Quasinormal modes

A perturbed black hole rings down in a discrete set of damped harmonics. Each tone is labelled by an angular number ℓ, an azimuthal number m, and an overtone index n. Its frequency depends only on the mass and spin of the black hole. Everything below is solved for live, as you move the sliders, in your browser!

time damping slice modes
Re ψ around the BH. The black sphere is the horizon, the rim is null infinity. Drag to turn.
The angular pattern, Re [−2Sℓm(θ) ei(mφ−ωRτ)].
What an observer at ι records: h+ and h×.
The radial profile on a hyperboloidal slice that reaches both the horizon and null infinity.
The complex frequency as the spin runs from 0 to 0.99: this mode, and its siblings of the same ℓ and n in faint.
How this is computed

Conventions: Geometric units G = c = M = 1, spin weight s = −2, and time dependence e−iωt + imφ. The frequency is obtained through Leaver's continued fraction (in the form of Cook & Zalutskiy 2014), coupled through the separation constant A(aω) to the angular equation, which is solved as a banded eigenvalue problem in spin-weighted spherical harmonics. Each spin sequence is continued from the Schwarzschild value.

The radial profile is obtained from Leaver's series Σ ak xk, with x = (r − r+)/(r − r−) running from the horizon (x = 0) to null infinity (x = 1). In the 3D view, a point at Boyer–Lindquist radius r is drawn at radius r/(r + 4M). The surfaces are Re ψ = ±0.3 of its peak. Animation time runs at 8 M per second.

The [Boyer–Lindquist] view shows the same mode at constant Boyer–Lindquist time t. Going from (τ, φ̃) to (t, φ) multiplies it by eiωh(r) + imr♯, where h behaves like r* at infinity and like −r* at the horizon. Since Im ω < 0 this factor grows exponentially towards both ends. Quasinormal modes blow-up, but that is a problem of the slicing.