Shear Scalar
Shear Scalar

Black hole symphonies: revealing the complexities in gravitational waves

General relativity is a non-linear theory and it is notoriously difficult to find analytical solutions in many physical situations, especially in the dynamical strong-field regime. Black hole mergers are excellent examples. The process by which two black holes merge to form a remnant black hole is non-perturbative and one might expect non-linearities to be especially important for describing the merger.

Nevertheless, it has been found previously that linear perturbation theory works surprisingly well and is able to describe the emitted gravitational wave signal close to the merger. The question is then: where are the non-linearities? Furthermore, what is the relation between the strong field region near the horizon and the observed gravitational wave signal? 

This question has been addressed by PhD student Ariadna Ribes Metideri along with other researchers at Radboud University: Daniel Pook-Kolb, Béatrice Bonga and Badri Krishnan in collaboration with researchers at the University of Utrecht, and the Perimeter Institute for Theoretical Physics and the University of Guelph in Canada. The work is published in Physical Review Letters.  

Quasinormal modes

After a black hole merger, the newly formed object evolves into a standard black hole by emitting gravitational waves characterized by a set of complex frequencies known as quasinormal modes (QNMs). These quasinormal modes are somewhat similar to the musical tones emitted by a violin (or any other instrument). The beauty is that these QNMs are remarkably simple and black hole perturbation theory predicts the exact frequencies of these QNMs. Previously, only linear perturbation theory was used to analyze the QNMs of black holes. In this new work, the authors showed the presence of QNM frequencies predicted by second-order perturbation theory in the gravitational wave flux infalling into the horizon. "I am very proud that Ariadna managed to find results even better than we had imagined!" says Béatrice Bonga, Assistant Professor at Radboud University. "Initially, the team had found one second-order QNM, but Ariadna kept pushing the analysis and managed to show that multiple second-order QNMs are present. This shows the importance of non-linearities in the aftermath of black hole mergers."

Remarkably, these second-order modes turn out to be in close correspondence with similar evidence of quadratic modes found in the gravitational wave signal at infinity. Badri Krishnan, Professor at Radboud University, adds to this that "these results indicate that we can infer detailed properties of black hole horizons from gravitational wave observations. This is very exciting for future gravitational wave observations."

Shear Scalar
The image shows cross-sections of the dynamical horizon of one of our simulations stacked on the time axis to visualize its world tube. The value of the squared shear scalar is shown as color.
data model
Mismatch between the l=2 data of one of our simulations and a model for the shear with three tones, in which two frequencies are fixed to the GR predictions, and the third one is varied. The lowest mismatch occurs around the quadratic tone 20\times 20, t
Literature reference

Publication: Nonlinear Ringdown at the Black Hole Horizon
Neev Khera, Ariadna Ribes Metidieri, Béatrice Bonga, Xisco Jiménez Forteza, Badri Krishnan, Eric Poisson, Daniel Pook-Kolb, Erik Schnetter, and Huan Yang
Phys. Rev. Lett. 131, 231401 – Published 6 December 2023

Contact information

About person
Prof. B. Krishnan (Badri) , Dr B.P. Bonga (Béatrice) , A. Ribes Metidieri (Ariadna)