So far I have seen two distinct ways to derived the conservation law $$\nabla_{a}T^{ab}=0$$, one from diffeomorphism of the action, the other one uses variation principle with the metric fixed. Both methods assumes that the fields are on-shell.

For example, from the fact that the Minkowski spacetime metric is fixed by translation and Lorentz transformations, we can derive the conservation laws of momentum and angular momentum via the second method. But we can also derive the conservation law of energy-momentum tensor from arbitrary diffeomorphism, for instance, the Polyakov action in string theory.

These two approaches seem very different. What is the real meaning of the conservation law?


It is important to specify which stress-energy-momentum (SEM) tensor, we are talking about.

  1. The fact that the Lagrangian density ${\cal L}_m$ has no explicit spacetime dependence leads (via Noether's 1st theorem) to a conservation law for the canonical SEM tensor.

  2. Diffeomorphism invariance of the matter action $S_m[\Phi^A;g_{\mu\nu}]$ leads (via Noether's 2nd theorem) to a conservation law for the Hilbert SEM tensor.

For details and how they are related, see my Phys.SE answers here & here.


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