Why is the stress-energy tensor symmetric?

[dEdt]

If we use the "flux of 4-momentum" definition of the stress-energy tensor, it's not clear to me why it should be symmetric. Ie, why should $T^{01}$ (the flux of energy in the x-direction) be equal to $T^{10}$ (the flux of the x-component of momentum in the time direction)?

 

[Bill_K]

If we use the "flux of 4-momentum" definition of the stress-energy tensor, it's not clear to me why it should be symmetric. Ie, why should $T^{01}$ (the flux of energy in the x-direction) be equal to $T^{10}$ (the flux of the x-component of momentum in the time direction)?

No, T⁰¹ is the flux of energy all right, but T¹⁰ is the momentum density. And they are equal because that's what momentum IS: momentum is the flux of energy. Thus in the center of mass frame, the total 3-momentum is zero.

 

[stevendaryl]

No, T⁰¹ is the flux of energy all right, but T¹⁰ is the momentum density. And they are equal because that's what momentum IS: momentum is the flux of energy. Thus in the center of mass frame, the total 3-momentum is zero.

You seem to be saying that it is true by definition. But there is an argument that it must be symmetric that seems like it's not true by definition.
http://en.wikipedia.org/wiki/Cauchy...2.80.99s_stress_theorem.E2.80.94stress_tensor

Also, another article in Wikipedia suggests that if there is a nonzero spin density, then that implies a nonsymmetric stress-energy tensor:
http://en.wikipedia.org/wiki/Spin_tensor

 

[Bill_K]

Also, another article in Wikipedia suggests that if there is a nonzero spin density, then that implies a nonsymmetric stress-energy tensor: http://en.wikipedia.org/wiki/Spin_tensor

No, both of these may be referring to the "canonical" stress-energy tensor. It's not to be confused with the stress-energy tensor we use in General Relativity T^μν ≡ 2 δL_mat/δg_μν, which is guaranteed to be symmetric.

 

[stevendaryl]

No, both of these may be referring to the "canonical" stress-energy tensor. It's not to be confused with the stress-energy tensor we use in General Relativity T^μν ≡ 2 δL_mat/δg_μν, which is guaranteed to be symmetric.

Okay, the subject is a little murky to me. But this article
http://en.wikipedia.org/wiki/Belinfante–Rosenfeld_stress-energy_tensor
shows the relationship between the symmetric stress-energy tensor used in GR and the canonical tensor found from Noether's theorem, in the presence of particles with nonzero intrinsic spin.

 

[bcrowell]

See http://physics.stackexchange.com/questions/68564/why-is-the-stress-energy-tensor-symmetric

My explanation in ch. 9 of this book may help: http://www.lightandmatter.com/sr/