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Another question is, because the stress-energy tensor present in the EFEs is only for the matter(I mean, you know, sources!) present in the region, and because there will be some energy-momentum exchange between the matter and space-time, we should have [itex] \partial_\mu T^{\mu \nu} \neq 0 [/itex]. Is this right?

But then I see that actually [itex] \nabla_\mu T^{\mu \nu}=0 [/itex], which seems strange to me.

If we say this is a conservation law for the stress-energy tensor, then this should be in contradiction with the above paragraph and also we should ask why making the derivative covariant makes a non-conserved quantity conserved, which seems non-sense to me! And it can't be true because making the derivative covariant still doesn't account for the gravitational energy!

But if we say this isn't a conservation law for stress-energy tensor, we should ask what's its meaning?(Actually I see it being called a conservation law!)

I'm really confused about it and can't find a way out. I need urgent help!

Thanks