Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 35D (General Relativity)

The symmetrized covariant derivative equation is ${\nabla }_a{\xi }_b+{\nabla }_b{\xi }_a=\psi g_{ab}$. Lower ${\xi }_b=g_{bc}{\xi }^c$ and expand both derivatives. Using the Christoffel symbols of the metric and their symmetry, the coefficient of ${\xi }^c$ simplifies byHenceThis is the coordinate formula for the Lie derivative ${\mathcal{L}}_{\xi }g=\psi g$, proving equivalence with the Conformal Killing vector field condition.

Apply the product rule for the Lie derivative to $\tilde{g}=e^{2\varphi }g$:The prescribed transport equation makes this zero, so $\xi$ is a Killing vector field for $\tilde{g}$. This is a conformal rescaling that absorbs the original infinitesimal dilation.

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.