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### Relativistic Lorentz force

Posted: Mon Mar 07, 2016 8:05 pm UTC
I feel like this is a stupid question, but I'm blanking on the justification for why the Lorentz force makes sense in terms of special relativity. Please note that I am well aware there exist mathematical formalism for this. I'm looking for a more conceptual explanation.

Suppose you have a positive point charge moving in the +y direction. There is a uniform magnetic field pointing in the +z direction for all (x,y,z). The Lorentz force exerted on the particle would act according to the right hand rule (initially, this would be the +x direction). This causes the particle to undergo circular motion on the x-y plane.
However, in the particle's frame of reference, it is not moving at all. It is stationary at the origin with a uniform magnetic field all around it. The Lorentz force would therefore not exist. I'm guessing how the particle "sees" the magnetic field is crux of the problem, but there isn't an intuitive reason for why this would be the case. You can just as well say the magnetic field is sliding past the particle, but the field is symmetric across all space so to what extent is it moving with respect to the particle?

### Re: Relativistic Lorentz force

Posted: Mon Mar 07, 2016 10:09 pm UTC
Is this the "A moving magnetic field looks like an electric field" thing?

When changing the reference frame from a moving particle in a uniform magnetic field to a frame with a stationary particle, you get a stationary particle in an electric field, which naturally accelerates.

### Re: Relativistic Lorentz force

Posted: Mon Mar 07, 2016 10:32 pm UTC
SuicideJunkie wrote:Is this the "A moving magnetic field looks like an electric field" thing?

Probably, but why? The magnetic field at (0,0,0) and the magnetic field at (0,dy,0) are identical. How does it magically become electric field if the field isn't changing?

### Re: Relativistic Lorentz force

Posted: Mon Mar 07, 2016 10:33 pm UTC
Yeah it's a pretty interesting effect. The magnitude (and presence of) and direction of electric and magnetic fields turns out to be frame dependent.

Here's an example that gets to the heart of the issue.

There's a train moving with velocity v in the x direction with respect to an observer standing still on the ground looking at the train. I'm standing on the train holding a charged particle in my hand. In my reference frame the charge has a radial electric field, but there are no moving charges around so there are no magnetic fields to speak of. The observer on the ground, however, sees a moving charge so she infers the presence of magnetic fields curling around the x axis as the train and particle move.

Who has the correct description? Is there a magnetic field or isn't there? You can put another charge in the vicinity of the train and both myself and the observer will predict equations of motion for this test charge. Turns out that after correct for special relativity we will both get the same answers. This is sort of what people mean when they special relativity is "built in" to electromagnetism.

Really cool in my opinion!

### Re: Relativistic Lorentz force

Posted: Mon Mar 07, 2016 11:47 pm UTC
Cradarc wrote:I feel like this is a stupid question, but I'm blanking on the justification for why the Lorentz force makes sense in terms of special relativity. Please note that I am well aware there exist mathematical formalism for this. I'm looking for a more conceptual explanation.

Suppose you have a positive point charge moving in the +y direction. There is a uniform magnetic field pointing in the +z direction for all (x,y,z). The Lorentz force exerted on the particle would act according to the right hand rule (initially, this would be the +x direction). This causes the particle to undergo circular motion on the x-y plane.
However, in the particle's frame of reference, it is not moving at all. It is stationary at the origin with a uniform magnetic field all around it. The Lorentz force would therefore not exist. I'm guessing how the particle "sees" the magnetic field is crux of the problem, but there isn't an intuitive reason for why this would be the case. You can just as well say the magnetic field is sliding past the particle, but the field is symmetric across all space so to what extent is it moving with respect to the particle?

You have to consider what moving charges are actually generating that field to make sense of things in the point charge's own rest frame. This is a lot easier if you're dealing with a straight wire, though in this case the field won't be uniform. But if the wire is lying along the y-axis, and we assume (by the usual convention which was dreamed up before people realized current is made of moving electrons) that the current is made up of moving positive charges, then if these charges are moving in the -y direction, by the right-hand rule for line current, the field will point in the +z direction in the half of the x-y plane where x takes positive values, so we can assume the external charge initially lies in that half.

The key thing to keep in mind when trying to understand how this fits with relativity is that when you induce a current in a wire, every positive charge that's induced to move leaves behind a negative charge which remains at rest relative to the wire (like negatively charged electrons and the positively charged ions they leave behind, but again with the charges reversed because of stupid convention), and the wire remains neutral overall in its rest frame. Then if you have an external positive point charge moving in the +y direction, when you transform into that charge's own frame, in this frame the negative charges which were at rest in the wire's frame are now in motion, so due to Lorentz contraction the distance between them shrinks. Likewise, the speed of the positive charges which were moving in the wire's frame becomes even larger in this frame, since in the wire's frame they were moving in opposite directions; this means the distance between the positive charges shrinks by a greater Lorentz factor. Thus, in this frame the density of positive charges in the wire is greater than the density of negative charges, so in this frame the wire has an overall positive charge at any given instant, and this is why the external positive charge is repelled from it, in this case getting pushed in the +x direction. A similar example with some helpful illustrations can be found here, in the section "Magnetism as a Consequence of Length Contraction".

If you really want to consider an example with a uniform field, we again have to account for that field in terms of moving charges. We could explain that in terms of the particle lying within an infinitely long solenoid where the distance between loops of the wire is idealized as zero, so you just have positive charges on the surface of the cylinder that are traveling in circles (around circular cross-sections of the cylinder), and again there would be negative charges at rest relative to the cylinder to balance them out and make sure the cylinder is overall neutral in its own rest frame. In this case, in the cylinder's rest frame there will be a uniform magnetic field inside the cylinder and no electric field. If the long axis of the cylinder is parallel to the z-axis in its own rest frame, and the charge on the surface is circulating counter-clockwise when viewed from "above" (i.e. looking down at a cross section from a vantage with a higher z-coordinate), then by the right hand rule for charge moving in a circle, the magnetic field will point in the +z direction. But if the particle inside is moving in the +y direction in the cylinder's rest frame, then if you transform into the particle's own rest frame, it no longer lies within a cylinder with circular cross-sections, instead due to to Lorentz-contraction the cross-sections would be ovals, and meanwhile the circulating positive charges at different points on the oval would have different velocities in this frame due to relativistic velocity addition and thus different degrees of Lorentz contraction between different nearby circulating charges, while the negative charges on the oval would all be moving at the same velocity, so I assume if you did the math you'd find again that the oval is not electrically neutral in this frame, and perhaps that its charge varies with angle. You'd have to do some fancy math to integrate the electric force from all the different parts of the oval on the point charge, but I'm sure the end result would be that just as with the straight wire, there'd be a net electric force in the +x direction.

### Re: Relativistic Lorentz force

Posted: Tue Mar 08, 2016 12:12 am UTC
This sort of thing is the topic of the most brilliant of brilliancies, chapter 5 of Purcell's book. He is a hero.

### Re: Relativistic Lorentz force

Posted: Tue Mar 08, 2016 5:39 pm UTC
Hypnosifl, your explanation was really helpful. I think the crucial thing is to not consider the magnetic field as its own entity, but rather as a phenomenon produced by moving charges. The particular movement of those charges become distorted when you change reference frames.

### Re: Relativistic Lorentz force

Posted: Tue Mar 08, 2016 7:31 pm UTC
Cradarc wrote:Hypnosifl, your explanation was really helpful. I think the crucial thing is to not consider the magnetic field as its own entity, but rather as a phenomenon produced by moving charges. The particular movement of those charges become distorted when you change reference frames.

Yes, this is the key thing. Once you move into special relatively, you can't really just think of "turning on" a constant magnetic or electric field the way you can in statics, because the field itself is frame-dependent. So you have to think about how you're generating the field, and how this changes with things in motion.

### Re: Relativistic Lorentz force

Posted: Tue Mar 08, 2016 7:35 pm UTC
You also can just work with the fields, but you have to combine them into the Faraday tensor so they can be transformed covariantly. This is a fun tensor.

### Re: Relativistic Lorentz force

Posted: Tue Mar 08, 2016 8:08 pm UTC
The fact that electromagnetic fields behave counter-intuitively under Special Relativity would be why Albert Einstein titled his paper on SR "On the Electrodynamics of Moving Bodies".

### Re: Relativistic Lorentz force

Posted: Tue Mar 08, 2016 8:20 pm UTC
doogly wrote:You also can just work with the fields, but you have to combine them into the Faraday tensor so they can be transformed covariantly. This is a fun tensor.

Are there tensors that are not fun?

### Re: Relativistic Lorentz force

Posted: Mon Jun 13, 2016 7:00 pm UTC
FWIW my understanding of this is that magnetic fields don't really exist, there's only electric fields with changes that propagate at the speed of light.

In other words the equation that defines the curl of B in terms of the current and the rate of change of E; is this the definition of the relationship between the E and B fields, or is it the definition of the B field in terms of the E field. I put it to you that it's the latter.

It's a bit like angular momentum. There's not a completely separate 'angular momentum' from 'linear momentum', you pretty much just get angular momentum whenever linear momentum is different between two bodies; just like there's not a separate B field from the E field; and in fact the maths are somewhat analogous.

If you go through the implications of an E field propagating at a finite speed, really, really, really carefully, out pops magnetic fields and relativity, time dilation, lorentz contractions and lack of simultaneity, everything.

### Re: Relativistic Lorentz force

Posted: Mon Jun 13, 2016 9:08 pm UTC
Sure, but then it's a little odd to privilege E; you can say "tehy're both really the same," but "B is just E with a different PoV" is a little odd

### Re: Relativistic Lorentz force

Posted: Mon Jun 13, 2016 10:48 pm UTC
Not really, the E field has a scalar potential associated with it, and we're knee deep in electric monopoles.

I think the apparent symmetry between the E field and the B field is just because they're the same underlying field.

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 4:18 am UTC
Yup, electric monopoles are everywhere (elrctrons and protons and their antimatter counterparts), but magnetic monopoles appear to be nowhere to be found.

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 7:35 am UTC
Wolfkeeper wrote:
Not really, the E field has a scalar potential associated with it, and we're knee deep in electric monopoles.

I think the apparent symmetry between the E field and the B field is just because they're the same underlying field.
The magnetic field has a vector potential associated with it. Why is a vector potential somehow less fundamental than a scalar potential? The monopole argument is somewhat more valid, but then again practically everything has a magnetic dipole moment anyway so to conclude that B is not as fundamental as E from that is also a bit of a leap, let alone the statement that "B does not really exist". (And actually, fuck the whole "does not exist" business some people raise about things in physics anyway. If multiple ostensibly different formulations are mathematically equivalent, claims to the "real existence" of objects from either formulation become utterly meaningless.)

Of course, the tensor formulation is far more elegant than any expression in terms of B and E anyway. You have one four-potential Aµ, you impose gauge invariance, and out comes all of classical electrodynamics in a perfectly Lorentz-invariant formulation with very little effort.

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 12:41 pm UTC
d*F=J for lyyyyyyyyyfe

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 2:33 pm UTC
Link wrote:The magnetic field has a vector potential associated with it. Why is a vector potential somehow less fundamental than a scalar potential?

In principle, yes. In reality, no. Which direction does the magnetic vector point in? Why is it at 90 degrees to the electric field that is generating it? Understand that the magnetic field vector is generated by the electric field; it's pretty much the angular momentum of the electric field.

The monopole argument is somewhat more valid, but then again practically everything has a magnetic dipole moment anyway so to conclude that B is not as fundamental as E from that is also a bit of a leap, let alone the statement that "B does not really exist". (And actually, fuck the whole "does not exist" business some people raise about things in physics anyway. If multiple ostensibly different formulations are mathematically equivalent, claims to the "real existence" of objects from either formulation become utterly meaningless.)

Of course, the tensor formulation is far more elegant than any expression in terms of B and E anyway. You have one four-potential Aµ, you impose gauge invariance, and out comes all of classical electrodynamics in a perfectly Lorentz-invariant formulation with very little effort.

Yes, but why is that tensor valid? What is that tensor really saying?

My point is not that it can only be understood in the terms I have outlined, but that you can explain everything from that very, very simple initial starting point. Sorry, but a tensor is not that simple, and just being able to solve an equation and get the right answer doesn't mean you understand it.

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 6:04 pm UTC
Wolfkeeper wrote:Yes, but why is that tensor valid? What is that tensor really saying?

My point is not that it can only be understood in the terms I have outlined, but that you can explain everything from that very, very simple initial starting point. Sorry, but a tensor is not that simple, and just being able to solve an equation and get the right answer doesn't mean you understand it.

What would that "very, very simple initial starting point" be, then? Because the EM field tensor can be derived using symmetries (gauge and Lorentz symmetry, to be precise) only, without even introducing particles or forces or reference frames or any cruft like that.

I mean, sure, if you want to stick to your local pseudo-inertial reference frame and are only interested in well-behaved classical low-energy experiments in said frame, then there is some merit to E and B. But once you learn to think in a way that is free from coordinate systems or reference frames, you'll soon find that E and B are a godsawfully poor choice of objects to describe anything. And if you're going to use special relativity to claim E exists and B doesn't, then frankly, that's not even wrong.

FWIW, I think we should start introducing tensors in high school. Because they're fucking elegant!

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 6:34 pm UTC
The B/H field doesn't exist, it's just the E field. You know the phrase 'electromagnetism'? That's why that's phrase is really used, there's only one field, the electromagnetic one, and really it's the electric field, we know that due to the quantum mechanics underneath running the whole show, and there's electric charges, +1,-1 etc. doing that.

You can certainly express it in different ways, but otherwise if I ask you how you know the universe obeys Lorentz, you can't tell me. You can't tell me why those particular tensors are correct; you could show me experimental verification of it, but all I need to show you is that a) normal matter is held together by electric forces b) changes in the electric field travel at the speed of light and I can derive those tensors and magnetic fields from that, relativity, everything.

That's the problem with electromagnetism education, virtually nobody really understands it.

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 8:07 pm UTC
Yes, there is only one electromagnetic field, and it's usually called Fμν in the literature.

From a philosophical point of view, all we can ever claim to know about anything in physics is based on two cornerstones: correspondence to empirical evidence, and mathematical consistency. The universe obeying Lorentz invariance satisfies both, as does the existence of magnetic fields, electric fields, electric charges, electromagnetic field tensors, the whole shebang. Claiming any particular subset of those is "real" while the others aren't is ultimately completely arbitrary, is the point I'm trying to make.

The other point I'd like to make is that if I asked you to give me the three-loop correction to a fermion propagator in QED using only the electric field, you (and indeed anyone else) would probably be crying for tensors in no time at all.

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 10:07 pm UTC
Fμν is just a restatement of Maxwell's equations. And it's not in any sense wrong, it just includes the fictitious magnetic forces, in a similar way that changing to a rotating reference frame includes fictitious inertial forces.

It would just be an astonishing coincidence if there actually was another dimension of space, to give magnetic fields, that had exactly the same properties as a fictitious field that can be derived from the electric field.

### Re: Relativistic Lorentz force

Posted: Tue Jun 14, 2016 11:47 pm UTC
They'd be more analogous to fictitious forces if they only showed up in non-inertial frames, but this is not the case. Any sort of moving source, boosted frame, or a dipole point source would give them to you.

I'm not sure what you're claiming. That there's a preferred frame for looking at electromagnetic fields?

### Re: Relativistic Lorentz force

Posted: Wed Jun 15, 2016 7:56 am UTC
Wolfkeeper wrote:It's a bit like angular momentum. There's not a completely separate 'angular momentum' from 'linear momentum', you pretty much just get angular momentum whenever linear momentum is different between two bodies; just like there's not a separate B field from the E field; and in fact the maths are somewhat analogous.

Sure, but why imply that linear momentum is more fundamental than angular? You can easily go the other way. Intrinsic angular momentum aka spin is an important property of all quantum particles, and most fundamental particles (not counting the Higgs boson) have non-zero spin. Linear momentum is merely orbital (i.e., non-intrinsic) angular momentum with zero curvature (IOW, infinite radius of curvature).

### Re: Relativistic Lorentz force

Posted: Thu Jun 16, 2016 2:29 pm UTC
doogly wrote:I'm not sure what you're claiming. That there's a preferred frame for looking at electromagnetic fields?

There's no such thing as a magnetic field, there's only electric fields, that produce magnetic effects. There's no 'magnetic field' dimension of space, but there is an electric field.

You can consider this point as either trivial or deep; as we know that there are quantised electric charges as monopoles, but there's no known quantised magnetic charge monopoles, and I would suggest to you that the equations are not evidence for their existence either.

### Re: Relativistic Lorentz force

Posted: Thu Jun 16, 2016 4:04 pm UTC
I think you are tiptoeing off into a deep end. Of course there is no "magnetic field dimension of space," neither is there an electric dimension to *space.*

If you like to think of the fields in geometric terms (and who wouldn't! these terms are the best!) the whole electromagnetic shebang is a U(1) fiber bundle. This is why you get dF=0 for free, since it's "just" the Bianchi identity.

### Re: Relativistic Lorentz force

Posted: Thu Jun 16, 2016 4:49 pm UTC
While I'm not up to speed on U(1) I'm saying since you get the same results form a single electric potential, there's not really a 4 vector, that's just a mathematically clean way of describing how all the charges are interacting, but it is ultimately fictitious.

Another analogy, my house's power supply is single phase, but it's common to express the AC voltage and current as a complex number. But the voltage coming out of the wall is clearly only a real quantity. We do that only because it's more mathematically tractable. I think all physicists know that, but I'm not so sure they know that magnetic fields are not real, in the same way that imaginary voltages are not real.

Anyway, I think you get my point; you probably disagree with it, but if so, I think that proves my point about not all physicists really understanding even these basic topics!

### Re: Relativistic Lorentz force

Posted: Thu Jun 16, 2016 5:11 pm UTC
It's like you read the glorious and holy Purcell Ch 5 but came away with a weird metaphysics and a weirder superiority complex.

But yes, I think the U(1) story helps clear things up. It shows you what the coordinate-free and intrinsically geometrical picture of what's going on looks like. Same with the differential forms I've been pushing (they're all the same story.) If you say, oh, four vector version of A is (phi,A), it looks like you just smashed some meaningfully distinct things together for convenience. It makes the four vector look like a trick. Instead, the full gauge story is what makes sense. It's the lil splits into some local rest frame that are the cheap trick - and they wear out quick.

I think we've also chosen words like "imaginary" and "real" that are a disservice.

### Re: Relativistic Lorentz force

Posted: Thu Jun 16, 2016 10:34 pm UTC
Yes. We do not mean "not real" in the sense of "having no existence outside of our minds", but rather in the same sense that we call "centrifugal force" a "fictitious force"--the entity being described has actual existence, but it is an energent property of something deeper.

### Re: Relativistic Lorentz force

Posted: Thu Jun 16, 2016 11:29 pm UTC
That's not the sense in which "imaginary numbers" are, either. Some types of numbers, like counting numbers, are one-dimensional. Other types of numbers are two-dimensional. Some things are best modeled with one-dimensional numbers, others with two-dimensional. There's nothing complicated about that; the two are equally "real" (however "real" you wish to treat numbers as).

### Re: Relativistic Lorentz force

Posted: Fri Jun 17, 2016 10:26 am UTC
Yes, "imaginary numbers" were unfortunately named by people who thought that a Real Number had to correspond to a quantity of physical objects--you can measure out "x" of a material substance and look at it, but you could not do the same for "i" of it.

The name "complex numbers" is a much better term for describing them.

### Re: Relativistic Lorentz force

Posted: Fri Jun 17, 2016 12:01 pm UTC
Imaginary numbers are one thing, real numbers are another and complex numbers contain both. A number composed of any value* of real (including zero) 'added to' any value* of imaginary (including zero).

In any suitable context, 4 is complex (also real), -iπ is complex (also imaginary), 0 is complex (and both).

* I thought I'd specify "any real value", but it doesn't actually matter if you feed it imaginary/complex values, except for some simplification being advised. Just avoid going into hypercomplex territory, for that way madness lies...

### Re: Relativistic Lorentz force

Posted: Fri Jun 17, 2016 12:17 pm UTC
But if they're not part of the complex numbers, imaginary numbers are super dumb. They're just real numbers that you aren't allowed to multiply with an extra label at the end. That is when they look like a really artificial structure. C though is super good.

And maybe something similar happens with B? Like for one thing, it's a "pseudovector." Or so a footnote in Purcell told me. I asked my professor freshman year to elaborate, he had never heard of such a thing. What is the deal with these guys? They're actually super normal. There's nothing pseudo about them. But they're also not vectors. 2-forms can map onto 1-vectors because we live in flat space, but it's somewhat awkward and can easily have you thinking that they are somehow less "fundamental" if the edges show. It's the mapping that is a bit off, not the object itself though.

Re "fictitious" forces:
The thing I want to emphasize is that B is not something that just shows up in non-inertial frames. An experiment can distinguish inertial and non-inertial, so if you want to call the non-inertial frame dependent guys "fictitious" or "less fundamental," you at least have a coherent sentence. I've never had a unicorn toss around the tiny bones in my inner ear and I'd reserve strong sounding fighting words like "fictitious" for things like them, but at least it makes sense and you can separate out the categories. Not so with B.

### Re: Relativistic Lorentz force

Posted: Fri Jun 17, 2016 12:47 pm UTC
doogly wrote:But if they're not part of the complex numbers, imaginary numbers are super dumb. They're just real numbers that you aren't allowed to multiply with an extra label at the end. That is when they look like a really artificial structure.

Think of them as an alternate 'type' of variable in a programmng language. The string "12345" might look like the value 12345, but there's often good reasons to not try to multiply "12345" by 12345 without acknowledging the fact that it's a different thing and that you must apply an offcial conversion to it, like a StrToInt(), to keep you on the straight and narrow regarding what you're shoving where. (The fact that I use Perl a lot, a language with one of the most promiscuous type-handling scheme that I've ever used, makes me both bless (NPI!) and curse the other languages that don't let me do "1" x 9, and then go on to treat that directly as a number, then concatenate the string ".999" to it and potentially still treat it as a number!)

And a pure imaginary number that pops out as a solution to a root should not be treated as 'basically a real', forgetting its (ahem!) true roots... If it is a fully complex root (non-zero, either component), you get the clue, but you shouldn't be tempted to forget the 'imaginariness' of a number just because it's 'real'y zero.

### Re: Relativistic Lorentz force

Posted: Fri Jun 17, 2016 3:37 pm UTC
I think you have missed my point, so I will state it with fewer sentences:
Imaginary numbers is an incredibly boring and worthless set, and is only interesting insofar as it is a subset of the field of complex numbers. The only properties it can have on its own as a group are isomorphic to the real numbers under addition.

### Re: Relativistic Lorentz force

Posted: Mon Apr 08, 2019 11:06 am UTC
Wolfkeeper wrote:If you go through the implications of an E field propagating at a finite speed, really, really, really carefully, out pops magnetic fields and relativity, time dilation, lorentz contractions and lack of simultaneity, everything.

This is effectively what I am trying to do, can you give me a reference where someone else has done this?

Thanks

Lamaan

### Re: Relativistic Lorentz force

Posted: Mon Apr 08, 2019 3:27 pm UTC
lamaan wrote:
Wolfkeeper wrote:If you go through the implications of an E field propagating at a finite speed, really, really, really carefully, out pops magnetic fields and relativity, time dilation, lorentz contractions and lack of simultaneity, everything.

This is effectively what I am trying to do, can you give me a reference where someone else has done this?

Thanks

Lamaan

Chapter 5 of Purcell, it's in the thread, it's the greatest.

### Re: Relativistic Lorentz force

Posted: Wed Apr 10, 2019 8:55 am UTC