# building expression for momentum from stress energy tensor

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)

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.