Geodesic Deviation — Jacobi Equation

Neighboring geodesics · curvature causes convergence or divergence

Surface

Curvature Info

SurfaceSphere
Gauss curv K+1
BehaviorConverge
Jacobi eq.J''+KJ=0
SolutionJ=sin(t)

About

The Jacobi equation D²J/dt²+R(J,T)T=0 governs how nearby geodesics diverge. For constant curvature K:
• K>0 (sphere): J∝sin(√K·t) — geodesics converge, meet at antipodal points.
• K=0 (flat): J∝t — linear divergence.
• K<0 (hyperbolic): J∝sinh(√|K|·t) — exponential divergence = chaos.
This is the geometric heart of tidal forces in GR.