Stress-energy tensor components for a perfect fluid

Question

The stress-energy tensor for a perfect fluid is given by $$T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}-pg_{\mu\nu}$$ where U is the 4-velocity. The matrix components of the SEM are written as $$T_{\mu\nu}=diag(\rho,p,p,p).$$ Will this always be the components of the stress energy tensor despite having different components for the metric in each case or do they change according to the formula?

Answer 1

Consider your stress-energy tensor: $$ T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}-p g_{\mu\nu} $$ This is a legit tensor written in terms of other tensors (the scalar fields $\rho$ and $p$, the metric and the velocity field): the RHS is the constitutive relation for $T_{\mu\nu}$ in terms of your macroscopic fields ($U_{\mu}$, $\rho$...). You can use more complex constitutive relations (possibly involving more macroscopic fields) for dissipative fluids, see e.g. this and this questions and links here.

Note that $\rho$ and $p$ are scalars by construction, because are the physical properties of the matter in the local reference frame comoving with the local matter element (more precisely: this is the rest frame defined by the conserved current in your system, whatever it is, typically the baryon current).

If you want the energy in the local comoving reference frame, you just have to calculate $T_{\mu\nu} U^\mu U^\nu$.

This representation of $T_{\mu\nu}$ is tensorial, meaning that it's always valid (for any spacetime and fluid configuration). The second representation that you're proposing is valid only in the reference frame of the fluid (aka, a frame that is instantaneously comoving with a local fluid element). Locally, in this frame, the velocity is represented by $U^\mu \sim (1,0,0,0)$. Therefore, $T^{00}= \rho$, and so on... (you find the diagonal matrix representation). Again, this is valid for every metric, but in a specific "tetrad" of the fluid element.

See also: Wiki article, Stress-energy tensor for a perfect fluid in GR, Stress-energy tensor. Why this general form?, What is a "perfect fluid", really?.