 Solution

In Minkowski spacetime, the dilation field $\xi =t{\partial }_t+x{\partial }_x+y{\partial }_y+z{\partial }_z$ has ${\partial }_a{\xi }^c={\delta }_a^c$, so ${\mathcal{L}}_{\xi }\eta =2\eta$ and $\psi =2$. On $t>0$, take $\varphi =-\log t$, for which $\xi (\varphi )=-1$. Part (b) gives the explicit Lorentzian metricwith the requested Killing vector field components. The restriction $t>0$ makes the logarithm smooth and the conformal factor nonzero.