# Why is the stress-energy tensor symmetric?

1. Feb 12, 2014

### 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)?

2. Feb 12, 2014

### Bill_K

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.

3. Feb 12, 2014

### stevendaryl

Staff Emeritus
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

4. Feb 12, 2014

### Bill_K

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 Tμν ≡ 2 δLmat/δgμν, which is guaranteed to be symmetric.

Last edited: Feb 12, 2014
5. Feb 12, 2014

### stevendaryl

Staff Emeritus
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.

6. Feb 12, 2014

### bcrowell

Staff Emeritus