Asymmetric stress tensor gives asymmetric stress-energy tensor?

[bcrowell]

I'm sure there's a trivial explanation for this, but it's escaping me.

The space-space components of the stress-energy tensor are interpreted as the 3x3 stress tensor. But WP claims that the symmetry of the stress tensor need only hold in the case of equilibrium:

"However, in the presence of couple-stresses, i.e. moments per unit volume, the stress tensor is non-symmetric. This also is the case when the Knudsen number is close to one, [...] or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers." -- http://en.wikipedia.org/wiki/Stress...m_equations_and_symmetry_of_the_stress_tensor

How can this be reconciled with the Einstein field equation's prediction that T must be symmetric (since the Einstein tensor is symmetric by definition)?

Obviously polymers don't violate GR! Fundamentally, a polymer is made out of nonrelativistic matter, i.e., dust in relativistic parlance, interacting through electromagnetic fields. Dust and EM fields both have symmetric stress-energy tensors.

 

[Dale]

The space-space components of the stress-energy tensor are interpreted as the 3x3 stress tensor. But WP claims that the symmetry of the stress tensor need only hold in the case of equilibrium

The wikipedia entry for the stress energy tensor has a "warning" that may be relevant. I have noticed it before, but I am not certain what it means. Perhaps it will help:
http://en.wikipedia.org/wiki/Stress–energy_tensor#Identifying_the_components_of_the_tensor

Note the warning immediately above the "Covariant and mixed forms" section.

 

[bcrowell]

The wikipedia entry for the stress energy tensor has a "warning" that may be relevant. I have noticed it before, but I am not certain what it means. Perhaps it will help:
http://en.wikipedia.org/wiki/Stress–energy_tensor#Identifying_the_components_of_the_tensor

Note the warning immediately above the "Covariant and mixed forms" section.

Interesting, thanks!

Here's the warning:

Warning: 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.

Most likely this was written by C.H.

I assume "comoving" refers to an inertial frame that is instantaneously comoving with the c.m.? "Momentum convective term" isn't transparent to me; maybe this would refer to momentum transport? (To me, "convective" refers to thermodynamics.) If there is momentum being transported into or out of the system, it's not in equilibrium; equilibrium is one of the conditions stated for the 3x3 stress tensor to be symmetric.

I'm just getting vague glimmers here, still not clear on what the resolution is.

Maybe it has to do with the interaction of the system being described by the stress tensor with an external body, which is what's transporting momentum into it. Then maybe the description of the full system, including the external body, restores symmetry?

 

[Dale]

I'm just getting vague glimmers here, still not clear on what the resolution is.

Same here. Sorry I cannot be more helpful.

 

[Ilmrak]

I don't know if this helps but, if you have translational and rotational symmetry, you can use the angular momentum tensor to symmetrize the (Noether) stress-energy tensor via Belinfante's method (http://en.wikipedia.org/wiki/Belinfante–Rosenfeld_stress-energy_tensor).
The symmetrized tensor turns out to be the right one to couple to geometry in GR.

Ilm

 

[atyy]

Another reference for Ilmrak's solution is Tong's notes Eq 1.44-1.48.

 

[bcrowell]

I don't know if this helps but, if you have translational and rotational symmetry, you can use the angular momentum tensor to symmetrize the (Noether) stress-energy tensor via Belinfante's method (http://en.wikipedia.org/wiki/Belinfante–Rosenfeld_stress-energy_tensor).
The symmetrized tensor turns out to be the right one to couple to geometry in GR.

I think the solution to my problem is probably much simpler than that. My problem doesn't have anything to do with spin, and in the absence of spin, there is no distinction between the Belinfante and Noether (canonical) stress-energy tensors.

 

[atyy]

I think the solution to my problem is probably much simpler than that. My problem doesn't have anything to do with spin, and in the absence of spin, there is no distinction between the Belinfante and Noether (canonical) stress-energy tensors.

Try the discussion in Tong. The canonical stress-energy tensor is Eq 1.44 which may not be symmetric. If the theory has an action, the stress-energy which enters the field equations is given by Eq 1.48.

 

[bcrowell]

Try the discussion in Tong. The canonical stress-energy tensor is Eq 1.44 which may not be symmetric. If the theory has an action, the stress-energy which enters the field equations is given by Eq 1.48.

But I'm talking about examples like polymers and cornstarch. (You can make a non-Newtonian fluid by mixing cornstarch and water.) I don't think the intrinsic spin content of cornstarch is relevant here! Without spin, the canonical stress-energy tensor is the same as the Belinfante version, and therefore an asymmetric canonical T is the same as an asymmetric Belinfante T -- which would falsify GR.

I refuse to believe that cornstarch falsifies GR ... :-)

This is why I started this as a separate thread rather than posting in the preexisting one at https://www.physicsforums.com/showthread.php?t=615574

 

[Ilmrak]

I could be very wrong, but I think that Belinfante simply shows that any stress energy tensor can be put in a symmetric form by adding to it an appropriate super-potential containing the angular momentum tensor.

I think moreover that the "spin" content of angular momentum isn't necessary the same thing of the intrinsic spin of the system considered, cause it can be arbitrary modified (even cancelled) with a superpotential.
Infact if it was then any spin zero field would have a symmetric stress-energy tensor which isn't true (maybe).

Ilm

 

[KnownUniverse]

Actually, the energy momentum tensor (EMT) does not have to be symmetric. If the metric tensor is not symmetric then neither is the EMT. Einstein tried to relate the non-symmetric part to the electromagnetic field, but this never panned out. Recently it has been shown that the antisymmetric part of the metric tensor is the potential of a spin field-torsion (see http://arxiv.org/pdf/1207.5170.pdf). So everything is tied together, spin, non-symmetry and physics. A similar question appeared on the Blog of The Unknown Universe http://www.theunknownuniverse.net.

 

[atyy]

But I'm talking about examples like polymers and cornstarch. (You can make a non-Newtonian fluid by mixing cornstarch and water.) I don't think the intrinsic spin content of cornstarch is relevant here! Without spin, the canonical stress-energy tensor is the same as the Belinfante version, and therefore an asymmetric canonical T is the same as an asymmetric Belinfante T -- which would falsify GR.

I refuse to believe that cornstarch falsifies GR ... :-)

Tong's Eq 1.48 will produce a symmetric stress-energy tensor as long as cornstarch has an action. Do we have an action for cornstarch? I'm now googling relativistic cornstarch ...