= Conformal preservation of null geodesic paths
{title2=$\widetilde g=\Omega^2g$}
The difference of the two <Levi-Civita connections> is $\widetilde\Gamma^a{}_{bc}-\Gamma^a{}_{bc}=\delta^a_b\partial_c\log\Omega+\delta^a_c\partial_b\log\Omega-g_{bc}g^{ad}\partial_d\log\Omega$. Along a <null geodesic> with tangent $k$, the last term vanishes and the remaining acceleration is $2k(\log\Omega)k$. It is parallel to the tangent, so changing the <affine parameter> removes it. Thus a positive <conformal factor> preserves unparametrized <null geodesics>, although it generally does not preserve timelike or spacelike geodesics.
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