Affine parameter change under conformal rescaling ($d\tilde{\tau }/d\tau =C{\Lambda }^2$)

If $\tilde{g}={\Lambda }^{2}g$, a null geodesic with affine parameter $\tau$ for $g$ has affine parameter $\tilde{\tau }$ for $\tilde{g}$ when $d\tilde{\tau }/d\tau =C{\Lambda }^2$ along the curve. Indeed the conformal change of the Levi-Civita connection gives ${\tilde{\nabla }}_{V}V=2V(\log |\Lambda |)V$; rescaling $V$ by $d\tau /d\tilde{\tau }$ cancels this tangential acceleration. Constant $C>0$ preserves the direction of parametrization. Agreement of unparametrized null geodesics does not imply agreement of their affine parameters or completeness.