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Conservation of Particle Number

  1. Nov 20, 2012 #1
    Because E=mc^2, when we increase the energy of a system we can introduce new particles. What about accomplishing this simply via a change of coordinates via Special (or even General) Relativity?

    If I have a system of one electron sitting still (E = E0), then I change to a coordinate system moving at near the speed of light, is it possible that particles could pop out or will there only ever be one electron just with increased energy? I'm not talking about particle collisions, but strictly from the change in energy coming from the new coordinates...
  2. jcsd
  3. Nov 20, 2012 #2


    Staff: Mentor

    E=mc^2 is for a particle at rest. If you transform then the particle is no longer at rest.
  4. Nov 20, 2012 #3
    Well, actually that's not true. m changes when it's in motion, as does E. Regardless of that, you're not answering my question. Since the energy of the particle in the moving reference frame is significantly higher, do particles appear? Is the particle number conserved under coordinate change or not? I would expect, IF particles are created, that they come in particle / antiparticle pairs, but I'm not sure if the total particle number (operator N) is conserved or not.
  5. Nov 20, 2012 #4


    Staff: Mentor

    Actually, it is true. Energy and momentum form a four-vector. The invariant norm of the four-vector is mass. The timelike component of the four-vector (energy) is only equal to mc^2 in the rest frame.

    There is an old concept of relativistic mass, which is no longer used. That is probably what you are thinking of.

    Also, you need to distinguish between frame invariance and conservation. Particle numbers are not conserved, but they are frame invariant.
  6. Nov 20, 2012 #5


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    Adding energy to a system via a physical process is a different thing totally from adding energy to a system via changing coordinates.

    In spite of this, I believe it is true that photon number in particular is not conserved in General Relataivity. (It's conserved just fine in SR).

    The effect responsible is called Unruh radiation.

    This is on my list of thigns I want to understand better someday.
  7. Nov 20, 2012 #6
    Yes, and like I said, it's not relevant to my question. Thank you however for the commentary. I'll make a note to not use relativistic mass anymore.

    Semantics. Ok, sounds like the answer to my question is that particle number in invariant under coordinate change. So, despite the extra energy, no new particles will appear simply because we are in a moving reference frame compared to the other.

    That would mean though that a frame falling in a gravitational field would ALSO show in-variance in particle number.
  8. Nov 20, 2012 #7


    Staff: Mentor

    Locally, yes, but I am not sure that holds over regions where curvature becomes significant.
  9. Nov 20, 2012 #8
    Sure that would make sense since curvature means the space is filled with non-zero energy ( Energy Momentum tensor non-zero ). That, I would guess, acts like an interaction and thus particles can be created as the interaction occurs.
  10. Nov 20, 2012 #9


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    A good idea. The fact that photon number is conserved when you have a linear Lorentz boost pretty much demonstrates to total irrelevance of the issue in any event.

    While the particle number is invariant under some coordinate changes (i.e. Lorentz boosts), it is not in general invaraint under ALL coordinate changes.

    Specifically, an eternally accelerating observer WILL see "the vacuum" turn into a sea of thermal particle radiation, which explicitly demonstrates the non-conservation of particle number.

    See for instance Unruh's paper: http://prd.aps.org/abstract/PRD/v14/i4/p870_1
    Or the wiki article for an overview:

    You'll find similar remarks in some GR textbooks, for instance Wald, though I'm a bit too lazy to track them down at the moment.

    It's not particularly clear if that is true or not, sorry. The results are subtle enough that you have to be very careful not to jump to conclusions about cases that seem similar at first glance.

    http://www.springerlink.com/content/d582g4784283x603/fulltext.pdf "On the Physical Meaning of the Unruh Effect" and http://arxiv.org/abs/hep-ph/0610391 "On the relation between Unruh and Sokolov--Ternov effects" discuss the rather interesting case of an electron orbiting in a magnetic field as an Unruh-type detector.

    The second reference of the two above says the following:

    I'm not sure how to reconcile the statements made here with the one's in the abstract of Unruh's paper, specifically
    "it is shown that a geodesic detector near the horizon will not see the Hawking flux of particles. ".

    In any event , the question here is mostly QFT rather than classical GR. You might get better answers in another forum.
    Last edited: Nov 20, 2012
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