Momentum Vectors in Bondi coordinates

In the Bondi-Sachs formalism, we can define the notion of ‘retarded’ time via a coordinate transformation of the usual Minkowski metric
$$
d s^{2}=eta_{mu nu} d x^{mu} d x^{nu}=-left(d x^{0}right)^{2}+left(d x^{1}right)^{2}+left(d x^{2}right)^{2}+left(d x^{3}right)^{2},
$$

defining the retarded time as $u = x^0 – r$, where $r = sqrt{left(x^{1}right)^{2}+left( x^{2}right)^{2}+left(x^{3}right)^{2}}$. We can then define Bondi coordintes $(u,r, theta,phi)$ by further performing the spherical coordinate transformations
$$
x^{1} pm i x^{2}=r e^{pm i phi} sin theta, quad x^{3}=r cos theta.
$$

When it comes to a momentum four-vector, we can transform it to the spherical polar coordinate system by taking
$$
p^mu = (E,p^1,p^2,p^3)rightarrow (E, |textbf{p}| cos varphi sin theta, |textbf{p}| sin varphi sin theta, |textbf{p}| cos theta).
$$

Can we also do a transformation to the energy, to define some sort of retarded energy, and if so what is its physical interpretation?

EDIT: It seems as if we can define either retarded or advanced energy, corresponding $omega_+ = E+|textbf{p}|$ or $omega_-=E-|textbf{p}|$. These only both seem valid in for massive momenta, since we can write
$$
omega_+ = frac{E^2 – |textbf{p}|^2}{E-|textbf{p}|} = frac{m^2}{E-|textbf{p}|},~~~~~omega_- = frac{m^2}{E+|textbf{p}|}.
$$
In the massless limit, we find that $E = |textbf{p}|$, assuming the principle square root, meaning $omega_+ = 2E$ and $omega_- = 0$. I’ve no idea what this means, but it does agree with the parameterization in the massless case.