In the Newmann-Penrose formalism , Weyl scalar $\psi_{4}$ is given by $\psi_{4} = C_{a b c d} n^{a} \bar m^{b} n^{c} \bar m^{d}$, where $C_{a b c d}$ are weyl tensor components and $n^{a}$, $\bar m^{b}$, $n^{c}$, $\bar m^{d}$ denotes null tetrads. If we are dealing with gravitatinal waves or gravitational radiation near null infinity, we expand $\psi_{4}$ in terms of spin weighted spherical harmonics having spin weight -2. If I am right, this is because $\psi_{4}$ has a spin weight -2. Why it is so ? Can anyone give an explanation or proof of why $\psi_{4}$ has a spin weight -2 ?
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The spin weight encodes how a quantity transforms under a rotation of the tetras leaving $l^a$ and $n^a$ invariant. Because $\psi_4$ is contains two factors of $\bar{m}^a$, one easily infers its spin weight to be -2.
Alternatively, you can derive the Teukolsky equation for $\psi_4$ and confirm that its angular part is the differential equation for spin weight -2 spheroidal harmonics.
