[Paper Review] Classification of electromagnetic fields in general relativity and its physical applications

Nikolai V. Mitskievich|ArXiv.org|Feb 24, 2008

Pulsars and Gravitational Waves Research5 references3 citations

TL;DR

This paper proposes a physically motivated classification of electromagnetic fields in general relativity into three types—electric, magnetic, and null—each subdivided into pure and impure subtypes, based on invariants $I_1$ and $I_2$. It introduces reference-frame-based methods to eliminate spurious field components (e.g., removing magnetic fields in pure electric fields) and demonstrates that only pure null fields propagate at speed $c$, while all others propagate slower. The key contribution is a modernized Rainich–Misner–Wheeler duality approach enabling the construction of new exact Einstein–Maxwell solutions from seed solutions, including three new types of rotating charged black holes with Kerr–Newman geometry.

ABSTRACT

The simplest electromagnetic fields' (general- as well as special-relativistic) classification is formulated which is based on physically motivated ideas. According to this classification these fields can belong to three types (electric, magnetic and null), each of them being split in pure and impure subtypes. Only pure null type field propagates with the fundamental velocity $c$, all other fields have the propagation velocity less than that of light. The reference-frame-based methods of elimination of alternative three-fields (e.g., magnetic in the electric type case) are given for pure subtypes; for pure null type the generalized Doppler effect takes place instead. All three types of impure fields are shown to be {\bf E}-{\bf B}-parallelizable. Thus such an elimination in pure non-null and parallelization in all impure cases mean transformation to the reference frame co-moving with the electromagnetic field in which the Poynting vector vanishes. The methods we propose modernizing the Rainich--Misner--Wheeler approach, also permit to construct new exact Einstein--Maxwell solutions from already known seed solutions. As examples, the Kerr--Newman and Liénard--Wiechert solutions are considered, three ``new'' types of rotating charged black holes (with the same Kerr-Newman geometry) are presented, and new physical effects are evaluated. PACS 2008 Numbers: 04.20-{\bf q}, 04.20.Ex, 04.40.Nr, 04.70.Bw

Motivation & Objective

• To develop a physically grounded classification of electromagnetic fields in general relativity that applies uniformly across special and general relativity.
• To resolve ambiguities in field interpretation by introducing reference-frame-based elimination of spurious field components (e.g., magnetic field in pure electric fields).
• To demonstrate that only pure null fields propagate at speed $c$, while all other fields propagate slower, and to define their true propagation velocity.
• To extend the Rainich–Misner–Wheeler duality method to generate new exact Einstein–Maxwell solutions from known seed solutions, including new types of rotating charged black holes.
• To clarify misconceptions in field propagation velocity, particularly in superpositions like Liénard–Wiechert fields, by showing their true velocity is less than $c$ and that Poynting vector vanishing in co-moving frames is physically meaningful.

Proposed method

• Classification of electromagnetic fields based on the two invariants $I_1 = \mathbf{E}^2 - \mathbf{B}^2$ and $I_2 = \mathbf{E} \cdot \mathbf{B}$, with field types determined by the signs and values of these invariants.
• Use of reference frames co-moving with the electromagnetic field to eliminate non-physical field components (e.g., removing $\mathbf{B}$ in pure electric fields) by choosing appropriate observer congruences.
• Application of generalized duality rotation (in the spirit of Rainich–Misner–Wheeler) to transform seed Einstein–Maxwell solutions into new solutions of the same spacetime geometry but different field types.
• Derivation of the propagation velocity of electromagnetic configurations using the Poynting vector and energy flux, with the velocity defined as $v = \frac{\text{Poynting vector magnitude}}{\text{energy density}}$, ensuring physical consistency.
• Explicit construction of co-moving frames for impure fields where $\mathbf{E}$ and \mathbf{B}$ are parallel, enabling vanishing of the Poynting vector and simplifying physical interpretation.
• Analysis of superpositions such as Liénard–Wiechert fields and plane waves with uniform magnetic fields to show that their true propagation velocity is less than $c$, contradicting naive linear superposition assumptions.

Experimental results

Research questions

• RQ1 How can electromagnetic fields in general relativity be classified in a way that reflects their physical propagation and observer-dependent structure?
• RQ2 What is the true propagation velocity of a composite electromagnetic field such as the Liénard–Wiechert solution, and how does it differ from the speed of light?
• RQ3 Can duality rotation applied to a seed Einstein–Maxwell solution generate new exact solutions with different field types while preserving the spacetime geometry?
• RQ4 In what reference frame does the Poynting vector of a non-null electromagnetic field vanish, and how is this frame constructed for both pure and impure field types?
• RQ5 Why do standard linear superpositions of electromagnetic fields fail to correctly represent the true physical velocity and energy flow of the composite field?

Key findings

• Only pure null electromagnetic fields propagate at the speed of light $c$; all other fields, including pure electric and magnetic types, propagate at velocities strictly less than $c$.
• For pure electric and magnetic fields, there exists a global reference frame co-moving with the field in which the magnetic or electric field component, respectively, vanishes, enabling a physically unambiguous interpretation.
• The generalized Doppler effect fully characterizes pure null fields in their co-moving frames, replacing the need for field component elimination.
• Impure fields (with $I_2 \neq 0$) are always E-B-parallelizable, meaning that in a canonical reference frame, $\mathbf{E}$ and $\mathbf{B}$ are parallel, and the Poynting vector vanishes.
• The Liénard–Wiechert field is classified as pure electric, and there exists a global co-moving frame in which its Poynting vector vanishes everywhere outside the source worldline, a fact previously overlooked.
• A superposition of a plane electromagnetic wave and a uniform magnetic field in vacuum is an exact solution that propagates with a velocity less than $c$, and this velocity is correctly computed via the ratio of Poynting vector magnitude to energy density, not via naive linear averaging.