Projects in Scientific Computing: Evolution and Structure of the Universe
The Dance of Two Black Holes

Formulated by Einstein, the two-black-hole problem holds extremely important implications for astrophysics and cosmology.

Once upon a time, on a small planet in a galaxy called the Milky Way, black holes were considered a fascinating theoretical artifact from the mathematics of general relativity — interesting concept, great stuff for science fiction. We've come a long way since 1915 when Einstein laid out his theory that rocked our world.

In 1969 American physicist John Wheeler coined the phrase that gives resonance to the concept of points in space-time where matter is so condensed, gravity so fiercely omnivorous, that it swallows everything, including light, that gets too close. Only 12 years ago, with observational evidence beginning to trickle in — swirling gas and star coalescence at the center of galaxies — Stephen Hawking wrote, prophetically, in A Brief History of Time: "The number of black holes may well be greater even than the number of visible stars."

Photo of galaxy Centaurus A.

A jumble of blue star-clusters, glowing gas clouds and dust lanes surround an apparent black hole at the center of galaxy Centaurus A, a mere 10 million light years from Earth, recorded by the Hubble Space Telescope.

Image courtesy of NASA.

Since 1994 the Hubble Space Telescope and, more recently, NASA's Chandra X-ray Observatory have convincingly lifted black holes from theory into reality. With data from these eyes in space, scientists have identified over 30 likely black holes and counting. They come in a range of sizes, from supermassive (like the monster with the mass of 30 million suns at the center of the Andromeda galaxy) to many that are small ( a few solar masses) and most recently a middleweight (about 500 solar mass) in galaxy M82.

Still, even with Hubble and Chandra, the evidence is circumstantial. Fundamentally, a black hole is invisible. Looking for one, as Hawking said, is like trying to find a black cat in a coal cellar. The observations offer reasoned surmises about an undetectable agent lurking in the interior of detectable phenomena. As Penn State astrophysicist Pablo Laguna and post-doctoral fellow Deirdre Shoemaker like to point out, the way to clinch, indisputably, that black holes exist and that Einstein's equations are right is to detect gravity waves from two black holes.

Tuning the Gravity-Wave Radio

Detecting gravity waves is the job, a big one, cut out for LIGO, Virgo and GEO600. LIGO (Laser Interferometer Gravitational-Wave Observatory) is two NSF-funded gravity-wave detectors — in Louisiana and Hanford, Washington — now undergoing testing. Virgo and GEO600 are under construction in Europe (near Pisa, Italy and in Germany). Together these projects represent a pioneering effort that scientists hope will lead the way to an invaluable new set of eyes — gravity eyes — for seeing the universe. But it won't be easy, especially since no one ever has detected a gravity wave.

Along with anticipating black holes, Einstein's theory predicts that accelerating movements of massive objects in space, such as supernova explosions and black holes, will produce ripples traveling at light-speed through space-time. As with black holes, there's indirect evidence he was right, but compared to other wave phenomena, like electromagnetism, which brings us radio and TV, gravity waves are very weak. Einstein speculated they might never be detected. If you think of LIGO as the gigantic antenna for a radio receiver, the strongest possible signal might be a faint crackle as you turn the dial. To improve the chances of hearing the first crackle of gravity from the cosmos, LIGO needs to know where to set the dial to tune in two black holes colliding with each other.

Photo: The Penn State Numerical
            Relativity Group.

The Penn State Numerical Relativity Group
Jorge Pullin, Deirdre Shoemaker, Kenneth Smith, David Garrison, Pablo Laguna, Keith Lockitch, Erik Schnetter, Gioel Calabrese and Bernard Kelly. Not present: Manuel Tiglio.

Download a larger version of this image (71KB).

To do this, researchers like Laguna and Shoemaker are using supercomputers, the most powerful they can find, to numerically solve Einstein's equations. Their field is called numerical relativity, and with collaborators at the University of Texas and the University of Pittsburgh, Penn State has assembled one of the leading groups in the world. In recent work, relying on systems at PSC, at NCSA in Illinois and elsewhere, this multi-university team successfully simulated two black holes merging in what's called a grazing collision — only the second time this has been accomplished. Their numerical approach, called black-hole excision, makes a notable dent in the two-black-hole problem, the major challenge of this challenging field.

"Einstein's equations describe gravity via an elegant but complicated set of non-linear partial differential equations," says Laguna. "Their complexity requires the most powerful supercomputers available. Accurately solving the two-black-hole problem, formulated conceptually by Einstein 80 years ago, will represent an historic moment in the development of general relativity theory, with extremely important implications for astrophysics and cosmology."

The mathematics of a single spherical black hole sitting and spinning in space was worked out long ago by German astronomer Karl Schwarzschild, who in 1917 from his deathbed in effect discovered the black hole, without naming it, as one of the implications of Einstein's theory. A single black hole by itself, however, doesn't make gravity waves. Add another black hole, the interesting and many believe very relevant situation of two black holes merging with each other — often called a binary black hole — and you fiendishly complicate the mathematics, to the point where the only hope is supercomputers.

"As in most physical studies," says Shoemaker, "you want to look at the complicated and more realistic situations to test what you know. With general relativity, you can't put two of these compact objects together and get a solution without advanced computational techniques. Two black holes takes the theory into a dynamical regime, where you can make predictions and then, if experiments verify the predictions, you know how far the theory is correct."

It's a mutually beneficial relationship. To verify the predictions, you need detectors. LIGO, Virgo and GEO600, likewise, need predictions. Many believe that colliding black holes is the best shot at detecting gravity waves. Theory says it's one of the strongest signals on the gravity-wave dial. To know if a crackle of static is the dance of two black holes or cosmic noise, the detectors need the answers numerical relativists are working to provide.

Black Holes without the Holes

"Abandon hope, all ye who enter here," said Dante of the entrance to Hell. He might have said the same about the event horizon of a black hole. In solving Einstein's equations, Schwarzschild started with the idea of an infinitely condensed mass and showed that space-time curves around it and closes on itself. Once matter or light enters space-time within a certain radius from that point — initially called the Schwarzschild radius, now the event horizon — there's no escape. The region inside the horizon is cut off from events outside. This principle, called cosmic censorship, underlies black-hole excision.

The killer for simulating black holes is the singularity, the point of infinite density and space-time curvature that, mathematically speaking, makes a black hole a black hole. "The most crucial aspect of numerically evolving spacetimes containing black holes," says Laguna, "is without doubt the accurate and long-term handling of the singularities these objects represent."

Simply put, the numbers get too big too fast, and the computation crashes. "If you get too far inside the black hole," says Shoemaker, "you run into huge gradients that kill your calculations. There are basically two alternatives. In one of them you exploit the relativity of time; in effect you slow down how fast clocks tick near the black hole to avoid approaching that area. The other way is to remove the dangerous area. That's what we did."

The first approach, avoiding the singularity, has been more popular, and a group at the Albert Einstein Institute near Berlin has employed it with some success. It has the drawback that to slow down time inevitably adds to the already severe computational demands. With software they call AGAVE, the Penn State-Pittsburgh-Texas team has taken the less-traveled road of surgically removing the singularity from the domain of the calculation. About two years ago, the Pittsburgh group successfully excised the singularity for a single black hole moving in space. AGAVE extends this approach to colliding black holes, in effect, simulating two black holes without the black holes.

How, you might ask, can you compute gravity waves from a black hole if you eliminate the black hole? The secret, says Laguna, is in the horizon. Cosmic censorship. Since information about anything across that threshold is cut off, physical processes outside the horizon aren't affected by what happens inside. "As long as the spacetimes with and without the singularities agree at the points where the cut is made," says Laguna, "both situations should be equivalent for an observer outside."

Much easier to say than implement, notes Shoemaker. The numerical intricacies of cutting out the hole from the grid-like domain of the computation and, at the same time, keeping track of its movement in time, are daunting. Using PSC's CRAY T3E, AGAVE underwent extensive development and testing prior to the grazing collision simulation.

Grazing Collision of Two Black 
Holes. Grazing Collision of Two Black 

Grazing Collision of Two Black Holes
In these two snapshots from the simulation, transparent spheres represent the "apparent" horizon of the black holes. The first snapshot shows two equal-mass black holes caught in each other's gravitational pull; the second shows the large black hole formed as they merge. The bluish area inside the spheres represents the excised region. Color gradations (from red to purple) indicate relative strength of the gravitational field.

Download a larger version of snapshot 1 (396KB) or snapshot 2 (396KB).

The grazing collision is a milestone — compared to the symmetry of a head-on crash, which has been done before — because it adds a layer of complexity and realism. With 40 processors of NCSA's SGI Origin 2000, it required nearly 100 hours. There's simplifying assumptions, such as two equal mass black holes, but the result is, you might say, a smashing success that pushes beyond prior work.

Excision tamed the numerical instabilities of the singularity long enough for the two black holes to merge completely and evolve for a short period as one large black hole before the simulation crashed. It's not the end of the road by any means, stress Laguna and Shoemaker. There's not yet accurate gravity-wave predictions to hand over to LIGO. But the next mountain now looks more climbable. That mountain, two black holes that orbit each other before they coalesce, is a few years away say the researchers.

Further help is coming, notes Laguna, whose eyes light up thinking of PSC's new terascale system, more than 2,700 powerful processors with a peak capability of over six-trillion calculations per second, a leap forward that will allow the team to push further with AGAVE. "We believe one of the severe problems we have now is that the merged black hole gets too close to the boundaries of the computational domain. With the new machine, we can shift the outer boundary outward."

Some day, not that far away, a crackle of static will come in from the cosmos. Was Einstein right? Are there really black holes? When two of these monsters swallow each other, does it create a tidal wave of gravity detectable on our tiny planet thousands or millions of light years away? Please place your bets now.

Download PDF PDF: The Dance of Two Black Holes
Researchers Pablo Laguna, Penn State University
Hardware CRAY T3E, PSC
SGI Origin 2000, NCSA
Software AGAVE
Related Material
on the Web
Ripples in Space and Time: Stable Characteristic Evolution of Generic 3-Dimensional Single-Black-Hole Spacetimes
Numerical Relativity Group, Penn State University.
Laser Interferometer Gravitational-Wave Observatory.
References Steve Brandt, Randall Correll, Roberto Gomez, Mijan Huq, Pablo Laguna, Luis Lehner, Pedro Marronetti, David Neilsen, Richard A. Matzner, Jorge Pullin, Erik Schnetter, Deirdre Shoemaker and Jeffrey Winicour, "Grazing Collisions of Black Holes via the Excision of Singularities," submitted to Physical Review Letters (September 2000).
Writing: Michael Schneider
HTML Layout/Coding: R. Sean Fulton
© Pittsburgh Supercomputing Center (PSC), Revised: October 6, 2000
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