Null affine parameter under a conformal rescaling ($d\tilde{\lambda }={\Omega }^{2}d\lambda$)

If $g$ is rescaled to $\tilde{g}={\Omega }^{2}g$, a null geodesic retains its unparametrized path. For its original affine tangent $k$, the changed connection gives ${\tilde{\nabla }}_{k}k=2k(\log \Omega )k$. Rescaling the tangent by ${\Omega }^{-2}$ cancels this term and proves the displayed affine-parameter relation, up to a constant multiplier. A conformal boundary may therefore be reached at finite unphysical affine parameter while physical affine parameter is infinite.