This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: Strike a bell, and it rings with a pitch and a fading that tell you about the bell: its size, its shape, the metal it is made of. When two merge, the newborn black hole shivers and sheds gravitational waves in a brief, dying chord, and since 2015, gravitational-wave detectors have been listening.
The notes of that chord, which physicists call quasinormal modes, depend only on the black hole's mass and spin and on the law of gravity itself. Change the law, and the chord changes. My colleague Davide Batić and I, both mathematicians at Khalifa University in Abu Dhabi, wanted to know how the chord changes when space has more dimensions than the three we see, and when Einstein's equations receive a correction suggested by string theory.
We report the answer in a paper published in Physical Review D. Along the way, we met something we had not been looking for: two quite different kinds of waves that ring at exactly the same notes. Several attempts to unite gravity with quantum physics, string theory first among them, need extra dimensions of space.
At low energies, some string theories add a term to Einstein's equations built from the curvature of spacetime, called the Gauss–Bonnet term. In our four-dimensional spacetime, this term leaves the gravity equations unchanged; it only comes alive when there are more dimensions. So we studied nonrotating black holes in dimensions from five to 26, the upper end being the number required by bosonic string theory.
We disturbed them in three ways: with a scalar field, the simplest kind of wave, and with two families of ripples of spacetime itself, one that twists space and one that stretches and squeezes it. Each disturbance feels an effective barrier around the black hole, and the tones are set by that barrier's shape. The standard shortcut for computing them, the WKB approximation, treats the barrier as a single smooth hump.
With the Gauss–Bonnet term switched on, some barriers develop deep dips and intricate shapes, and the shortcut can no longer be trusted. We used a Chebyshev spectral method instead. It maps the whole region outside the black hole onto a finite interval, expands the wave in polynomials, and turns the problem into a large matrix eigenvalue problem, which we solved with 300 significant digits.
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