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Show d^4k is Lorentz Invariant

    • Thread Starter

    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, more generally, 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, if this is the first step required ?and how do I go about it? Many thanks in advance.

    (No time dilation, length contraction argument please, I need to make use of what is given in the question - thank you).

    (Original post by xfootiecrazeesarax)
    Show that d^4k is Lorentz Invariant
    I'm not sure what you mean by d^4k - is this working in momentum space or something?

    Anyway, I've seen this done for the usual 4-volume element by Jacobians. You have under a Lorentz transformation:

    d^4 x'= dx' dy' dz' dt' = |\mathcal{L}| dx dy dz dt

    where |\mathcal{L}| is the determinant of the Jacobian of the Lorentz matrix. So if you can show that this is 1, then the volume elements are the same in both coord systems and I don't think that is too tricky, though I'm not sure I've ever done it myself.
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Updated: May 19, 2017
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