Finding stress energy tensor of a rod under tension

[demonelite123]

I am given a rod with mass per unit length $\mu$, cross sectional area A, and under a tension F. I am also told that the tension F is uniform over the cross sectional area A. Find the stress energy tensor inside the rod.

I know that for the stress energy tensor $T^{00}$ gives the energy density, $T^{0i}$ gives the energy flux through a i-surface, $T^{i0}$ gives the i-momentum density, and $T^{ij}$ gives the i-momentum flux through a j-surface.

I have that $T^{00} = \mu / A$ and that $T^{0i} = T^{i0} = 0$ since the rod is stationary so the particles have no net momentum. As for $T^{ij}$, if i assume that the rod is lying lengthwise along the x axis, then $T^{11} = F/A$ since the momentum flux is the momentum per unit time per unit area or in other words the force per unit area. Then it seems to me that $T^{ij} = 0$ for all other entries since the tension acts along the x direction.

Therefore the stress energy tensor only has 2 nonzero entries on the main diagonal, and the rest of the entries are 0. Am I missing something here? It seems too simple a form to have and I am not sure if i took everything into account.

 

[Nate810]

The stress energy tensor inside the rod is: T^{00} = \mu / A T^{ij} = 0 (for all i,j except for i = j = 1)T^{11} = F/A