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4-velocity length under a force

  1. Jan 11, 2010 #1
    4-velocity "length" under a force

    In relativity we have [tex]u^\mu u_\mu=c^2[/tex], which is just another way of saying that [tex]p^\mu p_\mu=m^2c^2[/tex] or [tex]E^2=m^2c^4+p^2c^2[/tex].

    This is relatively (no pun!) easy to see for a free particle. But if we have a vector potential acting on the particle, is the above still true? Or do we have instead that

    [tex](p_\mu +qA_\mu)(p^\mu+qA^\mu)= constant[/tex]

    ?
     
  2. jcsd
  3. Jan 11, 2010 #2
    Re: 4-velocity "length" under a force

    Ok. I think I have it. It is actually

    [tex](p_\mu -qA_\mu)(p^\mu-qA^\mu)= m^2c^2[/tex]

    but p here is the canonical conjugate momentum

    [tex]p_\mu =u_\mu+qA_\mu[/tex]

    so when you put that in, you still get [tex]u_\mu u^\mu=c^2[/tex].

    I'd still love to hear if you experts concur.
     
  4. Jan 12, 2010 #3

    clem

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    Re: 4-velocity "length" under a force

    This is correct. That is how vector potentials affect the otion of particles.
    For a 4-scalar potential S, you would let m-->m+S.
     
  5. Jan 13, 2010 #4
    Re: 4-velocity "length" under a force

    Really? that would be weird. This would be identical to situation in which the particle's mass depended on its position in spacetime--its actual rest mass, not its "relativistic mass".
     
  6. Jan 13, 2010 #5

    clem

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    Re: 4-velocity "length" under a force

    No, m is still the invariant mass(which is equal to the rest mass). It is the energy E that changes with position.
     
  7. Jan 15, 2010 #6
    Re: 4-velocity "length" under a force

    Isn't energy the same as p^0? And isn't that modified already by the 0th component of the vector potential?

    We're going off on a tangent here, but I'm curious.
     
  8. Jan 15, 2010 #7

    clem

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    Re: 4-velocity "length" under a force

    Yes, E=p^0. With 4-vector [tex]A^\mu=(V,{\vec A})[/tex] potential
    and scalar potential S, the equation is
    [tex](E-V)^2=({\vec p}-{\vec A})^2-(m+S)^2.[/tex]
     
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