The ADM mass is expressed in terms of the initial data as a surface integral over a surface S at spatial infinity: M:=−18πlim where \sigma_{ij} is the induced metric on S, k=\sigma^{ij}k_{ij} is the trace of the extrinsic curvature of S embedded in \Sigma (\Sigma is a hypersurface in spacetime containing S). and k_0 is the trace of extrinsic curvature of S embedded in flat space.
Can someone explain to me why ADM mass is defined so. Why is integral of difference of traces of extrinsic curvatures important?
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