Past exam of the general relativity II course of the University of Oxford / 2017 / 2 / a / ii / Solution

Put $\omega =\log |\Lambda |$, which is smooth because $\Lambda$ is nowhere zero. The conformal change of the Levi-Civita connection isThis follows directly by differentiating ${\tilde{g}}_{ab}=e^{2\omega }{g}_{ab}$ and using ${\tilde{g}}^{ab}=e^{-2\omega }{g}^{ab}$ in the Christoffel symbol formula.

Let $V=dx/d\tau$ be a geodesic tangent for $g$, with $\tau$ an affine parameter. Since ${\nabla }_{V}V=0$,For a null geodesic the second term vanishes. The remaining acceleration is tangent to the same curve and can be removed by changing its parameter. Explicitly set $\tilde{V}=fV$, with $f=d\tau /d\tilde{\tau }$. ThenIt vanishes when $f=C^{-1}{\Lambda }^{-2}$, orThis affine parameter change under conformal rescaling shows that conformal preservation of null geodesic paths means preservation of unparametrized curves; the original affine parameter usually changes.

For a non-null geodesic, the component of ${\mathrm{grad}}_g\omega$ orthogonal to $V$ generally survives. A change of parameter can remove only the tangential component of acceleration. As a concrete example use Minkowski spacetime,The curve $(t,x,y,z)=(\tau ,0,0,0)$ is a timelike geodesic for $g$, but along itThis acceleration is not parallel to ${\partial }_t$, so the curve is not a geodesic of $\tilde{g}$ under any reparametrization. Special non-null curves may still be preserved, for example when $\Lambda$ is constant; nullness is the general guarantee.