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My understanding of GR is that curvature of spacetime reflects the density of energy-matter. Does the curvature itself have energy? Or if energy is assigned to curvature it simply reflects the energy density at that point?

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The gravitational field can indeed be assigned an energy. Unfortunately though whereas for, say, the EM field you can define an energy density at a point ($\bf{E}^2+\bf{B}^2$), for the gravitational field you can't do this. - Whichever way you define the energy in terms of the Christoffel symbols, you run into the problem that you can make them, and hence the energy, vanish at a point be choosing an appropriate frame.

So people have come up with non local energy definitions for the gravitational field- ADM energy, Bondi energy etc. all of which involve integration over spacetime regions.

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I'm a little unsure of terminology here - my training is more mathematical than physical - I think your local means at a point; and non-local means over a small spacetime region. Is that right? –  Mozibur Ullah May 13 '13 at 14:41
    
@MoziburUllah: no, local means a spacetime region - even if it becomes arbitrarily small. –  zhermes May 13 '13 at 14:53
    
@Zhermes: yes, thats what I would normally suppose; but twistor59 said non-local. Are you suggesting his terminology is a mistype? –  Mozibur Ullah May 13 '13 at 14:56
    
@MoziburUllah, sorry, the additional criteria is that 'local' refers to a region which resembles flat space-time. –  zhermes May 13 '13 at 15:02
    
@zhermes: so his non-local means non-flat or not necessarily flat? –  Mozibur Ullah May 13 '13 at 15:03
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