How black hole spectroscopy can put general relativity to the test

General relativity has passed every test that multiple generations of researchers have thrown at it. Those tests include the three that Albert Einstein proposed when he introduced the theory in 1915 as well as repeated precision experiments performed in the lab and through astronomical observations.

The 2015 direct detection of gravitational waves from the merger of two black holes¹ marked the opening of a new avenue for testing general relativity. The discovery occurred four decades after Russell Hulse and Joseph Taylor Jr confirmed the existence of gravitational waves indirectly by observing their effect on the orbit of a binary pulsar,² and it offered the promise of unprecedented access to the strong-field regime of gravity. The now hundreds of mergers that have been spotted by the Laser Interferometer Gravitational-Wave Observatory (LIGO) in the US, Virgo in Italy, and KAGRA in Japan involve black holes and neutron stars that are colliding at extremely high velocities and are subjected to some of the universe’s strongest gravitational fields. If there are cracks in general relativity, then tests involving strong gravity present the ideal means of finding them.

And there is good reason to look for those cracks. Einstein’s theory does not seem to meld with quantum mechanics, and it does not explain the mysterious phenomena of dark matter and dark energy. To solve such lingering questions, various modified theories of gravity have been proposed.

In the strong-field regime, researchers may find deviations from general relativity by studying the nonlinear nature of gravity: how tiny spacetime perturbations, in the form of gravitational waves, interact with themselves to produce more perturbations. The observable effects of that nonlinearity are too small to have been detected in previous tests. By using a technique called black hole spectroscopy to analyze how nonlinear effects are manifested in gravitational-wave observations of black hole mergers, researchers can devise powerful tests of general relativity—and perhaps find hints of how to modify the theory to answer some of the biggest open questions in physics.

Black hole ringdown

As two black holes begin to merge, they whip up linear perturbations in spacetime that interact to form nonlinear perturbations. The nonlinear effects are so complex that they can be faithfully modeled only by solving the Einstein equations on a supercomputer³ (see the article by Thomas Baumgarte and Stuart Shapiro, Physics Today, October 2011, page 32). Eventually, when the black holes have merged, the object that forms is expected to be reasonably well described by an exact solution of Einstein’s field equations. Found in 1963 by Roy Kerr, who is shown in figure 1, the solution describes a rotating black hole in isolation, and it depends only on the black hole’s mass and its angular momentum, or spin⁴ (see the article by Remo Ruffini and John Wheeler, Physics Today, January 1971, page 30).

A sweet spot for studying nonlinearities occurs just after the merger, when the newly formed object is settling down into a Kerr black hole. The signal emitted during that phase, which lasts for about a millisecond for the events detected by LIGO, Virgo, and KAGRA, is similar to the sound waves emitted by a bell struck by a hammer, and thus it is called the ringdown.⁵ The ringdown portion of the gravitational-wave signal from a prototypical black hole merger is shown in figure 2.

Ringdown waves can be modeled using black hole perturbation theory. Assuming that the remnant object is described almost exactly by the Kerr solution, then the perturbation consists of small deviations with respect to the Kerr spacetime. Researchers keep track of the size of the deviations with a bookkeeping parameter, ε.

Most perturbation-theory work on black hole ringdown over the past half century has been done at linear order, with all terms of order ε² and higher omitted. Derived using a formalism developed in 1973 by Saul Teukolsky, the solution of the Einstein equations at linear order in ε yields a ringdown waveform that consists of a superposition of quasi-normal modes: decaying oscillations of spacetime that give way to the stationary final black hole solution.^6–8

The beauty of general relativity is that the frequencies at which quasi-normal modes oscillate and the rates at which they die out are determined exclusively by the mass and spin of the final black hole. As a result, measuring one quasi-normal mode frequency—which combines a real component (the oscillation frequency) and an imaginary one (the damping rate)—allows researchers to determine the black hole mass and spin. Then general relativity dictates what the rest of the oscillation spectrum for that black hole must be.

As a result, researchers can test general relativity by obtaining measurements of at least two quasi-normal mode frequencies: If general relativity holds, then the two frequencies must be consistent with the same black hole properties. Suppose that after inferring the mass and spin of the final black hole, researchers observe one or more additional frequencies that are not in the spectrum predicted by general relativity. The most likely explanation is some combination of instrumental, modeling, and astrophysical effects. But if significant deviations consistently appear that cannot be explained by any such effects, then they could be evidence for an alternative theory of gravity.

Experimental black hole spectroscopy

The observational program in which we use quasi-normal mode frequencies to identify the parameters of black holes is called black hole spectroscopy. It became an experimental science with the historic LIGO detection of September 2015 (see Physics Today, April 2016, page 14). At least a dozen signals from the black hole mergers detected via some combination of LIGO, Virgo, and KAGRA observations through 2020 have ringdown waves that are loud enough for researchers to confidently identify at least one complex quasi-normal mode frequency and thereby infer the mass and spin of the final black hole.⁹

But we still have a ways to go before we can use black hole spectroscopy to conduct robust tests of general relativity. Even the loudest gravitational-wave events observed to date have small signal-to-noise ratios in the ringdown, so it can be difficult to confidently measure the two or more quasi-normal modes that are required to test general relativity.

Next-generation gravitational-wave detectors should ameliorate some difficulties of ringdown analysis. Planned upgrades to LIGO, Virgo, and KAGRA should reduce noise in the signals that the observatories pick up, and a new detector is slated to be built in India. Future ground-based detectors, such as the Einstein Telescope in Europe and Cosmic Explorer in the US, should have even higher sensitivity.^10,11

There are also plans for a space-based gravitational-wave observatory, which is illustrated in figure 3. Whereas current observatories detect signals emitted by merging black holes with tens or hundreds of solar masses, the Laser Interferometer Space Antenna (LISA) would observe lower-frequency spacetime ripples that are generated by coalescing black holes that are millions of times as massive as the Sun. The signal-to-noise ratio for ringdown signals from those supermassive black holes is expected to be orders of magnitude larger than that for current signals. Several quasi-normal mode frequencies could be measured in a single merger event observed by LISA.

Although the anticipated dramatic enhancements in detection sensitivity would aid black hole spectroscopy research, they would not eliminate the theoretical challenges. Observing a louder signal makes smaller effects—including but not limited to nonlinearities—more visible. A crucial next step is to develop the ability to characterize those small effects accurately. Part of the current theoretical effort is understanding how and when the nonlinear quasi-normal modes at order ε² or higher in black hole perturbation theory will become significant in gravitational-wave observations.

Powerful tests with an improved ringdown model

Recent simulations of black hole mergers have shown conclusively that nonlinear quasi-normal modes are present in the waveforms and that their amplitudes can be similar to those from linear quasi-normal modes.^12–14 The results imply that accurately characterizing the gravitational-wave signal—and therefore the black hole source—requires more than the linear-based analyses we have relied on to date. It necessitates furthering our understanding of how both linear and nonlinear quasi-normal modes depend on the intrinsic parameters, such as mass ratio, spins, and eccentricity, of the binary progenitor. Researchers are exploring those connections using a range of analytical and numerical techniques.

Another challenge with detecting quasi-normal modes is that they decay exponentially as a function of time. A promising way to move forward is to try pushing the validity of perturbation theory to earlier in the merger by including more nonlinear effects. For example, although the mass and spin of the black hole is still evolving at early stages of the merger because of the effects of gravitational radiation, we could potentially model those effects by including third-order contributions in perturbation theory. An additional priority is identifying which quasi-normal modes are present in gravitational-wave signals and which are not.^15,16

Once researchers get a handle on some of those outstanding questions, they can start planning even more stringent tests of general relativity. Because the frequencies of the nonlinear quasi-normal modes during the ringdown are the sum or difference of the frequencies of the linear modes, we can calculate the frequencies of the nonlinear modes from those of the linear ones. The nonlinear quasi-normal mode frequencies that are excited are restricted by selection rules that are similar to those that apply to atomic transitions in quantum mechanics. We can thus test whether the measured nonlinear quasi-normal mode frequencies in detected gravitational waves match the predictions of general relativity.

We do not have to stop there. From general relativity, we can compute the relative amplitudes of the linear and nonlinear quasi-normal modes.^17,18 The modes’ relative amplitudes depend mainly on the spin of the remnant black hole, and they seem to be only mildly dependent on the initial conditions of the perturbation. We therefore can also predict the nonlinear mode amplitudes and phases from the linear modes that are sourcing them and compare the results with observations. Significant theoretical work is focused on understanding how the amplitudes of the nonlinear modes depend on the initial conditions of the merger. The effort is somewhat analogous to computing transition probabilities in quantum mechanics.

The future of black hole spectroscopy

Much of the current work in black hole spectroscopy involves developing a picture of how nonlinearities are excited in general relativity. But to test alternative theories of gravity, we will also have to understand nonlinear ringdown in each of those theories. Do we still expect the nonlinear frequencies to be the sum of the linear mode frequencies? How will the relative amplitudes of the nonlinear modes change because of deviations from general relativity? Those questions have yet to be answered.

Yet more theoretical work remains to be done to address other important questions. For example, will the presence of nonlinearities in general relativity inhibit our ability to see deviations from the theory in the quasi-normal mode spectrum?

Our best bet is to shore up the theoretical research of black hole spectroscopy so that when, in the 2030s, LISA is slated to start observing supermassive black hole mergers, we will be ready to use the measured nonlinear quasi-normal modes to perform tests of general relativity. The payoff will be a striking experimental confirmation of Einstein’s general relativity in the strong gravity regime or, perhaps, the observation of deviations that could point to new directions in quantum gravity and cosmology.

We thank Lia Halloran for allowing us to use her painting for the opening spread. She is represented by the gallery Luis De Jesus Los Angeles and is a professor and the art department chair at Chapman University.

References

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13. M. H.-Y. Cheung et al., Phys. Rev. Lett. 130, 081401 (2023).
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Emanuele Berti is a professor in the William H. Miller III Department of Physics and Astronomy at Johns Hopkins University in Baltimore, Maryland. Mark Ho-Yeuk Cheung and Sophia Yi are PhD students working with Berti.