Geodesic Deviation & Tidal Forces

Geodesic Deviation Equation (Jacobi equation) describes how nearby geodesics separate due to spacetime curvature: D²ξᵘ/dτ² = −Rᵘ_νρσ u^ν ξ^ρ u^σ where ξ is the separation vector, u is the 4-velocity, and R is the Riemann tensor. In Newtonian gravity, this reduces to the tidal force: d²ξᵢ/dt² = (∂²Φ/∂xᵢ∂xⱼ)ξⱼ. Near a black hole (Schwarzschild): radial geodesics converge (tidal compression), transverse ones diverge (spaghettification). A gravitational wave stretches and squeezes a ring of test masses alternately in + and × polarizations.