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Let's say I have a stress-energy tensor with the the following non-zero components: the diagonal components $T^{00}, T^{11}, T^{22}, T^{33} $ and $T^{10}=T{01}$. I know that the energy density is just equal to $T^{00}$. My problem is, how do I write an expression for pressure if the quantities $T^{11}, T^{22}, T^{33}$ are not equal.

Also, does the fact that there are non-zero off diagonal terms imply that Im dealing with an imperfect fluid (I'm under the impression that its a perfect fluid only if the non-zero components are just the diagonal terms)

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Generally, the idea of a single value for pressure only makes sense when the spatial part of the energy-momentum tensor is proportional to the identity. When those are not equal, the best you can do is to say that the fluid is feeling different pressures in different directions.

Plus, the fact that you have a nonzero $T^{01}$ does not mean that you don't have a perfect fluid, since this component (or equivalently the $T^{10}$ components) are related to the momentum density of the fluid; thus, you are simply looking from a reference frame in which the fluid is moving in the $x$-direction. If you had off-diagonal components in the purely spatial part, on the other hand, then that would mean you didn't have a perfect fluid -- because a perfect fluid by definition does not resist to shear stresses, and that's precisely what the purely spatial components $T^{ij}$ are.

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