= Solution
Put $\omega=\log|\Lambda|$, which is smooth because $\Lambda$ is nowhere zero. The <conformal change of the Levi-Civita connection> is
$$
\widetilde\Gamma^a{}_{bc}-\Gamma^a{}_{bc}
=C^a{}_{bc}
=\delta^a_b\partial_c\omega+\delta^a_c\partial_b\omega
-g_{bc}g^{ad}\partial_d\omega.
$$
This follows directly by differentiating $\widetilde g_{ab}=e^{2\omega}g_{ab}$ and using $\widetilde g^{ab}=e^{-2\omega}g^{ab}$ in the <Christoffel symbol> formula.
Let $V=dx/d\tau$ be a <geodesic> tangent for $g$, with $\tau$ an <affine parameter>. Since $\nabla_VV=0$,
$$
\boxed{
\widetilde\nabla_VV
=2V(\omega)V-g(V,V)\operatorname{grad}_g\omega.
}
$$
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 $\widetilde V=fV$, with $f=d\tau/d\widetilde\tau$. Then
$$
\widetilde\nabla_{\widetilde V}\widetilde V
=f^2\left(2\frac{d\omega}{d\tau}+\frac{d\log f}{d\tau}\right)V.
$$
It vanishes when $f=C^{-1}\Lambda^{-2}$, or
$$
\boxed{\frac{d\widetilde\tau}{d\tau}=C\Lambda^2,\qquad C>0.}
$$
This <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 $\operatorname{grad}_g\omega$ orthogonal to $V$ generally survives. A change of parameter can remove only the tangential component of acceleration. As a concrete example use <Minkowski spacetime>,
$$
g=-dt^2+dx^2+dy^2+dz^2,\qquad
\Lambda=e^{kx},\quad k\ne0.
$$
The curve $(t,x,y,z)=(\tau,0,0,0)$ is a <timelike geodesic> for $g$, but along it
$$
\widetilde\nabla_{\partial_t}\partial_t=k\partial_x.
$$
This acceleration is not parallel to $\partial_t$, so the curve is not a <geodesic> of $\widetilde g$ under any reparametrization. Special non-null curves may still be preserved, for example when $\Lambda$ is constant; nullness is the general guarantee.
Back to article page
