= Orthogonality of proper-time deviation vectors
{title2=$\frac{d}{d\tau}g(S,T)=0$}
For a <geodesic variation> of unit <timelike geodesics>, <proper time> normalization holds across the entire variation: $g(T,T)=-1$. <Metric compatibility> and $\nabla_TS=\nabla_ST$ then give $d g(S,T)/d\tau=\tfrac12S(g(T,T))=0$. 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.
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