Interacting Kerr-Newman Electromagnetic Fields

TL;DR

The paper analyzes the G→0 limit of the Kerr-Newman solution, known as the electromagnetic magic field, and develops a field-only description in flat space with a nontrivial two-sheeted topology and a ring singularity. It formulates a Lagrangian framework using the complex potential and computes the self-energy and interaction energy for one and two magic fields, respectively, revealing a finite single-field Lagrangian and spin-induced corrections to the Coulomb potential. By examining two canonical configurations (up-down and side-by-side) of interacting magic fields, it derives closed-form interaction energies and their asymptotic expansions, uncovering potential wells and equilibrium points tied to the spin parameters. The results strengthen the electron–Kerr–Newman analogy through multipole coincidences and spin-topology insights, and they point to broader implications for classical spin interactions and the double-copy relation between gravity and electromagnetism.

Abstract

In this paper, we study some of the properties of the $G \to 0$ limit of the Kerr-Newman solution of Einstein-Maxwell equations. Noting Carter's observation of the near equality between the $g = 2$ gyromagnetic ratio in the Kerr-Newman solution and that of the electron, we discuss additional such coincidences relating to the Kerr-Newman multipoles and properties of the electron. In contrast to the Coulomb field, this spinning Maxwell field has a finite Lagrangian. Moreover, by evaluating the Lagrangian for the superposition of two such Kerr-Newman electromagnetic fields on a flat background, we are able to find their interaction potential. This yields a correction to the Coulomb interaction due to the spin of the field.

Figures (11)

• Figure 1: Values of the spin parameter $a/M$ vs. mass (by order of magnitude) for objects with sizes ranging from the electron to the solar system. The value for a CD disk is calculated from the value $10^{18}$ for a 33 rpm vinyl LP record given by Dietz and Hoenselaers DietzHoen. The astronomical values are calculated from Allen2002.
• Figure 2: Electric and magnetic components of the magic field viewed in Euclidean space. The rainbow color-coding demonstrates the fields strengths, with red representing the extreme values and purple the negligible ones. The red points on the $\chi$-axis display the singular ring $\mathcal{R}$ intersections, and the line segment between them displays the intersection of the branch-cut disk $\mathcal{D}$ spanned by the ring. The fields have azimuthal symmetry around the $z$-axis. The apparent discontinuity of the field on the disk spanned by the singular ring is removed by considering the analytic continuation of the field, cf. section \ref{['sec:AnalyticContinuationDef']}.
• Figure 3: The whole Riemann surface $\mathcal{V}$ of the electromagnetic magic field: (a) the first sheet $\mathcal{V}_1$, (b) the second sheet $\mathcal{V}_2$, (c) the first gate $\mathcal{D}_1$ between the upper part of the first sheet $\mathcal{V}_1^+$ and the lower part of the second sheet $\mathcal{V}_2^-$, and (d) the second gate $\mathcal{D}_2$ between the lower part of first sheet $\mathcal{V}_1^-$ and the upper part of the second sheet $\mathcal{V}_2^+$. The red points represent the ring singularity $\mathcal{R}$. The blue and orange line segments that separate the upper and lower halves in (c) and (d) are identified respectively.
• Figure 4: Curve $\gamma$ circling around the ring singularity $\mathcal{R}$ in double-sheeted Riemann surface $\mathcal{V}$. For $0\le\alpha\le2\pi$, the curve $\gamma$ is in the first sheet $\mathcal{V}_1$ starting from the orange point $\alpha=0$. It enters the second sheet $\mathcal{V}_2$ from the black gate $\mathcal{D}_2$ at $\alpha = 2\pi$ depicted by the blue point on the gate. For $2\pi\le\alpha\le4\pi$, it is in the second sheet $\mathcal{V}_2$ and enters back to the first sheet from the gray gate $\mathcal{D}_1$ and closes at the starting point. The depicted curve is an $\epsilon$-constant curve, $\epsilon$ being the semi-radius of the curve defined in \ref{['eq:ring_centered_coords1']}.
• Figure 5: Electric and magnetic components of the magic field through the gate $\mathcal{D}_1$ of the Riemann surface $\mathcal{V}$ of the complex potential. The rainbow color-coding demonstrates the fields strengths, with red representing the extreme values and purple the negligible ones. The red points on the $\chi$-axis display the singular ring $\mathcal{R}$ intersections, and the line segment between them displays the intersection of the branch-cut disk $\mathcal{D}$ spanned by the ring $\mathcal{R}$. The fields have azimuthal symmetry around the $z$-axis. There is no discontinuity of the field on the disk spanned by the singular ring. Similar graphs can show the fields structure in the other gate $\mathcal{D}_2$.