Why must the EM stress-energy tensor be symmetric?

[michael879]

Can someone please walk me through (or provide a link that does) the derivation of the EM stress-energy tensor? I get all the concepts I'm just a little confused on some of the details. Basically, you have the definition of the stress energy tensor in terms of the lagrangian, and the condition that $\partial_\mu T^{\mu\nu} = 0$. What you end up with is an expression that can have anything added to it as long as its derivative remains 0. This is how you generally make the stress-energy tensor symmetric. What I'm confused about is WHY it has to be symmetric, and what prevents you from adding arbitrary constants to it? Is there some condition I'm missing?

 

[michael879]

ok to add a little more detail to what I'm looking for, I'm trying to derive the E&M stress energy tensor for an SU(N) gauge field in a "material" (i.e. the constitutive relations relating E and B to D and H are undefined). This is a trivial exercise, since the stress-energy tensor is easily derived from the lagrangian. My problem is in getting an expression like the U(1) stress-energy tensor in free space, which is typically made to be symmetric. If I could just understand why it has to be made symmetric, and what the E&M stress-energy tensor is in some material, it would help a lot (generalizing to an SU(N) field is easy).

So basically I'm looking for:
1) a detailed derivation and explanation of the free-space E&M stress-energy tensor
2) the E&M stress-energy tensor without the assumption of a linear material (i.e. undefined constitutive relations)