A Two questions on the Stress-Energy Tensor

[zenmaster99]

I am trying to get a good feel for the Stress-Energy tensor, but I seem to be hung up on a few concepts and I was wondering if anyone could clear up the issues.

First, when I look at the derivation of the Stress-Energy tensor for a perfect fluid (of one species, say), the 00 entry can be written as a sum over each "particle" in the fluid:
$$T^{00}=\sum_kmu_k^0u_k^0=m\sum_k\gamma_k^2=m\sum_k\left(1+{v_k^2\over1-v_k^2}\right)=m\sum_k\left(1+\gamma_k^2v_{k,x}^2+\gamma_k^2v_{k,y}^2+\gamma_k^2v_{k,z}^2\right)$$
$$=\sum_km+\sum_kmu_k^1u_k^1+\sum_kmu_k^2u_k^2+\sum_kmu_k^3u_k^3$$
In the rest frame of the bulk, I can divide by a small 3-volume element and, assuming things are isotropic, I find
$$T^{00}=\rho_0+3P$$
where $\rho_0$ is the rest-mass density and $P$ is the pressure. Is this correct? If so, is it useful? Clearly, I can't use this for photons.

My second question involves the Lorentz scalar ${T^\alpha}_\alpha$. Setting $T^{00}\equiv\rho$, when I actually find this using $T^{\alpha\nu}\eta_{\nu\alpha}$, I get $-\rho+3P$ instead of $+\rho+3P$. What am I doing wrong?

I ask this because I'm guessing that this scalar $T$ must be a useful quantity, but I don't know what, exactly. It's implicated in the Trace Energy Condition... but if I'm getting a negative value for a dust ($P=0$), then I can't possibly satisfy $T\ge0$.

Any and all insight is greatly appreciated. Much thanks in advance.

ZM

 

[Orodruin]

Clearly, I can't use this for photons.

For a photon gas, $T^{00}=3p$ in the rest frame, so the expression has the correct limiting behaviour.

The sign in the trace condition depends on your metric convention.