A 3+1 computational scheme for dynamic spherically symmetric black hole spacetimes: Initial data - art. no. 104007

This is the first in a series of papers describing a 3 + 1 computational sc heme for the numerical simulation of dynamic spherically symmetric black ho le spacetimes. In this paper we discuss the construction of dynamic black h ole initial data slices using York's conformal-decomposition algorithm in i ts most general form, where no restrictions are placed on K (the trace of t he extrinsic curvature) and hence the full 4-vector nonlinear York equation s must be solved numerically. To construct an initial data slice, we begin with a known black hole slice (e.g, a slice of Schwarzschild or Kerr spacet ime), perturb this via some Ansatz (e.g, the addition of a suitable Gaussia n to one of the coordinate components of the 3-metric, extrinsic curvature, or matter field variables), apply the York decomposition (using a further Ansatz for the inner boundary conditions) to project the perturbed field va riables back into the constraint hypersurface, and finally optionally apply a numerical 3-coordinate transformation to restore any desired form for th e spatial coordinates (e.g, an areal radial coordinate). In comparison to o ther initial data algorithms, the key advantage of this algorithm is its fl exibility: K is unrestricted, allowing the use of whatever slicing is most suitable for (say) a time evolution. This algorithm also offers great flexi bility in controlling the physical content of the initial data, while placi ng no restrictions on the type of matter fields; or on spacetime's symmetri es or lack thereof. We have implemented this algorithm for the spherically symmetric scalar field system. We present numerical results for a number of asymptotically flat Eddington-Finkelstein-like initial data slices contain ing black holes surrounded by scalar field shells, the latter with masses r anging from as low as 0.17 to as high as 17 times the black hole mass. In a ll cases we find that the computed slices are very accurate: Using 4th orde r finite differencing on smoothly nonuniform grids with resolutions of Delt a r/r approximate to 0.02(0.01) near the perturbations, and Gaussian pertur bations yielding scalar field shells with relative width sigma/r approximat e to 1/6 and with about 2/3 the black hole mass, the numerically computed e nergy and momentum constraints for the final slices are less than or simila r to 10(-8)(10(-9)) and less than or similar to 10(-9)(10(-10)) in magnitud e respectively. For similarly perturbed numerically computed "warped" vacuu m slices, the Misner-Sharp mass function is independent of position, and th e 4-Riemann quadratic curvature invariant RabcdRabcd is equal to its (posit ion-dependent) Schwarzschild-spacetime value, to within relative errors of less than or similar to 10(-5)(5 X 10(-7)) and less than or similar to 3 X 10(-4)(5 X 10(-5)) respectively. All these errors show the expected O((Delt a r)(4)) scaling with grid resolution at. Finally, we briefly discuss the e rrors incurred when interpolating data from one grid to another, as in nume rical coordinate transformation or horizon finding. We show that for the us ual moving-local-interpolation schemes, even for smooth functions the inter polation error is not smooth. If interpolated data is to be differentiated, we argue that either the interpolation order should be raised to compensat e for the non-smoothness, or explicitly smoothness-preserving interpolation schemes should be used. [S0556-2821(98)06218-3].