When Black Holes Merge, They'll Ring Like a Bell
by Brian Koberlein

When two black holes collide, they don’t smash into each other the way two stars might. A black hole is an intensely curved region of space that can be described by only its mass, rotation, and electric charge, so two black holes release violent gravitational ripples as merge into a single black hole. The new black hole continues to emit gravitational waves until it settles down into a simple rotating black hole. That settling down period is known as the ring down, and its pattern holds clues to some of the deepest mysteries of gravitational physics.


Gravitational wave observatories such as the Laser Interferometry Gravitational-Wave Observatory (LIGO) have mostly focused on the inspiral period of black hole mergers. This is the period where the two black holes orbit ever closer to each other, creating a rhythmic stream of strong gravitational waves. From this astronomers can determine the mass and rotation of the original black holes, as well as the mass and rotation of the merged black hole. The pattern of gravitational waves we observe is governed by Einstein’s general relativity equations, and by matching observation to theory we learn about black holes.

General relativity describes gravity extremely well. Of all the gravitational tests we’ve done, they all agree with general relativity. But Einstein’s theory doesn’t play well with the other extremely accurate physical theory, quantum mechanics. Because of this, physicists have proposed modifications to general relativity that are more compatible with quantum theory. Under these modified theories, there are subtle differences in the way merged black holes ring down, but observing those differences hasn’t been possible. But a couple of new studies show how we might be able to observe them in the next LIGO run.


The modified Teukolsky equation. Credit: Li, Dongjun, et al

In the first work, the team focused on what is known as the Teukolsky Equation. First proposed by Saul Teukolsky, the equations are an efficient way of analyzing gravitational waves. The equations only apply to classical general relativity, so the team developed a way to modify the equations for modified general relativity models. Since the solutions to both the Teukolsky and modified Teukolsky equations don’t require a massive supercomputer to solve, the team can compare black hole ring downs in various gravitational models.

The second work looks at how this would be done with LIGO data. Rather than focusing on general differences, this work focuses on what is known as the no-hair theorem. General relativity predicts that no matter how two black holes merge, the final merged black hole must be described by only mass, rotation, and charge. It can’t have any “hair”, or remnant features of the collision. In some modified versions of general relativity, black holes can have certain features, which would violate the no-hair theorem. In this second work, the authors show how this could be used to test general relativity against certain modified theories.

LIGO has just begun its latest observation run, so it will be a while before there is enough data to test. But we may soon have a new observational test of Einstein’s old theory, and we might just prove it isn’t the final theory of gravity after all.

Reference: Li, Dongjun, et al. “Perturbations of spinning black holes beyond General Relativity: Modified Teukolsky equation.” Physical Review X 13.2 (2022): 021029.

Reference: Ma, Sizheng, Ling Sun, and Yanbei Chen. “Black hole spectroscopy by mode cleaning.” Physical Review Letters 130.2 (2023): 141401.

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