Understanding Gauss' Law for GR in Point Mass Clouds

[stevendaryl]

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.

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?

 

[atyy]

Gauss's law is a particular case of what is nowadays called Stokes's theorem. Stokes theorem is used in deriving the Komar mass http://arxiv.org/abs/gr-qc/0703035 (see the remark after Eq 7.81, which refers to Wald's Eq 11.2.10).

A quick google shows that Stokes's theorem is also used in the proof of the "positive-mass" or "positive-energy" theorem.

 

[WannabeNewton]

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?

Well Gauss's law let's you compute the total mass inside a volume given the gravitational field on the boundary of that volume so it's different from what you stated. Regardless, the answer is "sort of" because there is a Gauss's law that relates the energy-momentum in a volume to the gravitational field on the boundary of that volume in the limit as the boundary goes to the asymptotically flat region of a given asymptotically flat stationary space-time but the gravitational field, being given only by first order derivatives of the metric, will not explicitly involve the Weyl tensor.

What we can do, for stationary asymptotically space-times, is with a bit of tensor calculus and Stokes' theorem show that $-\frac{1}{8\pi}\int_{S^2} \epsilon_{abcd}\nabla^c \xi^d = 2\int _{\Sigma}(T_{ab} - \frac{1}{2}Tg_{ab})n^a \xi^b dV$ where $S^2$ is a 2-sphere taken in the asymptotically flat region, $\Sigma$ is the surface it bounds, and $\xi^a$ is the time-like Killing field. Note that the volume integral involves not only $T_{ab}$ but also $g_{ab}$; this is necessary because $T_{ab}$ in and of itself does not contain any information about the (quasi) local gravitational energy density. For a derivation of the above see Wald section 11.2; the key relation is $\nabla_{[l}(\epsilon_{mn]cd}\nabla^c\xi^d) = \frac{2}{3}R^{e}{}{}_{f}\xi^f\epsilon_{elmn}$.

For non-stationary space-times, in particular those which contain gravitational waves in the far-field region, things get much more complicated and one must resort to the ADM formalism.

EDIT: See also: http://en.wikipedia.org/wiki/Bel–Robinson_tensor and exercise 4.6 in Wald.