Suppose that you have a region of space with no fields and the only matter is in the form of a cloud of point-masses.(adsbygoogle = window.adsbygoogle || []).push({});

On the one hand, within the cloud, the stress-energy tensor vanishes almost everywhere (except right at the point-mass, in which case it is infinite). So the Einstein tensor is zero almost everywhere, and the spacetime curvature is completely described by the Weyl tensor.

On the other hand, it seems that within the cloud, we should be able to do some kind of coarse-graining, where we approximate the point-masses by a continuous mass-energy density. This would give us an approximate stress-energy tensor and an associated approximate Einstein tensor.

This leads me to think that there should be some kind of relationship between the two tensors, that would allow you to get some kind of average value of the Einstein tensor from the Weyl tensor. Is there an analog of Gauss' Law for GR, that would allow one to compute an average value of the stress-energy tensor inside a volume from the value of the Weyl tensor outside that volume?

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# Gauss' Law for GR?

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