Geodesics & Free Fall

In general relativity, freely falling objects follow geodesics — the straightest possible paths through curved spacetime. There is no gravitational force; only the geometry of spacetime guiding motion. Planets orbit stars not because a force pulls them, but because they're following the curves of spacetime carved by mass.

Orbits in general relativity are richer than Newton predicted. Elliptical orbits precess — the ellipse slowly rotates with each revolution. This precession of Mercury's perihelion was the first triumph of general relativity, resolving a 60-year anomaly in 1915.

Try it: Use the presets to see different orbit types — Mercury-like precession (the ellipse rotates), the innermost stable circular orbit (ISCO), and dramatic plunge orbits that spiral into the black hole. Toggle the Newtonian comparison to see why GR was needed.

The geodesic equation d²xμ/dτ² + Γμ_αβ (dxα/dτ)(dxβ/dτ) = 0 tells us exactly how free particles move. The Christoffel symbols Γ encode the "gravitational field" — really the connection coefficients of the metric. Step through the numerical integration to see each term at work.

The Schwarzschild effective potential V_eff(r) = (1 - r_s/r)(1 + L²/r²) has a crucial difference from Newton's: a barrier near the black hole that allows particles to plunge inward. In Newton's theory, angular momentum always provides a centrifugal barrier. In GR, the barrier has finite height — overcome it, and you fall in.

Spacetime has three types of geodesics: timelike (massive particles, ds² < 0), null (light, ds² = 0), and spacelike (ds² > 0, not physically traversable by matter or light). Each type follows different paths through the Schwarzschild geometry.