Interactive · strong-field gravity
How matter and light move around a rotating (Kerr) black hole. The hole’s spin points along +z and every orbit here is equatorial, so the orbital plane never tilts — what precesses is the periapsis, in-plane. The left panel is the orbit; the right is the effective potential that governs it — turning points are where the energy line meets the curve. Distances are in units of the mass M. Set a = 0 for Schwarzschild.
A massive particle starts at rest radially at r₀ — no radial velocity, but still carrying angular momentum — so r₀ is a turning point. Choose prograde (co-rotating with the hole) or retrograde: at nonzero spin the two are physically different orbits. Spinning the hole up pulls the prograde ISCO in from 6M toward M and pushes the retrograde one out toward 9M. The shaded region marks where no stable circular orbit exists — watch it shrink as you spin the hole up prograde, and swell retrograde. A start there is still a valid geodesic; it simply cannot settle onto a circle. A photon is deflected as it passes, and can whirl near the photon orbit before escaping or being captured. Marked radii are the horizon r₊ = M + √(M² − a²), the photon orbit, and the ISCO — all computed for the current spin and orbit sense. The zoom rail down the right edge of the potential panel rescales only that panel's vertical axis: at the top it frames the barrier and the turning points; drag it down to pull back and watch V plunge toward the horizon.
Source: geodesic-explorer.html · MIT licensed — self-contained, no dependencies; save the page and it still runs.