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 say we have a vector field $V$ defined on all of $\mathbb{R}^{3}$ such that $V=O(1/r^{3})$ in the limit as $r\rightarrow \infty$.
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 $\bigcup _{r}\bar{B_{r}}(x) = \mathbb{R}^{3}$ we get
$$\int_{\mathbb{R}^{3}}\partial _{i}V^{i}d^{3}x = lim_{r\rightarrow \infty }\int_{\bar{B_{r}}(x)}\partial _{i}V^{i}d^{3}x$$
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 integrals will eventually converge to zero.
This is all fine and dandy but my problem deals with a background flat spacetime with metric perturbation $(M,\eta _{ab} + \gamma _{ab})$ where $\eta _{ab}$ is the Minkowski metric and $|\gamma _{ab}| << 1$ as usual. We have a spacelike hypersurface $\Sigma $ of this manifold, the Landau Lifshitz pseudo tensor $t_{ab}$, which is divergence free, and the total energy $E = \int_{\Sigma } t_{00}d^{3}x$. I want to show that $E$ is time translation invariant.
Using the facts that $\partial ^{a}t_{ab} = 0, \partial _{0}E = -\partial ^{0}E$ we proceed as follows
$$\partial _{0}E = -\partial ^{0}E = -\partial ^{0}\int_{\Sigma }t_{00}d^{3}x = -\int_{\Sigma }\partial ^{0}t_{00}d^{3}x = \int_{\Sigma }\partial ^{i}t_{i0}d^{3}x$$
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 $\partial _{0}E = 0$. But here we do not have a prescribed metric $d$ on $\Sigma$ to make sense of closed balls as far as I can tell. 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.
Since $\Sigma$ is a submanifold of your spacetime you can define the induced metric which will give you a notion of distance.
So long as you original manifold is orientable, you can define an orientation of $\Sigma$ by restriction. See p111-112 of these notes for more details.
Putting these two together you can define a notion of integration on $\Sigma$ which is consistent with the usual divergence theorem.
Proving that the divergence theorem holds for arbitrary manifolds is nontrivial though! If you're interested in learning more I'd recommend reading Lee - Introduction to Smooth Manifolds.