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Schwarzschild Geodesics

Orbital Parameters

Presets

GR Field Equations

ds² = −(1−rs/r)c²dt²
    + dr²/(1−rs/r) + r²dΩ²

(dr/dτ)² = E²−(1−rs/r)(1+L²/r²)
dφ/dτ = L/r²
rs = 2GM/c²

Legend

Orbit
Event horizon
ISCO (r=3rs)
Photon sphere

About

In the Schwarzschild spacetime (non-rotating black hole), test particle geodesics precess — unlike Newtonian gravity where ellipses close. The precession of Mercury's perihelion (43"/century) was the first GR confirmation. Near the ISCO (r=6M), orbits become unstable. Below r=3M (photon sphere), even light cannot escape to infinity.