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Proof of invariance of dp1*dp2*dp3/E

  1. Oct 31, 2008 #1

    ala

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    If we introduce four-dimensional coordinate sistem with component of four-momentum on axis, then dp1*dp2*dp3 can be considered as zeroth component of an element of hypersurface given by p^2=m^2*c^2. Element of this hypersurface is parallel to 4-vector of momentum so we have that dp1*dp2*dp3/E is ration of components of two parallel vectors so it must be invariant under Lorentz transformation. (this proof is from Landau, Lifgarbagez - Fields)
    This proof seems ok but I can't see why I cannot apply it to show (wrongly) that dxdydz/t is invariant.

    Thanks in advance.
     
  2. jcsd
  3. Oct 31, 2008 #2

    clem

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    Science Advisor

    The proof for momentum and energy uses the constraint E^2=p^2+m^2.
     
  4. Nov 1, 2008 #3

    ala

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    Yes, that is equivalent to p^2=m^2*c^2 (p is 4-vector of momentum). So for x, I can watch hypersurface given by equation x^2=c*\tau (x is 4-vector, and \tau proper time). So the rest is same and conclusion is worng (namely that dxdydz/t is invariant)
     
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