Electromagnetic helicity flux density for radiative systems

TL;DR

This work formalizes the electromagnetic helicity flux density as a distinct radiative observable, defined at future null infinity ${\cal I}^+$, and clarifies its relationship to magnetic and optical helicities through CS structures. By deriving the general flux-density expressions from a broad current and detailing the point-particle case, the authors show that the helicity flux can reveal a handedness projection $\mathbf n\cdot(\mathbf v\times\mathbf a)$, with the total helicity often vanishing for simple trajectories. A comprehensive multipole expansion demonstrates that the radiative helicity flux is governed by cross terms between electric and magnetic multipoles, with dipole results recovering known power formulas and helicity requiring simultaneous excitation of both multipoles. The paper further analyzes four toy-models to illustrate angular distributions, and discusses astrophysical applications to pulsars, providing quantitative estimates and suggesting observational routes via polarization measurements. Overall, the EM helicity flux density provides a rich, angle-resolved diagnostic of radiative sources with potential implications for astrophysical systems and beyond.

Abstract

We show that the helicity flux density is distinguished from magnetic helicity by analysing Hopf solitons. The electromagnetic (EM) helicity flux and the magnetic helicity are Chern-Simons terms at different hypersurfaces. We find the helicity flux density for a point charge moving with an acceleration, extending the Liénard-Wiechert angular distribution of radiant power. We also derive the multipole expansion of the helicity flux density, generalizing the Larmor's formula for the radiant power. These formulae have been applied to discuss the helicity flux density in several toy models such as circular and helical motion as well as soft bremsstrahlung. We also comment on the potential applications of the EM helicity flux density to pulsar systems.

Figures (7)

• Figure 1: This figure schematically illustrates a method for measuring the EM radiative helicity flux density within the Penrose diagram of Minkowski spacetime. An EM wave, generated by a source, propagates through the bulk and interacts with a detector (e.g., a magnetic dipole) positioned near $\mathcal{I}^+$, inducing a precession of the dipole. In the diagram, the magnetic dipole is depicted as a circular loop of electric current, with an arrow indicating its direction according to the right-hand rule. The dashed line represents the worldline of the detector at a constant radius $r=R_0$. A similar setup, employing a free-falling gyroscope as the detector to measure the gravitational helicity flux density, can be found in Seraj:2022qyt.
• Figure 2: Two different approaches to cover the entire Minkowski spacetime
• Figure 3: Angular distribution of the rescaled helicity density $\frac{\overline{dH}}{d\Omega}$ for a circular motion.
• Figure 4: The "phase space" to determine the locations of the extreme values of the helicity density ($0 < v < 1$). The signs in the brackets $(\bullet,\bullet)$ indicate the sign of ($f_+$, $f_-$)
• Figure 5: The purple line is used to indicate the locations of the minimal values of the helicity density.