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.

This post imported from StackExchange Physics at 2014-03-22 17:20 (UCT), posted by SE-user Luboš Motl