and the subscript notation indicates this implicit relationship.

Energy-momentum tensor
----------------------

The radiation flux will be described as an incoherent “null dust" with energy-momentum tensor $$\label{Tmunu}
  T^{\mu\nu}=\Phi^2 p^\mu p^\nu \,,$$ where $\Phi^2(r,\theta)$ scales the radiation energy density. The latter has to be fixed by the conservation law ${T^{\mu\nu}}_{;\nu}=0$ after choosing a certain profile on some surface stretching across the rays; in our case, it is natural to choose the equatorial profile $\Phi^2_{\rm eq}=\Phi^2(r_{\rm eq},\theta\!=\!\pi/2)$. For an incoherent radiation flux, this implies $$0=(\Phi^2 p^\mu p^\nu)_{;\nu}
   =\Phi^2 {p^\mu}_{;\nu}p^\nu+p^\mu(\Phi^2 p^\nu)_{;\nu}
   =p^\mu(\Phi^2 p^\nu)_{;\nu} \;,$$ because the photon congruence is geodesic: ${p^\mu}_{;\nu}p^\nu=0$. Hence for the particular “vertical" flux chosen in the previous section, ${T^{\phi\nu}}_{;\nu}=0$ is satisfied trivially due to $p^\phi=0$ (in the equatorial plane, ${T^{r\nu}}_{;\nu}=0$ also holds automatically, because $p^r=0$ there), while the other components reduce to a single common condition $$\label{Div(Phi2.p)=0}
  (\Phi^2 p^\nu)_{;\nu}=0
  \quad \Longrightarrow \quad
  (\Phi^2)_{,\nu}p^\nu=-\Phi^2 {p^\nu}_{;\nu}$$ which says that the evolution of $\Phi^2$ [*along the photon congruence*]{} is tied to the latter’s expansion ${p^\nu}_{;\nu}$; somewhat more explicitly, $$(\Phi^2)_{,r}\,p^r+(\Phi^2)_{,\theta}\,p^\theta
  =-\frac{\Phi^2}{\sqrt{-g}}
    \left[(\sqrt{-g}\,p^r)_{,r}+(\sqrt{-g}\,p^\theta)_{,\theta}\right].$$ An even more explicit equation follows using equation (\[photon-momentum\]) to substitute for $p^i$. In doing so, one has to realize that the differentiation is performed in a [*general*]{} direction, not just along 