Beltrami Fields

Ingenio Eugenio (Beltrami)
Ingenio Eugenio (Beltrami)

“It’s not a fragrant world”
Raymond Chandler

Recently, our tribal leader has expressed his will for a trip into the depths of the remote past. There lies a hidden treasure of secrets covered under tons of malodorous Danish Blue otherwise known as the Copenhagen Interpretation.

Having had the same feeling for years, I asked a million loyal Mongol bots to complete their “Seek and Loot” mission among old and dusty library shelves, forgotten drawers and even locked chests. The loot is in, and all prisoners have been interrogated. Just as I expected, their testimony reveals that we have perhaps come full circle.

Readers may recall the disarray caused to the old guard of 19th century electrodynamics when they realized the conundrum caused by a circulating electron. Why did it not radiate and fall in upon the nucleus? This gave rise to the famous Atomic Stability problem.

Even today, many still think that this problem has no solution in classical electrodynamics. However, people of the “Old Kingdom” were unaware of the richness of certain classical solutions of Maxwell equations. The key is a “Radiation Cancellation” condition, wherein the field is dynamic but not dissipative.

The first example involved an homogeneous, isotropic sphere of charge that is unable to radiate when in purely radial oscillation. Subsequently, Langevin examined the case of a wall of charge in motion. Progress remained slow until 1948 when Bohm and Weinstein published a general study of Non-Radiating Distributions in the Physical Review.

The first to notice the importance of such Radiation-less Conditions for Quantum Mechanics was Goedecke with his landmark 1964 paper:

Classically Radiationless Motions and Possible Implications for Quantum Theory

where we seem important connections made with quantum theory.

Finally, in 1973, the seminal paper of Devaney and Wolf entitled:

Radiating and Nonradiating Classical Current Distributions and the Fields They Generate

saw the light of day. Here a magnificent truth is finally revealed in all its glory: an important general formula with which every radiating current must abide.

It is not surprising that Devaney and Wolf reached their conclusion in an effort to solve a practical problem unrelated to difficulties with QM. They focused on the task of recognizing and accurately identifying a radiation source. This requires the solution of an inverse problem of great significance to radar research and other such applications.

Decades later, two more pioneers, Marengo and Ziolkowsky continued this early work and recovered the full condition for a non-radiating current. It is quite suggestive to take a look at the specific formula appearing in p.3346 as eq.(4) of this paper which describes a condition on the current.

Perhaps the most important, the most significant, characteristic of this formula is a category of current flows that simply fail to obey the given condition. Such currents give rise to a host of possible Non-Radiating solutions!

The origin of this neglected possibility is to be found in 19th Century Hydrodynamics, but does not appear widely known, as judged by the contemporary Wiki consensus: see Non-Radiation Condition. The missing link consists of the Maren-Ziolkowsky work combined with the concept of a Force-Free Magnetic Field

It is then sufficient to take the formula for the eigenfield of the rotation operator, the “curl”, and instead of applying it to the magnetic field, apply it directly to the current sources.  All sorts of strange things will then start coming out of the hat. One will be forced to conclude that, under such circumstances, there will be a myriad of strange, tangled current flows giving rise to non-radiating, non-stationary charge volumes when the rotation eigenvalue is not a constant number.

What we are facing here is clearly a case of general conditions being reduced and studied only as special cases. Such oversights in science can happen. However, it is perhaps embarrassing that no mention, and no credit, is given to the father of the “force-free” condition, although he was well known in his own time. One has to dig under tons of old literature to rediscover the work of Eugenio Beltrami in the general classification of the different types of hydrodynamic flows.

It was Beltrami who explored eigen-fields of the rotation operator, which first appeared in his 1889 “Considerationi Hidrodynamiche”, Rend. Reale Ist. Lombardo 22. Some translations of extracts of these works are listed here:

The Importance of Eugenio Beltrami’s Hydroelectrodynamics

On the Mathematical Theory of Electrodynamic Solenoids

Considerations on Hydrodynamics

In our view, such vector fields might properly be called Beltrami Fields. They constitute eigen-fields of the rotation operator with a non-constant eigenvalue. As such, they can never satisfy the radiation condition of Devaney-Marengo-Ziolkowsky!

Evidently, it may be worth reexamining certain assumptions. Especially when one knows of other works in Solar Heliodynamics and Magneto-Hydrodynamics (MHD) where such general solutions of Beltrami equations lead to “Compartmentalised” solutions.

These are conceptually quite close to the apparent “quantisation” conditions only with some new ingredients that may perhaps serve as a model for an extended non-linear theory including the essential ingredient of “mass” as a self-interaction term as already proposed by our tribal leader.

In the end, I would not be surprised to find some similarity between such controversial proposals as “Spherical Lightning” and protons and electrons.

At least to this, I have had a minor contribution:

A Transmission Line Model for the Spherical Beltrami Problem

As for the rest, perhaps we must wait for the Shamans to fly away from their present obsession with branes, strings and other sundry instruments of mind haze.

[Editor’s note: An interesting and heartfelt post. Theo is encouraging us all to re-examine the classical works of Eugenio Beltrami and perhaps widen our field of vision. I would add one thought to amplify what he is perhaps alluding to at the conclusion. Locating self-consistent solutions to the Maxwell-Dirac and Einstein-Maxwell-Dirac equations has, so far, resisted all attempts by a most capable and committed group of mathematicians. However, we do know that static solutions with a net charge are not possible. Since an electron is charged and has a permanent magnetic moment the commentary above does seem pertinent. We are seeking a current configuration that is dynamic but non-radiating. In this respect, Theo has pointed us toward a very interesting and neglected avenue of inquiry.]

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