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In constant sectional curvature $\kappa$, a deviation vector orthogonal to a unit timelike geodesic obeys $D^{2}S=\kappa S$, where $D={\nabla }_T$. Therefore $K=g(DS,DS)-\kappa g(S,S)$ is constant, since its derivative is $2g(DS,D^{2}S)-2\kappa g(S,DS)=0$. Its squared length $f=g(S,S)$ satisfies $\ddot{f}=2K+4\kappa f$. In four spacetime dimensions, $\kappa =R/12$.

For a geodesic variation $F(s,\tau )$, the deviation vector $S=F_{\ast }{\partial }_s$ describes infinitesimal separation at equal parameter values. With $T=F_{\ast }{\partial }_{\tau }$, commuting parameter derivatives give $[S,T]=0$. A torsion-free connection then gives ${\nabla }_{T}S={\nabla }_{S}T$. For an affinely parametrized variation, geodesic deviation states ${\nabla }_T^{2}S=R(T,S)T$ with the convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$.

For a geodesic variation of unit timelike geodesics, proper time normalization holds across the entire variation: $g(T,T)=-1$. Metric compatibility and ${\nabla }_{T}S={\nabla }_{S}T$ then give $dg(S,T)/d\tau =\frac{1}{2}S(g(T,T))=0$. Consequently an initially orthogonal deviation vector stays orthogonal. The common normalization is essential: an arbitrary family of affinely parametrized curves with different tangent norms does not imply this conclusion.

Let $x^a(\lambda ,s)$ be a smooth family of affinely parametrized null geodesics. The tangent is $U=\partial x/\partial \lambda$, and the deviation vector is $Z=\partial x/\partial s$, comparing nearby rays at the same affine parameter. The two parameter derivatives commute, soThus the separation vector is Lie-transported along the null geodesic congruence; it is not required to be parallel-transported. Differentiating once more gives the geodesic deviation equation. With curvature convention $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$,For optical measurements one uses transverse deviations with $U\cdot Z=0$ and identifies longitudinal additions proportional to $U$. The scalar $U\cdot Z$ is constant along affine null rays by the displayed transport relation, so an initially transverse deviation remains transverse.

Describe the geodesic variation by a smooth map $F(s,\tau )$, with each curve $\tau \mapsto F(s,\tau )$ a timelike geodesic parametrized by proper time. Its tangent vector is $T=F_{\ast }{\partial }_{\tau }$ and its deviation vector is $S=F_{\ast }{\partial }_s$. Thus $S$ measures the first-order separation of neighboring members at the same value of proper time.

Mixed coordinate derivatives on the parameter surface commute. Consequently, along the variation,For an embedded two-dimensional variation this is the usual Lie bracket of vector fields on its image; more generally the same identity is read along the map $F$ using its commuting parameter derivatives. Since the Levi-Civita connection is torsion-free, it follows that ${\nabla }_{T}S={\nabla }_{S}T$.

Use precisely the stated convention for the Riemann curvature tensor, and let $D={\nabla }_T$. The geodesic equation says ${\nabla }_{T}T=0$ because proper time is an affine parameter. The commuting variation and the torsion-free connection giveThus the geodesic deviation equation isHere $(R(X,Y)Z{)}^a=R^a{}_{bcd}{Z}^b{X}^c{Y}^d$. Declaring this index convention fixes the sign: writing $R(S,T)T$ instead would require a minus sign. A deviation vector satisfying this equation is a Jacobi field.