# Show that d^4k is Lorentz invariant

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1. May 15, 2017

### binbagsss

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?

2. May 15, 2017

### DuckAmuck

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

3. May 15, 2017

### binbagsss

huh? using the relevant equations please?
don't think i'd get the marks if I plucked something out of the air ...