Analytical event horizons of merging black holes

But since I wish to make a point, it wouldn't be wise for me to leave
up to the reader to retrieve his own copy of the journal because, if
your desk is like mine, your copy must be buried beneath one of those
stacks of paper. Or, unfortunately enough, it could even have been
filed so ingeniously that you are sure now not to be able to find it.
Let me instead save you the trip to the library; to start with, let me
just borrow from the article the main features of event horizons:
(*i*) The horizon is generated by null rays that continue
indefinitely into the future. (*ii*) The null generators may
either continue indefinitely into the past, or meet other generators
at points that are thereby considered as their starting points. (*
iii*) The cross-sectional area of the horizon increases
monotonically to a constant at late times. We might as well say that
the event horizon is a null hypersurface that does not self-intersect,
which is formed by following each null geodesic in a bundle of finite
area into the past just so far up to the point where it meets another
geodesic of the bundle. In fact, if we have an expanding null
hypersurface of finite area at late times, which generically does
self-intersect in the past, we might as well regard as an event
horizon the piece of the null hypersurface that lies to the future of
the crossovers, and regard the crossovers as the boundary of the
horizon.

In this context the ``seam'' along the inside of the trouser legs is a crossover line where the generators are terminated. The computer simulation [2] of the horizon provided deep insight into the nature of this boundary of the event horizon, distinguishing the caustic points (where neighboring rays meet) from the simple crossover points (where distant rays intersect without focusing).

The news is that *another* pair of pants has recently been
released. It looks pretty much like the original, up to smooth
deformations. My point is, however, that this newest pair of pants is
not the product of numerical integration, but is the embedded picture
of an analytical event horizon. There is now an analytical expression
for the intrinsic metric of the event horizon of merging eternal black
holes.

The new pair of pants was constructed by Luis Lehner, Nigel Bishop,
Roberto Gómez, Béla Szilágyi and Jeff Winicour [3], of the
University of Pittsburgh relativity group, which has traditionally
sustained an interest in null hypersurfaces (tell me about it). The
recipe for making this event horizon calls for all sorts of
ingredients available in the pantry of the characteristic formulation
of the Einstein equations. Surprisingly, perhaps, it does not call for
a spacetime metric. Surprisingly, because one might think that, since
the metric is needed in order to find geodesics, the horizon could
only be known *a posteriory* of finding the spacetime metric.

The key to this remarkable work is to understand that the event
horizon can be used as partial data for constructing the spacetime
metric. From this point of view, the metric will be known *a
posteriori* of finding the horizon! And the horizon is found by
solving only constraint equations, namely, equations interior to the
horizon itself.

More precisely, the horizon is regarded as one of two intersecting
null hypersurfaces that jointly act as the initial surface for
evolution in double null coordinates. In this case, the conformal
metric of the null slice constitutes free data. The authors choose
the conformal structure so that the 3-metric of the horizon is
where *h*_{ij} is the pullback of the
Minkowski metric to a self-intersecting hypersurface which is null
with respect to the Minkowski metric. (An example of such a
self-intersecting hypersurface is the hypersurface traced in four
dimensions by the imploding wavefront of an ellipsoid in 3-space.)
The conformal factor is then determined by the projection
*n*^{a}*n*^{b}*R*_{ab}=0 of the vacuum Einstein equations along the null
generators *n*^{a} of the hypersurface. This is an *ordinary*
second-order differential equation for
that determines the
dependence of
on the affine parameter *u* along one null
geodesic. Apparently, finding the solution is quite simple. The
freedom is huge, but the authors point out that
relates to
the cross-sectional area of the light beam, and thus its asymptotic
behavior is fixed by the condition that the area must be finite at
late times .
Furthermore, the behavior of the area
element at the boundary of the horizon is determined by the property
of the boundary of containing either caustic points or plain
crossovers, which is also used in restricting the behavior of
.
The requirement that the Weyl curvature must be regular
provides further tips for the integration. The intrinsic geometry
of the horizon is thus found explicitly in terms of two
angular coordinates
labeling the light rays, and the
affine parameter *u*, acting as a time.

It is rather instructive to see how the figure arises. The pair of
pants is constructed by stacking up 3-dimensional Euclidean embeddings
of 2-dimensional surfaces obtained by slicing the horizon with
constant-*u* hypersurfaces. Actually, the figure corresponds to a
case of symmetry of revolution, so that one dimension can be ignored,
but this is exactly as in the case of the ``computational'' pair of
trousers of the *Science* article. Also, strictly speaking, the
calculation represents the fission of two white holes, but time
reversion allows for its interpretation in terms of the merger of two
black holes. At no time does the conformal geometry used as data
exhibit more than one hole. However, the horizon obtained by
integrating the single Einstein equation does have two holes at early
affine times, and just one hole at late affine times. The authors
attribute these interesting features to the richness of the Einstein
equations; still, a good deal of foresight on their part must have
helped bring them to light.

**References:**

[1]
R. A. Matzner, H. E. Seidel, S. L. Shapiro, L. Smarr,
W.-M. Suen, S. A. Teukolski and J. Winicour, Geometry of a Black Hole
Collision, *Science* **270**, pp 941-947 (1995).

[2]
Please check the *Science* article for references to several authors
that contributed computational results collected in the article.

[3]
L. Lehner, N. T. Bishop, R. Gómez, B. Szilágyi and J. Winicour, Exact
Solutions for the Intrinsic Geometry of Black Hole Collisions, gr-qc/9809034