where ξ^μ is the separation vector between nearby geodesics, u^μ = dx^μ/dτ is the
4-velocity, and R^μ_{νρσ} is the Riemann curvature tensor.
Tidal forces in Schwarzschild: near a mass M, the Newtonian tidal tensor
∂²Φ/∂xⁱ∂xʲ gives radial stretching ~ +2GM/r³ and transverse squeezing ~ −GM/r³.
The ratio −2:1:1 satisfies ∇²Φ=0 (vacuum). A freely falling body is
spaghettified: stretched along the radial direction and compressed transversely.
Gravitational waves produce alternating stretch/squeeze in ×/+ polarizations,
encoded in R¹₀₁₀ = −(1/2)ḧ₊ — the TT-gauge Riemann component.
LIGO detects this as differential arm-length change ΔL/L = h/2.