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Number Crunchers

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.

Meeting the Grand Challenge

Early Milestones

Supercomputers to the Rescue

Adding a New Dimension

Number Crunchers: Present and Future

March to the Teraflop and Beyond

But You Need Codes to Crunch With!

Meeting the Grand Challenge

The theory of general relativity anticipates the existence of black holes and that disturbances, such as infalling matter, cause these black holes to emit gravitational waves. The Grand Challenge Alliance, a collaboration of astrophysicists, physicists, and computer scientists from nine universities, is seeking to solve the Einstein Equations in their full generality. This type of problem is presented when two rotating black holes, spinning around each other, finally collide. It's a Grand Challenge problem in part because the complexity of the equations demands new developments in both computer technology and software, and because of the fundamental importance of general relativity in physics and cosmology.

Early Milestones

Numerical Relativity Timeline

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.

Supercomputers to the Rescue

The plight of numerical relativists was relieved somewhat by the release of the first Cray supercomputer in 1978, and the field began to mature significantly. At last the scientists had the computational power to find accurate, numerical solutions to Einstein's field equations in two dimensions, although their "recipes" or computer codes were only just beginning to mature.

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.

Movie of a Schwarzschild Black Hole

Movies of Distorted Black Holes

Movie of a Rotating Black Hole

Movies of Colliding Black Holes

Adding a New Dimension

Now, for the first time, researchers had at their disposal enough computer memory to break away from the restrictive assumptions about symmetry and time they had been forced to make with the old machines. This meant that they could begin to think seriously about solving Einstein's equations in three dimensions -- an enormously complex problem -- in their full generality. These "general" solutions will not only allow numerical relativists to develop an accurate model for the collision of two black holes and the gravitational waves they emit, but will also open the door to solving other important problems in physics and cosmology.

Movie of Gravitational Waves in 3-D

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."

Number Crunchers: Present and Future

Several types of advanced machines are making three-dimensional simulations of Einstein's equations possible. Harnessing the combined power of these machines through high-speed networks c onstitutes an important goal of Grand Challenge research. However, yet more powerful machines may be needed to produce highly accurate solutions to the problem of two black holes rotating and spiralling abou t each other, ending in a collision.

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.

March to the Teraflop and Beyond

Several research efforts are underway to make teraflop computers a reality by the year 2000. Pioneering work in numerical relativity applications by the NCSA Relativity Group and other teams participating in t he Grand Challenge Alliance will help computer designers achieve this goal. However, the development of teraflop computing will depend on further advances in microprocessor-based technologies and comp uter networks.

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

Larry Smarr, NCSA/Univ. of Illinois, on-camera, continued
Movie/Sound Byte
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.

But You Need Codes to Crunch With!

Microprocessor-based computers, high-end computers and high speed networks are all very fine, but they won't deliver teraflop performance without sophisticated software. Codes must be built that are compatible with different machines and which can tap the ir raw power to the hilt. Developing advanced relativity codes will take collaboration between the best minds in numerical relativity and computer science. Progress also demands new tools and techniques to visualize the res ulting avalanche of data.

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

NCSA. Last modified 11/9/95.