Transverse modulation in electrovac Brinkmann pp-waves: Maxwell consistency and curvature universality
Galin S. Valchev
TL;DR
The paper addresses whether a weak transverse modulation 1+γ f(x,y) of a plane electromagnetic wave can be realized within aligned electrovac Brinkmann pp-waves. It proves a Maxwell-consistency theorem: in the aligned null sector, F_{ui} must be divergence- and curl-free on the transverse plane, so F_{ui} = ∂_iΦ with Δ⊥Φ = 0, forcing the envelope to be harmonic and introducing only a residual harmonic datum Ψ_H for gauge completion. Consequently, to order γ the electromagnetic source T_{uu} becomes universal, lacking explicit dependence on the transverse profile f under standard boundary conditions, and the Brinkmann profile h has a cycle-averaged isotropic r^2 term plus an oscillatory breathing term at frequency 2ω for non-circular polarization; genuinely transverse structure is encoded in harmonic (vacuum) data rather than the electrovac source. The authors further show that Kerr–Schild linearity allows superposition with an arbitrary co-propagating vacuum gravitational pp-wave, analyze near-axis curvature and test-particle dynamics, and conclude that modeling localized beams within strict electrovac pp-waves requires currents, non-null components, or more general Kundt/gyraton geometries.
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 $1+γf(x,y)$. 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 $\mathcal O(γ)$, 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 $F_{ui}=\partial_iΦ$ with $Δ_\perpΦ=0$. 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 $\mathcal O(γ)$ 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 $\mathcal O(γ)$: the familiar cycle--averaged isotropic $r^2$ term plus an isotropic oscillatory correction at frequency $2ω$, 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.