Energy momentum tensor - off diagonal terms

In summary: 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.Ok, I get it.
  • #1
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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?
 
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  • #2
Herculi said:
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.
 
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  • #3
Orodruin said:
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?
 
  • #4
Why do you think ##p_3## is non-zero? Is the rod moving?
 
  • #5
Orodruin said:
Why do you think ##p_3## is non-zero? Is the rod moving?
No, but the waves are propagating, no?
 
  • #6
Herculi said:
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.
 
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  • #7
Orodruin said:
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?
 
  • #8
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.
 
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1. What is the energy momentum tensor?

The energy momentum tensor is a mathematical object that describes the distribution of energy and momentum in a given space. It is a key concept in the theory of general relativity and is used to describe the curvature of spacetime.

2. What are the off diagonal terms in the energy momentum tensor?

The off diagonal terms in the energy momentum tensor represent the flow of energy and momentum between different points in space. These terms are important for understanding how energy and momentum are distributed and transferred in a given system.

3. Why are the off diagonal terms important?

The off diagonal terms in the energy momentum tensor are important because they provide information about the interactions and exchanges of energy and momentum between different parts of a system. They can also reveal asymmetries or imbalances in the distribution of energy and momentum.

4. How are the off diagonal terms calculated?

The off diagonal terms in the energy momentum tensor are calculated using the stress-energy tensor, which is a mathematical object that describes the energy and momentum density of a system. The off diagonal terms are obtained by taking the cross product of the momentum vector with the energy flux vector.

5. What is the significance of the off diagonal terms in physics?

The off diagonal terms in the energy momentum tensor have significant implications in physics, particularly in the study of general relativity and cosmology. They are used to describe the behavior of matter and energy in the universe, and can help us understand the dynamics of systems such as black holes and gravitational waves.

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