A black hole rings like a bell when something massive falls into it. This ring is gravitational waves, and its overtones tell us about the shape of the horizon. Recently, in beyond-Horndeski gravity theory (an extension of general relativity), black holes with primary scalar 'hair'—an additional field that does not vanish even at rest — were discovered.
The authors calculated how such a field changes the spectrum of quasinormal modes (damped oscillations) and Hawking radiation. It turns out that the fundamental tone, associated with the light ring, changes moderately. But higher overtones are very sensitive: frequencies shift, overtones rearrange, and echoes appear — additional wave packets after the main signal. The reason is a tiny deformation of space right at the horizon, created by the scalar field.
Even more interesting is the behavior of radiation transmission (the greybody factor). In some parameter ranges, resonant tunneling occurs: waves pass through the barrier not monotonically but with characteristic bursts. As a result, Hawking radiation starts to oscillate: its intensity depends on frequency not smoothly but with bumps and dips. Such behavior could be sought in data from gravitational-wave observatories and in future observations of primordial black hole evaporation.
Interestingly, naked singularities (solutions without a horizon) in this theory turned out to be quantum-mechanically singular: they lack proper dynamics. Nature seems to forbid their existence — another bridge between classical gravity and quantum theory.
as written by the authors arXiv:2606.16125
Recently, a new family of asymptotically flat black-hole solutions endowed with primary scalar hair has been discovered in beyond-Horndeski gravity. We study in detail the quasinormal modes spectra, graybody factors, and Hawking radiation of this class of black holes. We demonstrate that presence of primary scalar hair leaves characteristic imprints on the ringdown properties, shifts the quasinormal frequencies, inducing overtone rearrangements, and rise of echoes. While the fundamental modes associated with the light-ring are affected moderately, higher overtones are highly sensitive to the small near-horizon deformation produced by scalar field. In certain parameter regimes, the graybody factors exhibit resonant-tunnelling behavior, which leads to an oscillatory frequency dependence of the Hawking emission rate. Thus, both black-hole spectroscopy and Hawking radiation may provide complementary and distinctive probes of the beyond-Horndeski gravity. Additionally, we demonstrate that the corresponding naked singularites are quantum mechanically singular and do not admit a well-defined dynamics.
Higher overtones of a black hole can be hundreds of times more sensitive to the shape of the horizon than the fundamental tone — roughly like being able to tell from the sound of footsteps in a corridor whether a door is ajar by a millimeter.
In Greg Egan's novel 'The Light of Distant Stars,' signatures of scalar fields around black holes are used to test theories of gravity — much like in this work.