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

TL;DR

This paper investigates the effects of transverse modulation in electrovac Brinkmann pp-waves, demonstrating that under Maxwell constraints, the electromagnetic contribution to the wave profile is universal at first order, with residual freedom linked to harmonic transverse data.

Contribution

It proves that transverse electromagnetic modulation in Brinkmann pp-waves is constrained by Maxwell equations, leading to a universal first-order profile and clarifies the conditions for superposing vacuum gravitational waves.

Findings

Transverse electromagnetic modulation must satisfy divergence and curl-free conditions.

First-order electromagnetic profile is universal and independent of the modulation function.

Residual Maxwell-admissible freedom is characterized by harmonic transverse data.

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.