Spin and Helicity Related to EM and Gravitation
(c) Robert Neil Boyd

Below is a conversation concerning Spin and Helicity related to EM and Gravitation between R. Stirniman and R. M. Kiehn.

[STIRNIMAN said]:

The conventional recipe for helicity density (AdotB) does not always work. A recipe can be derived, based on the amount of twist in the field, which is nearly the same, and in many examples is exactly the same as AdotB.

Suggested example problems:

Helicity density in the helicoidally polarized E parallel to B standing wave. (Constant helicity everywhere in the wave.) Requires, either: addition of electrical field helicity (Ae.D), or a helicity recipe based on twist in the field.

[RMK says]:

First: Be aware that I do not like to change Maxwell's formulation of EM as written in terms of the partials of E and B (scalar components related to forces) and in terms of D and H (density components related to sources). WHY? Because, from a topological point of view, the two statements of topological - not geometrical - constraint, F-dA=0 (A is a 1-form) and J-dG=0 (J is a N-1 form density, or charge-current density), yield Maxwell's PDEs for any variety (choice of coordinates or "frame of reference") of N>3 dimensions. No geometric constraints of metric, connection or gauge are utilized.

The idea is that for these two topological axioms the Maxwell PDEs are the same in format and valid in every coordinate representation, independent from the choice of any gauge constraint. The Maxwell PDEs are also independent of number of variables used to describe the space of interest! For example, F-dA = 0 leads to dF = 0, and when N >3, to a system of many, many PDEs. However, the first 4 PDEs are always in the same precise format of the Maxwell-Faraday induction PDE equations (using partial derivatives for all abstract coordinates, not total derivatives).

The equations involve only the first 4 variables and are uncoupled to the other variables of the set. The idea is that the Maxwell Faraday PDEs form a nested, universal, set whose format is the same for any choice of 4 or more coordinate variables. These PDEs do not depend upon the imposition or constraint inherent in a choice for a metric, and do not depend upon a choice for a connection, and therefore do not depend upon a choice for some geometrical equivalence class (Lorentz - Galilean - etc.)

AS such, the Maxwell PDEs should to be treated as topological rules, not geometrical rules.

The PDEs involving E B D and H are valid in any reference system: Lorentzian, Galilean , or any other equivalence class of diffeomorphisms. It is when a constraint is imposed, forcing a relationship between the sets E B and the sets D H, that the lack of validity in any reference system is destroyed.

This remarkable topological robustness of these nested sets of PDEs makes me suspect of any attempt to change the concepts of such partial differential (not covariant differential) systems of equations.

Constitutive (geometrical) constraints imposed upon these topological equations can be useful, but select out subsets of the general ideas.

It has long been known that it is those singular solutions to the PDE equations (not the equations themselves, but those singular solutions which represent propagating discontinuities in the E and B fields) that generate the impact and dominance of the Lorentz equivalence class. The Lorentz group is the only linear group that will preserve such a given discontinuity, by mapping one discontinuity of fields into another discontinuity. Other mappings will iron out the discontinuities and can make them disappear. So when a "signal" is defined as a propagating E and/or B field discontinuity, the "signals" appear as "signals" with respect to the all observers who are members of the Lorentz equivalence class of observers, but not to all members of, say, the Galilean equivalence class of observers. In the linear Lorentz equivalence class, the propagation speed of the discontinuity is finite and greater than zero: (C).

It is also NOT so well known that there is a non-linear group that preserves the concept of a signal as a propagating discontinuity. It is the Fractional Projective or Moebius transformation. However, the speed of propagation of the propagating discontinuity is not a constant in such an equivalence class, and can be bigger or smaller than C of the Lorentz group. In fact it can be infinity (See Luneberg or Fock)

Now with respect to helicity concepts. There are two species of "helical" structures in EM theory. They are distinct, but have many similarities. One specie is related to Faraday rotation, and it is non-reciprocal. The other specie is related to Optical Activity and is reciprocal. Most USA textbooks (such as Jackson) treat the Faraday rotations and the Optical Activity rotations of a plane of polarization in similar manner. Indeed that is possible. But when the partially rotated beam is reflected in a mirror, the linear polarization state returns to its original position for the Optical Active experiment (reciprocal), but ratchets through a second partial rotation for the Faraday case (non-reciprocal).

There are two classes of "twisted" fields in EM theory. Each has a space-like component and a time-like component. The first class is a vector field that I have called topological torsion, A^F. It is the dominant field in the Faraday effect. IN 4D, the space-like part is the vector: E x A + B phi. The time-like part is the scalar, A dot B. A finite value of "helicity" = A dot B is sufficient ( but not necessary) for the existence of topological torsion, A^F (physical dimensions = angular-momentum/impedance)

The second class is a vector density that I have defined as topological spin, A^G. IT is the dominant field in Optical Activity. Its space-like part is the current-density: A x H +D phi. The time-like part is the density: A dot D. A finite value of A dot D is sufficient ( but not necessary) for the existence of topological spin. (physical dimension = angular momentum)

In each case, the time-like part can be zero, but the space-like part need not be zero. These constructions have topological significance. THE ABOVE STATEMENTS DO NOT REQUIRE MODIFICATIONS OF OR CONSTRAINTS PLACED UPON THE ORIGINAL FORM OF MAXWELL's EQUATIONS in UNIVERSAL PDE FORMAT, OR BETTER Said, AS DEDUCED FROM THE TOPOLOGICAL AXIOMS, F-DA= 0 and J-dG=0.

The divergence of the topological torsion is NOT necessarily zero, but is zero if E dot B = 0.

Div(A^F) = F^F = 2 (E dot B) dx^dy^dz^dt

When the divergence is zero, the integrals of the 3-form A^F over a closed domain can lead to values whose ratios are rational (quantized) invariants (a conservation law). Evolution in the direction of the topological torsion 4 vector is thermodynamically irreversible.

Also, the divergence of the topological spin is not necessarily zero. It is zero when twice the (Magnetic energy density, B dot H, minus the electric energy density (D dot E)) is equal to the interaction energy between the potentials and the charge currents.

(A dot J - rho phi). Div( A^G) = F^G - A^J ={ ( B dot H - D dot E ) - (A dot J - rho phi) }dx^dy^dz^dt

When the divergence is zero, the closed integrals of A^G yield rational (quantized) angular momentum. Again these are topological results that require nothing more than the original Axioms, F-dA=0, J-dG=0, and do not depend upon any modifications of the Original Maxwell Equations.

I hope this adds some clarity to the following comments of Stirniman

[STIRNIMAN said]:

Helicity density in the helicon/stellarator type helical coil (twisted magnetic field). In cylindrical coordinates the source (coil) is symmetrical in the theta and z directions. Helicity density can vary only in the radial direction. Proper formulation, results in a radial component in the magnetic vector potential. In the conventional (defacto Lorentz gauge) definition of the magnetic vector potential, with all components of the vector potential parallel to current elements, there is no radial component in the vector potential.
Example presented by G.E. Marsh, Chapter 2 of textbook, Advanced Electromagnetics, "Helicity and Electromagnetic Field Topology". Marsh presents an example (although non-physical) of a magnetic field which clearly exhibits twist and flux linkage, but using the recipe AdotB, the field has zero helicity density.

In analysis of helical EM fields, issues with the Lorentz gauge figuratively jump right off the page. Yet, similar issues exist in nearly all EM field problems.

Some non-helical examples:

One of the first experiments conducted with free ions in a magnetic field is presented in the text: Characteristics of Electrical Discharges in Magnetic Fields, Published 1949. Experiment done during the Manhattan project. Quote from page 3: "Yet the main problem is not why the discharge is constricted, but rather, in accordance with the small electron Larmor radius, why it was not constricted a great deal more. It seemed clear that a hitherto unsuspected mechanism was causing electrons to move across the field more easily than expected." Ion diffusion and pressure forces are analyzed, and are orders of magnitude too small to account for the observed effect. No alternative is presented.

The "unsuspected mechanism" in the above EXPERIMENT is resolvable by looking at a more foundational problem:

What is the force between two relatively moving electrical charges?

The conventional solution to the above is the induced electrical field due to the Lorentz force vXB, along with the E field of the adjusted scalar potential (Lienard-Wiechart potential). Many papers have been published about this, pro and con. Idiosyncracies abound. Most notably: the force found depends on the reference frame adopted, and a more obvious problem -- one can find a reference frame in which the induced E field has a non-zero divergence. Induction fields by definition have zero diverence. Also, the resulting mass flow (momentum density) fields, and spin density fields which result from this solution do not obey the Maxwell-like gravitational field equation: curl(J) = 2p. There is also the annoying issue of the 4/3 ratio between mass-energy in the field, and momentum in the field.

These arguments have been ongoing since the beginnings of electrodynamic theory. Akademiks look no further. The Lorentz gauge formulation of electrodynamics is established law. But, for an alternative view it might be useful to start here:

[RMK says]:

Some of these paragraphs above seem to come straight out of Phipps "Heretical Verities" and use the Hertz recipe for modifying Maxwell equations to make them "valid" in Galilean reference systems. I prefer not to use these methods, which from a topological viewpoint impose certain constraints limiting the domain of validity.

[STIRNIMAN said]:

Moon and Spencer, "On Electromagnetic Induction", Journal of the Franklin Institute, Sep 1955 p213.

Moon and Spencer, "Some Electromagnetic Paradoxes", Journal of the Franklin Institute, Nov 1955 p373.

Moon and Spencer demonstrate a variety of idiosyncrasies in simple experiments relating to magnetic forces and the Lorentz force law. An alternative force law is presented which works in all experimental examples. This force law is presented with the comment -- "The non-Maxwellian approach subsumes the whole of electrodynamics under a single equation."

Well, not exactly. Moon and Spencer's result can be found directly from Maxwell's equations, by using full time derivatives, rather than partial derivatives, and eliminating the artificial addition of the Lorentz force.

[RMK says]:

Perry Moon was a very interesting man who I knew at MIT long ago. However, the Lorentz force is (to me) not an artificial addition to Maxwell theory. It arises naturally when questions of topological evolution are asked. How can one describe topological change in physical systems? (Note that Hamiltonian dynamics does not permit topological evolution). Given an 1-form of Potentials, A, and any direction field V representing an evolutionary flow, the equations of topological evolution can be deduced from Cartan's magic formula, describing how exterior differential forms propagate down flow lines generated by a vector field.

L(V)A = i(V)dA + d(iV)A) => Q.
L(V)A = W + d(U) => Q.

This expression does not depend upon the geometrical ideas of metric or upon a connection! It is a dynamical equation that can describe topological evolution, not only geometrical evolution. It has been shown that the equation is equivalent to the first Law of Thermodynamics. (see my Website for downloads of publications). It can be shown that the 1-form of virtual work when constrained to zero, W= 0, defines what is called the extremal vector field V in the Calculus of Variations. For the equivalence class of A and V such that the W = dH, Cartan proved that all such motions V were Hamiltonian, and therefore topology is preserved by such processes.!!!!

Using this dynamical equation of Cartan, it is easy to prove (using the Maxwell axioms) that the closed integrals of the field intensities (F=dA) are deformable topological invariants for any evolutionary process, V. (= Conservation of Flux). Also the closed integrals of the current density (J = dG ) are also deformable topological invariants for any evolutionary process, V (Conservation of Charge currents). These integrals define deformable coherent structures in the EM field.

Now back to the concept of the Lorentz force. The first term in Cartan's magic formula is defined as the Work 1-form, W = i(V)dA. When the 4D spatial coefficients of this 1-form are worked out in EM format, the result is always of the format,

E + v x B,

to within a factor. In other words, the concept of the Lorentz force is a derived result, based upon the dynamics of topological evolution of the defining exterior forms, and is not arbitrarily imposed on the electrodynamic system. The Lorentz force format is a derived concept. The Lorentz force has nothing to do with metric, gauge, connections, or the Lorentz group. Even more remarkably, if the symbols are changed, one recognizes that the coefficients of i(V)dA = W becomes the Lagrange Euler convective derivative of hydrodynamics. Hence there exists a Maxwell-Faraday induction law for fluids, as well.

[STIRNIMAN said]:

Perhaps some have noticed that in ALL electromagnetic problems, once we have made our adjustments to the potentials, in order to make it work relativistically, as we "know" it should, all the information in the solution can always be found to exist soley in one of the potentials.

[RMK says]:

I am not sure of content of the above statement. To me, potentials are one thing, and charge current densities are another. The potentials can have measureable properties that are not exhibited by the E and B fields. There are experimental and theoretical instances of domains where there are potentials, charge currents (hence D and H), but zero E or B. (Type 1 and type 2 superconductors). There are also instances of domains where there are potentials, that generate E and B fields, but do not generate charge current densities, and yet D and H exist. (EM waves in a Lorentz vacuum)

Recently I stumbled on theoretical situations where associated with the potentials (A, phi) there are 3 dimensional hypersurfaces where the cubic curvatures of the hypersurface are precisely equal to the interaction energy density between the charge current densities and the potentials , (A dot J - rho phi).

There are equivalence classes of potentials , (A, phi), which are closed and do not produce any E or B fields. These closed contributions to the class of all potentials are the source of equivalence classes of "gauge theories". Similarly (in 4D) there are equivalence classes of 2-form densities (D and H) which are closed, and do not produce any charge current densities. Such Closed 2-form densities imply a different class of "gauge theories" which have not been studied very well. Constitutive equations are of this class, but have not been studied from the topological point of view (at least to my knowledge). I do not like to impose the constraints of choosing a gauge theory, until I know all about those things that do not depend upon such a choice.

[STIRNIMAN said]:

All righty then. Try this one. Write the ELECTRIC vector potential (Ae), for the static coulomb field. Peculiarly, this potential appears to have an axis of symmetry, although the static field does not. Align this axis with motion -- absolute or relative, assign this as the z (velocity) direction in cylindrical coordinates. Solve the vector wave equation for Ae.

(curl)(curl)Ae + (1/c^2)(d^2/dt^2)(Ae) = 0

Use FULL time derivative. If you are uncomfortable with the expression (v.del)(Ae), choose any reference frame, make the coordinate transformation z = -vt, and take the partial time derivative. The result is always the same.

Write the solution of the above wave equation as a series of terms:

Ae = Ae0 + Ae2 + Ae4 ....

Where Ae0 represents the zero order term (static coulomb field), Ae2 represents second order terms which depend on (v/c)^2, etc.

Solve for all of the higher order terms, for example: such that the second order term is found from the second time derivative of the zero order term. And the entire expression solves the vector wave equation. i.e.

(curl)(curl)Ae0 = 0
(curl)(curl)Ae2 = -(d^2t/dt^2)Ae0
(curl)(curl)Ae4 = -(d^2t/dt^2)Ae2
etc.

Next, solve for the D field.

D = curl(Ae).

Note that the first two terms of the expression for Ae, result in an E field (force field) which is identical to the expression for the velocity induced force, along with the static force, of Moon and Spencer. Neatly also appearing are: the Hooper effect, and Ampere's original law for the force between two current elements. The 4/3 field mass/momentum problem goes away. The forces found, work in any reference frame -- absolute and relative, and are equal and opposite between charges. Spin and momentum in the field obey the Maxwell-like gravitational field equation. And absolutely do so.

All of Maxwell equations, and more, can be found in the above. But -- always expressed as FULL time derivatives rather than partial derivatives, without the artificial addition of a separate force law, and always in the gauge where the vector potentials have zero divergence.

Source free field equations:

curl Ae = D = (coulomb field) - (e0)dAm/dt
curl Am = B = (mu0)(dAe/dt)
div Ae = 0
div Am = 0
div B = 0, (if e0 and c are vacuum constant, div H = 0)
div D = 0, (if e0 and c are vacuum constant, div E = 0)
D = curl Ae
B = curl Am
curl(E) = -dB/dt
curl(H) = dD/dt

[RMK says]:

The last 2 equations do not follow from the other equations in the set, unless certain geometrical constraints are imposed; namely - the vacuum constitutive relations.

[STIRNIMAN said]:

The force law on a charge can be most easily written as:

F = (E)q where (e0)E = curl(Ae).

It can also be written as a combination of the static coulomb field and the induced field. In no case does it exactly equal vXB. Ironically, in the case of a source field consisting of charge motion in a closed conductor circuit this force mimics vXB, except for a very small Hooper effect, which we choose to ignore. In the case of convection currents, i.e. moving isolated charges, the force is substantially different than vXB. And, in the case of linked or twisted current sources and twisted fields, the force is nearly identical to vXB in some cases, and substantially different in other cases.

One can also find Moon and Spencer's velocity force by looking at the magnetic field of a moving charge, and rather than using an A field having the divergence of the Lorentz gauge, write the A field with zero divergence, and solve for the induced E field as, E = -dA/dt. A factor of one-half turns up missing in Moon and Spencer's first velocity term, and their second velocity term works exactly right.

I believe, the following two equations might be most useful for folks to keep in mind for analysis and engineering of EM-Gravitational effects.

Lorentz Gauge = Bull Shit

Bull Shit In = Bull Shit Out

[RMK says]:

I am not sure I can accept the above paragraphs.

Re EM gravitational effects (of which I am not expert although I did explore, long ago, the possibility of measuring the effect of gravity on the polarization of a light beam):

A recent theoretical result, which I stumbled upon using Maple, states: Given certain sets of 4D homogeneous potentials, A, it is possible to construct examples of a global closed charge current density, J, and hence a set of excitation fields D and H, as well as a set of field intensities, E and B. IN such cases, the interaction energy density A^J is precisely the cubic curvature invariant of the 3D hypersurface defined by the homogeneous 4D potentials, A. This leads to the conjecture: As gravitational mass theory seems to be generated from quadratic curvature invariants, then this cubic electromagnetic curvature invariant, could, at very small curvatures, dominate the further evolution of a charged mass structure, and PREVENT THE GRAVITATIONAL COLLAPSE INTO A BLACK HOLE.

[Comment from Neil: You may recall I have suggested exactly the same thing on this forum in some of the debates we had regarding black holes.]

It is my opinion, that if electromagnetism is constrained by a gauge condition, a metric condition, or a constitutive condition , then the best that can be hoped for is a special case of a more general experimental and theoretical system. Such constraints single out special (but possibly observable and possibly useful) cases. Div A=0 is as much a gauge condition as any other. It may be that arbitrary constraints impose situations that are not admissible to experiment. Many other of my thoughts on EM can be found at www.cartan.pair.com, et seq Regards RMK.

Stirniman's Original Essay:

I agree. Important. Kiehn has done great work. There is a connection in EM and gravitation which results from spin and helicity in the fields. This is a good place to look.

Some kick-ass math and ideas are presented by Kiehn, which I do not have time to properly fathom. But, this much I do understand -- In analysis of rudimentary engineering examples of EM field helicity, the Lorentz gauge formulation of electrodynamics does NOT work. Further, the conventional recipe for helicity density (AdotB) does not always work. A recipe can be derived, based on the amount of twist in the field, which is nearly the same, and in many examples is exactly the same as AdotB.