Perfect fluids and the stress energy tensor

[black_hole]

Homework Statement

"Texbooks that describe perfect fluids are often a little unclear about what is being assumed. It may not be immediately obvious why can't the pressures be different in different directions? Let's examine this. Suppose T^αβ = diag(ρ,(1+ε)P,P,P) . Show that if one performs a rotation around the z axis by an angle phi that the stress energy tensor picks up off diagonal components of order εP."

What rotation matrix should I be using, it's coordinate dependent isn't it?

Homework Equations

The Attempt at a Solution

 

[Dick]

Homework Statement

"Texbooks that describe perfect fluids are often a little unclear about what is being assumed. it may not be immediately obvious why can't the pressures be different in different directions? Let's examine this. Suppose T^αβ = diag(ρ,(1+ε)P,P,P) . Show that if one performs a rotation around the z axis by an angle phi that the stress energy tensor picks up off diagonal components of order εP."

What rotation matrix should I be using, it's coordinate dependent isn't it?

Homework Equations

The Attempt at a Solution

Since they say rotation about the z axis by an angle phi, I think you can assume the spatial coordinates are cartesian. Just work with those.