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I'm getting a rather crazy looking result, but I'm beginning to think it may be right.
Unfortunately, I haven't been able to find any specific references on the topic to check my sanity level.
Basically, I'm finding that in relativistic terms, there are no pressure (or tension) terms in the stress-energy tensor of a rotating disk. (Perhaps I should say - there are not necessarily any such terms).
This is different from the engineering result. But I believe that the difference is due to the fact that in engineering, the stress-energy tensor is taken to be comoving with the disk. i.e:
http://en.wikipedia.org/wiki/Stress-energy_tensor
[tex]
\begin{array}{cccc}
rho(r) & 0 & p(r) & 0\\
0 & 0 & 0 & 0\\
p(r) & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
\end{array}
[/tex]
rho(r) and p(r) are two arbitary functions, representing the energy density and the momentum density.
Rigidity of the disk will impose a relationship between p and rho, uniformity of the disk will give us another constraint.
If this is NOT correct, then my understanding of the continuity equation [itex]\nabla_a T^{ab}[/tex] is wrong and needs to be fixed. The above stress-energy tensor satisfies the above equation - adding any radial tension terms would spoil this happy state of affairs.
Comments? References? Brickbats?
Unfortunately, I haven't been able to find any specific references on the topic to check my sanity level.
Basically, I'm finding that in relativistic terms, there are no pressure (or tension) terms in the stress-energy tensor of a rotating disk. (Perhaps I should say - there are not necessarily any such terms).
This is different from the engineering result. But I believe that the difference is due to the fact that in engineering, the stress-energy tensor is taken to be comoving with the disk. i.e:
http://en.wikipedia.org/wiki/Stress-energy_tensor
If we adopt a cylindrical coordinate system (t,r,theta,z) the stress-energy tensor is justWarning: In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress-energy tensor in the comoving frame of reference. In other words, the stress energy tensor in engineering differs from the stress energy tensor here by a momentum convective term.
[tex]
\begin{array}{cccc}
rho(r) & 0 & p(r) & 0\\
0 & 0 & 0 & 0\\
p(r) & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
\end{array}
[/tex]
rho(r) and p(r) are two arbitary functions, representing the energy density and the momentum density.
Rigidity of the disk will impose a relationship between p and rho, uniformity of the disk will give us another constraint.
If this is NOT correct, then my understanding of the continuity equation [itex]\nabla_a T^{ab}[/tex] is wrong and needs to be fixed. The above stress-energy tensor satisfies the above equation - adding any radial tension terms would spoil this happy state of affairs.
Comments? References? Brickbats?
Last edited: