Question

Τμ. = Σον δ(x – x (1)) = Σ c2P,"P δ(x – x(t)) = Τυμ, (7.19) E. η 1

Answer 1

Given the energy momentum tensor for the perfect fluid,
$T^{\mu\nu}=(\epsilon +P)\frac{U^\mu U^\nu}{c^2}-Pg^{\mu\nu}$
(i)
Contracting the above expresion with the metric, we get
  $T=g_{\mu\nu}T^{\mu\nu}=(\epsilon +P)\frac{g_{\mu\nu}U^\mu U^\nu}{c^2}-P g_{\mu\nu}g^{\mu\nu}$
And by using the fact the 4-velocity by definition has the norm
  $g_{\mu\nu}U^\mu U^\nu=c^2$
And also in four dimensions

Jug

So, we get
$T=(\epsilon +P)-4P$
  $\Rightarrow T=\epsilon -3P$

(ii)
In the local inertial frame, in which the fluid is at rest, the energy density $\epsilon$ is simply given by
  $\epsilon=\rho c^2$
And also, in the local frame, the 4-velocity is given by
$U^{\mu}\equiv (c,0,0,0)$
And locally, we have
  $g^{\mu\nu}\equiv \eta^{\mu\nu}=\text{diag}(1,-1,-1,-1)$
And so, we get in the local inertial frame,
$T^{\mu\nu}=(\rho c^2+P)\delta^{\mu}_{0}\delta^\nu_{0}-P\eta^{\mu\nu}$
And so, we get
  $T^{\mu\nu}=\text{diag}(\rho c^2,P,P,P)$
So, in this frame, the trace of the stress-tensor is given by
$T=\eta_{\mu\nu}T^{\mu\nu}=\rho c^2-P-P-P$
  $\Rightarrow T=\rho c^2-3P$
This is not exactly same as the answer (i). But, it is almost same in a sense only the energy density is given by the local energy density $\epsilon=\rho c^2$ .

(iii)
The stress-energy tensor of a system of particles as given in equation (7.19) in the question
$T^{\mu\nu}=\sum_{n}c^2 \frac{p_n^{\mu}p_n^{\nu}}{E_{n}} ~\delta(x-x_{n}(t))$
So, the trace in this is case means
$T=\eta_{\mu\nu}T^{\mu\nu}=\sum_{n}c^2 \frac{\eta_{\mu\nu} p_n^{\mu}p_n^{\nu}}{E_{n}} ~\delta(x-x_{n}(t))$
Now, for photons, the norm of the momentum is zero (i.e., photons are massless), i.e.,
  $\eta_{\mu\nu} p_n^{\mu}p_n^{\nu}=0$
And so, we get
$T=0$
Now, we use the result obtained in part (ii) combined with the above traceless condition (T=0) to get for photons as a perfect fluid
  $T=\rho c^2-3P=0$
$\Rightarrow \rho =\frac{3P}{c^2}$
This is the equation of state, i.e., the energy density as a function of the pressure.