Energy momentum tensor - off diagonal terms

[LCSphysicist]

Homework Statement

A rod has cross sectional area A and mass per unit length \mu. Write down the stress energy tensor inside the rod, when it is under a tension F.

Relevant Equations

.

Let's arrange the rod's axis parallel to the z axis.

$T_{00} = A/\mu$ (since it represents the energy density)

$T_{03}=T_{30} = \frac{F\sqrt{\mu / F}}{A}$ (It represents the flow of energy across the z direction)

$T_{33} = F/A$ (pressure)
It seems that $T_{33}$ i have got has the wrong sign, and that $T_{03} = T_{30}$ should actually be zero.
i am a little confused on both cases: Where does the sign at $T_{33} = (-F/A)$ comes from? And why are my reasoning involving $T_{30}$ wrogn?

 

[Orodruin]

T00=A/μ (since it represents the energy density)

I hope you mean the other way around …

I did not see any reasoning from you regarding $T_{03}$, just a statement that it is the energy current. Why should the energy current be what you quoted?

Regarding $T_{33}$, consider how T is defined.

 

[LCSphysicist]

I hope you mean the other way around …

I did not see any reasoning from you regarding $T_{03}$, just a statement that it is the energy current. Why should the energy current be what you quoted?

Regarding $T_{33}$, consider how T is defined.

Ok, $T_{00}$ indeed i wrote the inverse of the answer, sorry.
$T_{03} = dp_{3}/dV = \frac{F_3 dt}{dV} = \frac{F_3 dt}{A dz} = \frac{F_3}{A v} = \frac{F}{A \sqrt{F / \mu}}$
And $T_{33}$? Pressure $F/A$ at z direction, no?

 

[Orodruin]

Why do you think $p_3$ is non-zero? Is the rod moving?

 

[LCSphysicist]

Why do you think $p_3$ is non-zero? Is the rod moving?

No, but the waves are propagating, no?

 

[Orodruin]

No, but the waves are propagating, no?

Is there more to the problem than what you wrote in the OP? The OP says nothing about propagating waves.

 

[LCSphysicist]

Is there more to the problem than what you wrote in the OP? The OP says nothing about propagating waves.

No. But since this problem is from a special relativity book, i thought that rigid objects would be meaningless here, and we should consider the deformation, and so the wave.
But i think i got it, i am overthinking the problem, maybe?

 

[Orodruin]

Well, nothing says that the rod is rigid to deformation by new external forces. There just are no such forces mentioned. All I that is mentioned is a rod under tension. Presumably after any motion arising from the application of the forces providing the tension has dissipated and the rod has settled.