# Stress-energy tensors in GR

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?

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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.

Dale
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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.

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,

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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.
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.

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,

That is certainly about a perfect fluid disk.

Dale
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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.
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.

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.
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.

Dale
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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.

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 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.

Dale
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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.

Dale
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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.
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.

PeterDonis
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there are many possible metrics whose Eintein tensors and therefore their SETs have off-diagonal nonzero components.
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.

I was rather trying to find examples of those that are physically plausible and pass some realistic energy condition.
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.

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

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.
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.

Dale
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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
Yes, that is what they showed.

Dale
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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.
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.

PeterDonis
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2019 Award
this is AFAIK the most general form possible for the static interior Schwarzschild metric.
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.

Dale
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However, one can express the functions in different ways
Defenitely. And selecting the best form for a given application is quite an art.

There may be other possibilities as well; those are the ones that I've seen.
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.

PeterDonis
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I would appreciate any references you have using the others.
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.

PeterDonis
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But don't shear components appear when there is no isotropic pressure?
Not if the configuration is static, no. Shear implies relative motion of some parts of the object relative to other parts, and that's not possible in a static configuration.

If you look at the EFE components, they can be solved consistently with radial stress different from tangential stress (but the tangential stress must be the same in all tangential directions for spherical symmetry), but no off-diagonal SET components. The different radial vs. tangential stress just adds an extra term to the equation for ##dp / dr##; it doesn't add any further nonzero EFE components. See, for example, my PF blog post here:

https://www.physicsforums.com/blog.php?b=4149 [Broken]

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Dale
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You can have shear stress in a static configuration. The reason that it is diagonal is simply because the coordinate axes also happen to be the principal stress axes. If you have unequal principal stress in one coordinate system then you will have shear stress in other coordinate systems, and conversely for any configuration with shear stresses there exists a coordinate system where the stress tensor is diagonalized.

Physically, shear is the same as anisotropic pressure. It is just a matter of rotation to go from one to the other.

EDIT: at least, that is what I learned in my mechanical engineering courses

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PeterDonis
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Physically, shear is the same as anisotropic pressure. It is just a matter of rotation to go from one to the other.
Ah, yes, you're right, if the pressure is not isotropic then the SET will only be diagonal in a particular frame (in the cases we've been discussing, that will be the frame in which the matter is static and, as you say, the spatial axes are aligned with the principal stresses).

Ah, yes, you're right, if the pressure is not isotropic then the SET will only be diagonal in a particular frame (in the cases we've been discussing, that will be the frame in which the matter is static and, as you say, the spatial axes are aligned with the principal stresses).
How s this compatible with the spherical symmetry of the paper's static sphere? It looks as if it were not feasible to have anisotropy, no shear and angular invariance at the same time.
In the paper they remark that the conclusión only followed "if one accepts that the [static sphere's] radial pressure can be different from its transverse pressure." As if they were aware this is not easily acceptable.

It would be surprising that having available an imperfect fluid solution without pressure singularity and therefore no mass limit, all sources used the perfect fluid one from wich the Tolman–Oppenheimer–Volkoff limit for neutron stars is derived, no?

Dale
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It looks as if it were not feasible to have anisotropy, no shear and angular invariance at the same time.
You are correct, it is not feasible to have anisotropic pressure and no shear because anisotropic pressure is shear. The only difference is a spatial rotation.

This is assuming that the spatial-spatial part of the stress-energy tensor is the same as the stress tensor from mechanics.

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