Two neutron stars come together.
Visualizations done with Amira (amira.zib.de) by Werner Benger, Zuse Institute Berlin and Albert Einstein Institute.

By Adam Frank

All Grand Challenge problems are hard. They have to be. To define the cutting edge of computational science, a problem has to be big, it has to be complex, and it has to be very, very non-linear. Some Grand Challenge problems, however, are so fundamentally thorny they inspire a kind of awe in the minds of scientists. To gain membership in this class of problems, it helps to have a pedigree that descends directly from Albert Einstein.

It doesn't go quite far enough to say that general relativity is the study of gravity. When Einstein extended Isaac Newton's vision of gravity as an action at a distance between masses, he did not just replace one kind of theoretical force with another. With his General Theory of Relativity, Einstein changed the very stage on which physics was played. Gravity became not just a force but the shape of space-time, the geometry of the world's four-dimensional substratum. GR as it is called is a beautiful, elegant theory -- a towering intellectual achievement that addresses nature at its deepest and most profound existence.

In 1905 Albert Einstein wrote a paper that is today called the Special Theory of Relativity. He based his new theory on a reinterpretation of the classical principle of relativity, namely that the laws of physics had to have the same form in any frame of reference. larger pix

Oh yes, it is also a computational nightmare.

In all but the simplest cases, calculations in general relativity are next to impossible. Even simple, highly symmetric situations can require extraordinary effort. Take the more complicated process of a head-on collision of two black holes. If all you have is a pencil and paper, you can forget about it. Until recently, if all you had was the world's most powerful supercomputer, you still had to forget about it. That's a measure of computational general relativity's difficulty.

Luckily those difficulties haven't stopped Wai-Mo Suen, Ed Seidel and their colleagues. After great effort, their High-Performance Computing and Communications-funded project is toppling many of the barriers holding back the development of a general relativity numerical code, the kind that can take on all kinds of problems in all kinds of configurations.

The problem's problem

Wai-Mo Suen is a professor of physics at Washington University. Along with his collaborators (at institutions across North America and Europe), he has been attempting to build a numerical code that solves Einstein's field equations in their full generality. The development of such a tool is of more than purely theoretical interest.

Gravity waves are traveling ripples in the fabric of space-time. They are a natural consequence of Einstein's equations and should be created en masse in any violent process involving lots of mass. For years, physicists and astronomers have dreamed of directly studying these waves. Now they are about to get their chance. The Laser Interferometer Gravitational-Wave Observatory (LIGO), a gravity-wave telescope, is set to go online within a year. Consisting of finely tuned lasers traveling across many miles in underground tubes (one in Washington State and another in Louisiana), LIGO should be able to detect the space-time ripples of distant cataclysmic events. If the project is going to work, however, LIGO scientists must know the difference between laser vibrations from a passing truck and distortions of space-time from colliding black holes.

Photo Tom Treuter

Wai-Mo Suen, professor of physics, Washington University. larger pix

That is where Professor Suen and his team come in. "One of the major goals of this project," Suen explains, "is to predict the gravitational wave signature of mergers between binary compact objects such as pairs of black holes or neutron stars." Years ago scientists identified these mergers as the events that would appear most visible to a gravity-wave detector. While LIGO should be able to detect these kinds of events, the LIGO researchers need to know exactly how their wave form, the signal, will look. Giving LIGO observers those signatures has posed a special problem for GR theorists.

"Until now, it has been very difficult to keep the codes stable," says Suen. Even getting a compact object such as a neutron star or black hole to simply move across a computational grid has presented fundamental difficulties. The problem rests both with the mathematical complexity of 10 coupled, nonlinear partial differential equations of GR and the sticky physics embodied in those equations. "Our equations literally have thousands of terms in them," says Suen, "and the code has hundreds of large arrays." Clifford Will, one of Suen's collaborators and the chair of Washington University's physics department, emphasizes the underlying problem with the physics. "The issue," he explains, "is that space-time is part of the dynamic. You have to follow the physics of the event and the background on which the event is occurring."

Part of the difficulty rests with problems of coordinate systems. To track the physics of strongly curved space-time, one needs to name or label arbitrary points, giving them an identity that can be followed. This is done by laying down a coordinate grid. If the curvature changes because of the movement of some massive body, then the coordinate system must change as well. Sometimes instabilities occur in simulations that only involve the coordinate system. "If you have a neutron star propagating across a space," says Suen, "the coordinate lines can get twisted into spaghetti if you are not careful." Instabilities in the coordinates were just one kind of problem that Suen and his collaborators had to face as they attempted to follow the collision of compact objects. Other problems involved the representation of the objects themselves. "If we wanted to follow the collision of neutron stars," says Suen, "we would often have numerical difficulty at the star surface, which represented a sharp change in density gradient."

Visualization Werner Benger

Two neutron stars merge on the way to forming a black hole. larger pix

Faced with these difficulties, the Earth and Space Sciences (ESS) Grand Challenge team developed an entire suite of new codes to handle the GR equations. A series of recent successful simulations has shown their strategies, and those efforts have paid off.

A big crunch

"We just finished calculating the collision of two neutron stars," explains Suen, the enthusiasm apparent in his voice, "and they have already allowed us to answer at least one important question." When two neutron stars slam into each other, strong shock waves can form. The shocks heat the neutron star material to high temperature. It had been conjectured that the extra pressure supplied by the shocks would allow the merged configuration some extra time before it collapsed to form a black hole. "Some researchers have argued that the delay in collapse could be as long as 10 seconds," says Suen. Ten seconds is a long time in the world of GR and black holes. It allows the system to smooth out and become fairly spherical. A collapsing spherical mass will not produce any gravity waves. "That is bad news for LIGO," says Suen, "Our simulations allowed us to look at the process in detail and see what happens."

Visualization Werner Benger

Image in a numerical simulation showing two neutron stars colliding head-on. Rings of gas heated by the shock wave are spilling out from the collision center. The color represents the internal energy (temperature) of the gas (white is hottest). The merging neutron stars collapse into a black hole promptly because of the huge amount of compression in the process. The gridded mesh locates the position of a horizon of the newly formed black hole, from which nothing, not even light, can escape. larger pix

The new simulations followed the near-head-on collision of two neutron stars. While ESS researchers did see shock waves forming, they grew much slower than had been predicted. "What was really amazing," says Suen, "is that you could see the apparent horizon pop out, engulfing the shock waves." The apparent horizon is the surface that defines the edge of the newly forming black hole. It is a point of no return across which no information can escape to reach the outside world. "Since the shock waves get overrun by the horizon," says Suen, "they can't affect the rest of the collapse. Things don't get smoothed out. We see lots of gravity waves, and that is good news for LIGO."

Suen's enthusiasm comes in part from the importance of the results, but it is also driven by the anticipation of all that awaits them. "This is the first physical problem we have used this code to attack," he says. "There are many more waiting." Suen is quick to point out that real neutron star and black hole collisions will involve objects orbiting each other, not head-on collisions. "We have to be able to follow the merger of these bodies over many orbits," says Suen, "and that still presents problems for us."

Removing the obstacles

The dramatic progress Suen and his colleagues have made came through a concerted, coordinated effort of the international team. According to Ed Seidel, one of the ESS group leaders from the Max Planck Institute for Gravitational Physics and University of Illinois, it is the team's diverse specialties and expertise that have borne fruit. "These large collaborations are simply a necessity in attacking problems of this complexity."

Seidel sees advances coming from both the purely theoretical and the computational domains. "One of the things that allowed us to make progress was a new formulation of the Einstein Field Equations," he says. "It seems like a magic ingredient, but the new form of the equations allows the simulations to work much better." The development of the general code CACTUS (for solving thorny problems) using the new formulation was one key in the success of the neutron star collision simulations. "That is just one part of the story, however," says Seidel. "We are now developing the software tools that will allow our team to make progress by working closely, even if we are separated by a continent and an ocean."

The team's efforts include developing the ability to run a single simulation on widely separated platforms. Recently, colliding neutron star simulations have been performed, with one star being updated on a machine in Germany and the other star on a machine in the U.S. "If we combine this distributed approach with the ability to interactively steer the simulation," says Seidel, "we will be able to allocate more resources to the runs as needed."

As the international team continues honing its skills at animating Einstein's elegant but difficult child, the rewards they reap will increase. "It's a very exciting time," says Seidel. "We are really getting to the point now that we can begin addressing basic physics
issues."