# Quasinormal modes (QNM) We learned about normal modes in introductory classical mechanics: general oscillations can be decomposed into modes with specific frequencies. Usually, the systems are time-independent, and the boundary value problem is Hermitian. But the time translation symmetry is absent for dissipative systems, and we do not get normal modes. Black holes are dissipative almost by their very nature: things fall through the horizon. So to describe perturbations to a black hole as viewed from outside, we want the perturbations to decay. This is achieved by allowing the modes to have complex frequencies, with the imaginary part telling us how fast they decay. These modes are called **quasinormal modes** (QNM). They are not normalisable and do not form a complete set of basis. There are two main perspectives regarding what they are mathematically. First, they are eigenmodes of a dissipative system. Second, they poles in the [[0473 Retarded Green's function|retarded Green's function]]. (See below.) ## QNM as eigenvalue problem - BC at horizon: - $\Psi \sim e^{-i \omega\left(t+r_{*}\right)}, \quad r_{*} \rightarrow-\infty\left(r \rightarrow r_{+}\right)$ - i.e. nothing comes out of black holes - BC at spatial infinity: - $\Psi \sim e^{-i \omega\left(t-r_{*}\right)}, \quad r \rightarrow \infty$ - i.e. nothing enters the spacetime from infinity - wave equation: - $\frac{d^{2} \Psi_{s}}{d r_{*}^{2}}+\left(\omega^{2}-V_{s}\right) \Psi_{s}=0$ (radial) - goal: solve the wave equation with these BCs ## QNM as poles in the Green's function - first noticed in [[2001#Birmingham, Sachs, Solodukhin]] that QNM on BTZ correspond to poles of the [[0473 Retarded Green's function|retarded Green's function]] of CFT in thermal equilibrium - then shown in [[2002#Son, Starinets]] that poles of Green's function in AdS are exactly the QNMs - the idea is simple: the retarded Green's function is the ratio between the response and the source. In holography, changing the source amounts to changing the non-normalisable (slow falloff) part of the bulk field. The QNMs are defined to have Dirichlet boundary conditions, i.e., they die off at infinity so there is no non-normalisable part. So they give zero for the denominator and are therefore poles. ## Relation to the boundary theory - QNM spectra of the gravitational backgrounds give the location (in momentum space) of the poles of the retarded correlators in the gauge theory, supplying important information about the theory’s quasiparticle spectra and transport (kinetic) coefficients ## Analytic examples - massless scalar on Kerr - [[2013#Cvetic, Gibbons]] - rotating, higher derivative - [[CanoFransenHertog2020]][](https://arxiv.org/pdf/2005.03671.pdf) - spin 1/2, large frequency, analytic - [[2013#Arnold, Szepietowski]] - spin 3/2, large frequency, analytic - [[2013#Arnold, Szepietowski, Vaman]] ## Gauge invariance See [[2005#Kovtun, Starinets]], Sec. 3.3 for the gauge invariant perturbations of the metric and gauge fields. ## Refs - reviews - [[2009#Berti, Cardoso, Starinets (Review)]] - [[HatsudaKimura2021]][](https://arxiv.org/abs/2111.15197) - [[2025#Bolokhov, Skvortsova (Review)]]: analytic results - general understanding - QNM as a limit of normal modes - [[2024#Das, Porey, Roy]] - hyperboloidal approach - [[2024#Macedo, Zenginoglu (Review)]] - exact WKB - [[2025#Miyachi, Namba, Omiya, Oshita]] (see also [[0371 SW-QNM correspondence]]) - orthogonality - [[2025#Arnaudo, Carballo, Withers]] ## Related - [[0179 Pole skipping]] - [[0371 SW-QNM correspondence]]