Built on Facts

By the 1860s, the classical theory of electricity and magnetism was on a very solid theoretical footing. Maxwell’s equations describing the interplay of charges and currents with electric and magnetic fields were on paper by 1862, and with some changes in notation they’re the exact same today. Relativity wouldn’t be invented for another half-century or so, and that makes it all the more remarkable that Maxwell’s equations don’t actually need to be modified at all to work in a relativistic framework. Lorentz covariance is built right in, though it’s a bit hidden.

But Maxwell and Faraday and Ampere and the rest didn’t know that. There were some tantalizing hints though, and in fact it was the exploration of classical electrodynamics that led Einstein to the theory of special relativity. It’s entertaining to take a look at some of those hints, which are lurking right there in second-semester intro physics.

Consider a uniformly charged wire alongside a particle of charge q:

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(Apologies for the sloppy PowerPoint graphic, but it probably gets the gist across.) We know from freshman physics that (via Gauss’ law), the electric field generated by that charged wire is:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

The force experienced by the charge q in the field is:

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We’ll say the particle and the wire are both positively charged, so the force is repulsive and pointing radially outward from the while. For simplicity, we won’t bother with vector notation in this post, but do keep in mind that forces and fields are vectors and do have a direction that we have to pay attention to.

Now let’s start moving the wire and the particle to the right at a constant velocity v. Or equivalently, move ourselves to the left at a constant velocity v, leaving the wire and particle stationary in the lab frame. Physically, they are the same situation, and this ends up being a key part of relativity. A moving charged wire is a current carrying wire, since an electric current is just moving charge by definition. The current I is given by I = λv, since charge/time is the same thing as (charge/length)*(length/time). A current produces a magnetic field which wraps radially around the wire with a magnitude of

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So now we have electric and magnetic fields, like so:

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Now the force on a charged particle moving in a magnetic field is F = qvB, and in this case it’ll be directed radially inward, toward the wire.

Now hold on – when everything was standing still, the net force was qE, pointed away from the wire. When we changed nothing at all except sliding our own chair in the lab to the left at speed v, suddenly the net force is F = qE – qvB, which is something completely different. Substituting the expressions for E and B in, the net force is:

i-8d7070ce28f0af28047c7a5fb8246d53-4.png

What the heck? The force is an objectively measurable thing which gives the particle a specific acceleration. It can’t possibly be different depending on whether we’re moving or not. If classical electromagnetism makes such a prediction, surely the theory is wrong. Right?

Right – if we assume that all these charges and fields and currents and lengths are all the same in both frames. And that simply isn’t the case. You need relativity.

But it’s 1860, and we haven’t got relativity yet. How might we go about groping in the dark toward an answer? Well, we might postulate that fields aren’t the same in each frame. In the frame where the system is stationary, we have (say) the electric field E, while in the moving frame we have some different electric field E’, given by some coefficient E = αE’, where α is a function of v. If we assume the force is the same in both frames (it isn’t, but we don’t know that in 1860), we can look for that coefficient by solving:

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Which gives after a little algebra:

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If this initial groping-in-the-dark attempt at fixing classical E&M to work like our Gallilean intuition says it should is right, the E field in the moving should be slightly bigger in the moving frame than in the rest frame. Maybe the motion with respect to the ether somehow magnifies it, I dunno. In any case the correction factor is very small. If we’re talking about laboratory speeds in the m/s range, the correction factor is on the order of parts-per-quadrillion.

But is it right experimentally, if we could measure it? As it would turn out, no – but it’s close. It turns out that α should be the Lorentz γ factor, but at small speeds his γ and our α have the same order of magnitude (though we’re still off by a factor of 2, it turns out).

In any case our first attempt at a relativity theory is wrong – but closer to right than we were without it. Don’t be tempted to think that even people like Einstein had their brilliant ideas spring into being fully formed. Even the seemingly sudden great advances represent a lot of hidden hard work, tentative steps, and false starts.

Comments

  1. #1 Joshua Zelinsky
    July 8, 2011

    That’s really neat. Was this problem known in the 19th century or did this show up post Einstein?

  2. #2 Jim1950a
    July 9, 2011

    What about the moving charge in the particle? Does that also generate an E-field?

  3. #3 Russell
    July 9, 2011

    In some ways, the 19th century physicists were in much the same state as we are today. They had two great theories that worked great, but couldn’t be unified, because Newtonian mechanics was invariant across Galilean transformation, while Maxwell’s laws were not. We have QM and GM. The latest results are that if space and gravity show any granularity, it is at scales thirteen orders of magnitude less than the planck scale!

    http://rturpin.wordpress.com/2011/07/01/physics-theories-still-not-playing-well-together/

  4. #4 Roger
    July 9, 2011

    Maxwell knew about this problem, and Lorentz solved it, long before Einstein. After all, the factor is called the Lorentz factor, not the Einstein factor.

  5. #5 dr. dave
    July 9, 2011

    Why do “semi-relativistic” calculations always seem to come up short by a factor of TWO. This always confuses/amazes me.

  6. #6 Raskolnikov
    July 9, 2011

    The first one who came close to special relativity was Woldemar Voigt in 1887.

  7. #7 CCPhysicist
    July 10, 2011

    It isn’t remarkable that Maxwell’s equations don’t need to be modified. They were physically correct, so they have to be covariant. This simple fact is the reason that relativity is such a perfect example of paradigm shift. The only difference between Einstein’s work and that of Lorentz is that of their point of view.

    Lorentz felt he had to force the new theory (Maxwell) into the Galilean-Newton structure and did so with an interesting, yet very complicated theory. (The part where the self-energy of the electron produced a mass that varied as the shape of the electron varied, for example.) Einstein took the beauty of Maxwell’s equations a a sign that they are correct, and made Newton adapt to that reality.

    Read Einstein’s autobiography (or some of the Einstein bios) and you get a better example than the one you came up with. Einstein asked himself what light would look like if you traveled next to at c. The answer is it would be a standing wave that is not consistent with Maxwell’s equations. (Spatial curvature requires a time dependence that is not observed if the thought experiment is correct.) Something is wrong with Galilean relativity or radio waves can’t exist.

    @1, the problem was well known in the 19th century. I knew that someone had shown that Maxwell’s equations were invariant under something like the Lorentz transform, but couldn’t recall the name. Thanks to @6, you can use Wiki to get from him to a article about the history of the Lorentz transformation that gives those details. The important detail is that no one working on this problem, including Poincare, saw the simple solution: accept Maxwell over Newton.

    This was a bit easier than the Newtonian gravity, non-relativistic QM, relativistic QM, general relativity problems because Newton was fairly easy to fix. Indeed, the problem with Newton’s galilean formulation would have been experimentally obvious if high energy accelerators had been invented before relativity was discovered.

  8. #8 Roger
    July 10, 2011

    [@7] The important detail is that no one working on this problem, including Poincare, saw the simple solution: accept Maxwell over Newton.

    That is not true. Poincare accepted Maxwell’s equations and proved that they were covariant under the Lorentz group. He wrote about a new mechanics in which nothing goes faster than light. Poincare was ahead of Einstein on every aspect of relativity.

  9. #9 CCPhysicist
    July 10, 2011

    Good point @8, but my impression is that Poincare’ didn’t really believe it was needed in the way Einstein did. In addition, I was in error to say that Einstein accepted Maxwell over Newton in his papers. I was letting my belief that you can see evidence that the triumph of Maxwell’s theory drove his ideas interfere with what Einstein actually wrote in his paper.

    Einstein’s first postulate incorporates every valid physical law and his discussion implicitly incorporates the First Law through the idea of an inertial coordinate system. Newton’s basic laws are never rejected. Further, he took a sort of positivist approach by emphasizing an experimental result (the invariance of the speed of light) rather than a theoretical one (Maxwell was right). In the end, this makes his result more general, as his derivation applies even where Maxwell breaks down with photons and QED.

  10. #10 Roger
    July 10, 2011

    Poincare did not need Maxwell’s equations because he formulated relativity in terms of the geometry of spacetime, and he applied relativity to electromagnetism and gravity. Einstein did not understand that at all.

    Einstein did not incorporate every physical law. Just electromagnetism. Historians say that he disregarded experimental results. He just postulated what Lorentz proved, and Einstein’s approach was the least general.

  11. #11 Joakim
    July 11, 2011

    Lustiges Filmchen:

  12. #12 cases
    July 12, 2011

    Lustiges Filmchen: now that’s some creative beer ponging

  13. #13 CCPhysicist
    August 4, 2011

    @10:
    How can you seriously claim that the first postulate, that the laws by which physical systems change are not affected by which inertial coordinate system you choose, does not incorporate every physical law? There are no exceptions.

    As to whether proving what Lorentz postulated is better than proving what Einstein postulated is something one could debate forever, except Einstein did not prove that objects physically changed shape when in motion relative to a special, absolute rest frame. He proved that such assumptions about nature were not needed.

    And of course Einstein ignored some experimental results. He was not trying to fit data.

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