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

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

The phase isBecause the transverse-traceless gauge perturbation has no component alongonly the third term of the linearized Riemann curvature operator contributes to the required contraction:The derivatives and transverse polarization contraction areTherefore the Newman--Penrose scalar Psi4 isIts real and imaginary parts encode the plus and cross gravitational wave polarizations, up to the stated Riemann curvature tensor and complex null tetrad conventions.

Only the transverse coordinates occur in the perturbation, so its tensor components are unchanged by replacing $(t,z)$ with $(u,v)$. The plane gravitational wave in linearized gravity isEvery component depends on $u$ alone. The Minkowski wave operator in null coordinates therefore givesThus the field satisfies the vacuum Linearized Einstein equations.

Using the stated Christoffel symbols in the definition of the Riemann curvature tensor, the two nontrivial contractions areHenceand the vacuum Einstein field equations reduce toThis is the diagonal Rosen coordinates for a plane gravitational wave equation.

For the plus-polarized wave, setComparison of the transverse metric components gives, to linear order,orThereforeThe Vacuum Einstein equations are consequently satisfied at linear order. A real gravitational wave is obtained by taking the real part of the complex plane-wave notation.