Geodesics around a non-rotating black hole. Two scales matter: the event horizon at r = 2M and the photon sphere at r = 3M (where light orbits in unstable circles). Below the innermost stable circular orbit (ISCO) at r = 6M, no stable circular orbit exists — matter spirals in. The accretion disk's inner edge is exactly here, and the EHT image traced it.
Orbit (units of M)
Effective potential V_eff(r)
starting r/M 10.0
angular momentum L/M 3.80
energy E 0.970
speed 2.0
three radii to remember
The effective potential for a particle of unit mass and angular momentum L around a Schwarzschild black hole (mass M, in units c = G = 1):
V_eff(r) = (1 − 2M/r)(1 + L²/r²)
In Newtonian gravity, V_eff has a stable minimum for any L. In Schwarzschild, the −2ML²/r³ correction creates a maximum for small L and a minimum for L ≥ 2√3·M. Three radii emerge:
• Event horizon, r = 2M: V_eff → 0; once inside, no escape, the future is the singularity.
• Photon sphere, r = 3M: where light orbits in unstable circles. The "bright ring" in EHT images of M87 and Sgr A* is the multi-orbit accumulation of photons here.
• ISCO, r = 6M: the innermost stable circular orbit. For L = 2√3·M, the V_eff inflection is exactly here. Below this, no circular orbit is stable — matter from an accretion disk plunges through this radius into the hole.
Try the presets: the "bound elliptical" orbit has the most distinctive GR signature — instead of a closed ellipse (as Kepler would predict), the orbit precesses. This is the same 43"/century precession Le Verrier identified for Mercury in 1859, which classical Newtonian gravity couldn't explain. Einstein computed it correctly in 1915 — the first triumph of GR.