Conformal change of the Levi-Civita connection (${\tilde{\Gamma }}^a{}_{bc}-{\Gamma }^a{}_{bc}={\delta }_b^a{\omega }_c+{\delta }_c^a{\omega }_b-g_{bc}{\omega }^a$)

For metric tensors related by $\tilde{g}=e^{2\omega }g$, differentiating the rescaled metric tensor in the Christoffel symbol formula gives the displayed difference, where ${\omega }_a={\partial }_a\omega$ and the index on ${\omega }^a$ is raised with $g$. Consequently ${\tilde{\nabla }}_{V}V={\nabla }_{V}V+2V(\omega )V-g(V,V){\mathrm{grad}}_g\omega$. The last term vanishes on a null vector, proving conformal preservation of null geodesic paths; for a non-null geodesic its orthogonal component generally changes the path.