Black holes embody the ultimate abyss. They are the most powerful sources of gravity in the universe, capable of dramatically distorting space and time around them. When disturbed, they begin to “ring” in a distinctive pattern known as quasinormal modes: ripples in space-time that produce detectable gravitational waves.
In events like black hole mergers, these waves can be strong enough to detect from Earth, offering a unique opportunity to measure a black hole’s mass and shape. However, precise calculation of these vibrations through theoretical methods has proven a major challenge, particularly for vibrations that are rapidly weakening.
This inspired a team of researchers at Kyoto University to try a new method of calculating the vibrations of black holes. The scientists applied a mathematical technique called the exact Wentzel-Kramers-Brillouin, orexact WKB analysisto carefully trace the behavior of waves from a black hole out into distant space. While this method has long been studied in mathematics, its application to physics — especially to black holes — is still a newly developing area.
“The foundations of the exact WKB methodwere largely developed by Japanese mathematicians. As a researcher from Japan, I have always found this field intellectually and culturally familiar,” says corresponding author Taiga Miyachi.
This method allowed the research team to follow wave patterns in great detail, even in regions that are difficult to analyze with other existing methods. Their approach involved examining space near the black hole by extending it into the complex number domain, revealing a rich structure of the black hole’s geometry.
This included a mathematical phenomenon called Stokes curves, which designate where the nature of a wave suddenly changes. While previous studies have often overlooked the infinitely spiraling Stokes curves and paths that branch off from black holes, the research team incorporated these complex features into their analysis.
The findings revealed that the team had succeeded in developing a method that systematically and precisely captures the frequency structure of rapidly weakening vibrations. This demonstrates the power of the exact WKB method as a practical tool for bridging theoretical predictions with observational data.
“We were surprised at how complex and beautiful the underlying structure of these vibrations turned out to be. We found spiraling patterns in our mathematical analysis that had been missed before, and these turned out to be key in understanding the full picture of quasinormal modes,” says Miyachi.
This study makes it possible to analyze the “ringing sounds” of black holes across a wide range of theoretical models. Ultimately, this may help improve the precision of future gravitational wave observations and lead to a deeper, more reliable understanding of the true nature of our Universe and its geometry.
Looking ahead, the research team plans to extend their approach to rotating black holes and to explore the application of exact WKB analysis in studies related to quantum gravity effects.