# Physics related question on Divergence Theorem for “general” smooth manifolds

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.

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Hi unnamed user, welcome to Math.SE! Given the effort you put into typesetting the math, I'm positive that there is a very nice question here. However, it would be easier to find if you broke up your intimidating wall of text into readable paragraphs. Cheers! – Rahul Narain Feb 14 at 6:19
Cross-posted to physics.stackexchange.com/q/53933/2451 – Qmechanic Feb 14 at 15:55

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.