# Postulate: EVERY field has a magnetic equivalent.

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.)

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.

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.

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
Staff Emeritus
Gold Member

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
Mentor

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.

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

Is that what you're saying?

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.

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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
Mentor

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
Mentor

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?

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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.

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.

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.

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PeterDonis
Mentor

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
Mentor

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
Mentor

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
Staff Emeritus

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.

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
Mentor

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
Staff Emeritus

(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...

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.

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
Staff Emeritus

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 doesnt 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.