In this section we show how the methods detailed above can be applied
to pure, spherical Schwarzschild black hole. We consider both the
analytic Schwarzschild solution and a maximally sliced Schwarzschild
black hole evolved with the 2D, axisymmetric black hole code described
in Refs. [23, 14]. We use this important test
bed case to show the accuracy to which one can determine the location
of the horizon in a numerically evolved spacetime. Although the
numerical spacetime is geometrically the static Schwarzschild
spacetime, it is evolved with a maximal slicing condition which makes
the metric functions change in time. As discussed in
Ref. [20], such a time dependence makes even the
Schwarzschild spacetime quite difficult to evolve numerically for long
periods of time (beyond about t=100M with reasonable grid
parameters). As the coordinates fall in towards the hole the horizon
moves out in coordinate space, the lapse collapses, and large
gradients develop in the metric function near the horizon. (For more
details on these problems, see Refs. [20, 14].)
The advantage of using this as our first test bed case is that, on the
one hand, the numerical spacetime constructed with this code has many
of the properties and difficulties of a general numerically
constructed black hole spacetime. On the other hand, the spacetime is
really a Schwarzschild spacetime for which we know where the EH should
be for all time. In particular, in the numerical case the apparent
horizon (AH) and EH coincide We have accurate AH
finders [24] that can locate the AH, and thus in this
case the EH; on any given single slice of the spacetime we know both
horizons without needing to know the future or past of that slice. In
the analytic spacetime, the horizon is at r=2M. This provides us
with important accuracy checks on our methods of locating the EH
throughout the evolution.
In Figs. 2(a)- 2(d) we show results
for a spherical black hole spacetime. For the numerical spacetime in
Figs. 2(a), 2(b) and
2(d) we apply our horizon finder to the data obtained
in the evolution assuming neither spherical symmetry nor the fact
that the spacetime geometry is really Schwarzschild. At the final
time slice, the horizon-containing region is determined by examining
the lapse function and the radial metric function. For a spacetime
evolved with maximal slicing, the event horizon resides in a region
with a partially collapsed lapse function. In
Fig. 2(a), we show the lapse function 
at the
final time slice, t=100M. We take the horizon-containing region to
extend from 
to 
, with (o) labeling the
outer edge and (i) the inner edge of it. In
Fig. 2(b), the radial coordinate r of the two surfaces
(o) and (i) traced backwards in time is shown. At
t=100M, the two surfaces are separated in the radial coordinate by
3.4M. By t=85M, the two lines are separated by just one grid
zone, corresponding to a difference in r of 0.35M. By t=70M,
the two lines are no longer distinguishable, with a separation down to
1/10th of a grid zone, a difference in r of 0.03M. The
separation exponentially decreases down to 
grid zones
at t=0M. This rapid shrinking of the horizon-containing region is a
direct consequence of the divergence of null geodesics forward in time
shown in Fig. 1. We conclude that if the aim is to
locate the horizon to one grid zone accuracy, we have succeeded in
doing so for the times t=0M to t=80M. We emphasize that no
information about the apparent horizon is used in the process.
For the purpose of comparison, in Fig. 2(b) we have also
shown the trajectory of surfaces extremely far outside and far inside the
horizon-containing region. These surfaces are shown as dashed lines.
We see that the outer one converges quickly to the other test
surfaces, while the inner one is initially trapped in a region of
collapsed lapse. At t=40M, all the surfaces are practically
indistinguishable.
In Fig. 2(c), we show that the convergence is in fact
exponential. Here we plot, in the logarithmic scale, the maximum proper
distance of a photon on the null surface (o) from the horizon,
as a function of the Killing time t/M. The result is given by the
thick dashed line. We get a straight line in the logarithmic plot, with a
slope approaching 0.25, as shown in the inset. This is, as expected, the
analytic value, as can be easily deduced in the following.
Consider a null trajectory in the Schwarzschild geometry near the
horizon. The equations of motion are given by [17]
where 
is an affine parameter and b is an integration
constant. To leading order in 
,
Eqs. (8) and (9) can easily be integrated to give
for an outgoing photon. Its maximum distance from the horizon is
given by
showing that the exponent is 1/4 in units of t/M. The trajectory
given by Eq. (10) is plotted in Fig. 2(c) as
the dotted line. It is just barely distinguishable from the solid
line labeled as ``analytic'' in Fig. 2(c), which is
the trajectory obtained by integrating the full null geodesic equation
Eqs. (7), (8) and (9), without assuming r-2M to be
small. In turn the solid line lies right on top of the thick dashed
line representing the numerical backwards surface method in the
analytic spacetime, giving full support for the accuracy of the
method, at least in this simple case.
In Fig. 2(d), we show the evolution of the coordinate
locations of these surfaces in the first quadrant. The surfaces marked
(i), (o) and AH are the same surfaces as shown in
Fig. 2b. Here, we have evolved an additional,
nonspherical, surface. The location of this surface is given at
t=100M by the formula
with 
chosen to be the radial position of the apparent
horizon, with w=4 and A=0.2. We evolve this surface to
demonstrate that our initial trial surfaces need not have the same
angular dependence as the EH (in this case, spherical). In a general
dynamical black hole spacetime, it will not be possible to pick trial
surfaces having the same coordinate or geometrical angular dependence
as the EH to be traced out. Such trial surfaces are not necessary,
though. In the case shown in Fig. 2(d), where the trial
surface is quite non spherical, with part of the surface inside and
part outside of the EH, we see that the trial surface quickly
converges when traced backwards in time. All of the surfaces are very
close and almost completely spherical by t=70M. By t=50M, all the
surfaces are within 1/10th of a grid zone. We note, however, that a
sufficiently nonspherical surface may itself develop caustics,
particularly if it is initially far from the true EH. In
Fig. 2(e) we show the evolution (at times t=98.5M,
t=98.4M, t=98.3M, and t=98.2M from top to bottom, with the line
marked AH labeling the position of the AH and thus the true EH at
t=98.5M) of a highly distorted surface with the angular dependence
of Eq. (12) increased to w=16, the amplitude decreased to
A=0.1, and the center of the perturbation moved away from the
apparent horizon. We find that the surface method fails, with
numerical noise developing as the surface tries to cross itself. This
crossing in itself is not fatal to the surface method, but the
particular parametrization (5) of the surface cannot
describe this crossing. As we see below, the self crossing of the
surface can be handled with a proper parametrization, which is needed
when true caustics develop, as in the collision of two black
holes. We stress that when the black hole has returned to
quasi-stationarity at late times, one does not expect the EH to have
such a rapidly varying angular dependence and one would not pick such
surfaces as the outer or inner boundaries of the horizon-containing
region. We study such an extreme, contrived example only to explore
the limits of our methods.
We note that at late times the data representing the spacetime itself
as obtained in Refs. [20, 14] become inaccurate due
to the large spikes developing in the metric functions near the
horizon [20]. As these spikes become ever steeper during
the evolution, they become less and less well resolved, and therefore they
are not accurately modeled on the numerical grid. This lack of
accuracy in the spacetime itself is reflected in the calculation of
the area of the apparent horizon at late times, which increases rather
than remaining constant. Remarkably, this lack of accuracy at
late times, when the EH finding algorithm is started, does not
cause any difficulty in finding the EH at earlier times, as seen in
the figures. Therefore, not only are our algorithms able to find
accurately the true EH even with a poor initial guess for its
location, they are also insensitive to inaccuracies in the
spacetime data that inevitably occur at late times.
Finally, in Fig. 3, we show the tangential drifting
that can occur with the backward photon method, as discussed in
Sec. ii. Figure 3 also serves as an
illustration to the other comment we made above concerning the
``attractiveness'' of the EH to backward integrated photons, namely,
the attraction is only in the global sense. In this example the
tangential drifting is due to the choice of the initial direction of
integration. Two photons are traced backward (shown as dotted lines)
beginning at the exact location of the EH at t=100M, but with a 3%
error in the starting direction. That is, instead of being normal to
the EH ( 
, in obvious notation), we use 
. In Fig. 3, the
trajectories of these photons are shown with the corresponding times
marked. The radial coordinate are normalized by the position of the
EH, so that the EH is at 1 on the vertical (radial coordinate
) axis. We see that with the 3% error in the starting
directions of the photons, the photons move out of the EH when traced
backward in time. If these photons were taken as horizon generators,
this would introduce a small error (note the scale of the axis)
in the location of the ``EH'' for a period of time, as the photons are
gradually ``attracted'' back to the correct radial location after some
integration. However, the error in the tangential direction is
substantial, as we can see in Fig. 3, where the
horizontal axis is given in terms of 
in radians. In
particular, the two photons cross each other, creating an artificial
``caustic'' at t=98.0M on their way out from the EH. Although they
return to the correct radial EH location eventually, their
values change dramatically, making the trajectories very different
from those of the true horizon generators.