Hexadecapole at the heart of nonlinear electromagnetic fields

Abstract

In classical Maxwell’s electromagnetism, the monopole term of the electric field is proportional to r −2, while higher order multipole terms, sourced by anisotropic sources, fall-off faster. However, in nonlinear electromagnetism even a spherically symmetric field has multipole-like contributions. We prove that the leading subdominant term of the electric field, defined by nonlinear electromagnetic Lagrangian obeying Maxwellian weak field limit, in a static, spherically symmetric, asymptotically flat spacetime, is of the order O(r−6) as r→∞ . Moreover, using Lagrange inversion theorem and Faà di Bruno’s formula, we derive the series expansion of the electric field from the Taylor series of an analytic electromagnetic Lagrangian.