Lorentz Invariant Volume Element

[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.

 

[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|.