The curve $\gamma, x^{a}=x^{a}(\lambda)$, is a geodesic with affine parameter $\lambda$. Write down the geodesic equation satisfied by $x^{a}(\lambda)$.

Suppose the parameter is changed to $\mu(\lambda)$, where $d \mu / d \lambda>0$. Obtain the corresponding equation and find the condition for $\mu$ to be affine. Deduce that, whatever parametrization $\nu$ is used along the curve $\gamma$, the tangent vector $K^{a}$ to $\gamma$ satisfies

$$\left(\nabla_{\nu} K\right)^{[a} K^{b]}=0 \text {. }$$

Now consider a spacetime with metric $g_{a b}$, and conformal transformation

$$\tilde{g}_{a b}=\Omega^{2}\left(x^{c}\right) g_{a b}$$

The curve $\gamma$ is a geodesic of the metric connection of $g_{a b}$. What further restriction has to be placed on $\gamma$ so that it is also a geodesic of the metric connection of $\tilde{g}_{a b}$ ? Justify your answer.