Schwarzschild Geodesics
Particle and photon orbits determined by the curvature of spacetime near a non-rotating black hole
⚫ Schwarzschild Geodesics — General Relativity
In Newtonian gravity orbits close perfectly. General relativity adds extra curvature — orbits precess. The Schwarzschild metric around a mass M (non-rotating, uncharged) gives the geodesic equations:
(dr/dτ)² = Ẽ² − V²_eff(r) V²_eff = (1 − Rs/r)(1 + L̃²/r²)
where Ẽ = E/mc² (energy per unit mass) and L̃ = L/mc (angular momentum per unit mass). Rs = 2GM/c² is the Schwarzschild radius (event horizon).
- Stable circular orbit: minimum of V_eff at r = L̃²/(1 ± √(1−3Rs²/L̃²))
- ISCO (innermost stable circular orbit): r = 3Rs — smallest stable orbit
- Photon sphere: r = 1.5Rs — unstable circular photon orbit
- Plunging orbit: particle crosses horizon when Ẽ² > V²_eff(Rs)
Mercury's 43″/century perihelion precession was Einstein's first observational triumph — predicted perfectly by this simulation.