Counterfactual Physics: Lorentz Variance?

The theory of relativity takes its name from a very simple and appealing idea: that the laws of physics should look the same to moving observers as to stationary ones. “Laws of physics” here includes Maxwell’s equations for electricity and magnetism, which necessarily means that moving observers must see the same speed of light as stationary observers (Einstein included the constancy of the speed of light as a second postulate in his original relativity paper, but it’s redundant– the constancy of the speed of light is a direct consequence of the principle of relativity). This leads directly to all the observed weirdness of relativity– clocks running at different rates for different observers, moving objects shrinking, disagreements about the simultaneity of events, etc.

Of course, the notion of a single universal time also has a certain aesthetic appeal, which is part of why it was the default assumption of physics from the days of Galileo and Newton through to 1905. Having every clock in the universe, moving or not, tick at exactly the same rate would be simple and elegant in a manner similar to that of relativity. It would, however, require some drastic revisions of the laws of physics as we understand them.

The question is, what would need to change, and what would the consequences be?

You would clearly need to change the structure of Maxwell’s equations to accommodate a variable speed of light, but what would that do to, say, atoms and molecules, that are held together by electromagnetic forces? The speed of light shows up in things like the Rydberg constant that gives you the energy of atomic energy levels, so presumably these would get pushed around as the speed of an object changed. Would that mean, for example, that objects moving at too high a speed would literally fall apart as the energy levels determining the bonds between their component atoms shifted into a new configuration?

(I’m imagining a world that’s as similar to ours as possible, but has a variable speed of light, so things like the Michelson-Morley experiment would not give the negative result that they do in our world. I’m not looking for aether drag theories that use some baroque method to match those observations, but a version of reality in which those observations are consistent with a variable speed of light)

I don’t expect that anybody has put all that much thought into this– it would take a good deal of mathematical effort to reformulate E&M in a way that wasn’t consistent with relativity, and why would you do that? (Then again, there’s some amazingly abstract stuff on the arxiv, so who knows?) It’s kind of an interesting topic for idle speculation, though. So consider the comments section an open thread for idle speculation, or the posting of actual relevant information, should any exist.

(This post brought to you by Emmy, who asked “Would having all observers see the same time really be all that bad?” in chapter 3 of the book-in-progress.)

Comments

  1. #1 Raskolnikov
    September 8, 2010

    “the constancy of the speed of light is a direct consequence of the principle of relativity)”

    Careless statement there, what about galilean relativity. It certainly doesn’t imply the constancy of the speed of light. I know you are thinking of Lorentz invariance when saying that, but still.

  2. #2 Chad Orzel
    September 8, 2010

    The context of that sentence was a discussion of Einstein’s theory, so I thought it was clear that “principle of relativity” referred to the first principle in Einstein’s paper.

    Of course, I would say that the only real difference between Galileian and Einsteinian relativity is that physicists had learned about electricity and magnetism in the intervening 200-odd years. They have the same central idea– physics looks the same to an observer moving at constant speed– it’s just that the meaning of “physics” expanded to include Maxwell’s equations.

  3. #3 Eric Lund
    September 8, 2010

    You would clearly need to change the structure of Maxwell’s equations to accommodate a variable speed of light

    There is an entire branch of physics called plasma physics which deals with this problem. The difference is that plasma physics deals with the speed of electromagnetic waves in a medium and how that speed changes as a function of frequency. The constancy of the speed of light in vacuum is taken for granted. Since it’s the speed of light in vacuum that has to be constant according to Lorenz, one would expect that the speed of an electromagnetic wave in the actual medium might vary according to the observer’s relative velocity.

    How you get from this formulation to one where the speed of light in vacuum might depend on the observer’s frame of reference is not obvious to me, but I suspect that if it were possible to devise such a formulation self-consistently it would look quite a bit like plasma physics. You would be likely to run into lots of phenomena like wave dispersion in unexpected contexts.

  4. #4 Matt Leifer
    September 8, 2010

    Raskolnikov,

    I think what Chad means to say is:

    “Given that we know that the Maxwell equations are true empirically, the constancy of the speed of light is a direct consequence of the principle of relativity.”

    Whilst this is clearly true, I think it is the wrong way of thinking about it. I think we should say that the Maxwell equations are the way they are because of special relativity (in addition to other symmetry principles like U(1) gauge invariance) rather than saying that relativity is the way it is because of Maxwell’s equations. This is how we think about determining field theories these days, i.e. there is a list of principles that have to be satisfied, including Lorentz invariance, gauge symmetries and renormalizability. Taken together, these often uniquely determine the Lagrangian of the theory.

    The empirical correctness of Maxwell’s equations obviously provided the main motivation for taking special relativity seriously in the first place. Therefore, I can see why Chad (the experimentalist) might therefore want to take Maxwell’s equations as a given in understanding relativity. However, this is a historical accident and the existence of an invariant speed seems to be the more fundamental principle from which we should derive all of the more specific theories like Maxwell, Yang-Mills, etc.

  5. #5 Matt Leifer
    September 8, 2010

    Regarding the main question of the post, you might try doing a literature search for “luminiferous aether”. However, I don’t think you will find that much of the literature on the subject has been digitized :)

  6. #6 Chad Orzel
    September 8, 2010

    I think what Chad means to say is:

    “Given that we know that the Maxwell equations are true empirically, the constancy of the speed of light is a direct consequence of the principle of relativity.”

    That’s another way of putting it, yes. It’s close to the historical trajectory, too.

    Whilst this is clearly true, I think it is the wrong way of thinking about it. I think we should say that the Maxwell equations are the way they are because of special relativity (in addition to other symmetry principles like U(1) gauge invariance) rather than saying that relativity is the way it is because of Maxwell’s equations. This is how we think about determining field theories these days, i.e. there is a list of principles that have to be satisfied, including Lorentz invariance, gauge symmetries and renormalizability.

    Sure, if you want to be all theorist-y about it…

    Even phrased that way, though, you could imagine some different symmetry that would lead to different rules. I’m not sure exactly what that would be, but there’s presumably some symmetry you could use that would give time as a universal quantity, and imposing that symmetry would give you different rules for E&M. The question is, what would those rules look like, and what would that do to the rest of physics?

  7. #7 onymous
    September 8, 2010

    there is a list of principles that have to be satisfied, including Lorentz invariance, gauge symmetries and renormalizability

    Most of us wouldn’t include renormalizability in that list.

  8. #8 Eric Lund
    September 8, 2010

    The empirical correctness of Maxwell’s equations obviously provided the main motivation for taking special relativity seriously in the first place. Therefore, I can see why Chad (the experimentalist) might therefore want to take Maxwell’s equations as a given in understanding relativity.

    I will disagree here and say that Chad is entirely correct to do things in this order. Relativity as a concept goes all the way back to Galileo, as Chad @2 points out. Classical mechanics is quite simple under Galilean relativity, but Maxwell’s equations were found to be extremely messy under Galilean transforms. Thus it was necessary to come up with a different conception of relativity, and this was an active area of research in the late 19th and early 20th centuries. The correct transform bears Lorentz’s name because he was the first to show that Maxwell’s equations were indeed invariant under such transforms. Fixing classical mechanics to work under Lorentz transforms turned out to be a good deal easier than fixing Maxwell’s equations to work under Galilean transforms (propagation of EM waves in media was IIRC not understood at the time, and the same mathematical tools needed to treat that problem would be needed to make a Galilean invariant version of Maxwell’s equations).

    I suppose there might be pedagogical reasons for taking c to be a frame-independent constant, but there are other pedagogical reasons for deriving it as a consequence of Lorentz invariance. I took my Jackson course from a high-powered theorist who nonetheless adopted the approach Chad takes.

  9. #9 Raskolnikov
    September 8, 2010

    @Matt

    “Given that we know that the Maxwell equations are true empirically, the constancy of the speed of light is a direct consequence of the principle of relativity.”

    Yeah, that is about as deep as saying: given that the speed of light is a constant empirically, the constancy of the speed light is a direct consequence of the principle of relativity.

    For dog’s sake, Maxwell’s equations entail the constancy of the speed of light. They are Lorentz invariant. This was known long before Einstein came along. Poincaré knew it, Lorentz and Fitzgerald know it, Voigt knew it.

  10. #10 onymous
    September 8, 2010

    The question is, what would those rules look like, and what would that do to the rest of physics?

    It sounds like you’re looking for a Galilean-invariant gauge theory, but I think the main difficulty is that the concept of a massless particle doesn’t make sense in Galilean-invariant field theory. So it’s tempting to say that E&M would be forced to be short-range, but maybe there’s some sort of loophole. Probably this is discussed in the literature in various places….

  11. #11 Joseph
    September 8, 2010

    “you might try doing a literature search for “luminiferous aether”. However, I don’t think you will find that much of the literature on the subject has been digitized :)”

    You might have some good luck here at the OU History of Science collections. http://libraries.ou.edu/locations/?id=20

    (we’re trying to organize a tour of the collections with the HoS department here for the Midwest Solid State Conference, if anyone here is going. Confirmed; they do have some stuff on “luminiferous aether” e.g. “On the relative motion of the earth and the luminiferous Œther microform” by Albert A. Michelson and Edward W. Morley, he London, Edinburgh and Dublin Philosophical Magazine and Journal of Science. 5th Series, v. 24, no. 151 (Dec. 1887) If you want older, they have some cool stuff from e.g. Galileo and Darwin (some original Darwin notes, and books with e.g. notes in the margins in their hand and notes for future revisions of the book! My wife is in the HoS department and was surprised to find that the slightly singed book she had set in front of her one day was singed because it was saved from the Great Fire of London in 1666….)

  12. #12 onymous
    September 8, 2010

    On second thought I’m not satisfied with the “no massless particles” argument, since gapless excitations exist in condensed matter systems. It’s a fun exercise to think about what the closest Galilean analogue of relativistic E&M could be, I’ll give it more thought….

  13. #13 Sean Carroll
    September 8, 2010

    Count me among those who don’t think the constancy of the speed of light is implied by the principle of relativity. After all, the speed of sound is not the same to every observer, but nobody thinks that’s a violation of relativity. It’s just a reflection of the fact that sound moves through a medium, and any particular example of that medium has a rest frame. Pre-Einstein, that’s exactly how people thought E&M worked; the medium was the aether. Einstein’s great breakthrough was to show how you could get a consistent reconciliation of Maxwell’s equations and the principle of relativity by replacing aether with a constant speed of light.

  14. #14 Andrew Foland
    September 8, 2010

    Freeman Dyson gave a nice picture of what groups (including Lorentzian or Newtonian) could accommodate reasonable physics in his Missed Opportunities Lecture .

    In general, I think of c not as the speed of light, but as the speed of massless particles–this seems like a more likely fundamental basis for a universe than the particulars of any individual gauge interaction. Photons in vacuum happen to be massless. You generally recover the Newtonian limit if you let c go to infinity.

    For instance, you can write the Rydberg formula in terms of the coupling constant alpha (presumably unaffected) and c, and if you do so the Rydberg energy goes to infinity. Note that if you write the binding energy in terms of the electron charge c appears downstairs, but the apparent electron charge depends on the speed of light (sqrt(alpha*hbar*c)) whereas alpha is fixed. I think.

  15. #15 Raskolnikov
    September 8, 2010

    I just found this:

    http://www.springerlink.com/content/k3568758541j024x/

    I don’t like how they speak about nonrelativistic limit to designate the limit c to infinity. It’s just confusing the issue of the difference between the principle of relativity and the invariance of lightspeed again. Anyway, there you have it, electromagnetism with galilean relativity.

    PS: Doesn’t surprise Lévy-Leblond is involved. He’s also the guy who deduced a spin 1/2 equation from galilean invariance.

  16. #16 Moshe
    September 8, 2010

    Very quickly and naively (which is all I have time for), I’d think that a world where the electric field exists, but the magnetic field doesn’t, is an example of a world that obeys the principle of Gallilean relativity without being special relativistic. There may be other, less simple minded, limits.

    If true, I don’t think energy levels in the hydrogen atom would change all that much in such a world (though you may need to change units in order to keep them finite. Incidentally, these sort of theoretical speculations is the place where it is cleanest to talk about dimensionless quantities, such as energy ratios, i.e the pattern of energy levels as opposed to their absolute normalization).

  17. #17 Anonymous Coward
    September 8, 2010

    This may be a cop-out answer, but if you take a universe where the speed of light is infinite, you get synchronized clocks.

    So just take that limit in all your physical equations.

    Does this limit results in no magnetic fields as in Moshe’s (#16) suggested Galilean relativity world? I suspect so. Is it equivalent? I dunno. Like Moshe’s suggestion, I think this limit would keep the gross features of atomic structure intact, although the fine structure details would get all weird.

  18. #18 Roberto Baginski
    September 8, 2010

    Dear Chad,

    you are looking for either Hertz (see P. Moon et al. Physics Essays 7, 28 (1994) for instance) or Weber electrodynamics (see any of Assis’ papers listed here: http://www.ifi.unicamp.br/~assis/wpapers.htm). In both theories you will get Galilean invariance (universal time).

    In Hertz electrodynamics, there is no magnetic forces. Instead, a magnetic field creates an electric field that acts on the charged particles. Curiously enough, speed of light is still a constant…

    In Weber electrodynamics, there is really no field, just forces between particles.

    Good luck,
    Roberto

  19. #19 Kaleberg
    September 8, 2010

    Reading Dyson’s lecture on missed opportunities was fascinating. Some of the ideas are really far out, like a universe with absolute space instead of absolute time, except, I just read an article proposing something an awful lot like that just a few weeks ago.

    In the 19th century, the physicists were well ahead of the mathematicians, but now it seems that the string theorists are mathematicians ahead of the physicists.

  20. #20 Kaleberg
    September 8, 2010

    Speaking of counterfactuals, I looked up luminiferous aether in the journal Science and came across a fascinating paper on a repeat of the Michelson Morley experiment done in 1925, except that this experiment WAS able to measure the absolute motion of the earth and solar system! There is aether and there is aether drag, but there is also absolute motion. I’ll have to read it again, but it sure doesn’t correspond to conventional wisdom. I assume there was a follow up paper or two, or I may have accidentally found an internet gateway to an alternate universe.

    I’ve put a copy of the paper at: http://www.kaleberg.com/misc/1925experiments.pdf

    Please, tell me I misread it.