Stress Energy Tensor in Wald's General Relativity - Page 96

[latentcorpse]

This is page 96 in Wald's "General Relativity"

He says that the stress energy tensor of the ordinary matter in the universe is of the form

$T_{ab}=\rho u_a u_b$ where $\rho$ is the density of matter in the universe.

why does it take this form? also are $u_a$ and $u_b$ orthogonal vectors in the plane orthogonal to the tangent of the world line of an observer at that point?
or am i completely missing the point there?

anyway he then goes on to talk about how a thermal distribution of radiation at a temperature of about 3K fills the universe and for massless thermal radiation $P=\frac{\rho}{3}$ (where is this from - I've tried to find a formula for radiation pressure but can't find one). Anyway whilst, presently this doesn't contribute to the stress energy of the universe it was the dominant term in the beginning and so we should take $T_{ab}$ to be of the perfect fluid form

$T_{ab}=\rho u_a u_b +P (g_{ab} + u_a u_b)$
what is meant by the perfect fluid form?
where does the $P (g_{ab} + u_a u_b)$ term come from?

it says earlier (the previous page) that $h_{ab}(t)=g_{ab} + u_a u_b$ is the metric of either a sphere, flat Euclidean space or a hyperboloid on the surface $\Sigma_t$. does this help with understanding the above...

 

[otaniyul]

hi! wald is too much concise in that chapter. its chapter 5, right?

dude, i had studied it ealier, i will be brief and then you ask what is not yet clear later, ok?

yes, the vectors "u" are timelike vectors tangent to the hypersurface. the point here is to grab the definitions given. wald states that he is considering a spattialy homogenous and isotropic spacetime, and defines it as a spacetime folliated by homogenous and isotropic riemannian hypersurfaces. in doing so, he has stated that THERE IS A PREFERED COORDINATE SYSTEM, in which you take the integral lines of the vector field "u" as the curves of isotropic observers. elsewhere, your spacetime will obviously not be foliated in the right manner.

you can ellaborate on the form of the stress tensor then. you can conclude that the strees-energy tensor is like that taking only into account the symmetries of space time (the stress-energy tensor must have all the symmetries, since einstein's equation links that tensor with the geometry of the manifold). restricted to the hypersurface, it will be proportional to the metric, for the hypersurface is maximally symetric. the later interpretations of P and Rho come later with comparison to a distribution of particles. you can check it on wald chapter 4. the constriction 0 < P < rho/3 comes from the strong positivity condition for the stress energy tensor. The RHO and P are not given and don't arise from those symmetry statements alone, you must take a model into account for further calculations (wald does that later, for P = 0 and P = rho/3).

the riemannian metric h is stated later to be from a hyperboloid, flat space or sphere because those are the only maximmaly symetric 3d spaces, i.e. constant curvature space. wald gets into the constant spatial curvature using some hard to see passages...

I tried to study that same chapter a year ago, but without knowing too much gravitaion and geomtry, and i was also stuck. I will give you a sugestion. Get Weinberg's gravitation and cosmology, read the Chapter 13, on Symmetric Spaces, and some passages on the following cosmolody parts. That will be of much enlightment!

i hope it helped a bit...