A 3+1 Computational Scheme for Dynamic Spherically Symmetric Black Hole Spacetimes -- I: Initial Data

TL;DR

This paper introduces a flexible numerical scheme for constructing initial data in dynamic spherically symmetric black hole spacetimes, allowing for arbitrary slicing and demonstrating high-precision results in scalar field simulations.

Contribution

The authors develop and implement a novel 3+1 computational algorithm that handles nonzero, spatially variable extrinsic curvature in black hole initial data, improving flexibility over existing methods.

Findings

Achieved constraint violations below 10^{-8} in numerical simulations.

Demonstrated high accuracy with 4th order finite differencing near perturbations.

Analyzed interpolation errors, showing they are not smooth even for smooth functions.

Abstract

When using the black hole exclusion (horizon boundary condition) technique, $K$ is usually nonzero and spatially variable, so none of the special cases of York's conformal-decomposition algorithm 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 perturb a known black hole slice via some Ansatz, apply the York decomposition (using another Ansatz 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 (eg) restore 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. We have implemented this algorithm for the spherically symmetric scalar field system. We present numerical results for a number of Eddington- Finkelstein--like initial data slices containing black holes surrounded by scalar field shells. Using 4th order finite differencing with resolutions of $\Delta r/r \approx 0.02$ (0.01) near the (Gaussian) perturbations, the numerically computed energy and momentum constraints for the final slices are $\ltsim 10^{-8}$ ($10^{-9}$) and $\ltsim 10^{-9}$ ($10^{-10}$) in magnitude. Finally, we briefly discuss the errors incurred when interpolating data from one grid to another, as in numerical coordinate transformation or horizon finding. We show that for the usual moving-local-interpolation schemes, even for smooth functions the interpolation error is not smooth.