Initial Data for Dynamic Black Hole Spacetimes in 3 + 1 Numerical Relativity

York’s conformal-decomposition algorithm is widely used for solving the initial data constraints in 3+ 1 numerical relativity, but it’s usually used in one of several special cases, depending on restrictions placed on K (the trace of the extrinsic curvature). However, when treating black hole spacetimes via the black hole exclusion (horizon boundary condition) technique, K is usually nonzero and spatially variable, so none of the special cases apply, and the full 4-vector nonlinear York equations must be solved numerically. We discuss the construction of dynamic black hole initial data slices using this technique: We begin with a known black hole slice (e.g. a slice of Schwarzschild or Kerr spacetime), perturb this via some Ansätz (e.g. the addition of a suitable Gaussian to one of the coordinate components of the 3-metric, extrinsic curvature, or matter field variables), apply the York decomposition (using a further Ansätz for the inner boundary conditions) to project the perturbed field variables back into the constraint hypersurface, and finally optionally apply a numerical 3-coordinate transformation to restore any desired form for the spatial coordinates (e.g. an areal radial coordinate). In comparison to other initial data algorithms, the key advantage of this algorithm is its flexibility: K is unrestricted, allowing the use of whatever slicing is most suitable for (say) a time evolution. This algorithm also offers great flexibility in controlling the physical content of the initial data, while placing no restrictions on the type of matter fields, or on spacetime’s symmetries or lack thereof. We have implemented this algorithm for the spherically symmetric scalar field system. We present numerical results for a number of EddingtonFinkelstein–like initial data slices containing black holes surrounded by scalar field shells, the latter with masses ranging from as low as 0.17 to as high as