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The Building Blocks of Code

Numerical Relativity Timeline

One doesn't learn to play the piano by first tackling Beethoven's sonatas. Neither do numerical relativists attempt to solve Einstein's equations describing the most complex model: two asymmetric spinning black holes rotating around and spiraling toward each other until they collide. Numerical relativists have cut their teeth on a comparatively simple, spherical model: the Schwarzschild black hole.

Although Karl Schwarzschild analytically solved Einstein's equations for a spherically symmetric, stationary black hole in 1917, his solution demanded a very special choice of space and time coordinates. In computer simulations of black holes, these special "Schwarzschild" coordinates are not practical, even for purely spherical black holes. In other coordinate systems, even this spherical black hole will appear to be extremely dynamic and is quite difficult to study numerically. It wasn't until 1987, however, that numerical relativity researchers were able to employ then powerful supercomputers such as the CRAY X-MP and more accurate algorithms to solve the equations for the more general spacetimes of dynamic systems, such as colliding or spiralling black holes.

Although the Schwarzschild black hole is the simplest model, it's more than just a stepping stone to more complex systems: theory predicts that non-rotating black holes eventually settle down into Schwarzschild black holes, just as a gong struck with a mallet gradually stops vibrating. Moreover, it incorporates some of the most difficult features of more complex models.

Like all black holes, the Schwarzschild black hole has at its center a singularity, a region of infinite density and spacetime curvature. Since computer codes abhor singularities (they will crash if they encounter these "beasts"), any algorithms used to describe even the simplest black hole must somehow avoid infinity . And, again like all other black holes, the Schwarzschild black hole has a boundary called an event horizon. Nothing -- matter, light, information of any kind -- can escape from behind the horizon.

Numerical relativists continue to test new ideas and techniques on the Schwarzschild model before transferring them to more complex models. However, for all its virtues, the Schwarzschild black hole can be likened to the physicists' "spherical cow."

When asked how to increase the milk production of cows, theoretical physicists might answer, after much head-scratching and pages of calculations, "First, you start with a spherical cow." A real cow is too complicated, as is a real black hole. Physicists often resort to assumptions that simplify a problem, making it solvable. But the downside is that the solutions may not represent anything "real."

Because the mathematics of General Relativity do not allow spherical objects to emit gravitational waves, the Schwarzschild black hole excludes them. In the "real world," black holes are unlikely to be perfectly spherical: after all, the events surrounding their formation, evolution and interactions with each other or other large objects are likely to be very violent.

Scientists will only be able to prove the existence and compute the mass of black holes by measuring gravitational waves they emit; a model that marries gravitational waves and black holes is crucial. Clearly, computation of black hole spacetimes must be extended beyond simple spherical symmetry.

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Copyright 1995, The Board of Trustees of the University of Illinois

NCSA. Last modified 11/9/95.