A vector field $\xi^{a}$ is said to be a conformal Killing vector field of the metric $g_{a b}$ if

$$\xi_{(a ; b)}=\frac{1}{2} \psi g_{a b}$$

for some scalar field $\psi$. It is a Killing vector field if $\psi=0$.

(a) Show that $(*)$ is equivalent to

$$\xi^{c} g_{a b, c}+\xi_{, a}^{c} g_{b c}+\xi_{, b}^{c} g_{a c}=\psi g_{a b}$$

(b) Show that if $\xi^{a}$ is a conformal Killing vector field of the metric $g_{a b}$, then $\xi^{a}$ is a Killing vector field of the metric $\mathrm{e}^{2 \phi} g_{a b}$, where $\phi$ is any function that obeys

$$2 \xi^{c} \phi_{, c}+\psi=0 .$$

(c) Use part (b) to find an example of a metric with coordinates $t, x, y$ and $z$ (where $t>0)$ for which $(t, x, y, z)$ are the contravariant components of a Killing vector field. [Hint: You may wish to start by considering what happens in Minkowski space.]