# Why is the stress-energy tensor symmetric?

• dEdt
In summary, the conversation discusses the symmetry of the stress-energy tensor, specifically the question of why T^{01} and T^{10} should be equal. The explanation is that momentum is the flux of energy, and in the center of mass frame, the total 3-momentum is zero. The conversation also mentions the "canonical" stress-energy tensor used in General Relativity and its relationship to the symmetric tensor.
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)?

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.

## 1. What is the stress-energy tensor and why is it important?

The stress-energy tensor is a mathematical object used in the field of physics to describe the distribution of energy and momentum in a given space. It contains 10 components that represent the energy density, momentum density, and stress of a physical system. This tensor is important because it is a fundamental concept in general relativity and is used to describe the effects of gravity on matter and energy.

## 2. Why is the stress-energy tensor symmetric?

The stress-energy tensor is symmetric because it obeys the laws of conservation of energy and momentum. This means that the forces acting on a system must be balanced, and therefore the tensor must be symmetric to ensure that energy and momentum are conserved. Additionally, the symmetry of the tensor is a direct consequence of the symmetry of spacetime in general relativity.

## 3. How is the symmetry of the stress-energy tensor related to the principle of covariance?

The principle of covariance states that the laws of physics should be the same for all observers, regardless of their frame of reference. The symmetry of the stress-energy tensor is a manifestation of this principle, as it ensures that the equations of general relativity hold true for all observers regardless of their coordinate systems. Therefore, the symmetry of the tensor is essential for maintaining the consistency of the theory.

## 4. Can the stress-energy tensor be asymmetric in certain cases?

Yes, there are some situations where the stress-energy tensor may be asymmetric. This can occur in systems with non-conservative forces, such as electromagnetic fields, or in cases where the conservation laws are violated. However, the stress-energy tensor is generally expected to be symmetric in most physical systems, as it reflects the fundamental symmetries of the universe.

## 5. How is the stress-energy tensor used in practical applications?

The stress-energy tensor is used in a variety of practical applications, including astrophysics, cosmology, and engineering. It is used to describe the behavior of matter and energy in space, the effects of gravity on objects, and the evolution of the universe. In engineering, the tensor is used to model stress and strain in materials, and in fluid mechanics, it is used to calculate the distribution of pressure and momentum in a fluid.

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