Understanding Vaidya Metric & Pure Radiation Stress-Energy

[victorvmotti]

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?

 

[pervect]

I'd suggest understanding the null dust solution first. There's a Wiki article , unfortunately it's not that understandable.

You can start by imagining the stress-energy tensor of the solar wind, a stream of charged massive particles emitted by the sun. The solar wind consists of massive particles, however. To get to the null dust solution, you need to take the limit where the emitted particles having negligible rest mass. When you do this, you have a null dust, which streams outwards from the star at the speed of light. The stress energy tensor will be $T^{ab} = \rho \, u^a \, u^b$, formally the same as that of a pressureless perfect fluid with zero pressure, however u is a null vector rather than a timelike vector. In other words, the "velocity" of the fluid is c, the speed of light.

The PF thread https://www.physicsforums.com/threads/in-simplest-terms-what-is-null-dust.349020/ has some references that might be helpful.

Back to the Vaidya metric. It's basically the metric of a star that's loosing mass by radiation, which is modeled as an outgoing null dust.

Baez's article http://math.ucr.edu/home/baez/einstein/node3.html may also be helpful in understanding the stress-energy tensor as a "flow" of energy and momentum.

These flows are the diagonal components of a 4x4 matrix

called the `stress-energy tensor'. The components $T_{\alpha \beta}$ of this matrix say how much momentum in the $\alpha$ direction is flowing in the $\beta$ direction through a given point of spacetime, where $\alpha, \beta = t,x,y,z$. The flow of [PLAIN]http://math.ucr.edu/home/baez/einstein/img5.gif-momentum in the [PLAIN]http://math.ucr.edu/home/baez/einstein/img5.gif-direction is just the energy density, often denoted $\rho$. The flow of [PLAIN]http://math.ucr.edu/home/baez/einstein/img14.gif-momentum in the [PLAIN]http://math.ucr.edu/home/baez/einstein/img14.gif-direction is the `pressure in the http://math.ucr.edu/home/baez/einstein/img14.gif direction' denoted $P_x$, and similarly for y and z. It takes a while to figure out why pressure is really the flow of momentum, but it is eminently worth doing. Most texts explain this fact by considering the example of an ideal gas.

 

[victorvmotti]

Thanks very helpful.

How, presuming the metric, can we show that stress energy tensor is given by that equation and why the four velocity vector is nulll not timelike?

 

[pervect]

If you presume the metric, you can calculate the Einstein tensor $G_{ab}$ and use the fact that $T_{ab}$ is proportional to $G_{ab}$. But that's rather backwards.

There are several derivations of the stress-energy tensor of a swarm of particles, MTW has a derivation (you can find it online in google books), there's also one from MIT at http://web.mit.edu/edbert/GR/gr2b.pdf.

If you look at the MIT derivation, the result I mentioned before is in equation 19.

You should already know why the 4-velocity of light is a null vector. Another name for a "null interval' is "lightlike interval". I'm not sure offhand where to find a reference for something basic, this is from special relativity.