Zero divergence of energy-momentum tensor and gravitational energy

Question

Trying to teach myself general relativity and have just hit yet another confusion. I'm reading that in curved spacetime the energy-momentum tensor has zero divergence, ie

$$\nabla_{\mu}T^{\mu\nu}=0.$$

But that this doesn't imply the total conservation of energy and momentum as there is an additional source of energy (the gravitational field) that isn't included in the EMT. If that's the case, and if the EMT doesn't describe the total energy of system, how is it valid to use the tensor to describe various systems. For example, $$T^{\mu\nu}=0$$ for the Schwarzschild solution, or assuming spacetime is a perfect fluid in cosmology? How are these assumptions valid if they don't include the energy contribution of the gravitational field? Seems a bit of an elephant in the room-type situation. Or is that energy so small it can be ignored?

Thank you

Answer 1

The actual reason why one can't interpret the equation $$ \nabla_\mu T^{\mu\nu}=0 $$ as a global conservation law is that it uses covariant derivatives. If a law like that were valid with partial derivatives, you could derive such a law. But there's a covariant derivative which is one of the technical ways to explain that general relativity in generic backgrounds doesn't preserve any energy:

http://motls.blogspot.com/2010/08/why-and-how-energy-is-not-conserved-in.html

The text above also explains other reasons why the conservation law disappears in cosmology.

However, despite the non-existence of a global (nonzero) conserved energy in general backgrounds, the tensor $T_{\mu\nu}$ is still well-defined. As twistor correctly writes, it quantifies the contribution to the energy and momentum from all matter fields (non-gravitational ones) and matter particles. And if you can approximate the background spacetime by a flat one, $g_{\mu\nu}=\eta_{\mu\nu}$, which is usually the case with a huge precision (in weak enough gravitational fields, locally, or if you replace local objects that heavily curve the spacetime, including black holes, by some effective $T$, using a very-long-distance effective description), then $\nabla$ may be replaced by $\partial$ in the flat Minkowski coordinates and the situation is reduced to that of special relativity and the "integral conservation law" may be restored.

Answer 2

The stress-energy tensor describes the energy momentum content contributed by all the fields present with the exception of the gravitational field itself. However, just because it doesn't appear in the energy momentum tensor does not mean that gravity can't act, in some sense, as a source of gravity. The Einstein equations are nonlinear, and from this nonlinearity arises the possibility for gravity to "gravitate" in a consistent way.

Attempts to describe the energy-momentum of the gravitational field itself locally (as you would need to do if you wished to include a gravitational field contribution in the stress energy tensor) are known to run into problems. By locally I mean that we would try to write down a tensor field that encapsulates the gravitational field energy-momentum at each point.