Plunges

The plunge of a particle into a Kerr black hole is fixed by three conserved quantities: the energy E, the angular momentum L, and Carter's constant Q. Choose whether the plunge starts from infinity, or right inside the ISCO, and pick which mode you want to compute. Everything below is solved live in your browser.

from
Re [h(u, θ) eimφ] with u = t − r*. Drag to turn, scroll to zoom; The slider runs through Kerr–Schild time, and zero is when the particle crosses the horizon.
At this E and spin, the orbits inside the shaded region plunge; the others turn around and escape. Click to choose (L, Q).
The waveform of the mode, r hℓm(u)/μ: real part (solid), imaginary part (faint).
Frequency-domain waveform, |h̃ℓm(ω)|. Dashed: the fundamental quasinormal mode and its mirror.
How this is computed

Units G = c = M = 1. In Mino time λ (dτ = Σ dλ) the radial and polar motions decouple. A plunge has no radial turning point, so r falls monotonically. θ oscillates between θmin and π − θmin. It is followed through an angle χ, with cos θ = cos θmin cos χ; χ0 is its value at the start. Time and azimuth are integrated in ingoing Kerr coordinates, which cross the horizon smoothly. The equations are integrated with an adaptive Dormand–Prince method to relative accuracy 10−12.

“Infinity” is r0 = 1000M. From infinity (E ≥ 1), an orbit plunges when the radial potential R(r) stays positive all the way down to the horizon. At fixed L, raising Q only lowers R, so the plunging orbits are those below an edge Qc(L). On that edge the orbit would whirl forever on an unstable spherical orbit. The region is drawn in (L, √Q): for a non-spinning hole it is the half-disc L2 + Q < Lc(E)2.

From the ISCO, the constants are those of the innermost stable spherical orbit at inclination ι, where sin θmin = |cos ι|. The exact plunge has circled that orbit forever, which would put a line in the spectrum at mΩISCO. It is followed instead from three orbits before it leaves the ISCO (passes 0.97 rISCO), with its source switched on smoothly over the first of them. The animation runs in Kerr–Schild time v − r at 25M per second. Where the particle hardly moves inwards (|dr/dt| < 0.01, the slow whirl near the ISCO) it is fast-forwarded, by 0.01/|dr/dt| and at most ten times.

The radiation is computed in the frequency domain. For each frequency the Teukolsky equation is solved through the generalised Sasaki–Nakamura equation (Lo 2023), and the source is integrated along the orbit in Mino time (Drasco & Hughes 2006, with one misprint corrected). The strain of the mode is h(u, θ) = −2 ∫ dω Z(ω)/ω2 S(θ; ω) e−iωu, so its angular pattern changes in time as different frequencies dominate. The grid in ω has 256 points (512 from the ISCO) out to 1.8 times the quasinormal frequency, and is filled coarse-first.

A plunge from far away cannot simply start at a finite radius: the integral over the orbit would grow with it. The source is switched on smoothly instead, with a tanh step whose width is matched to each frequency, and the answer no longer depends on where.