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Why is the stress-energy tensor symmetric?

  1. Feb 12, 2014 #1
    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. jcsd
  3. Feb 12, 2014 #2

    Bill_K

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    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.
     
  4. Feb 12, 2014 #3

    stevendaryl

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    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
     
  5. Feb 12, 2014 #4

    Bill_K

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    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
  6. Feb 12, 2014 #5

    stevendaryl

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    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.
     
  7. Feb 12, 2014 #6

    bcrowell

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