What are the Maxwell equations of motion in retarded Bondi coordinates? I'm reading a paper about asymptotic symmetries at null infinity in electrodynamics. There, they had the following calculation:
The Maxwell equations $\nabla^{\nu} F_{\mu\nu} = J_{\nu}$ written in retarded Bondi coordinates are
$J_u = (\partial_u - \partial_r) F_{ru} - \frac{2}{r} F_{ru} - \frac{1}{r^2} \gamma^{AB} D_A F_{Bu}$
$J_r = \partial_r F_{ur} + \frac{2}{r} F_{ur} \frac{1}{r^2} - \gamma^{AB} D_A F_{Br}$
$J_C = (\partial_u - \partial_r) F_{rC} + \partial_r F_{uC}+  \frac{1}{r^2} - \gamma^{AB} D_A F_{BC}$
with $\gamma^{AB}$ as the components of the metric of $S^2$. The metric in retarded Bondi coordinates is $ds^2 = du^2 + 2dudr - r^2 \gamma_{AB} d\theta^A d\theta^B$ and $u=t-r$.
I'm in particular confused about the u and r components of the divergence. I don't get to the expression above.
 A: The u is the retarded time and r is a constant of the Minkowski space-time tensor. Remember they are simply harmonic coordinates and you have to treat them like separate systems of equations, since they are both hyperbolic and hyper surfaces. It is a wave equation with geometric units of 1. I recommend looking at the BMS group which has contra-variant metric equations. All you have to understand really is the r represents the root of the partial derivatives of the x,y,z planes. The reason they diverge is because null infinity or aleph infinity is cardinal only to a maximum metric space, in layman's terms, it can't go above 1, since the geometric unit is equal to 1. Hence it is forced to slowly diverge and converge around a fixed point, leading to a geodesic space curve, and the point at which this all occurs is hence known as the Bondi Coordinates. Same thing applies in electrodynamics, since the Klein-Gordon equations give the Bondi mass approximations, you can find the fixed point at which the divergence occurs. Its a bit like Riemann's zeta function to be honest!
