Transverse modulation in electrovac Brinkmann pp-waves: Maxwell consistency and curvature universality

Abstract

Electrovac pp--waves in Brinkmann form provide exact Einstein--Maxwell solutions for co--propagating null radiation. Motivated by lensing or scattering, one often ``modulates'' a plane electromagnetic wave by a weak transverse envelope $1+\gamma f(x,y)$. We show that, within the aligned null pp--wave ansatz ($A_v=0$, no $v$--dependence, $F_{xy}=0$) and enforcing the source--free Maxwell equations to $\mathcal O(\gamma)$, a generic profile $f(x,y)$ is incompatible with Maxwell: the transverse field $F_{ui}$ must be both divergence--free and curl--free on the transverse plane, hence $F_{ui}=\partial_i\Phi$ with $\Delta_\perp\Phi=0$. We give a minimal, polarization--agnostic gauge completion of the modulated potential and prove a cancellation theorem: under standard decay/regularity (or zero--mode) conditions that exclude additional harmonic transverse modes, all $\mathcal O(\gamma)$ dependence on $f$ drops out of $F_{ui}$ and therefore out of the electrovac source $T_{uu}$. Consequently, the electromagnetic contribution to the Brinkmann profile is universal at $\mathcal O(\gamma)$: the familiar cycle--averaged isotropic $r^2$ term plus an isotropic oscillatory correction at frequency $2\omega$, present only for non-circular polarisation. We isolate the residual Maxwell--admissible freedom as harmonic (holomorphic) transverse data and, by Kerr--Schild linearity, superpose an arbitrary co--propagating vacuum gravitational pp--wave, relating TT--gauge strain to Brinkmann amplitudes. Modelling genuinely localised beams, therefore, requires currents, non-null components, or more general Kundt/gyraton geometries.