A Two questions on the Stress-Energy Tensor

1. Jun 8, 2017

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

2. Jun 8, 2017

Orodruin

Staff Emeritus
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.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted