I am confused about what the components of the stress energy tensor actually represent and how they are representative of energy in spacetime.
I have seen multiple questions on here, and have re-read the Wikipedia page many times, they all explain the tensor as; energy density for the 00 component; momentum along the edges and finally the stress tensor in the middle, with pressure along the diagonal. The explanation accompanying this is usually something like- each component represents a different type of energy and they all share the same units for energy density.
Alternatively, I have seen explanations, specifically Susskinds General Relativity lecture 9 and other questions on this site, which talk about the components representing energy density, energy flux, momentum density or momentum flux, and which one is determined by the indicies of the tensor.
My question is, which of the explanations is used, or is it some combination of the two that I just don't understand?
Best Answer
One tries to encapsulate "numbers" into tensors because they are related (in a precise way) and it is more compact to write them as a whole object.
This tensor in particular is the result of applying Noether's Theorem to a general space-time transformation, it is in fact the conserved current density that is induced by the space-time symmetry. This means that you have conservation laws that can be expressed in an integral form, and the conserved objects that come from them are called energy, momentum, etc.
The indices or components (the $\mu$ and $\nu$) allow you to find the specific conserved quantities as you said, but you can observe directly where they come from when you look at the definition in terms of derivatives of the Lagrangian; it's no wonder they have those meanings in the end.
Edit: I think this answer gives extra details in the same vein.