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 Bäckdahl and Kroon [2011 Class. Quantum Grav. 28 075010]. The key difference is that for the Minkowski spacetime a necessary condition is that ∇ A A ′ κ ¯ A ′ ≠ 0 . In this note it is shown that if ∇ A A ′ κ ¯ A ′ = 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 .