A New Approach to Classical Electrodynamics: Covariant Second-Order Field Variations

This paper describes a formulation of classical electrodynamics without point charges and masses. Maxwell’s equations are assumed to be valid at all scales, and the Lorentz force is replaced as a fundamental law of nature by equations derived by a second order variation on a covariant “action” 2 4 16 n nk k jk i I i F A i d μcd F c d c π π Ω + = − Ω + Ω ∫ ∫ ∫ in which the usual term involving mass, 2 2 1 mc β − − ∑ , has been replaced by a covariant mass density 4 1 ik i k μ T u u c = associated with the fields. Fij and Tik are the usual field and energy-momentum tensors, and ui is the four-velocity associated with charge. Distributions having charge of a single sign are found to be stabilized by charge motion in the distribution's self-field, and they have finite extent, charge, and mass. The motion of these charged bodies in an imposed electromagnetic field is shown to obey the Lorentz force law, where the mass is equivalent to the total energy of the electromagnetic field integrated over all space. Ordinary classical electromagnetism is thus reproduced, and in addition, the classical structures of fundamental charged bodies are obtained. The singularities usual to dualistic theories disappear. Classical structures of the electron and proton are briefly discussed.


Introduction
Classical electromagnetism is a dualistic theory in which Maxwell's equations give the fields from the charges and their motions, and electrodynamics, i.e., the motions of the charges in the fields, is governed by the Lorentz force plus the radiation reaction. The charges have been taken as separate entities from the fields that cause their motion [1][2][3][4]. Charges have been taken to be singular points having infinite charge density, and attempts to identify the field energy with mass have yielded infinite masses.
The purpose of the present work is to formulate a new classical electrodynamics in which the charges arise naturally from the fields in such a way that the singularities of classical electromagnetism disappear. The usual Lorentz force law for the motion of charged bodies emerges from the theory as a derived law, and new electrodynamics equations describing static physical structures for fundamental entities, such as the electron and proton, are obtained.
We formulate a theory of electromagnetism without a priori reference to the structure of the charged bodies involved [3]. Charged entities and their properties must arise from the behavior of the fields.
The basic assumptions are: A covariant energy density E is equivalent to a covariant mass density µ = E /c 2 , where c is the speed of light in vacuum.

5.
The current density J = ρv can be written in terms of the charge density ρ and velocity v, and a given volume element contains charge of only one sign. 6.
All interactions are assumed to be purely electromagnetic. All physically observable properties, such as charge, mass, dipole moments, and angular momentum, are obtained from the electromagnetic fields. 7.
Stable charged bodies will be surrounded by an equipotential surface.
and F ik is the standard electromagnetic field tensor [1].
is the covariant four-vector for potential.

Derivation of the Fundamental Equations
We suppose that the Euler-Lagrange equations obtained from a variational principle provide physically meaningful "equations of motion", where L is a suitable Lagrange density and we alter the usual "action" for electromagnetic fields and particle motion [1,3] Taking µ from (3) and T ik from (5), one eventually arrives at the result (Appendix A) where the indices are 1,2,3,4. A second-order variation δI = 0 on the field variables yields the Euler-Lagrange equations Straightforward mathematical manipulations (Appendix A) yield four coupled, non-linear partial differential equations that are valid in and only in regions where ρ ≠ 0.
where is the purely antisymmetric Levi-Civita tensor. Equation (19) will be satisfied if This equation is a 4x4 matrix of twelve independent equations where the entries with l = j are identically zero. Equation (20) is valid in regions where ρ ≠ 0, and Maxwell's equations (a m = 0) hold if ρ = 0. The equations that explicitly involve time t (l = 4, j = 1, 2, 3) will be used in what follows here.

Motion of Charged Bodies
If ρ o does not depend explicitly on time t, then (21) is Integrate over a volume V = V o + V i . See Fig. 1.
The deformation of the charged volume when the body is accelerated is taken into account below. where S(t) surrounds V(t) and v S is the velocity of the surface. Equation (23) is then where v is understood to be the velocity with which the particular surface S 1 or S 2 moves. The integrals over S 1 cancel because of the opposite directions of the normals and because the fields are continuous across S 1 , so If the outer surface S 2 is taken to be fixed in time, then If S 2 is fixed at an arbitrarily large distance, the momentum ( ) P em j of a moving body is given by where U j is the velocity of the body's center of inertia. If a charged body is moving with velocity U j , then in regions where ρ = 0, the velocity of the fields associated with that body is U j .
If the linear momentum of a body at rest, i.e., U k =0, is to be zero, then we must have in which case, the three-momentum of a moving body is j, k = 1, 2, 3, or equivalently, Since µ is invariant under Lorentz transformations, M will be invariant when integrated over the proper volume 34 A New Approach to Classical Electrodynamics: Covariant Second-Order Field Variations (invariance means that if frames 1 and 2 are moving with relative velocity β 12 , then Hence, the temporal derivative of ( ) j, k = 1, 2, 3. Thus, the temporal rate of change of the momentum of an electromagnetic body is determined by the Lorentz force. In this equation, M includes the equivalent mass of all the fields. Since M is invariant, we can calculate M when the body is at rest.
where the double primes designate the velocities when the system is at rest.

A. Static equations
See (22). Solution of (36) and Maxwell's equations will yield contained and stable charge distributions.

B. Some static solutions
There are many approaches to solving (44). Two series solutions and one approximate solution are presented below.
The condition given in (30) is satisfied for the cylindrical static solutions found here, and f j in (34) is equal to the externally impressed fields. In addition: (1) There will be an equipotential surface (ϕ = const) that bounds the charged region of a static body. (2) If the electric fields of a charged body are to originate entirely from the charges within the body and if the body is to have mirror symmetry in the z = 0 plane, then the electric potential must be continuous and E z = 0 when z = 0. That is, Insert ϕ (β, ζ) = R (β) Z (ζ) into (44) with ζ = Cz and Z(ζ) = cos(kz) = cos(αζ), α = k/C The resulting O.D.E. is

Series expansion around β ≈ 0
Equation (47)  Equation (49) gives two independent solutions of (47), and the first three terms of these series are given below. where A j , η j , and κ are arbitrary constants. ζ = Cz, β = Cr, and the α j are values of α j = (k/C) j appropriate to the problem. Figure 2 shows a particular solution when α = 1 and .
To find charged regions enclosed by an equipotential, we set = constant. The boundaries of the spinning bodies are defined by rotating the equipotential contours shown around the ζ (or z) axis. The resultant spinning ring of charge has no temporal dependence and thus represents a stable charged body. Figure 3 depicts a generic shape, a spinning ring, applicable to all of the bodies derived here. The field outside the charged ring is obtained by solving Maxwell's equations in free space and matching the fields and potentials to those on the surface of the ring.
Persons familiar with forces and particles will ask what holds the charged shape together. Equation (36) shows that the electromagnetic forces balance out. Combining (4) and (36) shows that the mass and fields exchange momentum in a way that maintains a net momentum of zero within the spinning charged ring. The result is a static body.

Series Expansion Around β ≈ 1
Let R be a function of h = 1-β 2 . Then (47) becomes where the primes denote derivatives with respect to h. If If |h| < 1, these series converge for all finite values of α 2 ; however, when α 2 > 1, convergence is slow. The general solution of (44) is Curves for ϕ (7,h,0) = 1490R 0 -2888.6R 1/2 are shown in Fig. 4, with series terms to h 20 . Ordinate and abscissa are scaled properly for ϕ = -1.5, but E β has been multiplied by 4π/2000 and ρ by 4π/10 4 . The axis of rotation is at h = 1, off to the right. The abscissa/ordinate scale ratio is 125, so the height of each equipotential surface depicted is physically ≈ 11 times its width. ϕ maximizes at h ≈ 0.511, i.e., β ≈ 0.699.

Approximate Solution When ε = 1-β << 1
Rewrite (47) with R as a function of 1-β = ε << 1. Then, approximately, 1-β 2 ≈ 2ε, 1+β 2 ≈ 2(1-ε), and (62) Figure 5 shows the potential, electric field, charge density, and two closed equipotential curves in the βζ-plane for the potential given by (62). The constant κ = -0.1 is chosen to make ρ and ϕ zero at approximately the same value of ε. If ϕ and the equipotentials are shown on the proper scale, then E β should be multiplied by 72000 and ρ should be multiplied by 10 11 /4π. The axis of rotation is at ε = 1, far to the right in the figure.

C. Extrinsic properties of static charged bodies
The angular momentum density is an antisymmetric tensor The total angular momentum of a charged, static body is (64) because ζ j = 0 in V i . The angular momentum inherent to a stationary charged body will then be given by (65) The magnitude of L and its projection along one axis are invariant under Lorentz transforms.
The magnetic moment of the body is, in Gaussian units, Equivalently, The double primes are reminders that the velocities and fields are those of the body at rest. The connection between the charged body's angular momentum and magnetic moment is implicit because the integrals are over different volumes and the magnetic fields in V o are generated by the charge motions in V i .
The total charge on a static body will be where . Because ρ o and 2 m a are invariant, Q will also be invariant when integrated over the proper volume in different moving frames. Figure. 5. Potential, radial electric field, charge density, and equipotential curves for the potential given in (62) plotted versus ε = 1-β, β > 0.

Fundamental Bodies in This Field Electrodynamics
In principal, this classical field theory of electrodynamics can represent all charged bodies (electrons, protons, neutrons, etc.). The theory may provide a description; however, unlike quantum mechanics, it has not provided a unique catalogue of types or states.
The magnetic moment m and quadrupole moment D can easily be adequately (not accurately) estimated for a body with given charge. The spin and mass require machine computations and will be reported in a future paper.
Consider the electron as an example. The magnetic moment for these spinning charged rings is given by (66), which is approximately These examples strongly indicate that the electron can be represented by one of these spinning rings. There are six parameters (r o , α, A j , κ, C, and the value of the surrounding equipotential) to reproduce, in principle, the mass, charge, spin, quadrupole moment D, and magnetic moment of an electron, and there is even more flexibility if series having different α are summed.
Scattering experiments indicate that the proton's structure is complicated, so it may require a few spinning rings to represent the proton. For a single small ring of approximately the measured size of the proton, r o ≈ 8.7x10 -14 cm, as in (69), with a magnetic moment ≈ 1.41X10 -14 erg/G, 14

10
cm, gives β ≈ 0.309. Rough estimates of the spin and mass suggest that there are charge distributions circulating at radii considerably smaller than 8.7x10 -14 cm. If there were more rings, the charges on each ring would add to give the charge of the proton and the magnetic moments of each ring would add to give the proton magnetic moment, but the spin and mass must be calculated in detail from the resultant fields throughout the whole of space. These latter calculations cannot be done conveniently by hand.

Comparison with Other Work
The author is unaware of any previous second-order variational theory of classical electrodynamics or of any theoretical body comprising charge of a single sign and stabilized by electromagnetic interactions. There are many models of the electron, so we will compare with those.
The classical electron derived here is a spinning ring, not any of the toroids or helical entities postulated in earlier models of an electron [7][8][9][10][11][12]. Most models of the electron are ad hoc and focus on the connection between the model and quantum mechanics. Most, but not all, of these treatments are dualistic, with point particles and fields [4,12]. Some modifications of classical electrodynamics invoke two or more temporal or physical scales or modifications of Maxwell's equations or of the Lorentz force, sometimes involving gravity [4,8,13,18].

Future Work
Not all of the implications of a non-dualistic field theory of electrodynamics are addressed here. A few obvious topics are:


Computer calculations of the properties of these static bodies.  Time dependence using all twelve equations (20).  Radiation reaction can be described by (26) with (24) applied to (A k ) in (10).  A systematic characterization of the solutions of (47).  Solutions of the form ϕ = ϕ (β/r).  Scattering phenomena in very intimate collisions.  Incorporation of field electrodynamics into general relativity.

Summary
This paper presents a reformulation of classical electrodynamics in which the well-known Lorentz force law emerges as a derived equation and the structure of static bodies comprising charge of one sign is obtained. These static bodies involve only electromagnetic interactions. In this theory, charge is always associated with motion. It is the motion of the charge that produces the fields that stabilize charged bodies.
The assertion [1] that point charges must be used classically because extended bodies would distort at high speeds has been shown to be invalid. Extended bodies do distort, but when the fields are continuous, bodies still move as predicted by the Lorentz force law.
Consider the electron, which in standard electromagnetic theory is an enigmatic point held together by an unknown force and having infinite mass and charge density with a spin and magnetic moment arising from some unknown source. In the present construct, the electron is a spinning ring of charge held together by its internal velocity and electromagnetic self-fields. Those fields and the corresponding external fields yield finite mass and charge densities, which can be integrated to give finite mass and charge. Magnetic moment results from the motion of charge, and spin arises from the electromagnetic field produced by the motion of the charge.
It has been no part of this paper to relate this theory to quantum mechanics; however, see de Wet [19]. Also, Simulik and Krivsky [16][17][18] report that their modification of Maxwell's equations reproduces some quantum results. The quantum properties of the electron arise because its de Broglie wavelength is much larger than its size and is virtually independent of its structure.
The discovery that small charged bodies require no new strong forces to hold them together classically and knowledge of the classical structure of fundamental bodies, such as the electron, may facilitate understanding of the quantum nature of the electron and other fundamental bodies. A "particle theory" based on a field theory of electrodynamics would differ significantly from one based on the dualistic concept of distinct fields and particles.
So, applying (18) to L 2 yields identically zero, and the term J k A k in (14) contributes nothing (!) to the equations obtained from a second-order variation of this L.