Derivation of geodesic deviation equation

This is an advanced General Relativity reference, not a complete university lesson. It assumes covariant differentiation, the Riemann tensor and a smooth two-parameter surface of neighbouring geodesics.

Setup and convention

Let x^a(τ,s) be a smooth family of affinely parametrised geodesics. Define

We use

Changing this Riemann-tensor convention changes the final overall sign.

Because τ and s are coordinates on the two-parameter surface, the coordinate vector fields commute:

For the torsion-free Levi-Civita connection this gives

Each s = constant curve is an affinely parametrised geodesic, so

Covariant derivation

Differentiate the separation vector twice along the reference geodesic, using D/Dτ = ∇U:

The resulting geodesic-deviation equation is

With the opposite definition (∇c∇d-∇d∇c)V^a = -R^a bcd V^b, the right-hand side is instead preceded by a minus sign. A derivation that gives a sign without stating its curvature convention is therefore incomplete.

Interpretation and stopping point

ξ^a describes infinitesimal separation within the geodesic family, while D²ξ^a/Dτ² is the relative covariant acceleration. For a fixed reference geodesic, the equation is linear in ξ^a. Its solution also needs the initial separation and initial relative covariant velocity.

Zero curvature gives D²ξ^a/Dτ² = 0. In a parallel-transported frame along the reference geodesic, its components therefore have the form ξ^a = A^a + B^aτ, with constant A^a and B^a. The separation can change even in flat spacetime when the initial relative velocity is nonzero. Curvature determines the relative acceleration; it does not supply the initial conditions.

This reference stops before the general analysis of Jacobi fields, conjugate points and applications to Raychaudhuri’s equation. For the derivation and a worked sphere example, see David Tong’s General Relativity notes, section 3.3.4.