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8.6 Horizon pretracking

Schnetter et al. [133Jump To The Next Citation Point135Jump To The Next Citation Point] introduced the important concept of “horizon pretracking”. They focus on the case where we want to find a common apparent horizon as soon as it appears in a binary black-hole (or neutron-star) simulation. While a global (flow) algorithm (Section 8.7) could be used to find this common apparent horizon, these algorithms tend to be very slow. They observe that the use of a local (elliptic-PDE) algorithm for this purpose is somewhat problematic:

The common [apparent] horizon […] appears instantaneously at some late time and without a previous good guess for its location. In practice, an estimate of the surface location and shape can be put in by hand. The quality of this guess will determine the rate of convergence of the finder and, more seriously, also determines whether a horizon is found at all. Gauge effects in the strong field region can induce distortions that have a large influence on the shape of the common horizon, making them difficult to predict, particularly after a long evolution using dynamical coordinate conditions. As such, it can be a matter of some expensive trial and error to find the common apparent horizon at the earliest possible time. Further, if a common apparent horizon is not found, it is not clear whether this is because there is none, or whether there exists one which has only been missed due to unsuitable initial guesses – for a fast apparent horizon finder, a good initial guess is crucial.

Pretracking tries (usually successfully) to eliminate these difficulties by determining – before it appears – approximately where (in space) and when (in time) the common apparent horizon will appear.

8.6.1 Constant-expansion surfaces

The basic idea of horizon pretracking is to consider surfaces of constant expansion (“CE surfaces”), i.e. smooth closed orientable 2-surfaces in a slice satisfying the condition

Θ = E, (33 )
where the expansion E is a specified real number. Each marginally outer trapped surface (including the apparent horizon) is thus a CE surface with expansion E = 0; more generally Equation(33View Equation) defines a 1-parameter family of 2-surfaces in the slice. As discussed by Schnetter et al. [133Jump To The Next Citation Point135Jump To The Next Citation Point], for asymptotically flat slices containing a compact strong-field region, some of the E > 0 members of this family typically foliate the weak-field region.

In the binary-coalescence context, for each t = constant slice we define E∗ to be the smallest E ≥ 0 for which a CE surface (containing both strong-field regions) exists with expansion E. If E ∗ = 0 this “minimum-expansion CE surface” is the common apparent horizon, while if E ∗ > 0 this surface is an approximation to where the common apparent horizon will appear. We expect the minimum-expansion CE surface to change continuously during the evolution and its expansion E ∗ to decrease towards 0. Essentially, horizon pretracking follows the time evolution of the minimum-expansion CE surface and uses it as an initial guess for (searching for) the common apparent horizon.

8.6.2 Generalized constant-expansion surfaces

Schnetter [133Jump To The Next Citation Point] implemented an early form of horizon pretracking, which followed the evolution of the minimum-expansion constant-expansion surface, and found that it worked well for simple test problems. However, Schnetter et al. [135Jump To The Next Citation Point] found that for more realistic binary-black-hole coalescence systems the algorithm needs to be extended:

View Image

Figure 12: This figure shows the expansion Θ (left scale), and the “generalized expansions” r Θ (left scale) and r2Θ (right scale), for various r = constant surfaces in an Eddington–Finkelstein slice of Schwarzschild spacetime. Notice that all three functions have zeros at the horizon r = 2m, and that while Θ has a maximum at r ≈ 4.4 m, both rΘ and r2Θ increase monotonically with r.

Schnetter et al. [135Jump To The Next Citation Point] discuss these problems in more detail, arguing that to solve them, the expansion Θ should be generalized to a “shape function” H given by one of

H1 = Θ, Hr = hΘ2, (34 ) Hr2 = h Θ,
CE surfaces are then generalized to surfaces satisfying
H = E (35 )
for some specified E ≥ 0.

Note that unlike H1, both Hr and Hr2 are typically monotonic with radius. Neither Hr nor H 2 r are 3-covariantly defined, but they both still have the property that E = 0 in Equation (35View Equation) implies the surface is a MOTS, and in practice they work better for horizon pretracking.

8.6.3 Goal functions

To define the single “smallest” surface at each time, Schnetter et al. [135Jump To The Next Citation Point] introduce a second generalization, that of a “goal function” G, which maps surfaces to real numbers. The pretracking search then attempts, on each time slice, to find the surface (shape) satisfying H = E with the minimum value of G. They experimented with several different goal functions,

-- GH = H,-- GrH = h-H, (36 ) Gr = h,
where in each case the overbar ( -- ) denotes an average over the surface51.

8.6.4 The pretracking search

Schnetter’s [133] original implementation of horizon pretracking (which followed the evolution of the minimum-expansion CE surface) used a binary search on the desired expansion E. Because E appears only on the right hand side of the generalized CE condition (35View Equation), it is trivial to modify any apparent horizon finder to search for a surface of specified expansion E. (Schnetter used his TGRapparentHorizon2D elliptic-PDE apparent horizon finder described in Section 8.5.7 for this.) A binary search on E can then be used to find the minimum value E ∗.52

Implementing a horizon pretracking search on any of the generalized goal functions (36View Equation) is conceptually similar but somewhat more involved: As described by Schnetter et al. [135Jump To The Next Citation Point] for the case of an elliptic-PDE apparent horizon finder53, we first write the equation defining a desired pretracking surface as

H − H-+ G − p = 0, (37 )
where p is the desired value of the goal function G. Since H is the only term in Equation (37View Equation) which varies over the surface, it must be constant for the equation to be satisfied. In this case -- H −H vanishes, so the equation just gives G = p, as desired.

Because -- H depends on H at all surface points, directly finite differencing Equation (37View Equation) would give a non-sparse Jacobian matrix, which would greatly slow the linear-solver phase of the elliptic-PDE apparent horizon finder (Section 8.5.5). Schnetter et al. [135Jump To The Next Citation Point, Section III.B] show how this problem can be solved by introducing a single extra unknown into the discrete system. This gives a Jacobian which has a single non-sparse row and column, but is otherwise sparse, so the linear equations can still be solved efficiently.

When doing the pretracking search, the cost of a single binary-search iteration is approximately the same as that of finding an apparent horizon. Schnetter et al. [135Jump To The Next Citation Point, Figure 5] report that their pretracking implementation (a modified version of Thornburg’s AHFinderDirect [156Jump To The Next Citation Point] elliptic-PDE apparent horizon finder described in Section 8.5.7) typically takes on the order of 5 to 10 binary-search iterations54. The cost of pretracking is thus on the order of 5 to 10 times that of finding a single apparent horizon. This is substantial, but not prohibitive, particularly if the pretracking algorithm is not run at every time step.

8.6.5 Sample results

As an example of the results obtained from horizon pretracking, Figure 13View Image shows the expansion Θ for various pretracking surfaces (i.e. various choices for the shape function H in a head-on binary black hole collision). Notice how all three of the shape functions (34View Equation) result in pretracking surfaces whose expansions converge smoothly to zero just when the apparent horizon appears (at about t = 1.1).

View Image

Figure 13: This figure shows the expansion Θ for various pretracking surfaces, i.e. for various choices for the shape function H, in a head-on binary black hole collision. Notice how the three shape functions (34View Equation) (here labelled Θ, rΘ, and 2 r Θ) result in pretracking surfaces whose expansions converge smoothly to zero just when the apparent horizon appears (at about t = 1.1). Notice also that these three expansions have all converged to each other somewhat before the common apparent horizon appears. Figure reprinted with permission from [135Jump To The Next Citation Point]. © 2005 by the American Physical Society.

As a further example, Figure 14View Image shows the pretracking surfaces (more precisely, their cross sections projected into the black holes’ orbital plane) at various times in a spiraling binary black hole collision.

View Image

Figure 14: This figure shows the pretracking surfaces at various times in a spiraling binary black hole collision, projected into the black holes’ orbital plane. (The apparent slow drift of the black holes in a clockwise direction is an artifact of the corotating coordinate system; the black holes are actually orbiting much faster, in a counterclockwise direction.) Notice how, even well before the common apparent horizon first appears (t = 16.44 mADM, bottom right plot), the rΘ pretracking surface is already a reasonable approximation to the eventual common apparent horizon’s shape. Figure reprinted with permission from [135Jump To The Next Citation Point]. © 2005 by the American Physical Society.

8.6.6 Summary of horizon pretracking

Pretracking is a very valuable addition to the horizon finding repertoire: It essentially gives a local algorithm (in this case, an elliptic-PDE algorithm) most of the robustness of a global algorithm in terms of finding a common apparent horizon as soon as it appears. It is implemented as a higher-level algorithm which uses a slightly-modified elliptic-PDE apparent horizon finding algorithm as a “subroutine”.

The one significant disadvantage of pretracking is its cost: Each pretracking search typically takes 5 to 10 times as long as finding an apparent horizon. Further research to reduce the cost of pretracking would be useful.

Schnetter et al.’s pretracking implementation [135] is implemented as a set of modifications to Thornburg’s AHFinderDirect [156Jump To The Next Citation Point] apparent horizon finder. Like the original AHFinderDirect, the modified version is a freely available “thorn” in the External LinkCactus toolkit (see Table 2).


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