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- The 4 equations break down into two sets of equations
- The Momentum Constraint Equation ( vector equation)
- The Hamiltonian Constraint Equation (scalar equation)
The momentum equation can be solved with semi-analytical techniques for
2 black holes.
The Hamiltonian Constraint equation solver solves this constraint equation
along with the boundary conditions:
The boundary condition at infinity follows directly from asymptotic
and conformal flatness:
The boundary condition around the holes is, (
)
where 
and
(these are
Euclidean operators in the flat background space).
The first 3D solutions were found by three separate groups
- G. Cook (Cornell University)
- S. Klasky, M. Choptuik, R. Matzner (University Texas at Austin)
- M. Dubal, S. Oliveira, R. Matzner (University Texas at Austin)
This problem has been solved
The picture below shows isosurfaces of the conformal factor, psi,
and the extrinsic curvature. The gold surface is the isosurface
around psi=1.9. The purple surface is the isosurface around psi=1.6.
The isosurface of one of the extrinsic curvature components is shown in
green. Here the value used in the isosurface is at 0. The Data was
computed and visualized by S. Klasky, M. Choptuik, R. Matzner.
Scott Klasky
Wed Feb 28 10:19:33 EST 1996