|Jun9-12, 12:27 AM||#1|
Details of Stress-Energy-Momentum Tensor
I'm fairly familiar with the elements of the stress-energy-momentum, but I have some questions. Is it right to say the stress-energy-momentum tensor is symmetric? And if so, why does the wikipedia article show energy flux across the top row vector, and momentum density down the first column vector? Or am I messing up the meaning of 'energy flux?' Maybe they're the same?
As far as the 0,0 component goes.. I was watching this video lecture from MIT. You can find it by searching "Lec 3 | MIT 8.224 Exploring Black Holes" on youtube. And he uses the energy density as the 0,0 component. However, the wikipedia article states the 0,0 component is the relativistic mass density. I understand they only differ by a factor of c2 but which is correct to use?
Also, the element of 'energy density,' is it simply the addition of the mechanical energy and the electromagnetic energy? And the same question goes for the momentum density, and stresses. Do you simply add the mechanical and electromagnetic contributions?
|Jun13-12, 09:27 AM||#2|
These descriptions are all true, so that a logical consequence of them is that the energy flux happens to equal the momentum density (*c2), while these are 2 different concepts.
Usual conventions on tensors assume c=1 so that "mass" and energy are the same. If we don't put c=1, as the stress-energy tensor is defined as twice contravariant, its 0,0 component is the mass density; its 0,i and i,0 equal components are the density of momentum and the flux of "mass" where "mass" = energy * c2.
You can see this considering that for a twice contravariant tensor, the space and time components have the same magnitude when describing an object going at 1 m/s, as they are given by the tensor product of both parallel vectors (1,speed) and (mass, momentum) that are tangent to the movement of the object in space-time. The magnitudes of the flux of mass and density of momentum are ordinary, while the flux of energy would be huge at is is the energy of mass mc2 of the object that is moved.
(I have an introduction to tensors in my site : settheory.net)
The energy, is the sum of all possible forms of energies for all kinds of particles and forces. (We may argue that the "mechanical energy", beyond the energy of mass, is but a hidden form of electromagnetic energy)
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