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research33 min studyLesson 838 in Physics

Black-hole quasinormal modes

Interpret complex ringdown frequencies with specified time dependence and boundary conditions, and separate mode scaling from a full remnant inference.

Before you begin
On this page

The question for today

What you’ll understand

  • Relate a mode's complex frequency to oscillation frequency, damping time and quality factor.
  • State the black-hole wave boundary conditions and their physical meaning.
  • Track mass/redshift scaling and the limitations of fitting a few damped sinusoids.

After a sufficiently settled black-hole remnant is perturbed weakly, part of its response resembles a sum of damped oscillations. Quasinormal modes describe the characteristic complex frequencies selected by an ingoing horizon and outgoing infinity boundary problem. They are not ordinary normal modes of a closed energy-conserving cavity. Their frequencies constrain remnant properties under a gravitational theory and mode assignment, while amplitudes, start time, tails and detector response remain separate ingredients of an observed ringdown.

Boundary conditions make the spectrum complex

In an asymptotically flat black-hole perturbation problem, waves fall through the future horizon and escape toward infinity. The separated radial equation can be written in a wave-potential form using a tortoise coordinate, with the precise potential depending on spin and angular mode. The ingoing and outgoing boundary conditions allow energy to leave the exterior perturbation. Complex frequencies are consequently natural even for a stationary background.

Choose time dependence exp minus i omega t. A stable damped mode then has negative imaginary frequency. Write that imaginary part as minus the inverse amplitude damping time, not minus the inverse energy damping time. The real part divided by two pi gives the cyclic frequency. These conventions fix the sign of decay and prevent confusing angular frequency with hertz when converting a dimensionless tabulated eigenvalue to seconds.

ω=ωR−iτ,hmode∝e−t/τcos⁡(ωRt+ϕ),f=ωR2π,Q=ωRτ2=πfτ\omega=\omega_R-\frac{i}{\tau},\qquad h_{\mathrm{mode}}\propto e^{-t/\tau}\cos(\omega_Rt+\phi),\qquad f=\frac{\omega_R}{2\pi},\qquad Q=\frac{\omega_R\tau}{2}=\pi f\tau
The convention is e^(−iωt), τ>0 is an amplitude e-folding time in seconds and f is in Hz. Q is dimensionless. The real strain is a projected real part; a complex mode coefficient also contains source excitation and viewing information.↔ Scroll wide equations horizontally.

Mass supplies the time scale

For a vacuum Kerr black hole, dimensionless mode frequencies depend on dimensionless spin and angular/overtone labels, while total mass sets the overall inverse-time scale. In the Schwarzschild limit the fundamental quadrupolar gravitational mode has a familiar dimensionless complex eigenvalue. Multiplying mass by two halves its cyclic frequency and doubles its damping time at fixed dimensionless mode, leaving quality factor unchanged.

Cosmological propagation has the same dilation pattern: observer frequency decreases and damping time increases by one plus redshift. Ringdown therefore naturally measures redshifted mass unless additional information supplies the source frame. Rotation changes the dimensionless mode spectrum, so a single measured real frequency cannot determine both mass and spin. Even a measured damping time requires a sufficiently reliable mode and noise model to add the intended independent information.

GMω220c3≃0.37367−0.08896ifor Schwarzschild,f=0.373672π(GM/c3),τ=GM/c30.08896\frac{GM\omega_{220}}{c^3}\simeq0.37367-0.08896i\quad\text{for Schwarzschild},\qquad f=\frac{0.37367}{2\pi(GM/c^3)},\qquad \tau=\frac{GM/c^3}{0.08896}
The quoted mode is the fundamental gravitational ℓ=2 mode in the nonrotating limit; m is degenerate there. Mass is the remnant mass and GM/c³ is a time. The values are approximate eigenvalue data, not coefficients derived from a simple spring oscillator. The fundamental value is also tabulated directly as Mω in Kokkotas and Schmidt, table 1.↔ Scroll wide equations horizontally.

A ringdown fit is conditional on its signal model

A complete retarded perturbation response can include prompt behavior and late-time power-law tails as well as a quasinormal-mode-dominated interval. A finite mode sum is therefore not a complete representation of all times or arbitrary initial data. The merger-to-linear transition and the choice of fit start time matter, and nonlinear contributions or uncertain overtone content can bias an apparently precise parameter estimate.

Testing a remnant spectrum against a black-hole prediction requires identified modes, adequate signal-to-noise ratio and controlled waveform and detector systematics. Different angular harmonics project differently with inclination, while noise can imitate weak additional components. Multiple consistent modes can strengthen tests of a common mass and spin, but one damped sinusoid is not by itself a theorem about the remnant geometry or a model-independent demonstration of every assumption behind a no-hair test.

Work it through

Convert a dimensionless mode to detector-frame scales

Take a Schwarzschild remnant with observer-frame mass 60 solar masses. Use GM_sun/c³=4.9255 microseconds and the displayed eigenvalue to find f, τ and Q. State how f and τ change for 120 solar masses with the same dimensionless mode.

  1. The mass time is 60×4.9255 microseconds=295.53 microseconds=2.9553×10⁻⁴ s. Keeping this conversion explicit avoids inserting the mass in kilograms into a dimensionless table without the required constants.

  2. The cyclic frequency is 0.37367/[2π(2.9553×10⁻⁴)]≈201.2 Hz. The amplitude damping time is (2.9553×10⁻⁴)/0.08896≈0.003322 s, or 3.32 ms.

  3. The quality factor is 0.37367/(2×0.08896)≈2.10. Doubling observer-frame mass gives about 100.6 Hz and 6.64 ms with the same Q. This is a mass scaling, not evidence for a different angular mode.

The result

The 60-solar-mass mode has frequency about 201 Hz, amplitude damping time 3.32 ms and Q≈2.10. The quoted mass is already observer-frame, so no extra redshift factor should be applied again.

The living notebook

Explore the model

Interactive lab

Interactive lab · oscillator

Damping can remove oscillation in a different model

Reset and set both controls to Damping ratio ζ=0 and Natural frequency ω0=1; one Step gives time 0.5 and displacement cos(0.5)≈0.878. For a fresh comparison, Reset and set both controls to ζ=1 and ω0=1; one Step gives displacement 1.5 exp(−0.5)≈0.910 and critical damping. The initial conditions are x=1 and velocity=0.

Damping can remove oscillation in a different modelUnit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Current readings: Displacement x(t): 1; Velocity v(t): 0; Regime: underdamped. All parameters are adjustable with the labeled controls below.02.557.510−1.15−0.5700.571.15Time · sDisplacementunderdamped
Displacement x(t)Current time
Displacement x(t)
1
Velocity v(t)
0
Regime
underdamped
t = 0 s

Unit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe analytic positive-stiffness oscillator illustrates damping regimes only. It does not impose black-hole boundary conditions, calculate a quasinormal spectrum or model a merger; its transient is not the tabulated black-hole mode.

Pause & reason

Check your understanding

With time dependence exp(−iωt), which imaginary part describes a stable damped mode?

Your turn

Try it yourself

A fitted damped sinusoid has f=250 Hz and amplitude damping time τ=4.0 ms. Find Q and the amplitude-envelope fraction after 8.0 ms. Compare its squared-envelope fraction, without identifying that quantity as an exact instantaneous gravitational energy density.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Use Q=πfτ and t/τ=2. Squaring an exponential envelope doubles its decay exponent.

Compare with the solution

The quality factor is π(250)(0.004)=π≈3.142. At 8 ms the amplitude envelope is e⁻²≈0.1353 of its initial value, while its squared envelope is e⁻⁴≈0.01832. An amplitude e-folding time and a squared-amplitude e-folding time therefore differ by a factor of two. These algebraic envelope statements do not supply a unique local gravitational-wave energy density or establish which black-hole mode was observed.

Keep the idea

Take it with you

  • Quasinormal modes are an open-boundary perturbation spectrum with convention-dependent damping signs.
  • Mass and redshift dilate mode timescales, while dimensionless spin and mode labels change the spectrum.
  • Mode content, fit start time, tails and detector systematics limit ringdown inference.

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