Black Hole Geodesics
Trajectories of photons and massive particles integrated directly from the Schwarzschild geodesic equations. Light bends near the event horizon, orbits indefinitely at the photon sphere, or spirals inward past capture. Massive bodies trace rosette orbits with perihelion advance that makes Mercury look negligible. Every path is exact GR — no approximations.
Schwarzschild metric
For a non-rotating, uncharged mass M, Einstein's field equations are solved exactly by the Schwarzschild metric (G = c = 1):
ds² = −(1 − rs/r) dt² + (1 − rs/r)⁻¹ dr² + r² dΩ²
The Schwarzschild radius rs = 2GM/c² marks the event horizon — the causal boundary beyond which no trajectory can return. For the sun, rs ≈ 3 km; for a 10 M☉ stellar black hole, about 30 km. In this simulation M = 1 and rs = 2, so all distances are in units of M.
Geodesic equations
In the equatorial plane the two conserved quantities along any geodesic are the specific energy E and specific angular momentum L. The radial equation follows from the 4-velocity normalisation gμνuμuν = −κ (κ = 1 massive, κ = 0 photon):
ṙ² = E² − Veff(r)
Veff(r) = (1 − rs/r)(κ + L²/r²)
φ̇ = L / r²
Differentiating once gives the radial acceleration, which is what the integrator steps forward using 4th-order Runge-Kutta:
r̈ = −κM/r² + L²/r³ − 3ML²/r⁴
The last term — 3ML²/r⁴ — is the pure GR correction absent from Newtonian
gravity. It is repulsive at large L and r but dominant near the horizon,
responsible for the photon sphere and the non-closing orbits of massive bodies.
Special radii
Three characteristic radii define the geometry of the field. The event horizon at r = 2M is the point of no return. The photon sphere at r = 3M is where the GR correction exactly balances centrifugal repulsion, allowing photons in unstable circular orbits. The innermost stable circular orbit (ISCO) at r = 6M is the last orbit where a massive particle can remain without plunging — it sets the inner edge of accretion disks and is directly observable in X-ray binaries.
rhorizon = 2M
rphoton = 3M
rISCO = 6M
bcrit = 3√3 M ≈ 5.196 M
Photons with impact parameter b < bcrit are captured; those with b > bcrit are deflected. At b = bcrit exactly, a photon asymptotically approaches r = 3M, spiralling indefinitely around the photon sphere (in theory — numerically it orbits until the simulation halts it).
What to look for
Deflection — a fan of photons bending past the hole. Compare the angular deflection δφ ≈ 4M/b to the Newtonian prediction of 2M/b: GR predicts exactly twice as much. Eddington confirmed this ratio at the 1919 eclipse.
Capture — photons with b just below bcrit spiral in; those just above escape after many orbits. The transition is knife-sharp.
Photon ring — multiple photons straddling bcrit. The ones that nearly orbit the photon sphere pile up into a bright ring — the same feature seen in the Event Horizon Telescope images of M87* and Sgr A*.
Precession — a massive particle in an eccentric orbit traces a rosette rather than a closed ellipse. The perihelion advances by ≈ 6πM/((r₁+r₂)/2 − 3M) per orbit. For Mercury in the sun's field this is 43 arcseconds per century; here, with r₁ ≈ 7M, it is visible within a single orbit.
ISCO — the green circular orbit at r = 6M, right at the stability boundary. A gentle nudge inward would send it plunging to the horizon.
Plunge — a massive body launched from r = 8M with angular momentum below the ISCO threshold (L = 2.5 < LISCO = 2√3 ≈ 3.46). Without a stable turning point inside ISCO, it spirals directly into the event horizon.