Quasinormal modes of black holes and black branes

TL;DR

This review surveys quasinormal modes of black holes and black branes across asymptotically flat, AdS, and dS spacetimes, highlighting their mathematical structure, computation methods, and physical applications. It links QNMs to near-equilibrium dynamics via gauge-gravity duality, showing how poles of retarded correlators encode transport in strongly coupled theories. It also covers QNM spectra for astrophysical BHs, their detectability with current and future gravitational-wave detectors, and how ringdown measurements can test the Kerr hypothesis and no-hair theorems. The article also discusses analytic solutions, high-damping limits, and several modern developments including holographic hydrodynamics, higher-derivative gravity corrections, and analogue gravity concepts.

Abstract

Quasinormal modes are eigenmodes of dissipative systems. Perturbations of classical gravitational backgrounds involving black holes or branes naturally lead to quasinormal modes. The analysis and classification of the quasinormal spectra requires solving non-Hermitian eigenvalue problems for the associated linear differential equations. Within the recently developed gauge-gravity duality, these modes serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, in particular their transport coefficients, such as viscosity, conductivity and diffusion constants. In astrophysics, the detection of quasinormal modes in gravitational wave experiments would allow precise measurements of the mass and spin of black holes as well as new tests of general relativity. This review is meant as an introduction to the subject, with a focus on the recent developments in the field.

Figures (20)

• Figure 1: Four different physical processes leading to substantial quasinormal ringing (see text for details). With the exception of the infalling-particle case (where $M$ is the BH mass, $\mu$ the particle's mass and $\psi_2$ the Zerilli wavefunction), $\psi_{22}$ is the $l=m=2$ multipolar component of the Weyl scalar $\Psi_4$, $M$ denotes the total mass of the system and $r$ the extraction radius (see e.g. Ref. Berti:2007fi).
• Figure 2: Integration contour for Eq. (\ref{['originalu']}). The hatched area is the branch cut and crosses mark zeros of the Wronskian $W$ (the QNM frequencies).
• Figure 3: Percentage errors for the real (left) and imaginary part (right) of the QNM frequencies as predicted from WKB calculations. Thick lines: third-order WKB approximation; thin lines: sixth-order WKB approximation.
• Figure 4: Left panel: contour for calculation of the QNM frequencies in the complex-$r$ plane. The different regions are separated by the associated Stokes lines. Not shown in the plot are branch cuts from $r=-\infty$ to the origin and from $r=1$ to point A. Right panel: integration contour in the complex-$z$ plane, with $z\equiv r_*/2M-i\pi$. For more details see Ref. Motl:2003cd.
• Figure 5: Top: QNM frequencies for gravitational perturbations with $l=2$ (black circles) and $l=3$ (red diamonds). In both cases we mark by an arrow the algebraically special mode, given analytically by Eq. (\ref{['AlgSp']}); a more extensive discussion of this mode is given in \ref{['app:relpotentials']}. Notice that as the imaginary part of the frequency tends to infinity the real part tends to a finite, $l$-independent limit. Bottom: comparison of the $l=|s|$ QNM frequencies for scalar, electromagnetic and gravitational perturbations.