= Conformal change of the Levi-Civita connection
{title2=$\widetilde\Gamma^a{}_{bc}-\Gamma^a{}_{bc}=\delta^a_b\omega_c+\delta^a_c\omega_b-g_{bc}\omega^a$}
For <metric tensors> related by $\widetilde g=e^{2\omega}g$, 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 $\widetilde\nabla_VV=\nabla_VV+2V(\omega)V-g(V,V)\operatorname{grad}_g\omega$. 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.
Back to article page
