Understanding Vaidya metric and pure radiation stress-energy tensor

Question

I am following Vaidya metric and how it is related to pure radiation from Wikipedia.

But when it reaches the line where stress-energy tensor is equated to product of two four-vectors, I cannot follow from where they are assumed to be null vectors, and why if the stress-energy tensor is given in terms of null vectors, it must be related to the energy of massless particles, or alternatively to particles with relativistic velocities, both of which are definitions of radiation.

What should be the components of stress-energy tensor in a given set of coordinates to say that it is related to pure radiation?

Answer 1

If one generalizes the Schwarzschild metric to $m=m(u)$ instead of $m=\text{const.}$, where $u$ is a lightlike coordinate, the only non-vanishing component of the stress energy tensor is precisely $T_{uu}$, meaning there is no matter flux across $u=\text{const.}$ lines. For the outgoing Vaidya solution, consider $u$ constant on outgoing light rays. Then, a light ray escaping to infinity on $u=u_1$, feels a Schwarzschild spacetime with an effective mass $m(u_1)$.

So long as the stress-energy tensor can be decomposed as $T_{ab}= T(x^a)\, \delta^u_a\delta^u_b$ everywhere, then it is related to (your definition of pure radiation).

Some solutions of massless fields also admit lightlike quadri-velocites, but not necessarily everywhere. As an example, I refer you to a solution to a massless scalar field, in which the stress-energy tensor is not always related to lightlike quadri-velocity. (http://iopscience.iop.org/article/10.1088/0264-9381/11/5/012/pdf)