# Lorentz Invariant Volume Element

1. Oct 15, 2011

### Spriteling

So, the upper light cone has a Lorentz invariant volume measure

$dk =\frac{dk_{1}\wedge dk_{2} \wedge dk_{3}}{k_{0}}$

according to several sources which I have been reading. However, I've never seen this derived, and I was wondering if anyone knew how it was done, or could point me towards a source for it.

Cheers.

2. Oct 15, 2011

### Bill_K

The easiest way is to use the identity δ(ξ2 - a2) ≡ (1/2a)[δ(ξ+a) + δ(ξ-a)].

If you consider a 4-d integral ∫a(k) d4k which is a Lorentz invariant expression and apply it to a function concentrated on the light cone, a(k) = b(k) δ(k2), where of course k2 = k02 - |k|2, you get

∫b(k) δ(k2) d4k = ∫+b(k) (1/2k0) d3k + ∫-b(k) (1/2k0) d3k

where the first integral is over the future light cone, k0 = |k|, and the second integral is over the past light cone, k0 = -|k|.