the conformal metric. (Indices are raised with the inverse conformal metric.) Let D_a denote the covariant derivative build from the physical metric.

Choose maximal slicing (K=0), conformal flatness (\tilde g_{ab} is a flat metric) and \tilde M_{ab} = 0. With

   K^{ab} = \psi^{-10} {\tilde A}^{ab}

the momentum constraint D_a K^{ab} = 0 can be written as

   {\tilde D}_a {\tilde A}^{ab} = 0

This is solved by

   {\tilde A}^{ab} = {\tilde D}^a V^b + {\tilde D}^b V^a - \frac{2}{3} {\tilde g}^{ab} {\tilde D}_c V^c

if the vector V^a satisfies

   {\tilde D}_b{\tilde D}^b V^a + \frac{1}{3} {\tilde D}^a {\tilde D}_b V^b = 0

The Bowen-York solution is

   V^a = -\frac{1}{4r} (7P^a + n^a n_b P^b)  + \frac{1}{r^2} \epsilon^{abc}n_b S_c

which yields

  {\tilde A}^{ab}  = \frac{3}{2r^2} [P^a n_b + P^b n^a - (g^{ab} - n^a n^b) n_c P^c ] 
   + \frac{3}{r^3} [\epsilon^{acd}S_c n_d n^b + \epsilon^{bcd} S_c n_d n^a ]

In these expressions the components of P^a and S^a are constants in Cartesian coordinates. Also, r = \sqrt{(x^1)^2 + (x^2)^2 + (x^3)^2} is the coordinate distance from the origin, n^a = x^a/r is the radial normal vector, and \epsilon^{abc} is the alternating symbol. Indices on these quantities are lowered with the conformal metric.

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