Diagonal Stress-Energy Tensor What does it mean if the stress-energy Tensor ($T_{ij}$) is a diagonal matrix? (i.e. the only nonzero components are density energy $T_{00}$ and pressure $T_{11}$, $T_{22}$, $T_{33}$).
What can we say about this kind of universe?
ADD:
Where the density energy is constant and pressure is not constant.
 A: To write down the components of the stress-energy tensor you need to choose a coordinate system, and the values of the components will depend on what coordinate system you choose. So when you ask:

What can we say about this kind of universe

the question is really:

What can we say about this coordinate system

And the simplest interpretation is that your coordinates are comoving i.e. the average four velocity of the stuff in the universe in your coordinate system is $(c,0,0,0)$. I say average velocity because in the case of a gas or dust the individual molecules/particles may have non-zero spatial velocities but if these are random they will average to zero on a larger scale.
To understand this I recommend you read my answer to Intuitive understanding of the elements in the stress-energy tensor. This explains how to understand the stress-energy tensor starting with the stress-energy tensor for a point particles:
$$ T^{\alpha\beta}({\bf x},t) = \gamma m v^\alpha v^\beta $$
(where $v$ is the coordinate velocity, not the four velocity, and $v^0=c$).
If the velocities are random then all the $v^\alpha v^\beta$ terms will average to zero apart from the diagonal terms so we get a diagonal stress-energy tensor.
There's no special reason why either the energy density or pressure terms have to be constant in either space of time, but not that if they vary in space then the matter/energy distribution is likely to evolve with time into a form where the stress-energy tensor is no longer diagonal i.e. the stuff in your universe will acquire non-zero peculiar velocities.
The obvious example of a universe where the stress-energy tensor is diagonal in comoving coordinates is the FLRW universe, which approximately describes our universe.
