Geodesic Deviation — Tidal Forces & Riemann Tensor

The geodesic deviation equation D²ξ/dτ² = -R·u·u·ξ describes how nearby geodesics converge or diverge due to spacetime curvature. Visualize tidal stretching near a mass and the Jacobi field evolution.

Spacetime Geometry

Mass M (GM/c²): 1.0 Initial separation ξ₀: 0.5 Orbital radius r: 5.0 Tidal direction:

Curvature

R^r_trt (radial): —

R^θ_tθt (transv): —

Tidal acc: —

Separation ξ(τ): —

Geodesic deviation:
D²ξ^μ/dτ² = -R^μ_νρσ u^ν ξ^ρ u^σ

Schwarzschild tides:
δa_r = +2GM/r³ · ξ_r
δa_θ = -GM/r³ · ξ_θ

Geodesic Deviation — Tidal Forces & Riemann Curvature

Spacetime Geometry

Curvature