Merged Relativity+ / Farsight

[Farsight]

I think your arguments really do tend to suggest otherwise.

No, I really understand curved spacetime. And to be brutally frank, you don't.

Heaviside's interesting (at the time) speculations have of course been superceded by the approximation scheme in GR which I described above. Let's not muddy the waters with his outdated model, and just stick to gravitoelectromagnetism (henceforth "GEM", and I'll abbreviate plain electromagnetism to "EM").

No, let's not.

That aside, I find it very interesting that you cleave so tightly to the analogy which is only ever an approximation, but reject with such vehemence the one which is actually exact in the few situations where it applies. I have to wonder whether you have properly grasped either.

I have. And as I've said previously, space is not falling inwards in a gravitational field. We do not live on a Chicken-Little Earth. The sky is not falling in. The waterfall analogy is specious pseudoscience peddled by quacks.

Hmm, I think that calls your claim to understand the GEM-EM analogy into serious doubt. Did you realise that your sentence above is exactly synonymous with "I'm not fond of the concept of mass"?

It isn't. We are all familiar with the concepts of mass and charge. And we are all aware that a concentration of energy causes gravity.

ETA: And I have to ask this general question: why would your fondness for an idea have any relevance at all in a rational argument about its value?)

Because it's an abbreviated way of saying I take issue with gravitational charge for a number of reasons.

Ah, you think the differences I mentioned are irrelevant to the GEM-EM analogy.

No.

Let's start with the second difference. The fact that test masses all fall at the same rate in gravitational fields while test charges accelerate at different rates in electromagnetic fields is just as valid in GEM as it is in full GR. My apologies - I assumed you would already know this.

I know it. And I know exactly why test masses fall and why test charges accelerate.

Now for the first difference (the non-linearity of GR vs. the linearity of EM). This should serve to remind us that what underlies GEM is radically different from what underlies EM, and that the equations of GEM (unlike those of EM) are applicable only to small perturbations of a background field.

We know the mathematical approaches are different. What we are examining is the similarity between and electromagnetic field and a gravitomagnetic field.

An amusing aside: to be consistent with your comments on local flatness earlier in this thread (e.g. see the last paragraph of this post), perhaps you should now be proclaiming that "this means they don't apply at all!"

I said the principle of equivalence was an "enabling principle". It applies to an infinitesimal region. Which means it doesn't apply at all. A real gravitational field is not equivalent to accelerating through space, you cannot transform it away.

Both differences (and there are also many others) demonstrate that the GEM analogy is, like all analogies, limited. It would be unwise to take it too seriously, just as it would be unwise to take the waterfall analogy too seriously (a statement with which I'm sure you agree).

You've said nothing. Apart from a load of wishy-washy guff urging me to ignore Heaviside. As well as Maxwell and Minkowski and Einstein? Strewth.

As you are to Anders Lindman, so am I to you.

 

[Farsight]

... I'm hoping your knowledge of this particular paper, combined with your knowledge of vector calculus and willingness to educate us, will save me from concluding that Heaviside made a mistake at this critical step of his derivation.

No you aren't. You're just putting up a mathematical distraction that you're hoping will bog down the discussion and turn off other readers. Request declined.

 

[Farsight]

Here's a statement from Heaviside that may interest Farsight:

"It is not known that there are no magnetons, but rather that, if there be any, they are in pairs like the ions and cannot dissociate as far as can be concluded from div.B = 0. ... I think it wise not to be overhasty in rejecting terms in the circuital equations because magnetons do not exist".

What are magnetons? So-far-undiscovered particles which, if they exist, would cause div.B != 0? They're what we now call magnetic monopoles. Heaviside proceeds to write some electromagnetic wave calculations including the familiar monopole term.

No problem. Dirac was something of a Heaviside fan. But note that Heaviside said they are in pairs, and his overhasty was fine because it was circa a hundred and twenty years ago.

 

[Farsight]

I was hoping Farsight would take advantage of this opportunity to demonstrate his knowledge of vector calculus, but I guess he's been too busy promoting crackpottery in other threads.

How dare you accuse me of promoting crackpottery. I'm the one referring to Einstein and Minkowski etc and to the hard scientific evidence. You're the one pooh-poohing it like some Creationist. And for your information, the infinite universe is the crackpottery. It is at odds with big bang cosmology, and is abused by charlatan quacks who would tell you that there's an infinite number of W D Clingers out there tippy-tapping on their keyboards.

Farsight, as evidence for his mastery of electromagnetism and vector calculus, has often referred to his many quotations of a single sentence from page 558 of John David Jackson's Classical Electrodynamics, third edition. To return the favor he has done me by not responding to my request for the education he offered, I'll direct his attention to section 6.7 and equation (6.108), which is Jackson's statement of the electromagnetic equation Vorpal wrote above.

Come on Clinger, provide a link. Equation 6.108 in this edition is not in section 6.7.

With that background, let's return to the out-of-context sentence fragment Farsight quoted to demonstrate his appreciation of ctamblyn's remark that gravitomagnetism "is a sometimes-useful approximation and analogy, and definitely not something equivalent to GR in its full gory detail." In this expanded quotation from Heaviside's paper, I have highlighted the missing minus sign in red, highlighted Farsight's quote-mine in brown, and highlighted the more important parts of that paragraph in blue

And you're boring everybody to death.

In Heaviside's inexact (pre-relativistic) analogy, the vector field e is analogous to the electric field E, and the vector field h is analogous to the magnetic field H.

Yawn.

Farsight continues to reject the legitimacy of E and H, even to the point of saying they are not fields. It is therefore extremely difficult for me to understand how Farsight could accept Heaviside's analogy.

No it isn't. You know full well that the thing we call the electric field and the thing we call the magnetic field are but two aspects of the greater whole called the electromagnetic field. Here we go again, see wikipedia, and I quote: "Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field".

I hope Farsight will explain whether he rejects the legitimacy of Heaviside's e and h in the same way he rejects E and H. If he does reject e and h, I hope he will explain why he accepts an analogy based upon fields he rejects. If he does not reject e and h, then I hope he will explain why Heaviside regards e and h as analogous to the vector fields Farsight rejects.

I don't reject E and H. I reject people like ben m going around talking about them as if Jackson didn't say "one should properly speak of the electromagnetic field Fuv rather than E or B separately".

I doubt whether Farsight is any more willing to provide that requested explanation than he has been to demonstrate his mastery of vector calculus, but we can hope. This is, after all, an opportunity for him to teach us something about physics.

That it is. And I will refer you again to what Minkowski and Maxwell said about the screw nature of electromagnetism:

"In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect".

"A motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw".

You will of course dismiss that. Because as you are to Anders, so am I to you.

 

[Farsight]

Let's have a dedicated thread on this.

Testing, one two.

 

[Vorpal]

The equation W.D.Clinger refers to is (6.108) in both the 1975 second edition and the 1999 third edition. In the first edition you've linked (which I've never seen until today), it is (6.82) instead. In any case, it is exactly equivalent to the last equation I've posted in post #1558, except the fact that the first and second editions use Gaussian units, in which the Poynting vector has an extra factor of (c/4π).

 

[Farsight]

The screw nature of electromagnetism

Take a look at Minkowski’s Space and Time dating from 1908. Towards the back is this:

"In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect".

Note how he said the field, and referred to electric and magnetic forces. The electron doesn't have an electric field or a magnetic field, it has an electromagnetic field. See Wikipedia and note this:

"Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole - the electromagnetic field".

Also note the word screw. If you have a pump-action screwdriver you’ll appreciate that linear force is converted into rotational force. That’s like an electric motor: current flows along the wire, and the motor turns. When you use an ordinary screwdriver, rotational force is converted into linear force, and the screw is driven into the wood. That’s like a dynamo: turn the rotor, and current flows along the wire. We even have the right-hand rule which applies not just to electromagnetism, but to screw threads.

But you don’t usually hear about this "screw nature of electromagnetism". Instead you tend to read about the electric field E and the magnetic field B as if they’re separate fields rather than “two parts of a greater whole”. In John Jackson’s authoritative textbook Classical Electrodynamics you have to wait until section 11.10 before he says "one should properly speak of the electromagnetic field Fμv rather than E or B separately". By and large, physicists who are taught about electromagnetism think of E and B as fields rather than forces, and have no real concept of the Fμv electromagnetic field. They don't know how to visualize it. They can visualize the E and B easily enough. There's plenty of depictions out there. E is usually drawn with radial lines of force, and B is usually drawn with concentric lines. There's no problem visualizing them. But there's nothing that lets you visualize the "greater whole". However I think there is a way. A simple way. You just combine the radial and concentric lines. Like this:

Once you do this, you immediately appreciate the screw nature of electromagnetism in a visceral way. Other things start falling into place too. You read about the frame-dragging of gravitomagnetism and spot things like "there is a space-time vortex around Earth" and "if space is twisted". You appreciate why gravitomagnetism is an analogy of electromagnetism. You follow the lead back to Maxwell and you spot this: "a motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw". And you spot Maxwell’s page title too, which is "The Theory of Molecular Vortices". You appreciate Faraday all the more, and you realise why spinors are called spinors. You read about Dirac’s belt and you start to see things that were never in your textbooks. You can create an electron along with a positron in pair production. Out of light. And you can diffract the electron, because it has a wave nature. Like it’s light trapped by its own displacement current such that what was an electromagnetic field-variation now looks like a standing field. Like it's an optical vortex.

It fits. Especially since counter-rotating vortices attract, and co-rotating vortices repel. A cyclone is a vortex. If you could set down two cyclones next to one another they’d move linearly apart. If you could set down a cyclone near to an anticyclone, they’d move together. And if you could hurl the cyclone past the anticyclone, they’d swirl around one another too, like electrons and positrons do in positronium. They're dynamical spinors in frame-dragged space, and E and B denote the linear and rotational forces between them. The forces that result from electromagnetic field interactions, where it takes two to tango. Hence when you look at Wikipedia again, now you notice this:

"The electric field is a vector field. The field vector at a given point is defined as the force vector per unit charge that would be exerted on a stationary test charge at that point".

Yes, it makes sense, but you don’t hear much about the screw nature of electromagnetism. Or the geometry of electromagnetic systems. It ties in with topological quantum field theory, but not the Standard Model. People say the electron is a fundamental particle, and some even say it's a point particle, even though the electron is supposed to be a field excitation. Even though it's quantum field theory, not quantum point particle theory. For myself I think this needs to go into the Standard Model, and work is required "within the Standard Model", not "beyond the Standard Model". But it isn't easy persuading the sort of people who say the electron is surrounded by a cloud of photons popping in and out of existence. Spontaneously. Like worms from mud. As if hydrogen atoms twinkle, and magnets shine. So whilst we'll get there one day, it will be a while yet. And meanwhile, just like Max Planck said, science advances one funeral at a time.

 

[edd]

Vector fields with certain quite reasonable properties are 'Helmholtz decomposable' - they can be described as a curl-free field to a divergence-free field.
In Farsight's cartoon above the first is curl-free, and the second is divergence-free.

Farsight - are you saying the first is what is commonly called an electric field and the second a magnetic field (regardless of your dislike of separating the two - you are no doubt aware that others choose to do so rightly or wrongly)?

So you're saying the electromagnetic field is the sum of a divergence-free and a curl-free part that are the magnetic and electric parts respectively? Because it definitely looks like that is what you are trying to say, and if it isn't that I want to know what it is you are saying.

Note that I wouldn't say this myself - this much would be quite obvious to many already.

 

[Farsight]

OK thanks Vorpal.

All: I've got to go now I'm afraid. But I've started a new thread on the screw nature of electromagnetism. I hope it doesn't get moved or merged and that it receives your sincere attention. Please try to stay on topic there and focus on the essential message. Is it wrong? And if so why so? If you can't elucidate a reason, ask yourself if you've actually got one. And if you haven't, don't be too quick to dismiss what Minkowski and Maxwell said.

ETA:

Gotta go edd. I'll get back to you. Can we try to have different threads for different topic please.

 

[ben m]

However I think there is a way. A simple way. You just combine the radial and concentric lines. Like this:

What actual quantity are you represented by the lines in a, in b, and in c?

What central source has been depicted? If it's a point source rather than a line source, please show the two orthogonal sections to give a sense of the 3D structure.

 

[edd]

I've suggested that be merged back here. You might suggest otherwise. If any mod action happens I'll stick to where things end up of course.

 

[roger]

My browser seems to not be rendering the mathematical equations.

 

[W.D.Clinger]

poseur

... I'm hoping your knowledge of this particular paper, combined with your knowledge of vector calculus and willingness to educate us, will save me from concluding that Heaviside made a mistake at this critical step of his derivation.

No you aren't. You're just putting up a mathematical distraction that you're hoping will bog down the discussion and turn off other readers. Request declined.

You linked to a transcription of Heaviside's paper that contained a clear error in equation (9). Correcting that error was hardly a distraction. That error obscured the analogy between Heaviside's vector fields e and h and the electric field E and magnetic field H, which was the main point of the paragraph you had quoted.

That you do not understand the relevance of that error is telling. Had you not tried to dismiss my discussion of the error as "a mathematical distraction", you could have continued to pretend you understood Heaviside's analogy.

Farsight, as evidence for his mastery of electromagnetism and vector calculus, has often referred to his many quotations of a single sentence from page 558 of John David Jackson's Classical Electrodynamics, third edition. To return the favor he has done me by not responding to my request for the education he offered, I'll direct his attention to section 6.7 and equation (6.108), which is Jackson's statement of the electromagnetic equation Vorpal wrote above.

Come on Clinger, provide a link. Equation 6.108 in this edition is not in section 6.7.

As highlighted above, I specified the third edition because I know the numbering of equations and sections can change between editions.

Your reliance upon web links tells me you don't even own a copy of the book. No wonder you've been quoting only one sentence from only one of its 800+ pages.

Here's a low-resolution scan of page 259 from my own personal copy of the third edition:

How dare you accuse me of promoting crackpottery. I'm the one referring to Einstein and Minkowski etc and to the hard scientific evidence. You're the one pooh-poohing it like some Creationist. And for your information, the infinite universe is the crackpottery. It is at odds with big bang cosmology, and is abused by charlatan quacks who would tell you that there's an infinite number of W D Clingers out there tippy-tapping on their keyboards.

ETA: Despite your boasts (quoted below), you are quite confused about cosmology, relativity, and spacetime. Whether the universe is spatially infinite is a valid question that's independent of the multiverse hypothesis or Everett interpretations of quantum mechanics.

Those of us who have actually put in the time and effort needed to understand Einstein's mathematics can check for ourselves whether the FLRW solutions satisfy Einstein's field equations. They do. There are three main classes of FLRW solutions, in which the large-scale geometry of space (not spacetime) is (K=+1) spherical, (K=0) flat, or (K=-1) hyperbolic.

The "hard scientific evidence", as you put it, cannot at present distinguish between those three possibilities, which means the space we live in is close to being flat. Flat manifolds can have boundaries, but there is absolutely no "hard scientific evidence" for any such boundary; flat manifolds can also be finite in one or more directions without having boundaries, as in the case of a cylinder, but there is absolutely no "hard scientific evidence" for that possibility either; the "hard scientific evidence" available to us at present is consistent with the possibility that space (not spacetime) is (approximately) Euclidean, hence infinite.

That is not a crackpot opinion. That is the mainstream consensus, as can be seen by consulting standard references on relativity and cosmology (e.g. Hawking & Ellis, Misner/Thorne/Wheeler, Wald, Weinberg; you don't own any of those books either, do you?). When you dispute the mainstream consensus while presenting no evidence beyond your own authority and quote-mined snippets from physicists who are too dead to defend themselves against your misinterpretations of their writings, you are expressing a crackpot opinion.

I offered you an opportunity to demonstrate your understanding of freshman-level vector calculus and electromagnetism. You declined that opportunity, preferring to post in other threads.

When you finally deigned to respond here, you supported your opinion with your usual litany of sacred proof-texts and arguments such as:

And you're boring everybody to death.

Yawn.

Because as you are to Anders, so am I to you.

No it isn't.

No it isn't, edd. You're wrong.

No, I really understand curved spacetime. And to be brutally frank, you don't.

You've said nothing. Apart from a load of wishy-washy guff urging me to ignore Heaviside. As well as Maxwell and Minkowski and Einstein? Strewth.

As you are to Anders Lindman, so am I to you.

That's the kind of argument we'd expect from a poseur who has never put forth the effort necessary to understand freshman-level vector calculus and electromagnetism, let alone differential geometry and general relativity.

I did, however, see one tiny glimmer of intellectual progress:

I don't reject E and H. I reject people like ben m going around talking about them as if Jackson didn't say "one should properly speak of the electromagnetic field Fuv rather than E or B separately".

If you believe anyone here has been talking about the electric and magnetic fields as if they were unaware that the vector fields E and B taken together are equivalent to the electromagnetic tensor field F_μν, then you should cite examples of that.

ben m is certainly aware of that equivalence, as am I.

 

[Farsight]

Huh. I see we've been merged.

What actual quantity are you represented by the lines in a, in b, and in c?

I presume a b and c are the three items in this picture:

As for what actual quantities are being depicted, I struggle to say. Item a depicts radial "electric lines of force" around a charged particle. Item b depicts "magnetic field lines" as per the current-in-the wire. As we know the latter aren't quite lines of force because motion is around the field lines, and the concentric lines aren't quite apt for an electron because it has a dipole nature. But the "the greater whole" that is the electromagnetic field surely deserves some kind of depiction, and I think item c achieves that quite well.

What central source has been depicted?

The archetypal point particle, which I do not agree with. The electron is more like the "spinor" depicted in c.

If it's a point source rather than a line source, please show the two orthogonal sections to give a sense of the 3D structure.

Does not parse. It isn't a point source. Charge is topological. See this on physicsforums. Also see this spindle-sphere animation which relates to the centre of c but does not depict the associated frame-dragging.

 

[Farsight]

I've suggested that be merged back here. You might suggest otherwise. If any mod action happens I'll stick to where things end up of course.

I do suggest otherwise. It just isn't possible for me to have a sensible discussion with you and others about all the various aspects of physics and cosmology on a single thread. IMHO merging everything into a single thread isn't moderation, it's an underhand form of censorship, siding with orthodoxy, and siding against free speech in science. This is why I ask you to start a thread to discuss say dark energy. If you start a thread, it won't get merged with this one.

Vector fields with certain quite reasonable properties are 'Helmholtz decomposable' - they can be described as a curl-free field to a divergence-free field. In Farsight's cartoon above the first is curl-free, and the second is divergence-free.

Cartoon? I'm beginning to wonder if you are worthy of my time and attention.

Farsight - are you saying the first is what is commonly called an electric field and the second a magnetic field

Yes of course.

(regardless of your dislike of separating the two - you are no doubt aware that others choose to do so rightly or wrongly)?

Yes of course.

So you're saying the electromagnetic field is the sum of a divergence-free and a curl-free part that are the magnetic and electric parts respectively? Because it definitely looks like that is what you are trying to say, and if it isn't that I want to know what it is you are saying.

No, that isn't what I'm saying. Have you even read the screw nature of electromagnetism? What I'm saying is that the thing you think of as a charged particle's radial electric field isn't a field at all. It's merely a way of describing the force that results from electromagnetic field interactions. Like this:

 

[Farsight]

...That is not a crackpot opinion. That is the mainstream consensus, as can be seen by consulting standard references on relativity and cosmology (e.g. Hawking & Ellis, Misner/Thorne/Wheeler, Wald, Weinberg; you don't own any of those books either, do you?). When you dispute the mainstream consensus while presenting no evidence beyond your own authority and quote-mined snippets from physicists who are too dead to defend themselves against your misinterpretations of their writings, you are expressing a crackpot opinion....

Oh yawn, spare me from the magisterium.

Edited by kmortis: 

Removed personal comment

 

[edd]

Especially since counter-rotating vortices attract, and co-rotating vortices repel.

So if I set up a magnetic field that loops round in a clockwise direction, and set it down next to another source of a magnetic field looping round in a clockwise direction - lets say a pair of electric currents going in the same direction so their magnetic fields loop in the same direction - they'll repel?

Huh.

 

[W.D.Clinger]

Thank you, Farsight, for attempting to answer several of the critical questions people have been asking about your cartoons diagrams since 9 March 2010, the day you started this thread. The physicists will have much more to say about this, but I'll say a few of the more obvious things.

What actual quantity are you represented by the lines in a, in b, and in c?

I presume a b and c are the three items in this picture:

As for what actual quantities are being depicted, I struggle to say.

Your inability/refusal to say what is being depicted in those diagrams has been criticized throughout this thread. We appreciate that you are finally making an effort to answer that question.

Item a depicts radial "electric lines of force" around a charged particle. Item b depicts "magnetic field lines" as per the current-in-the wire.

That means the lines shown in your diagrams are directed—we should think of them as having arrowheads on them.

Your "current-in-the-wire" remark also implies symmetry within the plane but limited symmetry in 3-dimensional space. More on that in a moment...

But the "the greater whole" that is the electromagnetic field surely deserves some kind of depiction, and I think item c achieves that quite well.

The electromagnetic field does deserve depiction, but item c not only fails to do so accurately, it is downright misleading. That has been a common criticism of your diagrams since the day you started this thread. Please listen when the physicists explain to you why item c is misleading.

The electromagnetic tensor field is hard to depict within a 2-dimensional diagram because the electromagnetic field is 6-dimensional. To reduce those 6 dimensions to the 2 dimensions of a printed diagram, we try to take advantage of symmetries.

The electric field E surrounding an electron is symmetric under all rotations and reflections about the source. That's easy to depict: draw something like item a and tell your readers it would look the same under all rotations and reflections about the central point.

ETA: edd quickly pointed out (in post #1581) that my first sentence of the next paragraph is wrong. I have therefore edited this post to strike it out.​

The magnetic field B surrounding the electron is zero within the rest frame of the electron, so it's even easier to depict: Show your readers a blank diagram. What you have done in item b, however, is to show something like the magnetic field within a coordinate system in which there is a current running through your central point and orthogonal to the page. Your items a and b therefore depict different physical systems, which is already misleading.

When you combine depictions of unrelated systems into item c, your depiction loses contact with physical reality: There is no longer any single physical system you are even trying to depict.

You could fix that by revising your explanation of item a to say it shows the electric field E around a charged wire through which a current is also running. That would reduce the symmetry of your item a: It would look the same under all rotations and reflections whose axis (or plane of reflection) runs through the central point and lies orthogonal to your diagram, but would not look the same under rotations whose axis is not orthogonal to your diagram.

That would not solve all of the problems with your item c. As has been noted earlier within this thread, the electric field E and magnetic field B have different units. You can't just add them together, as the "+" sign in your diagram suggests.

For all of the above reasons (and for other reasons that will be explained to you momentarily by other people), the usual way to depict the 6-dimensional electromagnetic tensor field F_μν is to decompose it into fields of lower dimension. The usual way to do that is to decompose F_μν into two 3-dimensional vector fields E and B. The reasons for that particular decomposition become evident when you look at the Lorentz force law or at the components of F_μν. When written as a matrix, the components of F_μν have an obvious decomposition into a 3-vector that consists of the components of E together with a 3x3 matrix that contains all the components of B without any components of E.

Although you have criticized that decomposition, it has a lot going for it. Your main criticism, consisting of incessant quotation of a single sentence from Jackson's Classical Electromagnetism, has taken that sentence out of context. Indeed, you often omit the first part of that sentence: "But the fields are completely interrelated". That interrelationship between E and B must be taken into account whenever we transform from one coordinate system to another, which is the topic of Jackson's 11.10. The very first equation of that section, equation (11.146), is the same as Einstein's equation (8) from Die Grundlage der allgemeinen Relativitätstheorie. When you interpret the rest of Jackson's sentence to mean we shouldn't talk about E and B at all, as you have often done within this thread, you are just reminding us of how you got lost at Einstein's equation (3).

If it's a point source rather than a line source, please show the two orthogonal sections to give a sense of the 3D structure.

Does not parse. It isn't a point source. Charge is topological.

ben m's question is relevant, and you shouldn't try to escape the question by proclaiming your idiosyncratic semantics. You could answer the question by specifying the symmetries of your diagrams in 3-dimensional space, as I did above by mentioning the relevant symmetries, but it's probably going to be easier for you to answer ben m's question using the orthogonal sections ben m suggested.

 

[lpetrich]

The archetypal point particle, which I do not agree with. The electron is more like the "spinor" depicted in c.

Except that one can't very easily draw a picture of a spinor. A vector, maybe, but not a spinor.

Charge is topological. See this on physicsforums.

Electric charge is NOT topological. It's an interaction strength. "Topological charge" or a topological invariant is something very different. Winding number, for instance.

That link does not demonstrate that electric charge is a topological invariant.

 

[edd]

The magnetic field B surrounding the electron is zero within the rest frame of the electron, so it's even easier to depict: Show your readers a blank diagram. What you have done in item b, however, is to show something like the magnetic field within a coordinate system in which there is a current running through your central point and orthogonal to the page. Your items a and b therefore depict different physical systems, which is already misleading.

It isn't zero. The electron has a magnetic dipole moment. Of course as ben m has pointed out, a magnetic dipole doesn't look like item b, so there's still a valid criticism to be made.