★★★ Advanced

🕳 Schwarzschild Geodesics

Orbits around a non-rotating black hole in Schwarzschild spacetime. Watch GR perihelion precession, find the photon sphere (r = 1.5r_s), the innermost stable circular orbit ISCO (r = 3r_s), and launch plunging geodesics that cross the event horizon.

Energy Ẽ = 0.935 Ang. Mom. L̃/r_s = 3.46 r₀/r_s = 6.0

r/r_s = — v/c = — Precession = —°/orbit Orbit type: —

(dr/dφ)² = r⁴/L̃² [Ẽ² − (1−r_s/r)(1 + L̃²/r²)] | r_phot = 1.5r_s | r_ISCO = 3r_s

Schwarzschild Spacetime & Geodesics

In flat space (Newton) orbits are perfect ellipses that never precess. In Schwarzschild spacetime the effective potential (per unit mass) is V_eff(r) = (1−r_s/r)(1 + L̃²/r²), adding a GR term −r_sL̃²/r³ that makes close orbits unstable.

Key radii (r_s = Schwarzschild radius = 2GM/c²):

r = r_s — event horizon; nothing can escape
r = 1.5 r_s — photon sphere (unstable circular light orbit)
r = 3 r_s — ISCO: innermost stable circular orbit
r > 3 r_s — stable circular orbits exist

Mercury's famous perihelion precession of 43 arcsec/century is this same GR correction applied to the Sun's weak gravity field.