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Black Hole Geodesics

A Schwarzschild black hole rendered by the reused real-time lensing shader: the background star field is bent, the photon sphere glows as a ring, the shadow is a true black disc, and a Doppler-beamed accretion disc circles it. Below, an equatorial plane lets you fire test photons and massive particles whose geodesics are integrated from the exact orbit equation. The drama is the critical impact parameter b = 3 sqrt(3) M: a photon aimed just outside it loops the photon sphere and escapes, one just inside is captured. The effective potential V(r) shows why the innermost stable circular orbit sits at 6 M.

Figure 1. Ray-marched Schwarzschild (Kerr spin slider) black hole, with equatorial null/timelike geodesics integrated from the orbit equation d2u/dϕ2+u=3Mu2d^2u/d\phi^2 + u = 3Mu^2.

WHAT TO TRY

  • Watch the star field warp: the Schwarzschild metric bends every background ray, so stars near the line of sight smear into arcs around the hole. The lensing is the real-time geodesic integration.
  • Find the photon sphere at 1.5 Schwarzschild radii: light there orbits the hole, glowing as a bright ring at the edge of the black shadow. Inside it, nothing escapes.
  • Drag to orbit the camera: the shadow stays circular from every angle, and the Einstein ring of a background source closes up when the alignment is perfect.