How to obtain the transformation of Null Tetrad ?

[Karmerlo]

Currently, I meet with the so-called null rotation in my study. I cannot understand why it has a mathematical form like that? Is there anyone familiar with this? Can anyone give a lucid explanation of it or provide steps to derive it.

See the image above on the null transformation (in Newman-Penrose Formalism)

Thanks.

 

[Bill_K]

These are all the transformations that preserve the orthonormality relations among k, ℓ, m. For example, ask yourself, what tetrad rotation preserves ℓ? The most general form is

ℓ → ℓ
m → m + α k + β ℓ
k → k + γ ℓ + δ m + ε m*

and try to determine the coefficients:

k = k* ⇒ δ = ε*
ℓ·m = 0 ⇒ α = 0
k·m = 0 ⇒ β - ε
k·k = 0 ⇒ γ - ε ε*