Black Holes in Higher Dimensions Ring With a Hidden Symmetry That String Theory Breaks
Two Khalifa University mathematicians computed how black holes 'ring' in five to 26 dimensions and found two very different waves that share exactly the same notes, until a string-inspired correction is added.
Strike a bell and it rings at pitches set by its size, shape and metal. Black holes ring too. When two merge, the new black hole shudders and sheds gravitational waves in a short, fading chord, and detectors have been listening to those chords since 2015. The notes, which physicists call quasinormal modes, depend only on the black hole's mass, its spin and the law of gravity. Change the law, and the chord changes.
Mathematicians Denys Dutykh and Davide Batić of Khalifa University in Abu Dhabi set out to learn how that chord shifts if space has more than three dimensions, and if Einstein's equations pick up a correction suggested by string theory. Their results, published in Physical Review D, turned up something they were not looking for: two completely different kinds of waves that ring at exactly the same frequencies.
Several attempts to unite gravity with quantum physics, string theory first among them, require extra dimensions. At low energies some string theories add a term to Einstein's equations, called the Gauss-Bonnet term, that does nothing in our four-dimensional spacetime but comes alive in higher ones. The pair studied non-rotating black holes in every dimension from five to 26, the number required by bosonic string theory, and disturbed them three ways: with a simple scalar wave, with ripples that twist space, and with ripples that stretch and squeeze it.
Each disturbance meets a barrier around the black hole whose shape sets the tones. The standard shortcut for computing them, the WKB approximation, treats that barrier as one smooth hump. With the Gauss-Bonnet term switched on, some barriers develop deep dips and odd shapes, and the shortcut fails. So the team used a Chebyshev spectral method that turns the problem into a large matrix calculation, solved to 300 significant digits. That precision separates real tones, which stay put as resolution rises, from numerical "ghosts," which wander.
The surprise involved the simplest breathing vibration of a scalar field and the simplest twist of spacetime. Their barriers look nothing alike; one even dips below zero near the black hole. Yet in every dimension from five up, in Einstein's theory, they ring at exactly the same frequencies, overtones included. The researchers proved why: both barriers are built from the same underlying curve, one by squaring it and adding its slope, the other by squaring it and subtracting the slope. That is the structure of supersymmetric quantum mechanics, in which partner equations share one spectrum.
The authors stress a caveat. In Einstein's theory, this particular twist does not radiate by itself; it describes a black hole set gently spinning, so the coincidence lives in the equations rather than in a signal a detector could catch. Add the Gauss-Bonnet term and the symmetry breaks at once, with the two sets of tones drifting apart. "The symmetry belongs to Einstein's theory, not to its string-inspired extension," Dutykh wrote.
None of this can be heard directly. Lab tests of gravity at short range show any extra dimensions must be smaller than a few hundredths of a millimeter, so astronomical black holes ring as four-dimensional ones. But the method, and the warning that the usual shortcut can fail, carry over to four-dimensional theories where the Gauss-Bonnet term is coupled to an extra field, a coupling that gravitational-wave observations already constrain. The team has posted all its code on GitHub.
