I am trying to wrap my head around why the (3d) stress tensor is symmetric when rotational invariance is a symmetry of the system, and its relation to (what I understand to be) the reason for why the (4d) stress-energy tensor is symmetric. I have skimmed through quite a few different things, and my understanding (and usage of terminology) may be on very shaky ground. Fair warning. Also, I don't understand anything well enough to Math yet, so words (mostly) it shall be. Apologies. First, let me sketch out my own understanding so far.
1) First, the stress tensor. I know the usual (gedanken) argument. If the stress tensor is not symmetric, then, the shear forces on a given fluid element will such as to set an initially non-rotating fluid element into rotational motion, ie, angular momentum would not conserved.
The symmetry of stress tensor also must assume that there are no internal degrees of freedom that contribute to the angular momentum. If, for example, we had a fluid where spin was resolved (ie, we could assign a local spin density to the fluid), then, we should expect that the "mechanical" angular momentum + spin angular momentum should together be conserved. Here, by "mechanical", I mean the portion of the angular momentum density created by the momentum fluxes contained in the stress tensor. In this case, maybe we can admit an antisymmetric part in the stress tensor.
There is something not quite satisfactory about the above reasoning. For one, it relies on a global symmetry to say something about a local conservation law. Anyway.
2) Now, the stress-energy tensor. We know that for a given action that is invariant under a global Lorentz transform, in general, the conserved current corresponding to space-time translation invariance (via Noether) is NOT symmetric. This is known as the "Canonical" stress energy tensor. This may be a little surprising, because by 1), we have been led to believe that Global rotational symmetry of action => Symmetry of (local) stress tensor. Of course, we can symmetrize the stress-energy tensor (Belinfante). I don't quite understand what conditions are necessary for us to be able to do so.
But even assuming that we are always able to symmetrize the stress-energy tensor by the above procedure, this is not quite satisfactory. A more satisfactory way seems to be via the definition of stress tensor as the variation of the action with respect to the metric, which is symmetric (Einstein equation)
3) Now, I am not a general-relativist. If you tell me that the LHS has two indices, and the RHS has the same two indices, and LHS is symmetric under exchange of the two indices, and ask me, is the RHS symmetric as well? My answer would be yes, but I don't necessarily understand why the LHS is symmetric.
However, one thing seems to be clear to me. There is something called "general coordinate transforms". Invariance (of what?) under this set of transformations is inbuilt into the machinery of GR (and hence, implicit in the Einstein's equation). So, if we are deriving the properties of the stress-energy tensor based on its coupling with gravity, its symmetry is probably based on invariance under general coordinate transforms (GCT)
4) What are the GCT's we speak of? Local lorentz transforms. Say we have laid out a rod/clock grid on spacetime. Now, we imagine two sets of observers present at every spacetime-grid point, who are related by some (local) lorentz transform. We require that both, in their local coordinate systems, conclude that energy-momentum density is conserved. This gives us a suitably "local" version of the global lorentz invariance. Presumably, this should somehow lead to the symmetry of the locally defined stress-energy tensor. Is this true?
5) If yes, then coming back to our original stress tensor. Let us imagine two classes of observers. One sees the rigid rod grid (spatial) as a rectangular grid. The other sees curved grid lines, but the distances between the grid points, and the angles between the rods are the same as measured by the first (This is basically my intuition of Local rotation, analogous to local lorentz symmetry of (4) ). If we wish that both classes of observers conclude that momentum density is conserved, this should lead to symmetry of the stress tensor. Is this true?
6) I am also very curious about conformal/local scale transformations, and its (analogous) relation (if any) with fluids moving about on a curved surface. Say we have a fluid on 2d surface with a square grid drawn on it (fixed distance between grid points). We make a coordinate transform such that all angles are still the same, but the distances between grid points now acquires a spatial dependence. This is only possible if we gave the surface a "curvature". Local isotropy is preserved. So, we must still have a symmetric stress tensor. However, the curvature implies that now the coordinate derivative must be changed into a covariant one. Similarly, let us say we have a rod/clock grid laid out that appears static and rectangular to us. Now, imagine that we made a "coordinate transform" such that local lorentz invariance is still preserved, but proper distances/times between grid points changed. This implies addition of some source of energy/momentum, which "curved" spacetime. Because local lorentz invariance is preserved, we can still say the stress-energy tensor is symmetric, However now, to take account of the curvature, coordinate derivative must be changed to covariant derivative. Is this analogy correct?
Pedagogical references, linguistic suggestions, handwaving (of the "sounds physically plausible" variety) all welcome.
