Alternatively, assuming the event horizon in each slice to be a Strahlkörper in the manner of
Section 2.2, we can define a suitable level-set function by Equation (7
). Substituting this definition
into Equation (10
) then gives an explicit evolution equation for the horizon shape function,
Surfaces near the event horizon share the same “attraction” property discussed in Section 5.2 for
geodesics near the event horizon. Thus by integrating either surface representation (10) or (11
)
backwards in time, we can refine an initial guess into a very accurate approximation to the event
horizon.
In contrast to the null geodesic equation (8), neither Equation (10
) nor Equation (11
) contain any derivatives of the
4-metric (or equivalently the 3 + 1 geometry variables). This makes it much easier to integrate these latter equations
accurately11.
This formulation of the event-horizon finding problem also completely eliminates the tangential-drifting
problem discussed in Section 5.2, since the level-set function only parameterizes motion normal to the
surface.
For a practical algorithm, it is useful to integrate a pair of trial null surfaces backwards: an “inner-bound” one which starts (and thus always remains) inside the event horizon and an “outer-bound” one which starts (and thus always remains) outside the event horizon. If the final slice contains an apparent horizon then any 2-surface inside this can serve as our inner-bound surface. However, choosing an outer-bound surface is more difficult.
It is this desire for a reliable outer bound on the event horizon position that motivates our requirement (Section 4) for the final slice (or at least its strong-field region) to be approximately stationary: In the absence of time-dependent equations of state or external perturbations entering the system, this requirement ensures that, for example, any surface substantially outside the apparent horizon can serve as an outer-bound surface.
Assuming we have an inner- and an outer-bound surface on the final slice, the spacing between these two
surfaces after some period of backwards integration then gives an error bound for the computed event
horizon position. Equivalently, a necessary (and, if there are no other numerical problems, sufficient)
condition for the event-horizon finding algorithm to be accurate is that the backwards integration must have
proceeded far enough for the spacing between the two trial surfaces to be “small”. For a reasonable
definition of “small”, this typically takes at least of backwards integration, with
or
more providing much higher accuracy.
In some cases it is difficult to obtain a long enough span of numerical data for this backwards integration. For example, in some simulations of binary black hole collisions, the evolution becomes unstable and crashes soon after a common apparent horizon forms. This means that we cannot compute an accurate event horizon for the most interesting region of the spacetime, that which is close to the black-hole merger. There is no good solution to this problem except for the obvious one of developing a stable (or less-unstable) simulation that can be continued for a longer time.
The initial implementations of the “integrate null surface backwards” algorithm by Anninos et al. [7], Libson et al. [103
],
and Walker [162
] were based on the explicit Strahlkörper surface integration formula (11
), further restricted to
axisymmetry12.
For a single black hole the coordinate choice is straightforward. For the two-black-hole case, the authors
used topologically cylindrical coordinates , where the two black holes collide along the
axisymmetry (
) axis. Based on the symmetry of the problem, they then assumed that the event horizon
shape could be written in the form
This spacetime’s event horizon has the now-classic “pair of pants” shape, with a non-differentiable cusp
along the “inseam” (the axis
) where new generators join the surface. The authors tried two
ways of treating this cusp numerically:
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Caveny et al. [44, 46
] implemented the “integrate null surfaces backwards” algorithm for fully generic
numerically-computed spacetimes using the explicit Strahlkörper surface integration formula (11
). To
handle moving black holes, they recentered each black hole’s Strahlkörper parameterization (4
) on the
black hole’s coordinate centroid at each time step.
For single-black-hole test cases (Kerr spacetime in various coordinates), they report typical accuracies of
a few percent in determining the event horizon position and area. For binary-black-hole test cases
(Kastor–Traschen extremal-charge black hole coalescence with a cosmological constant), they detect black
hole coalescence (which appears as a bifurcation in the backwards time integration) by the “necking off” of
the surface. Figure 6 shows an example of their results.
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Caveny et al. [44, 45
] and Diener [60
] (independently) implemented the “integrate null surfaces
backwards” algorithm for fully generic numerically-computed spacetimes, using the level-set
function integration formula (10
). Here the level-set function
is initialized on the final slice
of the evolution and evolved backwards in time using Equation (10
) on (conceptually) the
entire numerical grid. (In practice, only a smaller box containing the event horizon need be
evolved.)
This surface parameterization has the advantage that the event-horizon topology and (non-)smoothness
are completely unconstrained, allowing the numerical study of configurations such as toroidal event
horizons (discussed in Section 4). It is also convenient that the level-set function is defined on
the same numerical grid as the spacetime geometry, so that no interpolation is needed for the
evolution.
The major problem with this algorithm is that during the backwards evolution,
spatial gradients in tend to steepen into a jump discontinuity at the event
horizon14,
eventually causing numerical difficulty.
Caveny et al. [44, 45] deal with this problem by adding an artificial viscosity (i.e. diffusion) term to
the level-set function evolution equation, smoothing out the jump discontinuity in . That is, instead of
Equation (10
), they actually evolve
via
Alternatively, Diener [60] developed a technique of periodically reinitializing the level-set function to
approximately the signed distance from the event horizon. To do this, he periodically evolves
In various tests on analytical data, Diener [60] found this event-horizon finder, EHFinder, to be robust
and highly accurate, typically locating the event horizon to much less than 1% of the 3-dimensional grid
spacing. As an example of results obtained with EHFinder, Figure 7
shows two views of the
numerically-computed event horizon for a spiraling binary black hole collision. As another example,
Figure 8
shows the numerically-computed event and apparent horizons in the collapse of a rapidly rotating
neutron star to a Kerr black hole. (The apparent horizons were computed using the AHFinderDirect
code described in Section 8.5.7.)
EHFinder is implemented as a freely available module (“thorn”) in the Cactus computational toolkit
(see Table 2). It originally worked only with the PUGH unigrid driver, but work is ongoing [61] to enhance
it to work with the
Carpet mesh-refinement driver [134, 131].
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