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Albert Einstein's General Theory of Relativity which describes gravity as the curvature of spacetime, is mathematically expressed as a set of 10 highly complex, coupled, nonlinear partial differential equations. Not only are they a mouthful, but they have proved impossible to solve using analytical methods for any but the simplest scenarios. Solving the equations for our three-dimensional universe that evolves through time requires numerical solutions, made possible only through very powerful computers. This is the domain of a young field called numerical relativity.

The history of numerical relativity is bound up not only with breakthroughs in theory but also with continuing advances
in the tools and techniques of computational science.

As early as 1964, Susan Hahn and Richard W. Lindquist tried to solve Einstein's equations for two colliding black holes.
Their pioneering work was doomed to failure for several reasons.
Appropriate methods for obtaining accurate and stable numerical solutions had not yet been developed for such a complex theory. Computer scientists didn't yet understand how to develop equations appropriate for
numerical solutions; the theory behind
black holes was not yet well defined; and the speed and memory
of their computers was severely limited.

By the middle 1970s, real progress had been made in algorithm development
and in the theoretical foundations of black holes. Bolstered by these advances,
Bryce DeWitt at the University of Texas-Austin and his then graduate student, Larry Smarr,
began to study the two colliding black hole problem again. This time, with the help of Kenneth Eppley, they enjoyed some success, showing that two
black holes colliding head-on would **merge** to form a single black hole.

Computer power was still far too limited to allow for accurate solutions, and their calculations took several days of computer time to complete. In spite of these obstacles, Dewitt, Smarr and Eppley were also able to demonstrate that the black hole resulting from the merger oscillates with a special frequency or "ringing mode." This discovery marked the beginning of the modern era of numerical relativity.

The introduction of increasingly powerful computers throughout the 1980s and into the early 1990s -- the CRAY X-MP, the CRAY-2, the CRAY Y-MP, and finally the CRAY C-90 -- made possible the study of models with more interesting physics. The scientists were able to study in more detail the Schwarzschild black hole which, although it lacks gravitational waves, exhibited some interesting features of more complex, more "real" black holes.

The work on the Schwarzschild black hole laid the foundation for later two-dimensional simulations and animations of **distorted**
black holes and, more recently, of **rotating** black holes (movie under construction). Both model systems bring together black holes and
the gravitational waves they would emit under different conditions.

The scientists also returned to the older problem of the head-on **collision** of two black holes and, with the aid of more powerful computers,
newer algorithms, and deeper theoretical analysis, they were able to compute and understand in detail the gravitational waveforms generated during this process.

For example, cosmologists can use the solutions to study the origin, nature and evolution of the universe in three dimensions; theorists will use them to explore the Einstein's equations more deeply; and astrophy sicists will use them to study a variety of astrophysical phenomena, such as the formation or collision of black holes and neutron stars, or the rotation of gigantic black holes thought to lurk at the center of so-called "a ctive galaxies."

Accurate solutions to these and other problems in generalized spacetimes (i.e. geometries that are unrestricted by symmetry) will require computers 100 times faster than the fastest computers available today (1
995). Such so-called **teraflop** computers will perform trillions of calculations (actually **floating point operations** or **flops**) every second.

For now, numerical relativists are restricted to tackling problems that the latest generation of supercomputers are able to solve -- problems that are generally much less interesting than the ones the scientists want to solve.

**Larry Smarr, NCSA/Univ. of Illinois, on-camera**

QuickTime Movie (2.3 MB);
Sound File (1.3 MB);
Text

QuickTime Movie (1.5 MB); Sound File (794K); Text

In the future, the researchers' questions should not be restricted by current technology, only by the limits of the theory. Such problems will probably require **petaflop** speeds (a quadrillion floating
point operations each second) with memory to match. Relentlessly, the speed of computers continues to increase, so this scenario is no pie-in-the-sky: expect it in the early 2000s.

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