Causal differencing of flux-conservative equations applied to black hole spacetimes

The apparent horizon boundary condition (AHBC) appears to be one of the fundamental techniques required for evolving black hole spacetimes using numerical techniques. In this paper, we present work on an AHBC in the context of the recently formulated Bona-Massó (BM) hyperbolic system for the Einstein Evolution equations, however in doing so, we present a technique which is generally causally correct for any first order flux-conservative set of PDEs. The idea of the AHBC was credited to Unruh by Thornburg [?]. The fundamental idea is that, rather than avoiding a singularity by taking slices which delay the infall of observers inside the horizon (eventually requiring an infinite force), one could take regular slices everywhere outside the horizon and place some stable boundary condition inside the apparent horizon, excising a portion of the numerical grid. Mathematically this is consistent because the apparent horizon is known to be inside the event horizon, and the interior of the event horizon is, by definition, causally disconnected from its exterior. The AHBC in numerical relativity was first shown to work by Seidel and Suen in Ref. [?]. The AHBC proposal by Seidel and Suen has two components. The first is to choose a shift condition which, after some evolution, locks the coordinates by freezing the position of the horizon and keeping radial distances between points constant. In spherical symmetry, this uniquely determines the shift everywhere. Additionally, to handle large shift terms, Seidel and Suen propose re-writing the finite difference representation of the ADM equations to obey the causal structure of the spacetime. This causal differencing is an important aspect of much AHBC work to date and is generally credited to Seidel and Suen (“causal differencing”) or Alcubierre and Schutz (“causal reconnection”) [?]. The idea proposed by Seidel and Suen is as follows. In the presence of a shift, the Einstein equations have additional terms in the evolution equations (due to the action of Lβ on γij and Kij). However, there is another coordinate system in which the shift is zero. Finding the coordinate system, in general, involves integrating a transformation function in time. The Seidel and Suen prescription is to finite difference in the transformed zero shift co-ordinates and then re-transform the finite difference representation to coordinates with a shift. Using these two techniques, Seidel and Suen proceed to demonstrate that they can accurately evolve Schwarzschild black holes and Schwarzschild black holes with infalling scalar fields for long periods of time. The followup to Seidel and Suen, a paper by Anninos, Daues, Massó, Seidel and Suen, [?] gave details of the Seidel and Suen causal differencing scheme and presented several shift choices. These shifts allow for coordinate regularity in the entire spacetime and provide horizon locking. Moreover, several of these shifts are extensible to full three dimensional cases, most notably the minimal distortion shift [?]. Using causal differencing and horizon locking shift conditions, Anninos et al. are able to evolve Schwarzschild black holes in spherical symmetry for 1000M with very small errors in the measure of the mass of the horizon, compared to 100% mass errors present in simulations without an AHBC around t = 100M . The first application of an AHBC to a hyperbolic scheme was that of Scheel et al. [?]. This scheme used a hyperbolic formulation due to York on a Schwarzschild black hole. The essence of the causal difference scheme was to decompose the time derivative into time evolution along the normal direction and spatial transport due to the shift. That is, they would evolve along the normal direction to some point no longer on their numerical grid, and then re-construct their numerical grid by interpolation. The Scheel et al. approach is roughly equivalent to our advective (non flux-conservative) causal differencer with interpolation after the step, as discussed below. The results of Scheel et al. were initially disappointing, as an instability arose on a short (10−100M) time scale. Very recent work [?] has removed this instability by adding constraints to the evolution equations (in a manner specific to spherical symmetry), and run times exceeding 10000M have been achieved.