Twistor initial data characterisation of pp-waves

This note gives a concise derivation of a twistor-initial-data characterisation of pp-wave spacetimes in vacuum. The construction is based on a similar calculation for the Minkowski spacetime in [Class. Quantum Grav. 28 075010]. The key difference is that for the Minkowski spacetime a necessary condition is that ${\mathrm{\nabla }}_A{}^{A^{\mathrm{\prime }}}{\bar{\kappa }}_{A^{\mathrm{\prime }}}\mathrm{\ne }0$. In this note it is shown that if ${\mathrm{\nabla }}_A{}^{A^{\mathrm{\prime }}}{\bar{\kappa }}_{A^{\mathrm{\prime }}}=0$ then the development is a pp-wave spacetime. Furthermore, it is shown that such condition propagates off the initial hypersurface, which, in turn, gives a twistor initial data characterisation of pp-waves.