### Physical Attraction

This will be old news to the physics geeks, but I still remember what a revelation it was, back in grad school, when the physicist Gary Horowitz told me why an electric current exerts a magnetic force on a moving charged particle. (This is the source of all magnetism; those magnets on your refrigerator have little electric currents flowing through them all the time.)

So imagine a wire, made of protons that stay still and electrons that drift rightward; that drift is what we call a current. And imagine a nearby charged particle—call it Fred—also traveling rightward.

Now relativity tells us that Fred is allowed to think of himself as stationary, and the protons (along with you and me) as drifting off to the left. Relativity also tells us that if passengers on a moving train say the cars are 100 feet apart, then an observer at the station will say they’re closer than that. In this case (according to Fred) you and I are the passengers moving with the train of protons, and if we say they’re an angstrom apart, then Fred says they’re closer. That means Fred sees more positive charge per inch of wire than we do. If Fred himself happens to be negatively charged, he’ll be drawn toward the wire.

As far as Fred is concerned, that’s a purely electrical force, but it’s a force that you and I can’t account for on electrical grounds. So you and I call it magnetism.

At the same time, Fred sees the electrons in the wire as slower-moving, and therefore farther apart, than you and I do, so he sees less negative charge per inch of wire than you and I do. According to Fred, then, the gap between positive and negative charge in the wire is even greater, which means he’s pulled in even harder, which you and I call even more magnetism.

If you’re geeky enough to care, it’s a nice exercise in relativity theory to show that the magnetic force is proportional both to the current (that is, the number of electrons per inch, times the speed of the electrons, as measured by you and me) and to Fred’s velocity. It just now took me three tries to get this right, but it’s very nice when it finally works.

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#### 28 Responses to “Physical Attraction”

1. 1 1 Harold

Brilliant! Thanks for that explanation. It is amazing how you can go from a couple of simple assumptions (constant light speed and laws independent of inertial frames), add a touch of genius, take it to its logical conclusion and explain such complex things as magnetism. You can see why Einstein thought that beauty was an essential part of a good theory. Now, why do opposite charges attract?

2. 2 2 Bennett Haselton

Does this mean that if Fred is stationary, then he’ll see more negative charge per inch of wire than positive charge, and will therefore be repelled from the wire? More generally, if he’s moving rightward at more than half the speed that the electrons are moving rightward, then he’ll see the protons “moving” faster than the electrons and he’ll be attracted to the wire, but if he’s moving rightward at less than half the speed of the electrons (or he’s stationary, or he’s moving leftward), then he’ll view the electrons as moving faster, and be repelled?

3. 3 3 Philip

Brilliant, indeed. A relatively (ouch) simple explication of magnetism that even I can grasp.

This insight may have heuristic value in the realm of political economy.

If I have this down right, as a stationary, positively-charged proton (say, as positive advocate for greater tax equity), I see you, a negatively-charged electron (say, a negative advocate for tax equity to any degree), as drifting right ward.

Likewise, of course, you will see me as drifting leftward.

Right so far?

And since there is no privileged frame of reference, we’re both stuck disagreeing with each other, I suppose.

BUT, I have the consolation of being a postive particle while you’re stuck being negative.

Now, a question arises: under special relativity
(i.e., “no privileged reference points”), are you allowed to say I’m the negative particle and you the positive?

4. 4 4 Steve Landsburg

Bennett:

Does this mean that if Fred is stationary, then he’ll see more negative charge per inch of wire than positive charge, and will therefore be repelled from the wire?

You’ve asked exactly the question that motivated the relativity puzzle I posted a couple of weeks ago. Here, I’m pretty sure, is the answer:

Then:

1) If the electrons all start moving at the same time (as measured in the lab), then they don’t contract in the lab frame—instead they expand in Fred’s frame. The ratio of measured lengths is still given by the Lorentz contraction, but in the lab frame, the electrons are just as far apart as they ever were, hence no net electrical charge.

2) If, more realistically, the electrons are pushed along by, say, a battery, which pushes one electron, which pushes the next, etc., then you’d expect the distance between the electrons to shrink in the lab frame. But the average distance between them *can’t* shrink, because neither the total number of electrons nor the total length of wire has changed. Somehow the rigidity of the wire exerts a force that stretches them out to the same average density (in the lab frame) as before. Thus we get the same result as in 1), though for a different reason: The electron density on the lab frame is the same as if they weren’t moving, and so must be less dense than this in Fred’s frame. And there is still no electric charge in the lab frame.

I am entirely sure of point 1) and almost sure of point 2). If a physicist wants to weigh in on this, I’ll be grateful.

5. 5 5 Bennett Haselton

“1) If the electrons all start moving at the same time (as measured in the lab), then they don’t contract in the lab frame—instead they expand in Fred’s frame. The ratio of measured lengths is still given by the Lorentz contraction, but in the lab frame, the electrons are just as far apart as they ever were, hence no net electrical charge.”

I’m confused, what’s Fred doing, in this example? I was asking if Fred is stationary. But if Fred is stationary, isn’t “Fred’s frame” the same as the “lab frame”?

(Actually, I can’t see how anything can ever “expand in your frame”. In your frame, things are either stationary (no contraction) or they’re moving (slight contraction)… How can they expand, in one’s own frame of reference? But, I never studied this after all.)

6. 6 6 GregS

I’m still chewing on (2), Steve. The paths of electrons in a real wire, driven by a battery, are parabolas. Without a battery driving them, they fly around in the soup of atoms and electrons and bounce off of the metal atoms, following approximately strait trajectories, but add the voltage and their paths are curved into parabolas. (Just like a ball, launched in a random direction, would follow a parabolic path back toward earth.) Call that “the rigidity of the wire at work” if you like. I haven’t convinced myself that this does or doesn’t change the density of negative charges yet, although it’s certain that the density of charges doesn’t change…that is, there EXISTS an explanation.

It would infuriate me to no end if someone objected on moral grounds to good, solid physics (I suppose the young earthers, and the now marginal flat earthers do that). Is it frustrating when people object on moral grounds to solid economic theory?

7. 7 7 Steve Landsburg

Bennett: Sorry, in 1) I meant that the electrons are uncontracted in the lab frame and hence expanded in *my* Fred’s frame (that is, the moving Fred’s frame). Your Fred’s frame is the same as the lab frame, of course.

8. 8 8 Steve Landsburg

It would infuriate me to no end if someone objected on moral grounds to good, solid physics (I suppose the young earthers, and the now marginal flat earthers do that). Is it frustrating when people object on moral grounds to solid economic theory?

I can give you a pretty unambiguous “yes”.

9. 9 9 Roger Schlafly

When you talk about an observer seeing a train, it is not clear whether you are talking about the straight Lorentz contraction, or how the train actually appears to someone watching the train, taking into account the finite speed of light from the train to the observer’s eyeballs. These two things are quite different, as the train will look more rotated than contracted. See http://en.wikipedia.org/wiki/Penrose-Terrell_rotation.

10. 10 10 Sean Abbott

‘If, more realistically, the electrons are pushed along by, say, a battery, which pushes one electron, which pushes the next, etc.,’

I always imagined the electrons being pulled by the battery not pushed. In a vacuum tube, for example, the electrons are pulled by the anode not pushed by the cathode.

11. 11 11 Steve Landsburg

Roger: I believe I have consistently used locutions like “seeing the train” to refer strictly to the Lorentz contraction.

12. 12 12 Jon Shea

My favorite physics professor worked through this for my freshman Modern Physics class. It was a major revelation for me too.

I was confused for a bit as to why you think that election density might increase when a wire is driven by a battery, but I think I understand now. You seem to be envisioning a bunch of electrons getting pushed by the battery, and then these electrons push the next ones, and so forth. As if electrons are a crowd of people and the battery is trying to push through them. In this model there would be a bunching. As you intuited, this is not how electricity is driven through a wire.

I drew up a quick sketch of how things look in a for a simple circuit which I think will explain what happens better than I can with words.

http://jonshea.com/surface_charges.pdf

The current in a wire is driven by an electric field in the wire. The electric field comes from a distribution of charges _on the surface_ of the wire. The surface charge distribution is exactly the distribution needed to create an electric field in the wire, parallel to the wire.

Where do the surface charges come from? They happen automatically. If electric field inside the wire is “wrong” then surface charges automatically redistribute to correct it. Imagine the electric field in the wire had some component that was not parallel to the wire, but was instead pointed towards the outside of the wire. This electric field would push charge carriers to the surface of the wire. Additional charge carriers would collect on the surface until their accumulated electric field exactly canceled out the internal electric field that pushed them to the surface.

This, while a beautiful result, doesn’t really help think about how relativity unifies electricity and magnetism. It’s probably easier if you think of the positive charges and negative charges as each being glued to long sticks which are moving with respect to each other. That way you don’t have to worry about how the charges might interact with themselves, and instead can just consider their effect on a test charge at some position.

13. 13 13 Philip

GregS-

Can you give me an example of a moral objection to solid economic theory? I don’t doubt that there are some, but I want to make sure I know what you mean.

Thanks.

14. 14 14 Bennett Haselton

I’m still confused as to how anything can ever be “expanded in your frame”. I thought that in a given person’s frame of reference, anything stationary in their frame would remain the same size, and anything moving in their frame would be Lorentz-contracted in their frame. Are these the wrong postulates to start with? (Even if the answer is just “Yes, these postulates are wrong”, it would be handy to know that :) )

15. 15 15 Steve Landsburg

Bennett: You have to be careful about what you mean by “Lorentz contracted”. If I am on the train and you are in the station, then the Lorentz contraction says that your measurement of the train (while it is moving relative to the station) is shorter than my measurement of the train (also while it is moving relative to the station). It says nothing at all about the relationship between your measurement of the train while it’s moving relative to the station versus your measurement of the train back at a time when it was standing still.

For me, the whole reason that this puzzle at first appeared paradoxical was that I failed to make exactly this distinction.

16. 16 16 Neil

Steve (and Bennett, who is having the same difficulty that I am)

Suppose you and I are in two different trains. Each of us measures the length of his train (say by sending a light signal down to the other end and measuring the time interval). We both find that the proper length (the length in our own inertial frame) of our own train is one mile.

We now look out and measure the length of the other each other’s train, and we are in relative motion. Each of us finds the length of the other other’s train as less than one mile. I am sure we are in agreement up to this point.

Suppose your train is passing me to my right. Perhaps I perceive myself as stationary, and you as moving, but of course it is relative. Now suppose I accelerate (if I decided I was moving, not you, I’d decelerate) in the same direction until you and I are travelling at the same speed (so we are now in the same inertial frame). Obviously, you still measure the proper length of your train as one mile. I say that I can accelerate my train without changing its proper length, so I still measure my own train at one mile. We measure each others trains, and now agree that both of our trains are one mile long.

Are these measurement examples consistent with your statement? Or are you claiming that I (the acclerating train) must now measure my train as longer than one mile? If so, you must measure my train as longer than one mile too.

17. 17 17 Steve Landsburg

Neil:

We now look out and measure the length of the other each other’s train, and we are in relative motion. Each of us finds the length of the other other’s train as less than one mile. I am sure we are in agreement up to this point.

This is absolutely correct.

I say that I can accelerate my train without changing its proper length, so I still measure my own train at one mile. We measure each others trains, and now agree that both of our trains are one mile long.

This is also absolutely correct, assuming you accelerate your train without changing its proper length, which you absolutely can do.

We measure each others trains, and now agree that both of our trains are one mile long.

Are these measurement examples consistent with your statement? Or are you claiming that I (the acclerating train) must now measure my train as longer than one mile? If so, you must measure my train as longer than one mile too.

I agree that in this example we both measure both trains to be one mile long. So up to here we agree on everything.

Here, though, is the point: The acceleration you describe will appear to ME as if the front and back of your train car started accelerating at different times.

IF, on the other hand, you accelerate in a way that makes it appear to ME that the front and back of your car accelerated SIMULTANEOUSLY (which is what I assumed in the original problem), THEN you cannot avoid stretching out your car.

18. 18 18 Neil

Steve: “Here, though, is the point: The acceleration you describe will appear to ME as if the front and back of your train car started accelerating at different times.”

I think we agree–I will think further and argue about it if I disagree.

Steve: “IF, on the other hand, you accelerate in a way that makes it appear to ME that the front and back of your car accelerated SIMULTANEOUSLY (which is what I assumed in the original problem), THEN you cannot avoid stretching out your car.”

1) Is this statement about the possibilities or impossibilities of acceleration schemes, or a statement about physical law? I recall something called Born rigidity, that required differential acceleration rates along a finite rigid object. Is this your point?

2) What about decelerations? What if I decided that instead of you moving to the right, it was me moving to left, and so I “decelerate” my train (in the way you say) in order to get into the same inertial state. Would I shorten my train? If not, what about a train that accelerates from a stop (relative to a stationary observer) into motion and then decelerates to a stop (in the way you say)? Is it not the same length to all in the end? Does nature distinguish accelerations and declerations? Surely not.

BTW, thanks for taking the time to debate this with me. I find it very interesting, and want to get to the truth of the matter–not to be argumentative.

19. 19 19 Steve Landsburg

Neil:

1) Is this statement about the possibilities or impossibilities of acceleration schemes, or a statement about physical law?

I’m not sure I understand this distinction. It is a statement about the relativity of simultaneity, the Lorentz contraction, and the other standard stuff of special relativity.

As for your questions in 2), the best way to answer these questions (and the best way to see that they are strictly questions about special relativity, not about anything else) is to draw the spacetime diagrams. Start with the assumption that, relative to me, you and your friend Fred are traveling leftward.

I say: I am standing still, you are moving leftward, my train has length 1 and your train has length 1/2. You and Fred say: I am moving rightward, you are standing still, my train has length 1/2, and your train has length 1.

At some point you change your speed to match mine. (Fred, however, keeps on going at your initial speed.) I say: You just decelerated to a speed of zero. Fred says: You just accelerated (rightward) up to my speed. (What you say depends on whether you think of yourself as having decelerated or accelerated, which is up to you.)

At this point everything depends on exactly how you accelerated/decelerated:

Scenario I: You accelerate/decelerate in such a way that *according to me* the left and right ends of your car decelerate simultaneously. Then *according to Fred* the left side of your car accelerates rightward before the right side. Everyone agrees that your car is now shorter than mine; they’re parked next to each other and we can easily verify this. But our explanations differ: *According to me*, my car has length 1 (as it always did) and your car has length 1/2 (as it always did). *According to Fred*, my car has length 1/2 (as it always did) and your car has length 1/4, because it got squashed when the back (the left) started accelerating before the front (the right). *According to you*, you used to see things Fred’s way, but now you see things my way.

Scenario II: You accelerate/decelerate in such a way that *according to Fred* the left and right ends of your car accelerate simultaneously. Then *according to me* the right end of your car decelerates before the left. Everyone agrees that your car is now longer than mine; they’re parked next to each other and we can easily verify this. But our explanations differ: *According to me*, my car has length 1 (as it always did) and your car has length 2, because it got stretched when the back (the right) decelerated before the front (the left). *According to Fred*, my car has length 1/2 (as it always did) and your car has length 1 (as it always did). And as for you, you used to see things Fred’s way, but now you see things my way.

Scenario III: You carefully manage your acceleration/deceleration in such a way that when you’re done, you and I agree that your train has length 1. This requires careful timing, but if you get the timing right, it will look to me like you stretched (from 1/2 to 1) because your right decelerated before your left, and it will look to Fred like you shrank (from 1 to 1/2) because your left accelerated before your right. And again, you’ll have switched your viewpoint from Fred’s to mine.

20. 20 20 Neil

Sigh.

Be it resolved that if a stick, measured as one meter long by a stationary observer, is put into motion in such a way that it remains one meter long to said stationary observer, then it must be the case that the stick has been somehow physically stretched so that its proper length is more than one foot. This has to be true, because the stick must be Lorentz contracted to the observer.

Be it also resolved that this is not Einstein’s rigid meter stick– it is a silly putty meter stick.

21. 21 21 Neil

I meant “proper length…more than one meter”, of course.

22. 22 22 Steve Landsburg

Neil: I agree with your “be it resolved”, but of course this way of putting it misses the entire point of the original puzzle, which is that a stick which accelerates “all at once” according to an observer in its initial frame must be one of the sticks you’re talking about. The fact that this was not obvious to me when I first thought about it, and was not obvious to you when you first thought about it, and was not obvious to other readers when they first thought about it, makes it, I think, a puzzle worth posing. Lots of excellent brain teasers are completely obvious once you’ve understood them.

23. 23 23 Neil

In all honesty, Steve, it never occurred to me to resolve relativity paradoxes by assuming a rubber yardstick.

24. 24 24 Steve Landsburg

Neil: You still don’t understand this. The resolution makes no assumptions beyond what’s given in the statement of the problem. If you’re willing to look at the diagrams I’m willing to answer questions about them. If you’re going to keep getting things wrong without making even that minimal attempt to understand them, then I can’t spend any more time on this.

25. 25 25 Neil

Excuse me, but what precisely have I got wrong? You seem to make a Minkowski diagram with a kinked world line (actually a physical impossibility) the ultimate arbiter. The diagram is a thinking aid, which is useful in some cases, and misleading in others.

26. 26 26 Steve Landsburg

Neil: You are fundamentally confused about what’s an assumption and what’s a conclusion.

You persist in thinking that I’ve *assumed* the train stretches. Instead, the assumption of simultaneous acceleration all along the train car forces me to *conclude* that the train stretches. The stretching is a *surprise*, not an arbitrary assumption.

This is the surprise: Assuming simulatneouos acceleration forces you to conclude that the train car stretches.

Of course that’s not at all a surprise once you understand it. But I’ve discovered that many people (including myself) are surprised by it at first. When we get to the point of understanding it, we’re not surprised anymore, and we’ve learned something. If you insist on dismissing it as unconditionally unsurprising (particularly after being one of the most vocal deniers for days on end), then you’re missing the entire point.

PS—The pole-in-the-barn paradox is also completely unsurprising once you understand it. But it’s still a neat example.

27. 27 27 Neil

Steve–I am not missing the point. You are considering an impossible simultaneous acceleration. The following example explains how a “real” simultaneous acceleration would work out. For a “real” simultaneous acceleration, the acceleration will start simultaneously for the stationary observer but not end simultaneously (because the train will then be moving relative to him.) It is long, so if you have got your mind made up that I just don’t understand, delete it. I don’t mean to clutter your blog. PS–I do enjoy your books.

KINEMATICS OF AN ACCELERATING TRAIN

A train consisting of all locomotives (so that all parts can accelerate simultaneously) sits stationary on a track and is measured by all to be L units long. For simplicity, assume just two locomotives. I sit in the front locomotive and synchronize with the engineer in the rear locomotive, say by means of light signals, an acceleration at a fixed rate for a short but finite interval. You, standing by the front of the train, have confederates at known distances along the track, including one at the rear of the train, who signal you with light flashes about observed events.

At a precise time, I send a light signal to the rear engineer to begin the acceleration. Knowing the speed of light and the length of the train, I begin the acceleration of the front locomotive at precisely the time that the rear engineer receives the light signal.

At the instant the front locomotive starts to move, you on the platform note the time. Your confederate at the rear of the train is instructed to send you a light signal the instant he sees the rear of the train start to move. Knowing the length of the train and the speed of light, you calculate (from the time you received the light signal from your confederate) that the rear of the train and the front of the train started moving simultaneously. (This is the last time you will see such simultaneity for the train.)

[On board the train, at the start of the acceleration, my engineer and I will SEE a strange optical illusion. It appears to me that the train is stretching (because at the instant I accelerate the front locomotive, the light signals from the accelerating rear locomotive haven’t reached me yet). It is an optical illusion because the engineer in the rear locomotive sees the train as shrinking. We can both measure the proper length of our train by means of timing a reflected light signal running the length of the train and conclude that the proper length remains L.]

In any case, once the acceleration stops (simultaneously in both locomotives because our clocks remain synchronized.) I see the train as shrinking for a while, whilst the rear engineer sees it as stretching. Once the light signals at the time acceleration stops have reached both me and the other engineer, we now both SEE, as well as MEASURE, our moving train at length L.

Out on the platform, something strange is happening. You have asked your confederates along the track to send you a light signal if an end of the train stops accelerating just as it passes them. Using this information, you establish that the front locomotive stops accelerating before the rear one (even though, according to your earlier measurements, they started to accelerate at the same time.) You now measure the train as shrinking as the rear engine continues accelerating while the front engine has assumes a state of constant motion. The train will continue to shrink until you measure all parts to stop accelerating and move uniformly. At this point, you will measure the length of the train with the help of your confederates, and it will be less than L by exactly the amount given by the Lorentz contraction. On board the train, I and my engineer will continue to measure the length of the train as L.

THE IMPOSSIBLE KINKED WORLD LINES OF YOUR DIAGRAM DO NOT PICK THIS UP. THE STARTING AND STOPPING OF AN INFINITE ACCELERATION OCCUR AT THE SAME TIME. WITH CARE, A PROPER MINKOWSKI DIAGRAM CAN BE DRAWN CONSISTENT WITH THIS GEDANKEN EXPERIMENT OF A FEASIBLE ACCELERATION.

28. 28 28 Steve Landsburg

Neil:

In any case, once the acceleration stops (simultaneously in both locomotives because our clocks remain synchronized.) I see the train as shrinking for a while, whilst the rear engineer sees it as stretching.

This is a red herring. You and the rear engineer are in the same frame; you therefore have to agree about the length of the train. I agree that there can be optical illusions going on, but those optical illusions have absolutely nothing to do with the problem; they are just a distraction. Your exposition would be much clearer without them (and without the confederates along the track, etc.)

Using this information, you establish that the front locomotive stops accelerating before the rear one (even though, according to your earlier measurements, they started to accelerate at the same time.)

Sure—IF the accelerations stop at the same time as measured on the moving train—but not at all if they stop at the same time as measured from the ground. Do you understand that you’ve added a new assumption here?

Your objection to the kinked world lines is a red herring; we can round out those corners and still tell essentially the same story. IF the two ends of the train perform their accelerations along the same time paths, as measured from the ground, then passengers on the train must see the train stretch. IF they perform their accelerations in some other way, then, as I’ve told you multiple times, the outcome can be anything at all depending on the choices that are made. Your example illustrates exactly that.

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