Multipole charge conservation and implications on electromagnetic radiation

TL;DR

The paper identifies a class of residual Maxwell gauge transformations in Lorenz gauge that carry nontrivial charges, which equal a fixed fraction of the electric multipole moments once the soft field contribution is included, making the total multipole charge conserved. Using the covariant phase space formalism, it derives a hard/soft decomposition of these charges and shows how the conservation laws constrain radiation from dynamical charged systems. It also extends the construction to magnetic multipoles via electric–magnetic duality and demonstrates that radiation is subject to an infinite set of constraints determined by the initial and final stationary multipole data. The work clarifies a deep connection between gauge symmetries and classical multipole structure, with potential implications for soft theorems, edge states, and extensions to gravity and black hole spacetimes.

Abstract

It is shown that conserved charges associated with a specific subclass of gauge symmetries of Maxwell electrodynamics are proportional to the well known electric multipole moments. The symmetries are residual gauge transformations surviving after fixing the Lorenz gauge, and have nontrivial charge. These "Multipole charges" receive contributions both from the charged matter and electromagnetic fields. The former is nothing but the electric multipole moment of the source. In a stationary configuration, there is a novel equipartition relation between the two contributions. The multipole charge, while conserved, can freely interpolate between the source and the electromagnetic field, and therefore can be propagated with the radiation. Using the multipole charge conservation, we obtain infinite number of constraints over the radiation produced by the dynamics of charged matter.

Figures (2)

• Figure 1: The dipole charge within a sphere of radius $R$, produced by a charged particle moving with constant velocity along the $z$ direction. The discontinuities occur when the particle enters and exits the integration surface.
• Figure 2: A distribution of charged matter (colored), which is stationary in far past and far future (regions I,III) and radiating in the region II. The charges are computed at two time slices $\Sigma_-$ and $\Sigma_+$ before and after the radiation phase, and are given by surface integrals over $S_\pm$ of radius $R$ where $S_\pm=\partial \Sigma_\pm$. These two surfaces are connected by a timelike hypersurface $\Sigma_B$ of constant radius (not drawn). Note that both $S_\pm$ reside in region I as $R \to \infty$.