Math may have caught up with the swirling mergers of black holes like the one in this simulation.


Theorist calculates the incalculable siren song of merging black holes

Just a month into a renewed observing campaign with a trio of detectors, physicists today announced they have spotted more gravitational waves—fleeting ripples in space set off when two massive objects such as black holes spiral into each other. The collaboration has now bagged 13 merging black hole pairs, as well as two pairs of neutron stars. But even as detections accumulate, one theorist has made an advance that could change how the team analyzes the signals and make it easier to test Albert Einstein’s theory of gravity, general relativity.

To interpret their signals gravitational wave hunters compare them to computer simulations. Now, Sean McWilliams, a theoretical astrophysicist at West Virginia University in Morgantown, has calculated an exact mathematical formula for the signal, or waveform, produced by two merging black holes.

“It’s a big step forward,” says Neil Cornish, a gravitational wave astronomer at Montana State University in Bozeman who was not involved in the work. “It’s going to allow for more accurate waveforms for doing analysis. But it also gives us more insight into what’s going on” in a black hole merger.

In 1916, Einstein predicted that as two stars orbit each other they’d radiate gravitational waves, although he figured the waves would be too feeble to detect. In 2015, physicists with the Laser Interferometer Gravitational-Wave Observatory (LIGO) spotted a burst of waves from two black holes that merged 1.3 billion light-years away, using their huge optical instruments in Hanford, Washington, and Livingston, Louisiana. The Virgo detector near Pisa, Italy, joined the hunt in August 2017, enabling the collaboration to triangulate to the sources of the events on the sky.

As two black holes spiral ever closer, they emit ripples in space that speed up. The waves’ intensity peaks as the two objects collide, and then peter out as the final, merged black hole undulates and settles down. To decipher the signal and determine the black holes’ masses and other parameters, scientists compare it to a catalog of simulated signals, a tack they have taken because of the complexity of the problem.

According to general relativity, gravity arises when mass and energy warp spacetime. And a black hole is the ultraintense gravitational field left behind when a massive star collapses to an infinitesimal point. So when two black holes swirl together, warping begets warping and renders the mathematics “nonlinear” and intractable.

Or so many scientists assumed. McWilliams says he has found a way to calculate the signal mathematically after all, as he reports in a paper in press at Physical Review Letters.

Two relatively simple formulas describe the peak and reverberation of gravitational wave signals like the first ones the Laser Interferometer Gravitational-Wave Observatory saw.

Caltech/MIT/LIGO Lab

The calculation involves special distances from the center of the black hole. Famously, nothing can escape a black hole if it draws closer than a characteristic distance called the event horizon. At a distance about 1.5 times that of the event horizon, the black hole’s gravity will bend passing light into a circular orbit, defining the “light ring.” A distance roughly three times that of the event horizon marks the limit for a massive object to maintain a circular orbit and not spiral in, a threshold called the innermost stable circular orbit (ISCO).

Previous attempts to calculate the exact waveform from a black hole merger relied on a standard mathematical transformation, turning the problem of two orbiting black holes into one of a single body spiraling in a funnel-shaped energy landscape. But within the ISCO, the body stops spiraling, forcing researchers to correct its path with numerical simulations. McWilliams realized he could avoid that problem by skipping to the final merged black hole. He then used general relativity to calculate how a tiny test mass spirals into and perturbs the final black hole, enabling him to calculate the radiated signal from the ISCO inward.

Once the test particle reaches the light ring, tracing its trajectory becomes mathematically untenable. But McWilliams says the physics there can be ignored for a simple reason: All the churning of spacetime within the light ring cannot escape to influence the spreading gravitational waves. Essentially, the black hole itself slurps up all the nasty nonlinearities. McWilliams provides a pair of formulas that neatly match the simulations. “I’ll be honest,” he says, “I was rather floored how well it agrees with the results of numerical relativity.”

Those formulas could prove valuable in tests of general relativity, McWilliams says, especially as black holes are objects made of pure gravitational energy, with no messy matter to get in the way. LIGO’s and Virgo’s observations have already confirmed general relativity’s accuracy to an unprecedented level, but researchers should be able to push further as they hone their instruments’ sensitivity. They’ll need more precise predictions of the waveforms from general relativity, McWilliams says, and the exact formulas should be more accurate than the numerical simulations.

Lionel London, a gravitational wave theorist and LIGO team member at the Massachusetts Institute of Technology (MIT) in Cambridge, isn’t so sure. McWilliams still has to rely on simulations to model the spiraling outside the ISCO, he notes, and that part of the signal is key to determining the masses of the initial black holes. The calculations also depend on certain simplifying assumptions, but do not provide estimates of the uncertainties carried with them, he says. The formulas are more of an “ansatz”—an educated guess at how the signal should look—than an exact solution to the problem, London says.

Cornish agrees it’s too early to replace numerical relativity. Still, he says, the formulas will be useful and should spur physicists to explain why black hole mergers seem to be simpler than they had anticipated. “There’s more to be learned.”

In the meantime, LIGO and Virgo researchers will have no shortage of signals. During the first month of their third observing run, they have detected five new candidate events, including three black hole mergers, a second neutron star merger, and a possible black hole-neutron star merger spotted last week. The mixed merger would be another gem for scientists, as they lack even good estimates of how often such things should occur. “Because it’s such an interesting astrophysical object, it’s generating a lot of excitement, which I think it deserves,” says Jessica McIver, a physicist and LIGO team member from the California Institute of Technology in Pasadena.

Still, the tantalizing signal is relatively weak. Researchers estimate that random noise should produce a similar spurious signal about once every 20 months, and there’s a 14% chance that it originated in terrestrial vibrations. “If you ask me, ‘Would you bet a coffee, your car, or your house on this?’ I would say, ‘I’d bet your car,’” says Salvatore Vitale, a physicist and LIGO member from MIT. To nail the case for the supposed mixed merger, astronomers would likely have to spot light and electromagnetic waves from it.