1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Necessity for action to be a Lorentz scalar

  1. Aug 12, 2014 #1

    On p.573 of Jackson 2nd Ed. (section 12.1), he says, "From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of motion are determined by the extremum condition, [itex]\delta A=0[/itex]."

    I agree that if the action is a Lorentz scalar, then that is sufficient to assure that the equations of motion are the same in all frames: Lorentz scalar, so Lorentz invariant, so action is minimized in all frames when it is minimized in one frame, since it is the same in all frames as it is in that one frame.

    However, Jackson seems to imply not only that it is sufficient, but that it is necessary as well. I do not see why this is the case - it seems that the action could vary from frame to frame, but still be minimized in all frames, resulting in the same equation of motion in all frames.

    Is inter-frame invariance of the action necessary, and if so why?

    Thanks very much for any help that you can give.

    -HJ Farnsworth
  2. jcsd
  3. Aug 13, 2014 #2


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    You are completely right! In order for the equations of motion to be invariant around the stationary point, not the action itself must be invariant but only its variation.

    Example: Take the electromagnetic potentials in Coulomb gauge and write down the corresponding action by working in the constraint with a Lagrange multiplier. This action is not Lorentz invariant, but the equations of motion, Maxwell's equations, are!
  4. Aug 14, 2014 #3
    Excellent, thank you very much!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook