Newton makes every orbit a closed ellipse — but Mercury's perihelion creeps forward 43 arc-seconds per century, which Newton cannot explain. Does General Relativity produce exactly that, with no fudge factor?
▶ Launch the interactive simulationThe Schwarzschild metric reduces (in the orbital plane) to Newton's orbit equation plus one relativistic term: u'' + u = GM/h² + (3GM/c²)u², i.e. a central force a = −[GM/r² + 3GM·h²/(c²r⁴)]. Integrate it with a symplectic leapfrog (energy conserved to ~1e-13), measure the per-orbit perihelion advance Δϖ, sweep the relativistic strength, and compare to the weak-field law Δϖ = 6πGM/(c²a(1−e²)). A Newtonian null control drops the relativistic term.
the orbit PRECESSES, exactly as Einstein predicted. Drop the relativistic term and the ellipse closes (Δϖ≈0, the Newtonian null); turn it on and the perihelion advances by Δϖ = 6πGM/(c²a(1−e²)) per orbit — recovered to a few % at weak field, the residual growing as the genuine higher-order term as the field strengthens. Fed Mercury's real numbers (a, e, period, G, M☉, c) the formula gives 42.99″/century, matching the observed 43″ that confirmed General Relativity in 1915. The live demo EXAGGERATES the relativistic strength so the rosette is visible (Mercury's real advance is 0.1″ per orbit); this is the orbit equation derived from the Schwarzschild metric, not a full numerical-relativity metric solve