Newman--Penrose scalar Psi4 (${\Psi }_4$)

For a null tetrad $(\ell ,k,m,\bar{m})$, one common curvature-sign convention defines ${\Psi }_4=C_{\mu \nu \rho \sigma }{k}^{\mu }{\bar{m}}^{\nu }{k}^{\rho }{\bar{m}}^{\sigma }$. In an outgoing wave zone it encodes the two transverse gravitational wave polarizations as one complex scalar.