Three quantum-inspired cores leave opposite fingerprints on the ringing of black holes

Every black hole hides a question at its center. Einstein's theory predicts that whatever falls in is crushed into a singularity, a point of infinite density where the theory itself breaks down. Most physicists expect quantum gravity to replace that point with something finite. But the center lies hidden behind the horizon. How could we ever learn what is there?

One answer is to listen. A disturbed black hole, for instance, one just born from a merger, rings like a struck bell, shedding gravitational waves in a few quickly fading tones that physicists call quasinormal modes. Each tone has a pitch and a fading rate, set by the shape of spacetime around the black hole. Build the center differently, and the black hole should ring differently.

Over the past year, my colleague Davide Batic and I, with Fabio Scardigli for the first two papers, computed these tones for three black holes whose centers are reshaped by ideas from quantum gravity. Our third paper, now published in Physics of the Dark Universe, completes the series, and the answers fall into a simple pattern: where gravity weakens at short distances, the black hole rings higher and longer; where it grows stronger, it rings lower and dies away sooner.

Gravity that changes with distance

In Newton's and Einstein's theories, one number, Newton's constant, fixes the strength of gravity. Several approaches to quantum gravity suggest that it depends on distance. In asymptotic safety, an idea due to Steven Weinberg, gravity weakens at the very shortest distances. Put such a running constant into Einstein's black hole, and its center changes.

In our first paper, in The European Physical Journal C, the constant follows the quantum correction to Newton's law, and gravity grows stronger near the center. The singularity survives, but it is spread over a sphere, the edge of a core of Planck density, still hidden behind the horizon.

In the second paper, in Physical Review D, we studied the black hole that Alfio Bonanno and Martin Reuter built from asymptotic safety in 2000. Gravity switches off at its center, a smooth core replaces the singularity, and a second, inner horizon appears. In the third, we followed the Hayward black hole, a simple model without a singularity whose form also arises in some asymptotic-safety constructions. One number sets the size of its core, and at a critical value, 32/27, its two horizons merge.

Listening with 200 digits

The usual shortcut, the WKB approximation, treats the barrier that waves meet around the black hole as a smooth hump. We used a Chebyshev spectral method instead: it maps the whole region outside the black hole onto a finite interval, writes the wave as a sum of polynomials and turns the problem into a large matrix eigenvalue problem, solved in 200-digit arithmetic. On Einstein's black hole, it gives back the known tones to six decimal places.

The extreme case needed care of its own: when the two horizons merge, the equations change character at the horizon, so we wrote a separate version of the problem to follow the Hayward black hole all the way.

Opposite fingerprints

The main gravitational tone tells the story. At its extreme point, the Hayward black hole rings 9.5% higher than Einstein's black hole of the same mass, and its tone fades 22.1% more slowly, so the ringing lasts 28% longer. The Bonanno–Reuter black hole at its own extreme point rings 12.0% higher and fades 22.3% more slowly. The Planck star of one Planck mass goes the other way: its main tone is 25.0% lower and fades 21.6% faster.

Two checks make me trust this contrast. Electromagnetic waves, whose tones depend on the geometry alone, go the same way: for the extreme Hayward black hole, their main tone rises by 8.0% and fades 17.9% more slowly. Gravitational waves also shake whatever the core is made of, which the geometry alone does not describe. We adopted one model of that response; with the other model used in recent studies, the pitch rises by about 6% instead of 9%, in the same direction.

The light ring decides

Why should a hidden center matter? Because the ringing is made outside, around the light ring, the distance at which light can circle the black hole. To a first approximation, the pitch of a tone is set by how fast waves go around this ring, and its fading by how quickly they slip off it.

Where gravity weakens near the center, it is also slightly weaker at the light ring, and the ring moves inward: for the extreme Hayward black hole, from 3 to 2.65 times the mass. Waves circle a smaller ring faster, so the pitch rises, and the orbit of light becomes less unstable, so they slip off more slowly. Where gravity grows stronger, as in the Planck star, the ring moves outward, from 3 to 4.14, and the pitch drops.

Small black holes, lasting lessons

Can such tones be heard? Not today. The core matters only when it is about as large as the black hole, which for a core set by quantum gravity means a few Planck masses, tens of micrograms. For a black hole as heavy as the sun, our first paper shows that the change would be smaller than one part in 10⁷⁶, so the black holes that detectors hear ring as Einstein predicted. Black holes this small could appear in the last instant of evaporating primordial black holes, if those were ever born.

What lasts, I believe, is the map. The trilogy shows which kind of core leaves which fingerprint: weaker gravity at the center makes a better bell, ringing higher and longer, while stronger gravity makes a duller one, lower and shorter. It also leaves precise tables of tones for others to build on, with our codes and spectra openly available. The center of a black hole may stay hidden forever, but its ringing need not keep the secret.

This story is part of Science X Dialog, where researchers can report findings from their published research articles. Visit this page for information about Science X Dialog and how to participate.

Denys Dutykh is an applied mathematician and Associate Professor of Mathematics at Khalifa University of Science and Technology in Abu Dhabi, United Arab Emirates, where he is also Acting Associate Dean of Graduate Studies of the College of Computing and Mathematical Sciences. Before joining Khalifa University in 2022, he spent 14 years as a researcher of the French National Center for Scientific Research (CNRS) at the University Savoie Mont Blanc. His research develops high-precision numerical methods for wave problems, from tsunamis and water waves to the vibrations of black holes. The work described here is part of a long collaboration with Davide Batic, a mathematical relativist at Khalifa University, on the spectra of black holes computed with spectral methods; the first two papers of the series were written with Fabio Scardigli of the Politecnico di Milano.