Spherically Symmetric Black Hole Spacetimes – II: Time Evolution

This is the second in a series of papers describing a 3+1 computational scheme for the numerical simulation of dynamic black hole spacetimes. In this paper we focus on the problem of numerically time-evolving a given black-hole– containing initial data slice in spherical symmetry. We avoid singularities via the “black-hole exclusion” or “horizon boundary condition” technique, where the slices meet the black hole’s singularity, but on each slice a spatial neighbourhood of the singularity is excluded from the domain of the numerical computations. We first discuss some of the key design choices which arise with the black hole exclusion technique: Where should the computational domain’s inner boundary be placed? Should a free or a constrained evolution scheme be used? How should the coordinates be chosen? We then give a detailed description of our numerical evolution scheme. We focus on a standard test problem, the time evolution of an asymptotically flat spherically symmetric spacetime containing a black hole surrounded by a massless scalar field, but our methods should also extend to more general black hole spacetimes. We assume that the black hole is already present on the initial slice. We use a free evolution, with Eddington-Finkelstein–like coordinates and the inner boundary placed at a fixed (areal) coordinate radius well inside the horizon. Our numerical scheme is based on the method of lines (MOL), where the spacetime PDEs are first discretized in space only, yielding a system of coupled ODEs for the time evolution of the field variables along the spatial-grid-point world lines. These ODEs are then time-integrated by standard finite difference methods. In contrast to the more common “space and time together” finite differencing schemes, we find that MOL schemes are considerably simpler to E-mail 1 implement, and make it much easier to construct stable higher order differencing schemes. Our MOL scheme uses 4th order finite differencing in space and time, with 5 and 6 point spatial molecules and a 4th order Runge-Kutta time integrator. The spatial grid is smoothly nonuniform, but not adaptive. We present several sample numerical evolutions of black holes accreting scalar field shells, showing that this scheme is stable, very accurate, and can evolve “forever”. As an example of the typical accuracy of our scheme, for a grid resolution of ∆r/r ≈ 3% near the horizon, the errors in gij (Kij) components at t = 100m in a dynamic evolution are ∼ 10−5 (3× 10−7) in most of the grid, while the energy constraint is preserved to ≈ 3× 10−5 of its individual partial-derivative terms. When the black hole accretes a relatively thin and massive scalar shell, for a short time 3 distinct apparent horizons are present, and the apparent horizons move at highly superluminal speeds; this has important implications for how a black-hole–exclusion computational scheme should handle the inner boundary. Our other results for the scalar field phenomenology are generally consistent with past work, except that we see a very different late-time decay of the field near the horizon. We suspect this is a numerical artifact. 04.25.Dm, 02.70.Bf, 02.60.Cb, 02.60.Lj Typeset using REVTEX 2