 Solution

An affine geodesic has constant squared tangent norm, but that constant depends on the scale of its parameter. WriteUsing $W=\xi$ in the line element givesThe affine normalization in a Kaluza-Klein geodesic reduction matters: the specific formula requested in the paper uses $\mathcal{N}=1$, namely $\lambda$ equal to the positive five-dimensional arclength. With that normalization,A timelike four-dimensional projection has $d{u}^2=c^{2}d{\tau }^2>0$, requiring $|\xi |<1$ in this normalization. At $|\xi |=1$ the projection is null and has no proper-time four-velocity. For an arbitrary affine parameter the first boxed relation, with $\mathcal{N}$ retained, is the general result.