Postulate: EVERY field has a magnetic equivalent.

[StephenofCaly]

Postulate: EVERY field has a "magnetic" equivalent.

Here is a topic which I thought might be important enough to merit its own thread. I apologize for fishing it up off another discussion thread.

Here is a thought experiment.

Two electrons, fired off near to each other in parallel at near-relativistic speed, seem to the observer (at the electron cannon) to repel each other slower than should be expected from their mutual repulsion in the electrical field. Two separate lines of thought thoroughly explain this phenomenon. One can choose either for satisfaction.

1) General relativity (Lorentz-Fitzgerald transformation) allows for the electrons' mutual repulsive affects to appear less robust, due to apparent time dilatation v$/$c$^{2}$. The point is that the Lorentz-Fitzgerald transformation can thoroughly explain the observation.

2) On the other hand, Heaviside's simplification of Maxwell's equations can also thoroughly explain the phenomenon, as the moving electrons each can be indepentently thought of as a current, which each generates its own magnetic field; and that magnetic field acts upon the other electron so as to show a contrary "attractive" force that lessens the electrical repulsive forces.​

Of course, an observer moving along with the electrons sees no velocity, and only sees the simple electromagnetic repulsion.

Since each explanation is true and sound, the thought experiment simply shows the interchangeability of "electrical" and "magnetic" effects in relativistic physics. They are well known to appear different only in certain circumstances due to the frame of reference. However, the usual frame of reference in which we do experimental physics in the laboratories of Earth, favor an apparent difference. Neither explanation is the true one, and the other somehow lesser.

The electron experiment makes one think that magnetism is properly classed as an "illusion;" that is a fair opinion, although really they are apparently separable only in our familiar frames and times of reference.

However, it becomes quickly obvious that for ANY particle that moves in any sort of differentiable field capable of producing a noncontact force, F=∇U, will show similarly apparent modulations of the force between the two particles, simply due to the Lorentz-Fitzgerald transformation. This decrease in action is also thoroughly explained by general relativity. ALL fields capable of producing force, whether known or unknown, recognized or unrecognized, will produce the relativistic effect.

It is only an accident of history that Maxwell's equations antedated Lorentz and Einstein, that we came up with the pure theory of magnetism, which can be considered not a fundamental thing in its own right, but rather a manifestation of electricity in motion. Had we offered general relativity somehow before understanding electricity and magnetism, the development of the concepts might have been different.

The strong force will have a "strong-magnetism" and gravity will have "gravito-magnetism," simply because the time-dilatation and mass effects of general relativity can be expressed in the field curl terms of Maxwell's equations.

Now that we have made things all tidy, there is a small mystery to this speculation as regards the gravitational field. Two electrons dashing off into the distance not only APPEAR to be separating more slowly, they also APPEAR to have more mass due to the relativistic energy-momentum equation.

What of null massive charges - say, two neutrinos - fired off in parallel? Would they have an increased force of attraction proportionate to their increased masses?

(I use all the physics & mathematics still left 30 years after graduating from MIT with a BS in chemistry, so please be kind.)

 

[Muphrid]

I think you assume magnetic effects should manifest in other phenomena without considering why they exist mathemaically. Further, both your 1 & 2 are required as far as I understand.

 

[StephenofCaly]

In response to your comments:

I think you assume magnetic effects should manifest in other phenomena without considering why they exist mathematically. ​

Actually, no. I'd found a thread here called "is there a magnetic field around a moving "single" charge?" as an example of what I agree with, and whence the magnetic fields arise, mathematically.

My point is that (1) and (2) are just fine, in same way the photon is just fine obeying (A) the photoelectric effect, as well as (B) the diffraction grating effect. A century ago, people puzzled over this "contradiction" asking which was "true." (A) observes nothing about the photon as a wave; (B) observes nothing about the photon as a particle.

It took some thought to realize that A & B have absolutely no contradictions in what they state about the photon. Similarly, (1) and (2) have no contradictions, but fully explain something without reference to the other.

In such an instance, one can use the idea of "duality" to say that the thing one is observing has both characters.

That's why I say that ANY field will produce a "magnetic" equivalent (a v$/$c$^{2}$ weaker effect analogous to magnetism.) The analogy is powerful, as it postulates the effect in all fields, whether discovered yet or no.

 

[Muphrid]

I see; I think you're saying that you think that according to GR any "magnetic" fields arise simply from the transformation laws between frames, whereas the full Maxwell equations suggest a different view, that such fields really do exist. Is this the apparent contradiction you're trying to delve into?

 

[bcrowell]

It doesn't make sense to call this a postulate. A postulate is something we assume without being able to prove it. What you're describing is, at least in the case of electromagnetism, something that can be proved based on more fundamental ideas.

There's an element of truth to what you're saying, but it's extremely sloppy and you're overgeneralizing. A counterexample to your claim is the Higgs field. The Higgs is a scalar, so it doesn't change under a Lorentz transformation.

 

[PeterDonis]

the thought experiment simply shows the interchangeability of "electrical" and "magnetic" effects in relativistic physics.

Not really. You are reading too much into what is actually an artifact of your particular scenario: the fact that there exists an inertial frame--the mutual rest frame of the electrons--in which the field is purely electric. There are EM fields for which that is not the case; and for those fields, the magnetic component cannot be interpreted the way you are interpreting it here.

 

[Lamarr]

gravitational fields also have gravitomagnetic effects as determined by General Relativity.

Is that what you're saying?

 

[StephenofCaly]

I greatly appreciate the replies of so many fine folks, and thank you.

To summarize several replies in one:

It doesn't make sense to call this a postulate. A postulate is something we assume without being able to prove it. What you're describing is, at least in the case of electromagnetism, something that can be proved based on more fundamental ideas.

There's an element of truth to what you're saying, but it's extremely sloppy and you're overgeneralizing. A counterexample to your claim is the Higgs field. The Higgs is a scalar, so it doesn't change under a Lorentz transformation.​

The postulate is that EVERY (smooth differentiable energy-related) field has a $\frac{v}{c^{2}}$magnetic analogy which can be demonstrated. The example is that of electromagnetism, which we understand so well, we can use as a bedrock example. Surely, the electromagnetic field and all its vicissitudes is fairly well understood, and is not an issue deserving of the term 'postulate.' But the proposition goes on from there.

I did raise the point that such things as mass would need to be handled a bit different than any other field concept. I raised it regarding gravitational attraction, but it would also be relevant with the suspected Higgs field, which requires force-carrying particles to have mass. As I am unfamiliar with the Higgs field or its energetics, I cannot offer a well-formed example of the matter.

Specifically to the comment. "There's an element of truth to what you're saying, but it's extremely sloppy and you're overgeneralizing" - I disagree. Early on in the 20th Century, it became necessary to go about the study of quantum physics indirectly. Certain mathematical constructs were offered that were logically coherent and clean. They were affirmed or rejected based upon the observable events which they predicted, in the manner of snow from clouds. If there is snow, that means that above there are snow clouds; and snow clouds go about a certain set of events.

The logical progression of thought became much more facile to the scope of human thought when it depended upon the expansion and development of a physical mathematics, and the handling of abstract concepts in mathematics, formerly thought to be intellectual pastimes, but ones which could "snow out" observables in the physical world.

I get a sense that "sloppiness" is used to describe my laying out of my thoughts through prose. Each of the concepts which I offered above has an identical description in the language of mathematical symbolism. But I did not offer that pattern of construction in my post.

The contrary epithet would be that you are asking for jargon - an identical offering of the concepts in mathematical form. Saying things in prose or mathematically can be fundamentally identical and precise; the tidiness or sloppiness is merely a subjective preference of the reader. Please feel free to offer the presentation in mathematical formalism, if it would make it more comfortable for another reader.

Similarly, "and you're overgeneralizing" simply begs the question entirely. A postulate, as per one online definition, is "to claim or assume the existence or truth of, especially as a basis for reasoning or arguing." I claim that EVERY field has a "magnetic" equivalent, and offer a few reasons for that claim. Stating that the postulate is over-general merely states a dislike for the claim.

One person offers - "gravitational fields also have gravitomagnetic effects as determined by General Relativity...Is that what you're saying? " Yes, but with a set of reasoning attached that I have not seen demonstrated as such. Whether my offering is simply an example of my own peculiar or dull thought process, it makes sense to me in a way that other examples have not.

I read, "there exists an inertial frame--the mutual rest frame of the electrons--in which the field is purely electric. There are EM fields for which that is not the case; and for those fields, the magnetic component cannot be interpreted the way you are interpreting it here." I have offered that frame of reference - and allowed for any other, as one would do when considering things in a relativistic sense. What would prevent the interpretation that I've used to be applied in other frames of reference?"

Muphrid gets right to my point. Please do have at it. I claim that there is no true "contradiction" other than our own stubbornness of thought, as we struggled with regarding the wave/particle duality of the photon.

I realize I'm being a little bit of a pest about the matter, but I think that nuisances are the open doors to knowledge. Remember that excepting the ultraviolet catastrophe, physics was entirely dull and circumscribed at the end of the 19th century - merely awaiting the filling-in of the details. Like Egyptology, perhaps. We might find the body of a previously unknown Pharaoh, but we would not come across any new ones. After Heaviside's presentation of Maxwell's Equations, there was metaphorically nothing new under the sun.

Twenty years later, it all was blown up into flinders. Then things got fun.

 

[Muphrid]

I asked because I wanted to fully understand what you were trying to get across. Now that you've agreed that I mostly understood you correcly, let me present a different viewpoint.

In most conventional lessons on physics, we deal with no more than scalar or vector fields. Tensor field are often introduced going into GR, but tensors of different ranks and signatures have wildly different physical interpretations. Let's stay concrete with EM theory. I pose to you this: that in Minkowski spacetime, there exists a combined electromagnetic field (usually denoted $F$) that is a field of oriented planes at every point in spacetime. These oriented planes are called bivectors. Just as vectors are oriented line segments, bivecors are oriented planes, with both magnitude and "direction".

In spacetime, there are six unit planes (4 directions times 3 other directions and divided by 2 for combinatoric reasons). We can choose a basis such that these planes are tx, ty, tz, yz, zx, xy. The tx, ty, and tz planes are the "electric" components, and the others are the magnetic components, but this distinction is arbitrary. EM theory need not and does not really distinguish between the two. We find that two electrons with the same velocity will only experience electric force in their own frame, yes, but only because these electrons must have timeike four-velocity. Particles with spacelike four-velocity could have purely magnetic interactions in some frame, but we rule out such things on other physical grounds. Nevertheless, it is often useful to talk about purely spacelike current densities, and we often do that when charges are moving in mostly opposing directions.

The descripion of the EM field in terms of $E,B$ is of course entirely equivalent to this description with the bivector $F$, but seeing them as parts of the same single force is, to me, much more elegant than to posit that for every force there must be a corresponding magnetic force. Here, the unified description of EM covers both aspects in one master stroke.

So, to me, it's somewhat peculiar to suggest that every force must have a magnetic equivalent. Fields capable of imparting forces must naturally be objects in spacetime, and the set of such objects is somewhat constrainted. There are scalar fields, vector fields, bivector fields, and duals to each of these. What does magnetic mean for scalar and vector fields (we've already covered the bivector case), and why should we distinguish these effects instead of considering them as part of a unified force?

 

[PeterDonis]

I read, "there exists an inertial frame--the mutual rest frame of the electrons--in which the field is purely electric. There are EM fields for which that is not the case; and for those fields, the magnetic component cannot be interpreted the way you are interpreting it here." I have offered that frame of reference - and allowed for any other, as one would do when considering things in a relativistic sense. What would prevent the interpretation that I've used to be applied in other frames of reference?"

You're missing the point. It's not a question of different frames of reference; it's a question of different kinds of EM fields.

Your scenario involves a very special kind of EM field: one which looks purely electric in *some* frame of reference. It's that fact which allows you to interpret the magnetic field that appears in other frames as a "relativistic effect".

But there are other kinds of EM fields in which there is both an electric *and* a magnetic component in *every* frame. Your interpretation does not cover that kind of field at all. So your interpretation does not generalize even to all EM fields; and therefore one would not expect it to generalize to other types of fields either.

 

[PeterDonis]

We find that two electrons with the same velocity will only experience electric force in their own frame, yes, but only because these electrons must have timeike four-velocity.

No, that alone is not enough. See the phrase I bolded; the key fact is that the two electrons have *the same* velocity. If they had different velocities, i.e., if they were in relative motion, the force between them would not be purely electric in *any* frame.

 

[atyy]

Isn't this a consequence of defining a force via the 4-force? http://en.wikipedia.org/wiki/4-force

The classical Yang-Mills fields and the linearized gravitational field can be split into "electric" and "magnetic" components
http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/ModaveI/Laurent2.pdf (Eq 21,22)
http://en.wikipedia.org/wiki/Gravitoelectromagnetism

I don't know about nonlinear gravity though. There is an interesting comment in http://mathpages.com/rr/s5-06/5-06.htm, which starts off with an important caveat "On a purely local (infinitesimal) basis, the phenomena of gravity and acceleration were, in Einstein's view, quite analogous to the electric and magnetic fields in the context of special relativity, i.e., they are two ways of looking at (or interpreting) the same thing, in terms of different coordinate systems."

BTW, is this similar to Muphrid's point of view?

 

[Muphrid]

No, that alone is not enough. See the phrase I bolded; the key fact is that the two electrons have *the same* velocity. If they had different velocities, i.e., if they were in relative motion, the force between them would not be purely electric in *any* frame.

My point was to contrast electrons versus, say, charged tachyons that could experience a purely magnetic force even between two comoving particles.

 

[Samshorn]

For ANY particle that moves in any sort of differentiable field capable of producing a noncontact force, F=∇U, will show similarly apparent modulations of the force between the two particles, simply due to the Lorentz-Fitzgerald transformation.

Indeed, but this isn't a new insight (and it isn't just a postulate). Any relativistic "force", which necessarily propagates at a finite speed, must entail velocity-dependent terms to avoid inconsistencies with aberration. This was understood and written about by Lorentz in 1904 and more explicitly by Poincare in 1905, who pointed out that it must also apply to the "force" of gravity and any other force, if the principle of relativity is to be upheld. This was needed to account for the absence of torque in the Trouton-Noble experiment (for example), and the same argument applies to any force, as Poincare noted. This explains why Laplace's calculation that gravity must propagate millions of times faster than light is not correct. A deeper understanding of the mechanism for the general velocity dependent terms in any force that satisfies relativity was given by von Laue in 1911 in terms of the energy and momentum flows associated with stresses when viewed from different frames. This is the explanation for the famous "Right Angle Lever paradox".

The key fact is that the two electrons have *the same* velocity. If they had different velocities, i.e., if they were in relative motion, the force between them would not be purely electric in *any* frame.

Hmmm... The Lorentz equation for the electromagnetic force on a charge q is q(E + v x B), so in a frame where v is zero the force on the change is entirely due to E. Conversely, at any point in an electromagnetic field we can apply a boost that transforms the magnetic components to zero at that point. (Of course, this single boost won't in general transform B to zero at ALL points, but I don't think that is what people are saying.) There are no magnetic monopoles (as far as we know), so all magnetic fields arise from current flows, i.e., the motions of electric charge. Even the magnetic dipoles of spinning electrons (quantum spin) are evidently Ampereian rather than Coulombian, as can be inferred from the energy of "flipping" them.

 

[StephenofCaly]

Thank you all for the fine replies here.

My purpose in posting this assertion is not to pretend to reveal something profound. It is rather to offer a concept in extremely simple terms. Peter gets at the very gist of the thing in the comment "Your scenario involves a very special kind of EM field: one which looks purely electric in *some* frame of reference." Indeed. It is a rare bird of a thought experiment, and designed to be so. It is unrealistically simple. It is almost uselessly simple. It is so simple, I can even understand it.

With this simple way of looking at things, I offer nothing that could be considered novel, or at least not after World War I. The Trouton-Noble experiment, along with the Michaelson-Morley experiment, rather trumped the idea of the aether. Lorentz, Einstein seem to have beaten me to the 1921 Nobel Prize in Physics, as did Michaelson a bit earlier. My chances for achieving such prize in the 20th century are vanishingly small.

I would love to pounce on "There are no magnetic monopoles (as far as we know)" but that's aside from the intent of this thread.

I thoroughly agree with the statement by Muphrid.
"I pose to you this: that in Minkowski spacetime, there exists a combined electromagnetic field (usually denoted F) that is a field of oriented planes at every point in spacetime. These oriented planes are called bivectors. . We can choose a basis such that these (bivectors) are tx, ty, tz, yz, zx, xy. The tx, ty, and tz planes are the "electric" components, and the others are the magnetic components, but this distinction is arbitrary. EM theory need not and does not really distinguish between the two. We find that two electrons with the same velocity will only experience electric force in their own frame, yes, but only because these electrons must have timelike four-velocity." Of course. The apparent "duality" of the two mechanisms of understanding in the little thought experiment unify through the handling of the electromagnetic field as an electromagnetic four-potential and calculating certain things by means of the electromagnetic field tensor.

Further on...​

"What does magnetic mean for scalar and vector fields (we've already covered the bivector case), and why should we distinguish these effects instead of considering them as part of a unified force?" Well, one fellow's fascination is another's dirty laundry. The goal of modern physics is the pursuit Occam's concept of parsimony in explanation - the Grand Theory of Everything - and mathematical rigor and simplicity allows for the more certain testing of falsifiability. One can expect that Reality as it stands is SO complex that it must arise from a small set of simple laws. We try to conceptualize these by simplifying apparent dualities and incongruities into single certainties. To stop waxing prolific, I think there is something to a general statement about differentiable potential fields in general which allows for some simplification. At least it's helping my mind, when I wander off the heights of amazement down the cliff of confusion - that's a daily occurrence, I regret.

 

[PeterDonis]

My point was to contrast electrons versus, say, charged tachyons that could experience a purely magnetic force even between two comoving particles.

Yes, but that would still be a special type of EM field. A fully general EM field, as I said, has both electric and magnetic components in *every* frame.

 

[PeterDonis]

The Lorentz equation for the electromagnetic force on a charge q is q(E + v x B), so in a frame where v is zero the force on the change is entirely due to E.

Yes, that works fine for one electron. Now consider two electrons in relative motion. Either one has a nonzero v relative to the other, so the Lorentz force on either one due to the other will have both an E and B component.

 

[PeterDonis]

Conversely, at any point in an electromagnetic field we can apply a boost that transforms the magnetic components to zero at that point.

Even for a completely generic EM field? I seem to remember that there is a "canonical" representation for a completely general EM field in terms of two wedge products:

$$F = E dt \wedge dz + B dx \wedge dy$$

How would you make the B term vanish with a boost?

 

[pervect]

Since E^2 - B^2 is an invariant, you can't transform a pure-B >0 into a pure E > 0, becuase the invariant can't stay constant , it would have to change sign.

 

[Samshorn]

Consider two electrons in relative motion. Either one has a nonzero v relative to the other, so the Lorentz force on either one due to the other will have both an E and B component.

Another way of saying it is that the force on each particle, in terms of its own rest frame, is purely due to the electric field at that location.

Even for a completely generic EM field?

No, only if E is perpendicular to B, as it is, for example, in an electromagnetic wave. You're right that for E not perpendicular to B, no single boost can make all the components of B vanish at the same time. The dot product of E and B is invariant (as is E^2 - B^2), so we can make B vanish only if it is perpendicular to E, so the dot product is always zero.

 

[PeterDonis]

Another way of saying it is that the force on each particle, in terms of its own rest frame, is purely due to the electric field at that location.

Hm, I see what you mean. I should probably look at the total EM field tensor produced by both electrons together.

No, only if E is perpendicular to B, as it is, for example, in an electromagnetic wave. You're right that for E not perpendicular to B, no single boost can make all the components of B vanish at the same time. The dot product of E and B is invariant (as is E^2 - B^2), so we can make B vanish only if it is perpendicular to E, so the dot product is always zero.

Ok, that's what I thought.

 

[pervect]

(I use all the physics & mathematics still left 30 years after graduating from MIT with a BS in chemistry, so please be kind.)

Were you perhaps thinking of this as a conjecture rather than a postulate? It makes more sense that way...

 

[StephenofCaly]

I note with pleasure that Pervect looks for a more acute definition in the philosophy of science, using the benchmarks of conjecture and postulate.

In the general convention in which I know these terms - with no clear mathematical implications - they are quite close. The distinction, as I know it, is that a conjecture is a statement that can be discretely dissected into a determined causal chain that links the statement to probative certainty. A postulate derives its claim to veracity in a more undefined manner, perhaps by its utility in application to the study of the order of things.
Dirac offered a postulate regarding the existence of magnetic monopoles, based on the understanding that electrical charge could be quantified if (but not iff) magnetic monopoles exist.

I go with probability here. I offer that any and every well-ordered potential energy field, no matter its source, has an observable relativistic effect that can be expressed by Heaviside's restatement of Maxwell's laws. The statement includes any sorts of forces which are tensors from the potential energy field, no matter the nature of the field. I offer that this explains gravitomagnetics as well as 'strong-magnetics,' which would be awfully awkward to demonstrate.

This way of looking at things may not be mathematically powerful, but it does offer the ability to construct a chain of causality consistent with that of other conjectures. I'll take the change to conjecture, as there is no apparent barrier to proof.

 

[StephenofCaly]

To consider the statement on magnetism:

Since E^2 - B^2 is an invariant, you can't transform a pure-B >0 into a pure E > 0, because the invariant can't stay constant , it would have to change sign.​

$\vec{B}$ = $\vec{B}$$\bot$ + $\vec{B}$ ‖ in terms of the apparent velocity of the charged particle.

$\vec{B}$² $\equiv$ $\vec{B}$•$\vec{B}$ = $\vec{B}$$\bot$•$\vec{B}$$\bot$ + $\vec{B}$ ‖ • $\vec{B}$ ‖

BUT:

The cross-product is distributive over addition, so
$\vec{v}$×$\vec{B}$ = $\vec{v}$×$\vec{B}$$\bot$ + $\vec{v}$×$\vec{B}$ ‖
=|$\vec{v}$| |$\vec{B}$$\bot$|

So the effective parallel component of $\vec{B}$ - does it cancel out or not?

 

[pervect]

I note with pleasure that Pervect looks for a more acute definition in the philosophy of science, using the benchmarks of conjecture and postulate.

In the general convention in which I know these terms - with no clear mathematical implications - they are quite close. The distinction, as I know it, is that a conjecture is a statement that can be discretely dissected into a determined causal chain that links the statement to probative certainty. A postulate derives its claim to veracity in a more undefined manner, perhaps by its utility in application to the study of the order of things.
Dirac offered a postulate regarding the existence of magnetic monopoles, based on the understanding that electrical charge could be quantified if (but not iff) magnetic monopoles exist.

Let me illustrate my thinking by a few examples. For postulates, we have Euclid's geometry, and things like the parallel postulate.

What happens with postulates , using Euclid's postulates as an example, is that one proves things from them after assuming them.

I don't see any such proofs coming out of your "postulate" , so it doesn't seem to me to be a good term for what you are saying.

Now, let's look at some conjectures. One that I'm familar with is the "hoop conjecture" by Thorne.


http://en.wikipedia.org/w/index.php?title=Hoop_Conjecture&oldid=454131955

he Hoop Conjecture, proposed by Kip Thorne in 1972, states that an imploding object forms a black hole when, and only when, a circular hoop with a specific critical circumference could be placed around the object and rotated. The critical circumference is given by:

[2 * pi * the schwarzschild radius]

So, it seems to me that what you are trying to say is much more similar to a conjecture than a postulate.

Unfortunately, it's not particularly clear how to test your conjecture, at least to me, the wording is a bit fuzzy.

One last related point:

We can say something that is not a conjecture. That something is that a Coulomb force law by itself isn't Lorentz invariant. Given that we expect physical laws to be Lorentz invariant (based on our current understanding of relativity), we don't expect an unmodified Coulomb force law to occur in nature.

 

[StephenofCaly]

What happens with postulates , using Euclid's postulates as an example, is that one proves things from them after assuming them.

I don't see any such proofs coming out of your "postulate" , so it doesn't seem to me to be a good term for what you are saying.

We can say something that is not a conjecture. That something is that a Coulomb force law by itself isn't Lorentz invariant. Given that we expect physical laws to be Lorentz invariant (based on our current understanding of relativity), we don't expect an unmodified Coulomb force law to occur in nature.

Your last sentence gets right to it, pervect. Let me restate my offering:

Hypothesis of Force Fields

1. A force is described as the tensor resulting from the differentiation of a scalar potential energy field. Movements of massive particles result from the sum of the forces thereupon, proportionate to their mass, no matter the character of the potential energy field.
2. Given that we expect physical laws to be Lorentz invariant (based on our current understanding of relativity), we don't expect an unmodified Coulomb force law to occur in nature.
3. Given that the Coulomb force law is not expected to have any unique behavior qua electromagnetism in a Lorenz transformation, one cannot state, based upon the timeline of movement, that the movement of those particles must arise from a specifically electromagnetic force.
4. Heaviside's restatement of Maxwell's equations, which we commonly refer to as "Maxwell's equations," suffice to explain the movement of electrical particles in the little thought experiment - without incurring any relativistic transformation. The two equations involving curl suffice to describe the movement of charge and its apparent contravailing "attractive force" from magnetics.
5. The Lorentz transformation alone and also Maxwell's equations alone serve to map the findings of the observer at rest with the charged particles, to any observation by an observer seeing the charged particles in motion.
6. If there are two independent ways of constructing a mathematical law of physics which produce identical observations, without conflict, they are likely two versions of one unique law, which can be simplified.

THEREFORE I OFFER what is truly a postulate:

• Particles moving in force fields described in (1) above, and which are affected in their movement by that force, can be equally well described the the Lorentz transformation or the two equations discussed in (4)
• In discussion of electromagnetism, the tautology described in (6) has been described to the point of tedium, as the phenomenon of "magnetism" is so familiar to the human observer.
• The tautology in (5) and the proposition in (6) are not, however, unique to electromagnetism (vid. 3) The 'curl equations' can be invoked for any motion in any field without the Lorentz transformation.
• Therefore, there must be an analogous concept similar to magnetism in electrodynamics that applies to any field described in (1), independent of the nature of the potential energy scalar from which it arises.

I do not propose that the magnetism is real or virtual here - it certainly is real (i.e. a separately observable concept) in electrodynamics, but not observed effectively yet in other forces. However, ANY field, arising from whatever source, which obeys (1), can be assigned a junior member, a 'magnetic' twin, in the manner of electromagnetism.

ONE EXCEPTION to the postulate is that of the gravitational field, and that is merely a quantitative, not a qualitative exception.

As explained before, the change in apparent mass of particles moving at high speeds introduces a truly second-order effect in the equations of motion which is not covered under the Maxwell's dynamics equations by analogy. I think it is interesting to see how the Maxwell's equations might be tweaked to cover such things. Proposition (3) above does not apply when discussing the gravitational field - one can observe the particle movement at a distance, and infer that it is a gravitational force that gives a certain component to the movement.

So, I rather beg to differ on the idea of a postulate. I believe that the lines of thought deriving from this valid observation have implications both in experimental and theoretical development. Perhaps the concepts are oblique to or simplistic in canonical descriptions of physical matter - the field is a highway, after all, and I am just four-wheeling it out in the intellectual boonies. However, I think I have all my wheels on the ground.

 

[atyy]

[*]A force is described as the tensor resulting from the differentiation of a scalar potential energy field. Movements of massive particles result from the sum of the forces thereupon, proportionate to their mass, no matter the character of the potential energy field.

To generalize, this would seem the definition of a gauge field.

http://michaelnielsen.org/blog/yang_mills.pdf (compare Eq 52 with Eq 53)

That may also explain why gravity at the nonlinear level is different - it is not a Yang-Mills gauge field.

 

[PeterDonis]

A force is described as the tensor resulting from the differentiation of a scalar potential energy field.

As it is stated, this is false even for electromagnetism; the EM field tensor, which is what appears in the covariant formulation of the Lorentz force equation, is the derivative (the exterior derivative) of a *vector* potential, not a scalar potential. So "scalar" needs to be replaced by "vector" in the above if it is going to apply to electromagnetism.

In the case of gravity, I'm not sure what the "potential" would be. Newtonian gravity can be formulated that way; but Newtonian gravity does *not* exhibit any "magnetic" phenomena. Atyy mentioned gauge theories, and General Relativity can be viewed as a gauge theory, but it's a more complicated gauge theory than electromagnetism or Yang-Mills theory (which basically generalizes electromagnetism to other gauge groups).

That said, there are definitely useful analogies between Maxwell's Equations for electromagnetism, and the Einstein Field Equation for gravity. MTW goes into this at some length; I believe other GR textbooks do to.

 

[Muphrid]

A force is described as the tensor resulting from the differentiation of a scalar potential energy field. Movements of massive particles result from the sum of the forces thereupon, proportionate to their mass, no matter the character of the potential energy field.

Yet in SR, we know that there is a potential four-vector, the EM field bivector, and the force arising from $f = F \cdot j$. How does that affect your statement here?

 

[Vanadium 50]

This has turned into a discussion of a personal theory.