Affine normalization in a Kaluza-Klein geodesic reduction

For $d{s}_5^2={\eta }_{ab}d{x}^{a}d{x}^b+(dy+A_{a}d{x}^a{)}^2$, with $A$ independent of $y$, an affine parameter $\lambda$ makes $\xi =dy/d\lambda +A_{a}d{x}^a/d\lambda$ and the tangent norm $\mathcal{N}=g_{AB}{\dot{x}}^A{\dot{x}}^B$ constant. The four-dimensional interval satisfiesConsequently the formula $d{\lambda }^2=d{u}^2/(1-{\xi }^2)$ 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.