= Affine normalization in a Kaluza-Klein geodesic reduction
For $ds_5^2=\eta_{ab}dx^adx^b+(dy+A_adx^a)^2$, with $A$ independent of $y$, an <affine parameter> $\lambda$ makes $\xi=dy/d\lambda+A_a dx^a/d\lambda$ and the tangent norm $\mathcal N=g_{AB}\dot x^A\dot x^B$ constant. The four-dimensional interval satisfies
$$
du^2=(\mathcal N-\xi^2)d\lambda^2.
$$
Consequently the formula $d\lambda^2=du^2/(1-\xi^2)$ assumes the additional normalization $\mathcal N=1$; affine parametrization alone does not set this value. With base <metric signature> $+---$ and $u=c\tau$, the projected curve is timelike only for $\mathcal N>\xi^2$. The normalized case gives $d\tau/d\lambda=\sqrt{1-\xi^2}/c$. The positive fibre term and the $+---$ base together have two positive directions; the same sign convention must be retained when interpreting the reduction.
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