Built on Facts

The Plot Thickens, and Radiates

Don’t think I’ve forgotten about the falling electron question from a while back. Short version: an accelerating charge radiates. So if you let an electron fall in a gravitational field, it should radiate. But a person (or detector) falling along beside it does not perceive the electron as accelerating and so the electron shouldn’t radiate in their frame. How to reconcile the perspectives?

I didn’t know. Still don’t. I got back to the university and asked a few grad student friends of mine. One had an interesting argument involving vacuum polarizability that the electron will radiate EM waves in one frame but not the other. We couldn’t hash out the details because we were in a restaurant, but even if he was right (I am not convinced) from the semi-classical EM wave perspective that still doesn’t reconcile things from the photon perspective. A counter falling beside the electron will either detect photons or it will not. The best my (very, very smart) friend could offer was to say that “quantum fields in curved space are very weird”, but he had no definite conclusion. I’ll try to ask some professors this week. If anyone here is a Ph.D. who knows something about general relativistic quantum electrodynamics, feel free to chime in!

That out of the way, here’s my classes this semester if anyone’s curious. I’m taking two, Classical Mechanics and E&M II (waves, basically). I’m teaching the non-calculus-based intro physics, which is definitely a nice easy assignment. One that’s quite fun to teach, as well. On this site I plan to go over concepts and problems from both the classes I’m teaching and the classes I’m taking.

Off topic and apropos of last Saturday’s post: Yesterday a bunch of grad students and I went to see Death Race. It was about like you’d expect. Anyway, there were posters for upcoming films, and what do you know: both Quantum of Solace and Defiance feature Daniel Craig exercising appropriate trigger discipline! This is a staggering rarity in film posters involving someone carrying a gun. Good for him.

Perfunctory politics: Biden? Ok. Very standard, very safe. Perhaps what he feels he needs. There was exactly one possible interesting choice – a particular senator from New York. Since she was not chosen (assuming the Biden news is accurate), I think McCain can breathe a sigh of relief. If McCain has any political savvy at all, he’ll pick Alaska governor Sarah Palin. I’m not holding my breath.

Have a great weekend! Sunday function is tomorrow, and we’ll be doing an alternate series approximation for a function whose power series fails to converge. Yes, it is EXCITING, darn it!

Comments

  1. #1 ppnl
    August 23, 2008

    The falling electron will feel tidal forces as it falls. I assume that it will radiate depending on those tidal forces.

  2. #2 andy.s
    August 23, 2008

    Tidal forces depend on the size of the particle in question.
    Would that be the Compton wavelength?

  3. #3 andy.s
    August 23, 2008

    Oh, and Matt, just out of curiosity, what texts are you using for CM & E&M. Goldstein and Jackson?

  4. #4 Fox
    August 23, 2008

    Undergraduate doubling in math/phys here, but does it matter that (assuming you’re not falling towards an infinitely flat gravitating plane) the observer and the electron will be moving closer together?

    That is, if they’re side by side. If one is above the other, I guess that gets to comment #1.

  5. #5 arby
    August 23, 2008

    Hmm, I musta missed the first falling electron post. Link, title? rb

  6. #6 Frederick Ross
    August 23, 2008

    Do you actually mean an electron or will you settle for a classical object? If you’ll settle for a classical object, you need section 90 of Landau and Lifshitz volume 2. Maxwell’s equations aren’t the same.

    If you want it in quantum, congratulations, no one knows. Your friend’s statement “quantum fields in curved space are very weird” is worse than he thinks. While you have L&L out, read the opening sections of volume 4. If you’re feeling cranky, you might walk around with Schwinger’s ‘Particles, Sources, and Fields’ under your arm, and snap at the field theorists who stop to caution you about it.

    Have fun with Jackson. You’ll want Schwinger’s E&M and L&L volumes 2 and 8 to hand. And volume 1 for classical, until you hit Hamilton-Jacobi, when something a little less terse may be useful.

  7. #7 Paulo Mauricio
    August 23, 2008

    I think you’ve forgotten the wired and almost magic Copenhagen’s Quantum Mechanics interpretation. How do you ‘see’ that the electron is falling beside you? Suposing you see it (you localize it), the electron (in your falling perpective will lost all certainty in momentum and energy.

  8. #8 Matt Springer
    August 23, 2008

    Here’s the first post on the subject.

    It’s such a basic-sounding question that it seems the answer should be obvious, but there you have it.

    Textbooks: We’re using Jackson for E&M, and L&L for mechanics. I have no idea why we’re using L&L as the primary text, but I’ll probably pick up Goldstein or one of the other classics too. And probably the L&L for E&M as well; might as well work on rounding out the collection since I’ve already got the quantum L&L.

    And finally, I don’t think tidal forces will be an issue because they can be made arbitrarily small. Either way you can recast the problem in terms of accelerating observers in free space and leave explicit gravity out of it completely, if desired.

  9. #9 arby
    August 23, 2008

    Thanks for the link, Matt. It turns out I had read it, but before there were any comments. That’s one of the negatives of using a reader, the post is often too fresh to have elicited any reaction. Great subject, I look forward to more. Thanks, rb

  10. #10 teramut
    August 23, 2008

    Why shouldn’t the observer see the electron radiate? You would expect to get different results for experiments in an accelerating frame. I think you are just confusing this with the case of inertial coordinate systems, where you would expect to get same results regardless of your coordinate system.

  11. #11 Alex Besogonov
    August 23, 2008

    As far as I remember, electron WILL radiate.

    But to a falling observer the _vacuum_ _itself_ will appear to be “warm”. And a falling electron will just appear to be in a thermal equilibrium with this warm background.

    See here: http://en.wikipedia.org/wiki/Unruh_effect

  12. #12 Matt Springer
    August 23, 2008

    Alex: A friend of mine mentioned that he thought the vacuum itself would radiate from the perspective of the accelerated observer, but he wasn’t sure of the details. I hadn’t had time to check especially since I’ve not yet familiar with the mathematical machinery needed, but I’m pretty sure this is the only explanation I’ve heard that could adequately conserve photon number in both frames.

    I think you have hit the answer. Thank you very much!

  13. #13 Carl Brannen
    August 23, 2008

    Maybe it will help to read the wikipedia article on Unruh radiation.

    Of course that’s not my explanation for the effect (i.e. that acceleration is equivalent to a gravitational field even for quantum mechanics). I think the gravitational field accelerates particles in a manner that is incompatible with the object feeling an acceleration due to a violation of relativity that is beyond the scope of this comment.

  14. #14 Winter Toad
    August 23, 2008

    A person who is accelerating alongside an electron is not in an inertial frame, the person can detect the acceleration. Let’s leave aside for now the idea of falling in a gravity field, because of the complications of handling that in a relativistic fashion, in which the particle is not accelerating at all, but is just falling freely along a locus in curved space.

    A person standing still while a water balloon accelerates past them will see the balloon as being distorted, since the balloon changes shape to transmit the force through to all of the water. A person accelerating beside the balloon will also see it distorted, but the person knows that they’re in an accelerating frame of reference, so this isn’t a problem.

    So, if I’m moving with an electron, and the electron is accelerated, perhaps by an electric field, but I also am accelerated to keep the same relative position with the electron, I will feel that big push, and I will see the electron radiate synchrotron radiation, and will not be surprised. Both the co-accelerating reference frame and some inertial outside observer will agree that the electron accelerated and that radiation was generated.

    A more interesting question might be this: an electron is, by virtue of an externally-applied force, held stationary in a gravitational field, and so is prevented from falling freely in four-space. Does the electron radiate? If so, where does the energy come from? If not, why not?

  15. #15 Alex Besogonov
    August 23, 2008

    Winter Toad:

    No, it’s more complex. Suppose that you’re sealed in a container with half-transparent walls (so you can’t see anything outside, but outside observers can see you). You can’t tell if you are flying in orbit (we’ll neglect tidal forces) or if you’re in a free-fall towards an event horizon of a black hole.

    However, there’s a difference for outside observers – they should see radiation from an accelerating observer.

  16. #16 Ron
    August 23, 2008

    What do you get if you take the electric field of a point charge at rest, and ask what that looks like to an observer under constant acceleration?

  17. #17 Alex Besogonov
    August 23, 2008

    Ron: You should see the charge radiating photons.

    Another interesting question: what happens with bosons of other fundamental forces?

  18. #18 Joel Eidsath
    August 23, 2008

    I think that ppnl may be correct. If the charged particle is radiating, it should actually experience a retarded acceleration due to Abraham-Lorentz. This is a caused by a self-force of the charged particle. It’s easier to conceptualize by imagining a charge distribution instead of a point charge.

  19. #19 Ron
    August 23, 2008

    Do Particles Exist discusses the notion that “the number of particles depends on the frame of reference in which you are measuring.”

    Free falling electric charge in a static homogeneous gravitational field also tackles the problem.

  20. #20 Winter Toad
    August 23, 2008

    Alex: the problem here is, in part, conceptual. You have a charge in free fall, and a co-falling observer. There is an observer who is constrained not to fall, by virtue of, say, standing on something. Who is accelerating? In classical physics, we’d say the falling observer and the charge are accelerating, in general relativity, we would say that the nominally stationary observer is accelerating because there is a force preventing him from following a free locus in 4-space. This specific problem was discussed in the Feynman Lectures on Gravitation, section 9.1.

    I’ll direct the interested reader to a newsgroup thread that discusses this: http://groups.google.com/group/sci.physics.research/msg/f07da51a03a37174

  21. #21 MartinB
    August 24, 2008

    My Ph.D. in physics is some time past, but I think this is a problem to be solved completely within the framework of special relativity, because the situation is not changed very much if the two observers are accelerating relative to each other without being involved in a gravitational field. I talked with my wife (who also has a PhD and tops it off with a great intuition) about it and we came to the following conclusion:

    If I’m next to an electron, I see its Coulomb field, a purely electrostatic field. If you are accelerating towards me and my electron, Lorentz transformation of this field will let you perceive an electromagnetic field, i.e., radiation. If I remember correctly, this kind of calculation can be done using Lennard-Wiechert potentials which are smewhere in the back of Jackson classical electrodynamics book.

    So far, there is no problem, we just see the field differently.

    The thing starts to get interesting when we’re trying to interfere with it. Imagine, for example, I would hold a mirror between me and you, so that the mirror (in your frame of reference) should reflect the photon back to me. In my frame of reference, the mirror gets an induced electrical charge and changes the electrostatic field. I still don’t see any radiation. You see radiation, but the radiation you see is now caused by the induced charges on the mirror, which are also moving in your frame and thus also create electromagnetic fields.

    If, on the other hand, you hold the mirror, then in your frame of reference the mirror reflects the light emitted by my electron. In my frame of reference, your mirror has electrical charges induced on it by my electron, and since these charges are accelerated towards me, I see radiation. So we can both agree that you back-reflect a photon towards me.

    I don’t think having the acceleration due to free fall changes anything – I still see only a Coulomb field, you see us accelerating towards you and see radiation.

  22. #22 JM
    August 24, 2008

    I found this http://arxiv.org/abs/0806.0464 which discusses the problem in terms of an electron in free fall around the earth, but asks “wouldn’t this be a perpetual motion machine?”

    It outlines a solution (apparently put forward by DeWitt and DeWitt in 1964). At a quick glance it seems their solution is pretty much as described by MartinB

  23. #23 Paul Murray
    August 25, 2008

    If a falling electron radiates, then the energy it loses by radiation should slow it down. Sort of like magnetic braking.

  24. #24 MartinB
    August 25, 2008

    To Paul Murray,

    yes, if it loses energy, this should slow it down. The trick is, in the rest-frame of the electron, the energy in the Coulomb field is static and stationary, but transforming it to the other frame you see radiation. That’s why I made a point of interacting with the radiation to see what happens to the energy.

    The problem of inertia due to radiation, BTW, is a serious problem of renormalization because the inertia is infinite. See Feynman Lectures vol. II, chapter 28 (I think).

  25. #25 Michele
    August 27, 2008

    See also gr-qc/9903052, “Classical roots of the Unruh and Hawking effects,” by Pauri and Vallisneri (ehm, that’s me, a while ago).

    Indeed, the conclusion is essentially MartinB’s.zz

  26. #26 Ian
    August 29, 2008

    “If McCain has any political savvy at all, he’ll pick Alaska governor Sarah Palin”

    Congrats on getting it right. Clearly he’s going after the young and the Hilarestless.

    It makes you wonder how much he thought she was adequate for the job (after screeching that Obama lacks experience! LoL!) and how much he’s desperately trying to get an edge and steal some of Obama’s thunder.

    Unfortunately, now the running mate for the vaccination doubter is a wannabe global warming denialist who’s in favor of “teaching the controversy”. Sad but true.

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