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.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)?
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.
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.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
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.
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.
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.
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.
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.
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.