Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 3 / a / Solution

The retarded and advanced null coordinates invert toThusand the Minkowski metric becomes

The coordinate basis transforms by the chain rule:or equivalentlyIn the $(z,t)$ diagram, ${\partial }_t$ points vertically upward and ${\partial }_z$ horizontally right. The vector ${\partial }_u$ points along the future-left null ray and ${\partial }_v$ along the future-right null ray; their factors of one half affect length in the coordinate drawing but not direction.

Using these derivative relations, the Minkowski wave operator in null coordinates is