Analytical event horizons of merging black holes

Simonetta Frittelli, Duquesne University

Probably all of us are familiar with the celebrated picture of the pair of pants that made it to the cover of the November issue of Science of 1995 [1]. This is the article where the geometry of the collision of two black holes is explained in rather lay terms (which has a definite appeal) and where an embedded picture of the event horizon of such a collision was generated by numerically integrating the light rays that generate the horizon.

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 $\gamma_{ij}=\Omega^2h_{ij}$ where hij 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 nanbRab=0 of the vacuum Einstein equations along the null generators na of the hypersurface. This is an ordinary second-order differential equation for $\Omega$ that determines the dependence of $\Omega$ 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 $\Omega^2$ 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 $u\to\infty$. 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 $\Omega^2$. The requirement that the Weyl curvature must be regular provides further tips for the integration. The intrinsic geometry $\gamma_{ij}$ of the horizon is thus found explicitly in terms of two angular coordinates $(\theta,\phi)$ 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.


[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

Jorge Pullin