# Divergence theorem over entire space on non euclidean spaces

I'm a physics major so bear with me here on the math. This is related to a problem from the textbook General Relativity - Wald. In classical electromagnetism if we have a vector field say $V$ defined on all of $\mathbb{R}^{3}$ such that $V\rightarrow 0$ , dropping off as $O(1/r^{3})$, in the limit as $r\rightarrow \infty$ then when calculating $\int_{\mathbb{R}^{3}}\partial _{i}V^{i}d^{3}x$ one usually takes a closed ball $\bar{B_{r}}(x)\subset \mathbb{R}^{3}$ and, using the fact that $\bigcup _{r}\bar{B_{r}}(x) = \mathbb{R}^{3}$ gives us, with an appropriate limit theorem, that $\int_{\mathbb{R}^{3}}\partial _{i}V^{i}d^{3}x = lim_{r\rightarrow \infty }\int_{\bar{B_{r}}(x)}\partial _{i}V^{i}d^{3}x$ to which we can then apply the divergence theorem to state that $lim_{r\rightarrow \infty }\int_{\bar{B_{r}}(x)}\partial _{i}V^{i}d^{3}x = lim_{r\rightarrow \infty }\int_{\partial \bar{B_{r}}(x)}V^{i}n_{i}d^{2}x = 0$ where the zero comes from the fact that the integral will drop off as $O(1/r^{2})$ as $r\rightarrow \infty$ so the sequence of surface integrals will eventually converge to zero.

This is all fine and dandy but my problem deals with a background flat space - time with metric perturbation $(M,\eta _{ab} + \gamma _{ab})$ where $\eta _{ab}$ is the minkowski metric and $|\gamma _{ab}| << 1$ as usual. We have a space - like hypersurface $\Sigma$ of this manifold, the Landau Lifshitz pseudo tensor $t_{ab}$, which is divergence free, and the total energy $E = \int_{\Sigma } t_{00}d^{3}x$. We must show that $E$ is time translation invariant. Using the facts that $\partial ^{a}t_{ab} = 0, \partial _{0}E = -\partial ^{0}E$ we proceed: $\partial _{0}E = -\partial ^{0}E = -\partial ^{0}\int_{\Sigma }t_{00}d^{3}x = -\int_{\Sigma }\partial ^{0}t_{00}d^{3}x = \int_{\Sigma }\partial ^{i}t_{i0}d^{3}x$. We are also given that $t_{\mu \nu }\rightarrow 0$ identically, dropping off as $O(1/r^{3})$, in the limit $r\rightarrow \infty$.

This is, of course, very similar in situation to the electromagnetic case and one would ideally like to use the divergence theorem to get the desired result that $\partial _{0}E = 0$ however here we do not have a prescribed metric $d$ on $\Sigma$ to make sense of closed balls as far as I can tell and even if there is some natural choice of metric $d$ for $\Sigma$ how will we know if the closed balls with respect to $d$ will be orientable? Thanks for any and all help and sorry if this was a bit long winded; it is my first post here so I'm not sure how it is meant to work :[; I just wanted to be thorough in explaining my issue. Thanks again.

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Cross-posted from math.stackexchange.com/q/303796/11127 – Qmechanic Feb 14 '13 at 15:56