# Electromagnetic radiation flux through null infinity

I encountered a problematic statement about electromagnetic radiation and I would be grateful if someone could shed some light on it.

The situation is the following:

Flat Minkowski space in 4D, with coordinates $(u,r,z,\bar z)$, where $(z,\bar z)$ are (complex) angular coordinates (whose exact form is not of interest here), $r$ is the usual radial distance and $u=t-r$ is the retarded coordinate. Future null infinity $\mathscr{J}^+$ is the surface at $r=+\infty$ with coordinates $(u,z,\bar z).$ The gauge is $A_r=0$ everywhere, and $A_u=0$ (on $\mathscr{J}^+$).

The problematic statement is the following:

radiation flux through $\mathscr{J}^+$ is proportional to $\int_{\mathscr{J}^+} {F_u}^z{F_u}_z$,

encountered in the article "New symmetries in massless QED" (here).

So my questions are:

1. How is the integrand related to radiation? The closest thing I can imagine is the $T_{uu}$ component of the stress-energy tensor (without the term $g_{uu}(F)^2=g_{uu}(B^2-E^2)$ using the approximation $B^2\sim E^2$. But, again, I can't manage to link it to radiation.

2. It looks to me that we're missing the volume element of the surface, which turns out to be $r^2 \gamma_{z,\bar z}$ with $\gamma$ a function of $(z,\bar z)$. Since this integral is later asked to be nonzero and finite to impose some asymptotic behaviour on potentials $A$, the $r^2$ should be important here. Am I right on this?

I would be very grateful for any help on this. This is my first question so forgive me if I've missed something. Suggestions are also welcome.

• Good question. Welcome to the site! – Emilio Pisanty Aug 2 '16 at 10:32

I think it's solved now. I'll write the reasoning as a reference.

Energy is the conserved quantity associated to invariance under time translation.
In that coordinates I would say that a time translation is represented by a vector parallel to $\partial_u$, since it is $u$ that has the temporal role here.
The conserved current associated to this translation (energy density current) is $T_{\mu \nu} X^{\nu}$ with $X$ parallel to $\partial_u$. Its flux through $\mathscr{J}^+$ gives the energy flux through that surface. The flux is:

$\int_{\mathscr{J}^+} * ( T_{\mu\nu} X^{\nu} dx^{\mu})$

which, using $dr=0$ on that surface, and the form of the metric, is written as

$\int_{\mathscr{J}^+} (-T_{uu} + T_{ur} )r^2 \gamma \, du dzd\bar z$

As in
$-T_{uu} + T_{ur} = -F_{u a}F_{z}^a - F_{ua}F_r^a -\frac{1}{4}(g_{uu}-g_{ru})F^2$
the term with $F_r^a$ is subleading and $g_{uu}=g_{ur}$, one obtains what originally asked for in the question, complete with the volume element that seemed missing from the quoted expression.