Figure 1: The merger of two black holes can set off vibrations in the resulting black hole. RIKEN researchers have analyzed these vibrations using a versatile mathematical technique. © MARK GARLICK/SCIENCE PHOTO LIBRARY
By applying a versatile mathematical technique to black holes, RIKEN cosmologists have uncovered new subtleties about how they vibrate1.
Black holes are some of the most intriguing objects in the Universe. Their gravitational pull is so strong that not even light can escape from their surfaces. Astronomers can only observe light produced indirectly by material falling into black holes.
Since 2016, they have had another means of investigating black holes-gravitational waves. These are ripples created in the fabric of space-time through cataclysmic events such as the merger of two black holes.
Immediately after merging, special oscillations known as quasi-normal modes are set up in the resulting black hole.
"Quasi-normal modes are frequency modes that are generated when a black hole experiences some major disturbance," explains Ryo Namba of the RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences.
These vibrations emit gravitational waves after a merger. While the higher overtones of quasi-normal modes of such gravitational waves are too weak for today's detectors, astrophysicists can study quasi-normal modes mathematically.
Initially, Namba was tackling a seemingly unrelated cosmological problem-how elementary particles formed in the early Universe. But after attending a math workshop, he realized that the mathematical method he was using for that problem might also apply to quasi-normal modes in black holes.
Called the exact Wentzel-Kramers-Brillouin (WKB) method, it was first developed more than a century ago, but has been subsequently revised by mathematicians, many of them Japanese. "I realized that this method is very powerful and that it doesn't only apply to particle production but can be used for other physical processes," says Namba.
Namba and collaborators at Osaka Metropolitan University, Kyoto University and Yukawa Institute for Theoretical Physics have now used the exact WKB method to calculate the frequencies of quasi-normal modes in black holes.
The solutions they obtained reproduced the results of other, more approximate methods. But they also revealed a new structure previously unseen.
Namba was relieved when the approach worked. "The exact WKB method is a powerful method, but there are many potential pitfalls when applying it to quasi-normal modes in black holes," he says. "I was very glad that we managed to obtain the correct result using it."
The project helped Namba overcome his wariness of singularities-points in space where physical quantities explode to become infinitely large.
"Previously, I was a bit scared of singularities because they're essentially points where normal physics breaks down," recalls Namba. "But the singularities turned out to have a rich structure and provided rich information into what is going on."
(Clockwise from top left) Ryo Namba, Taiga Miyachi, Hidetoshi Omiya and Naritaka Oshita have performed an exact WKB analysis of quasi-normal modes in black holes. © 2025 RIKEN
