Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Covariant field lines: a geometrical approach to electrodynamics

Yaron Hadad, Ido Kaminer, Avshalom Elitzur, Eliahu Cohen

Abstract

This paper revisits the geometric foundations of electromagnetic theory, by studying Faraday's concept of field lines. We introduce "covariant electromagnetic field lines," a novel construct that extends traditional field line concepts to a covariant framework. Our work includes the derivation of a closed-form formula for the field line curvature in proximity to a moving electric charge, showcasing the curvature is always non-singular, including nearby a point charge. Our geometric framework leads to a geometric derivation of the Lorentz force equation and its first-order corrections, circumventing the challenges of self-force singularities and providing insights into the problem of radiation-reaction. This study not only provides a fresh geometric perspective on electromagnetic field lines but also opens avenues for future research in fields like quantum electrodynamics, gravitational field theory, and beyond.

Covariant field lines: a geometrical approach to electrodynamics

Abstract

This paper revisits the geometric foundations of electromagnetic theory, by studying Faraday's concept of field lines. We introduce "covariant electromagnetic field lines," a novel construct that extends traditional field line concepts to a covariant framework. Our work includes the derivation of a closed-form formula for the field line curvature in proximity to a moving electric charge, showcasing the curvature is always non-singular, including nearby a point charge. Our geometric framework leads to a geometric derivation of the Lorentz force equation and its first-order corrections, circumventing the challenges of self-force singularities and providing insights into the problem of radiation-reaction. This study not only provides a fresh geometric perspective on electromagnetic field lines but also opens avenues for future research in fields like quantum electrodynamics, gravitational field theory, and beyond.
Paper Structure (16 sections, 43 equations, 4 figures)

This paper contains 16 sections, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Covariant field lines
  3. The curvature of a field line
  4. Consequences
  5. The correspondence between curvature and acceleration
  6. No singularities nearby charges
  7. Deriving an equation of motion from the curvature of the covariant field lines
  8. Discussion
  9. Acknowledgements
  10. Appendix
  11. Electric and Magnetic Field Lines
  12. Uniqueness of the Covariant Electromagnetic Field Lines
  13. On the Derivation of the Covariant Electromagnetic Field Line Curvature
  14. The Components of the Covariant Field Lines in the Rest Frame of an Accelerating Charge
  15. Action Principle for Electrostatic Field Lines
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Charges accelerate in the direction of straight field lines: Electromagnetic field lines are represented by curves (in blue), whose curvature is represented by the color and elevation of the surface (bright colors correspond to higher curvature/elevation). In this figure, three electrons (green) repel one another in the direction of the (red) arrows. The curvature is usually non-zero nearby each charge, yet the field lines pointing in the direction of the charge's acceleration are straight (zero curvature). Charges always accelerate in the direction of straight electromagnetic field lines, in analogy with the motion along a geodesic in a gravitational field in general relativity. The two peaks in the figure are singular points, which only occur where the total electromagnetic field is zero (since no field lines pass through these points).
  • Figure 2: Illustrations of electric, magnetic, and covariant field lines: The total electric, magnetic and covariant electromagnetic field lines are presented for an electron at rest (first column) and an electron at a relativistic velocity $v=4/5c$ out of the page (second column). The external field is a constant magnetic field upwards (no electric field), as can be seen in the second figure from the top (resting charge). For a charge at rest, the electric and covariant field lines coincide. Once the charge is moving, the electric field lines are denser in the covariant direction (well known), while the covariant field lines are both denser and also curve due to the magnetic force. The relativistic electron is expected to accelerate in the horizontal direction as the covariant field lines are flat in that direction.
  • Figure 3: Correspondence between charge acceleration and field line curvature (a) The field lines of a free charge at rest. (b) As the charge accelerates (red arrow), the field lines bend. (c) The field line curvature determines the acceleration of the charge. The charge always accelerates in the direction of straight field lines (black color represent zero curvature). Straight field lines are not unique, and there can be multiple straight field lines around the charge.
  • Figure 4: Useful notations: The world line of the charge $q$ in spacetime, where the null-vector $k^\mu$ connects the charge's retarded four-position $z^\mu (\tau_{\text{ret}})$ to an arbitrary event $x^\mu$. The retarded four-velocity of the charge $u^\mu$ is the tangent to the world line and is perpendicular to the retarded four-acceleration $a^\mu$.