Energy momentum tensor of a perfect fluid with vanishing density? I came across a stress tensor in general relativity that has the form:
$$T_{\mu\nu} = \left[ {\begin{array}{*{20}c}
   0 & 0 & 0 & 0  \\
   0 & w & 0 & 0  \\
   0 & 0 & { - w} & 0  \\
   0 & 0 & 0 & 0  \\
\end{array}} \right]$$
in other words, there are two components of pressure and zero energy density. Is such a thing possible? What would it physically mean? 
 A: Depending on the metric it may violate all or most of the positive energy conditions that have been proposed to insure the universe does not have negative mass. In all (or most) one of the conditions if $\rho$ $>=$ $|p|_i$ for each of the  three pressures.  
Depending on the metric, it could (it does in Minkowski spacetime) violate the strong energy condition, the weak energy condition and the dominant energy condition, used as assumptions in many GR theorems and results as non-physical. 
See the article at 
http://strangebeautiful.com/papers/curiel-primer-energy-conds.pdf
For the many energy conditions that are used, and violations shown in various solutions found that in most cases are deemed as non-physical. You might find some answer there
I do not know specifically what physics your stress energy tensor might represents but with no energy density and opposite pressures in two coordinate directions it may represent orthogonal excitations. It really depends also on the metric, you'd have to define a solution. I think clearly not Minkowski. Not immediately obvious what but I think you have some idea. If not Minkowski and a perturbation with the p's small, see what you get. 
