Why is the stress-energy tensor symmetric?

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Discussion Overview

The discussion centers on the symmetry of the stress-energy tensor, particularly in the context of its definition as the "flux of 4-momentum." Participants explore the implications of this symmetry, its relation to momentum and energy, and the distinctions between different types of stress-energy tensors, including those used in General Relativity and canonical forms.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants question why the components ##T^{01}## and ##T^{10}## should be equal, suggesting that the relationship is not immediately clear from the definition of the stress-energy tensor.
  • One participant asserts that ##T^{01}## represents the flux of energy and ##T^{10}## represents momentum density, arguing that they are equal because momentum is defined as the flux of energy.
  • Another participant introduces the idea that there are arguments for the symmetry of the stress-energy tensor that are not merely definitional, referencing the Cauchy stress tensor.
  • Concerns are raised about the implications of nonzero spin density on the symmetry of the stress-energy tensor, with references to Wikipedia articles discussing this topic.
  • Some participants clarify that the discussion may be conflating different types of stress-energy tensors, specifically distinguishing between the canonical stress-energy tensor and the one used in General Relativity, which is stated to be guaranteed symmetric.
  • A participant mentions the relationship between the symmetric stress-energy tensor in General Relativity and the canonical tensor derived from Noether's theorem, particularly in contexts involving particles with intrinsic spin.
  • External resources are shared, including links to articles and books that may provide further insights into the topic.

Areas of Agreement / Disagreement

Participants express differing views on the reasons for the symmetry of the stress-energy tensor, with no consensus reached regarding the implications of nonzero spin density or the definitions of the various forms of the tensor.

Contextual Notes

Participants note that the discussion involves complex definitions and relationships between different types of stress-energy tensors, which may lead to confusion regarding their properties and implications.

dEdt
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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)?
 
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dEdt said:
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.
 
Bill_K said:
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
 
stevendaryl said:
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 Tμν ≡ 2 δLmat/δgμν, which is guaranteed to be symmetric.
 
Last edited:
Bill_K said:
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.

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.
 

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