# Stress-energy tensors in GR

1. Feb 9, 2014

### TrickyDicky

I'm trying to find examples of stress-energy tensors from exact solutions of the EFE corresponding physically to matter-that leaves out all vacuum solutions(including electrovacuum and lambdavacuum) and pure radiation(null dust)-, I'm finding hard to find any other than the usual SET from perfect fluids, are there no stress-energy tensors describing matter that are not perfect fluids?

2. Feb 9, 2014

### Mentz114

There are solutions for rings, disks and shells of matter, but I'm not sure if they include interiors or are vacuum solutions. I'll have a look round.

3. Feb 9, 2014

### Staff: Mentor

With solid matter you can get pretty arbitrary SET since you can have shear stress. The most general form of Schwarzschild interior solutions can give you different radial and tangential stresses inside the matter.

4. Feb 9, 2014

### Mentz114

The RHS of the EFE for a matter solution will always contain at least a non-zero $T_{00}$, which describes static matter. The term 'perfect fluid' covers many configurations. I suppose the decomposition $T_{00}=\mu u_0u_0$ must exist if we want to describe this as perfect fluid. The density $\mu$ can be a distribution, and for discrete bodies some kind of localising function. I don't think any such solutions exist.

This is a good (earlyish) paper about disks,

Last edited: Feb 9, 2014
5. Feb 9, 2014

### TrickyDicky

All the Schwarzschild interior solutions are spherically symmetric perfect fluids AFAIK.

Can you point me to any exact solution with a SET that has shear stresses?
Theoretically in idealized conditions it is certainly posible or that is the conclusión I draw from MTW's exercise 22.7, but I can't find any examples.

6. Feb 9, 2014

### TrickyDicky

That is certainly about a perfect fluid disk.

7. Feb 9, 2014

### Staff: Mentor

Equation 1 here (http://arxiv.org/abs/gr-qc/9903007) is an example. It is the general form of a static spherically symmetric metric. As you can see in equation 5 the radial and transverse pressures can differ, in which case shear stresses are present.

8. Feb 10, 2014

### TrickyDicky

Thanks, that's a really interesting example.
In the conclusión it is asserted that" there is a regular static configuration of the static sphere whose radius R can approach the corresponding horizon size arbitrarily if one accepts that its radial pressure can be different from its transverse pressure "(i.e. an imperfect fluid example as I was seeking).
The problema is that its premise is precisely the acceptance of such a fluid in the interior of stars, wich is of course theoretically posible, there are many possible metrics whose Eintein tensors and therefore their SETs have off-diagonal nonzero components. I was rather trying to find examples of those that are physically plausible and pass some realistic energy condition.

In this particular case I don't know if this imperfect fluid SET is physically plausible, one remarkable feature(if I interpreted correctly the conclusión) seems to be that the matter described by such imperfect fluid cannot be made to collapse. I doubt that this is considered plausible in the mainstream view.

9. Feb 10, 2014

### Staff: Mentor

I agree with that, I wouldn't consider it a viable model of a fluid, but it certainly is a perfectly acceptable model of a solid. I am not sure what your concern is regarding energy conditions. I would use the same standard energy conditions.

10. Feb 10, 2014

### TrickyDicky

I believe that in this relativistic context the word fluid is used in a more generic manner that doesn't specify the matter phase, and in the astrophysical case is usually applied to plasmas.

11. Feb 10, 2014

### Staff: Mentor

Then I guess I am not sure what your remaining concern is. It is certainly a viable model of a solid (or "imperfect fluid") as long as $\lambda$ and $\eta$ are chosen to fulfill the standard energy conditions.

12. Feb 10, 2014

### Staff: Mentor

Sorry, I need to read more carefully. I believe this is your concern.

In their paper λ and η are free functions of r. If you assume a perfect fluid then they are no longer free, there are constraints such that choosing one fixes the other. Under those constraints there is no solution which is stable where the surface is less than 9/8 R where R is the Schwarzschild radius.

This paper was showing that if you drop that constraint and allow shear stresses then you can get stable configurations where the surface is arbitrarily close to R. They are not claiming that matter cannot be made to collapse, only that there exist non-collapsing solutions arbitrarily close to the horizon.

13. Feb 10, 2014

### Staff: Mentor

The SETs in the paper DaleSpam linked to are diagonal; they just have $T_{22} = T_{33}$ but $T_{11} \neq T_{22}$, i.e., the transverse stresses are different from the radial stress.

The paper doesn't appear to check this, and I suspect that the solutions it is talking about, that can be in static equilibrium with an outer radius arbitrarily close to the horizon radius, will violate at least some of the standard energy conditions. For example, the simplest case, where the radial pressure is zero, gives (from equation 11 in the paper)

$$F(r) = \frac{\rho m}{2 \left( r - 2m \right)}$$

where $F(r)$ is the tangential stress. But this means $F > \rho$ whenever $m > 2 \left( r - 2m \right)$, i.e., whenever $r < (5/2) m$, and of course this will be true for any of the static solutions of interest (the ones with an outer radius smaller than the limit of 9/4 m for a perfect fluid). And $F > \rho$ violates at least one energy condition.

14. Feb 11, 2014

### TrickyDicky

But don't shear components appear when there is no isotropic pressure?

15. Feb 11, 2014

### TrickyDicky

Yes, I'm aware of this. But once anisotropic stresses are allowed it seems like one can always find a configuration of stellar transverse and radial stresses that balances the radial matter pressure with the inward gravitational force, keeping its radius arbitrarily close to the Schwarzschild radius without reaching it, of course as long as the light speed c limit holds wich it should.
AFAIK all models of collapsing matter assume hydrostatic star pressure.

16. Feb 11, 2014

### Staff: Mentor

Yes, that is what they showed.

17. Feb 11, 2014

### Staff: Mentor

I agree. However, my point in posting the paper was not for the content of the paper itself but simply because this is AFAIK the most general form possible for the static interior Schwarzschild metric.

You can choose your λ and η to obtain any possible distribution of matter, including ones representing solids with shear stress, which was TrickyDicky's original desire. While it is true that some values of λ and η will violate energy conditions, it is also true that other values will represent any physically plausible spherically symmetric matter configuration.

18. Feb 11, 2014

### Staff: Mentor

Yes, in the sense that two independent functions are sufficient to generate all possible static, spherically symmetric metrics. However, one can express the functions in different ways: for example, in Schwarzschild coordinates, instead of $\lambda(r)$ one could just as easily use $m(r)$, the mass inside radius $r$, so that the second term in the metric becomes $dr^2 / \left( 1 - 2 m(r) / r \right)$. Or one could choose isotropic coordinates instead of Schwarzschild coordinates, so that the $e^{\lambda}$ term multiplies the entire spatial part of the metric, not just $dr^2$. There may be other possibilities as well; those are the ones that I've seen.

19. Feb 11, 2014

### Staff: Mentor

Defenitely. And selecting the best form for a given application is quite an art.

I would appreciate any references you have using the others. I find that it is helpful to see which forms are chosen for each paper.

20. Feb 11, 2014

### Staff: Mentor

The reference I'm most familiar with is MTW; all three forms appear in their discussion, but the one they use most often is the one in the paper you linked to.