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5.1 Integrating null geodesics forwards in time

The first generation of event-horizon finders were based directly on Hawking’s original definition of an event horizon: an event 𝒫 is within the black hole region of spacetime if and only if there is no future-pointing “escape route” null geodesic from 𝒫 to 𝒥 +; the event horizon is the boundary of the black hole region.

That is, as described by Hughes et al. [88Jump To The Next Citation Point], we numerically integrate the null geodesic equation

d2xa dxbdxc -----+ Γ abc------= 0 (8 ) dλ2 dλ dλ
(where λ is an affine parameter) forwards in time from a set of starting events and check which events have “escaping” geodesics. For analytical or semi-analytical studies like that of Bishop [31], this is an excellent algorithm.

For numerical work it is straightforward to rewrite the null geodesic equation (8View Equation) as a coupled system of two first-order equations, giving the time evolution of photon positions and 3-momenta in terms of the 3 + 1 geometry variables α, βi, gij, and their spatial derivatives. These can then be time-integrated by standard numerical algorithms8. However, in practice several factors complicate this algorithm.

We typically only know the 3 + 1 geometry variables on a discrete lattice of spacetime grid points, and we only know the 3 + 1 geometry variables themselves, not their spatial derivatives. Therefore we must numerically differentiate the field variables, then interpolate the field variables and their spacetime derivatives to each integration point along each null geodesic. This is straightforward to implement9, but the numerical differentiation tends to amplify any numerical noise that may be present in the field variables.

Another complicating factor is that the numerical computations generally only span a finite region of spacetime, so it is not entirely obvious whether or not a given geodesic will eventually reach 𝒥 +. However, if the final numerically-generated slice contains an apparent horizon, we can use this as an approximation: Any geodesic which is inside this apparent horizon will definitely not reach + 𝒥, while any other geodesic may be assumed to eventually reach + 𝒥 if its momentum is directed away from the apparent horizon. If the final slice (or at least its strong-field region) is approximately stationary, the error from this approximation should be small. I discuss this stationarity assumption further in Section 5.3.1.

5.1.1 Spherically-symmetric spacetimes

In spherical symmetry this algorithm works well and has been used by a number of researchers. For example, Shapiro and Teukolsky [141Jump To The Next Citation Point142Jump To The Next Citation Point143Jump To The Next Citation Point144Jump To The Next Citation Point] used it to study event horizons in a variety of dynamical evolutions of spherically symmetric collapse systems. Figure 2View Image shows an example of the event and apparent horizons in one of these simulations.

View Image

Figure 2: This figure shows part of a simulation of the spherically symmetric collapse of a model stellar core (a 5 Γ = 3 polytrope) to a black hole. The event horizon (shown by the dashed line) was computed using the “integrate null geodesics forwards” algorithm described in Section 5.1; solid lines show outgoing null geodesics. The apparent horizon (the boundary of the trapped region, shown shaded) was computed using the zero-finding algorithm discussed in Section 8.1. The dotted lines show the world lines of Lagrangian matter tracers and are labeled by the fraction of baryons interior to them. Figure reprinted with permission from [142Jump To The Next Citation Point]. © 1980 by the American Astronomical Society.

5.1.2 Non-spherically-symmetric spacetimes

In a non-spherically-symmetric spacetime, several factors make this algorithm very inefficient:

Because of these limitations, for non-spherically-symmetric spacetimes the “integrate null geodesics forwards” algorithm has generally been supplanted by the more efficient algorithms I describe in the following.


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