Strike a bell, and it responds with its own note. Likewise, a black hole disturbed by the fall of a star or another black hole begins to 'ring,' emitting gravitational waves. But instead of sound, it's the trembling of space itself, and instead of a single note, a whole set of decaying frequencies. These are called quasinormal modes.
To compute these frequencies, the authors took the trusty old WKB method (also known as the phase-integral method) and applied it to the equations describing perturbations around a black hole. WKB is like a walk through hilly terrain: you predict where you'll end up by knowing only the shape of the slopes under your feet. For ordinary quantum mechanics, the method works perfectly, but here it's trickier. The frequencies come out as complex numbers — with a real part (the tone) and an imaginary part (how quickly it fades).
The authors didn't stop at the usual WKB expansion. They computed the so-called quantum periods (or Voros symbols)—special integrals along closed paths in the complex plane that encode the entire spectrum. Then they applied Borel–Padé summation: a way to 'finish' a divergent series into a sensible number, like guessing the whole melody from the first few notes. The result was compared with numerical calculations for two types of holes: a charged one without rotation (Reissner–Nordström) and a rotating one without charge (Kerr). Both are in the extreme, maximal state, where the event horizon is about to disappear. The agreement was excellent.
This approach gives not just numbers but a deep understanding of the structure of the equations. In the future, it could be applied to more realistic black holes and possibly catch deviations from Einstein's theory of gravity — if they exist.
as written by the authors arXiv:2605.01321
We apply exact WKB analysis to the spectral problem arising in black hole perturbation theory. The boundary conditions for quasinormal modes lead to exact quantization conditions for the complex frequencies. To solve these conditions, one needs to evaluate the so-called quantum periods, or Voros symbols. For scalar perturbations of extremal Reissner--Nordstr\"om and Kerr black holes, we compute these quantities up to very high orders in the WKB expansion and perform Borel--Pad\'e resummation. The resulting resummed quantization conditions successfully reproduce the correct quasinormal mode frequencies with high precision.
An extreme black hole spins so fast that its surface at the equator moves at nearly the speed of light. If it spun just a bit faster, the event horizon would vanish, revealing a 'naked singularity'—a point of infinite density visible from outside.
In the movie 'Interstellar,' the heroes travel to the supermassive black hole Gargantua, which spins at nearly maximum speed — exactly the extreme case from this work.