“Gravity waves are the point of LIGO,” says Scott A. Hughes, a professor of physics at MIT’s Kavli Institute for Astrophysics and Space Research in Cambridge, Massachusetts. “The crossed lasers are designed to detect changes in space-time produced by passing gravity waves. But you have to know what to look for.”
When people started thinking about LIGO, they realized that the merger of two black holes would produce a strong signal. A binary system of two massive stars orbiting each other sets the stage for the merger. When the stars go supernova, they each leave behind a black hole corpse.
The black holes remain locked in orbit around each other. The closer they get, the more they disturb the space-time in their vicinity. The system generates gravity waves that ripple outward. The waves sap orbital energy from the black holes, which then circle each other ever more closely. They eventually merge into a single object in a cataclysm of distorted space-time and radiating gravity waves.
“LIGO should be able to see these mergers,” Hughes explains. The LIGO project set researchers on the black hole merger problem with a vengeance. For years, scientists have been trying to use general relativity to calculate the signals LIGO scientists should look for in their data. A match would mean LIGO has detected a black hole merger.
Death spiral
There are three phases to a black-hole merger. First, “inspiral” occurs as the black holes circle around each other on an ever-tightening orbit. Until they get close, slight modifications of Newtonian gravitational attraction work just fine.
At the middle phase of the merger, all hell breaks loose. Tracking two black holes as they spiral into collision demands the full-blown equations of general relativity with no tricks or simplifications. The merger also produces the strongest LIGO signal. The melding of two rapidly moving black holes shreds space-time and radiates a torrent of gravity waves from the scene.
Like the inspiral phase, the merger’s conclusion is relatively easy to calculate. “You end up with a single black hole,” says Hughes. “The event horizon of the merged black hole oscillates for awhile and radiates gravity waves until it stabilizes.” This final phase of a black hole merger is called the ringdown.
Accurately reproducing the details of a merger and its gravity wave signature is the grand challenge needed to satisfy LIGO. Earlier this decade, a merger was the prize everyone was gunning for. Winning that prize meant Einstein’s equations would have to be wrestled to the mat.
Einstein’s impossible legacy
It’s the complexity of relativity theory that makes numerical relativity so hard. To handle a problem like merging black holes, theorists have to solve 10 interwoven equations simultaneously. Take just a few steps into a calculation and you can end up with hundreds of terms — smaller pieces of equations — to follow. It’s like doing algebra and calculus in the middle of a tornado.
Theorists had to first think of ways to convert the equations of general relativity into a form that computers could swallow. “The first attempt at numerical relativity came in the 1970s,” says Joan M. Centrella, chief of the gravitational astrophysics laboratory at Goddard Space Flight Center in Greenbelt, Maryland. “It was really a heroic effort.”
Centrella understands such theoretical heroism. She and her research group have worked on numerical relativity simulations for 20 years. The first efforts focused on direct collisions of black holes. The calculations were crude, but they did point the way for a future generation of re-searchers. To make real progress, scientists would have to figure out how to simulate Einstein’s 4-D space-time on a computer and make it work for the most complicated situations.
In general relativity, space and time are intrinsically mixed. All objects in space-time, including you and me, have four dimensions. Each of us occupies and moves through the three dimensions of space and, simultaneously, a fourth dimension of time. That means each person’s life history is a four-dimensional object stretching from birth to death. The same principle applies to black holes.
Black hole weirdness
To make it possible to run the equations of general relativity in a computer, researchers had to develop a way to accommodate the complexities of four-dimensional objects. Then, they had to develop a way to visualize the data as 2-D and 3-D animations. This left another major challenge: simulating the black holes themselves.
Every black hole is surrounded by an event horizon, the point beyond which ordinary radiation cannot escape the object’s powerful gravity. The event horizon is the boundary between our universe and the weirdness inside.
Anything passing through the event horizon exits our world forever. It’s no surprise, then, that trying to simulate the interiors of black holes in computer simulations creates problems.
Inside of the event horizon is a place simulations can’t go. Yet, without a way of representing black holes in the model, there could be no numerical relativity. Over the years, two strategies emerged to treat the problem. “You can either cut the black holes out of the grid in a process called excising, or you can try and slow the evolution down near the hole and then insert a solution you already know, which is called puncture,” explains physicist Manuela Campanelli, director of the Center for Computational Relativity and Gravitation at the Rochester Institute of Technology (RIT) in New York.
Each choice has its problems, and neither worked well. The difficulties of computational relativity brought the field to a standstill. Only a few years ago, the situation appeared to be getting desperate. “The codes just crashed and crashed and crashed,” Centrella says. “You couldn’t even get an entire orbit out of them.”
The inherent mathematical nastiness of general relativity’s equations and the sheer difficulty of translating them into computer code rendered the models wildly unstable. Just a few steps into a calculation, the machines locked up by trying to divide by zero or some other impossibility.
Campanelli remembers it as a dark time. “People had really lost hope,” she says. “These were scientists who had spent years building their codes. All that work, all that mathematics. No one wanted to toss it away and start again.” Then, in the midst of the melancholy, everything changed. It was a day that numerical relativists are not likely to forget for a long time.