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5.2 Integrating null geodesics backwards in time

It is well-known that future-pointing outgoing null geodesics near the event horizon tend to diverge exponentially in time away from the event horizon. Figure 3View Image illustrates this behavior for Schwarzschild spacetime, but the behavior is actually quite generic.
View Image

Figure 3: This figure shows a number of light cones and future-pointing outgoing null geodesics in a neighborhood of the event horizon in Schwarzschild spacetime, plotted in ingoing Eddington–Finkelstein coordinates (t,r). (These coordinates are defined by the conditions that t+ r is an ingoing null coordinate, while r is an areal radial coordinate.) Note that for clarity the horizontal scale is expanded relative to the vertical scale, so the light cones open by more than ±45 ∘. All the geodesics start out close together near the event horizon; they diverge away from each other exponentially in time (here with an e-folding time of 4 m near the horizon). Equivalently, they converge towards each other if integrated backwards in time (downwards on the page).

Anninos et al. [7Jump To The Next Citation Point] and Libson et al. [103Jump To The Next Citation Point] observed that while this instability is a problem for the “integrate null geodesics forwards in time” algorithm (it forces that algorithm to take quite short time steps when integrating the geodesics), we can turn it to our advantage by integrating the geodesics backwards in time: The geodesics will now converge on to the horizon10.

This event-horizon finding algorithm thus integrates a large number of such (future-pointing outgoing) null geodesics backwards in time, starting on the final numerically-generated slice. As the backwards integration proceeds, even geodesics which started far from the event horizon will quickly converge to it. This can be seen, for example, in Figures 2View Image and 3View Image.

Unfortunately, this convergence property holds only for outgoing geodesics. In spherical symmetry the distinction between outgoing and ingoing geodesics is trivial but, as described by Libson et al. [103Jump To The Next Citation Point],

[…] for the general 3D case, when the two tangential directions of the EH are also considered, the situation becomes more complicated. Here normal and tangential are meant in the 3D spatial, not spacetime, sense. Whether or not a trajectory can eventually be “attracted” to the EH, and how long it takes for it to become “attracted,” depends on the photon’s starting direction of motion. We note that even for a photon which is already exactly on the EH at a certain instant, if its velocity at that point has some component tangential to the EH surface as generated by, say, numerical inaccuracy in integration, the photon will move outside of the EH when traced backward in time. For a small tangential velocity, the photon will eventually return to the EH [… but] the position to which it returns will not be the original position.

This kind of tangential drifting is undesirable not just because it introduces inaccuracy in the location of the EH, but more importantly, because it can lead to spurious dynamics of the “EH” thus found. Neighboring generators may cross, leading to numerically artificial caustic points […].

Libson et al. [103Jump To The Next Citation Point] also observed:

Another consequence of the second order nature of the geodesic equation is that not just the positions but also the directions must be specified in starting the backward integration. Neighboring photons must have their starting direction well correlated in order to avoid tangential drifting across one another.

Libson et al. [103Jump To The Next Citation Point] give examples of the numerical difficulties that can result from these difficulties and conclude that this event-horizon finding algorithm

[…] is still quite demanding in finding an accurate history of the EH, although the difficulties are much milder than those arising from the instability of integrating forward in time.

Because of these difficulties, this algorithm has generally been supplanted by the “backwards surface” algorithm I describe next.


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