1. Not finding help here? Sign up for a free 30min 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!

Show that d^4k is Lorentz invariant

  1. May 15, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that ##d^4k## is Lorentz Invariant

    2. Relevant equations

    Under a lorentz transformation the vector ##k^u## transforms as ##k'^u=\Lambda^u_v k^v##
    where ##\Lambda^u_v## satisfies ##\eta_{uv}\Lambda^{u}_{p}\Lambda^v_{o}=\eta_{po}## , ##\eta_{uv}## (2) the Minkowski metric, invariant.


    3. The attempt at a solution

    I think my main issue lies in what ##d^4k## is and writing this in terms of ##d^4k##
    Once I am able to write ##d^4k## in index notation I might be ok.
    For example to show ##ds^2=dx^udx_u## is invariant is pretty simple given the above identities and my initial step would be to write it as ##ds^2=\eta_{uv}dx^udx^v## in order to make use (2).
    I believe ##d^4k=dk_1 dk_2 dk_3 dk_4##?
    For example given a vector ##V^u = (V^0,V^1,V^2,V^3)## I don't know how I would express ##V^0V^1V^2V^3## as some sort of index expression of ##V^u## (and probably I'm guessing the Minkowski metric?). I would like to do this for ##d^4k##.
    Is this the first step required and how do I go about it?

    Many thanks in advance.
     
  2. jcsd
  3. May 15, 2017 #2
    Length contraction and time dilation "cancel" in a volume element.

    For simplicity, have the moving frame be moving along the x axis.
    dx = gamma dx'
    dt = dt' /gamma
    dy = dy'
    dz = dz'

    dxdydzdt = dx'dy'dz'dt' gamma/gamma = dx'dy'dz'dt'

    Same applies to momentum space with k1,k2,k3,k4
     
  4. May 15, 2017 #3
    huh? using the relevant equations please?
    don't think i'd get the marks if I plucked something out of the air ...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Show that d^4k is Lorentz invariant
  1. Lorentz Invariance (Replies: 15)

  2. Lorentz invariance (Replies: 1)

  3. Lorentz invariants (Replies: 14)

Loading...