Solution
= 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 $\phi=-\log t$, for which $\xi(\phi)=-1$. Part (b) gives the explicit <Lorentzian metric>
$$
\boxed{ds^2=t^{-2}(-dt^2+dx^2+dy^2+dz^2)}
$$
with the requested <Killing vector field> components. The restriction $t>0$ makes the logarithm smooth and the conformal factor nonzero.