= Conformal Killing geodesic identity
{title2=$d[g(X,U)]/ds=\lambda g(U,U)$}
Let $X$ be a <Conformal Killing vector field> with <conformal Killing factor> $\lambda$, and let $U$ be the <tangent vector> of a <geodesic> with <affine parameter> $s$. The <product rule>, <metric compatibility> and <geodesic equation> give
$$
\frac{d}{ds}g(X,U)=g(\nabla_UX,U)+g(X,\nabla_UU)
=U^aU^b\nabla_{(a}X_{b)}=\lambda g(U,U).
$$
Every <null geodesic> therefore has the <conserved quantity> $g(X,U)$. On a <timelike geodesic> with <proper time> as parameter, the derivative is $-\lambda$ in <metric signature> $(-,+,+,+)$. With a nonaffine parameter satisfying $\nabla_UU=fU$, an extra term $fg(X,U)$ appears; the conserved expression for a <null geodesic> is tied to its <affine parameter> normalization.
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