Logarithmic angle-dependent gauge transformations at null infinity

TL;DR

The paper addresses how logarithmic angle-dependent U(1) gauge transformations—previously identified at spatial infinity—manifest at null infinity in electromagnetism. It develops a detailed, coordinate-system-based analysis (notably via hyperbolic coordinates) to derive the explicit form of these transformations on A_mu near null infinity and to compute their conserved charges, including an explicit Goldstone-like boundary field. A central result is that, with relaxed fall-offs that allow A_r to contain log r terms, logarithmic transformations remain physical and yield finite, well-defined charges that are canonically conjugate to the standard O(1) angle-dependent charges; the work also clarifies parity-condition choices and matching conditions across spatial and null infinity, and discusses higher-dimensional extensions. The findings refine the understanding of asymptotic symmetries in electromagnetism and offer a robust framework for connecting spatial infinity data to null infinity, with potential implications for gravitational analogs and memory effects.Overall, the paper provides explicit constructions, charge formulae, and matching relations that extend the asymptotic symmetry program to include logarithmic, angle-dependent transformations at null infinity, clarifying longstanding misconceptions and setting the stage for further applications in gravity and holography.

Abstract

Logarithmic angle-dependent gauge transformations are symmetries of electromagnetism that are canonically conjugate to the standard $\mathcal O(1)$ angle-dependent $u(1)$ transformations. They were exhibited a few years ago at spatial infinity. In this paper, we derive their explicit form at null infinity. We also derive the expression there of the associated "conserved" surface integrals. To that end, we provide a comprehensive analysis of the behaviour of the electromagnetic vector potential $A_μ$ in the vicinity of null infinity for generic initial conditions given on a Cauchy hypersurface. This behaviour is given by a polylogarithmic expansion involving both gauge-invariant logarithmic terms also present in the field strengths and gauge-variant logarithmic terms with physical content, which we identify. We show on which explicit terms, and how, do the logarithmic angle-dependent gauge transformations act. Other results of this paper are a derivation of the matching conditions for the Goldstone boson and for the conserved charges of the angle-dependent $u(1)$ asymptotic symmetries, as well as a clarification of a misconception concerning the non-existence of these angle-dependent $u(1)$ charges in the presence of logarithms at null infinity. We also briefly comment on higher spacetime dimensions.