- #1
binbagsss
- 1,326
- 12
Homework Statement
Show that ##d^4k## is Lorentz Invariant
Homework Equations
[/B]
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.
The Attempt at a Solution
[/B]
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.