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
∇μTμν=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, Tμν=0
Thank you
Answer
The actual reason why one can't interpret the equation ∇μTμν=0
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 Tμν 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, gμν=ημν, 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 ∇ may be replaced by ∂ in the flat Minkowski coordinates and the situation is reduced to that of special relativity and the "integral conservation law" may be restored.
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