Divergence theorem question on hyperplanes

[WannabeNewton]

Hi guys, this is in regards to a problem from Wald from the section on linearized gravity. We have a quantity $t_{ab}$ very, very similar to the L&L pseudo tensor and have the quantity (a sort of total energy) $E = \int_{\Sigma }t_{00}d^{3}x$ where $\Sigma$ is a space - like hypersurface of a background flat space - time with perturbation. We must show this quantity is time translation invariant. We know, from previous calculations not shown here, that $\partial ^{a}t_{ab} = 0$ so in particular we know that $\partial ^{0}t_{00} = -\partial ^{i}t_{i0}$ where $i = 1,2,3$. We also have that $\partial _{0}E = -\partial ^{0}E$ and so 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 also know that $r\rightarrow \infty \Rightarrow t_{\mu \nu }\rightarrow 0 ,\forall \mu ,\nu$. Ideally, one would like to use the divergence theorem to get a surface integral over the boundary of this space - like hypersurface because, due to the boundary condition, the surface integrand will vanish identically on this boundary and therefore so will the surface integral thus giving us our desired result that $\partial _{0}E = 0$. Now my initial idea in order to do this was to somehow compactify $\Sigma$ but I'm not sure how to formalize this plus there is the issue of orientability. I've seen very hand - wavy arguments about taking a closed ball of some radius and taking the limit as this radius approaches infinity but I have not seen a proof that this works not to mention this perturbed background flat space - time doesn't have that extra metric structure pre - imposed so we would have to invoke a theorem allowing us to place some metric on the manifold. Any and all help is appreciated, thank you!

 

[WannabeNewton]

Well following the technique from electromagnetism in $\mathbb{R}^{3}$: by using the fact that we can endow $\Sigma$ with some metric $d$, we can claim that, by virtue of the respective limit theorem, if $\bar{B_{r}(x)}\subseteq \Sigma$ is a closed ball (with respect to this metric) then $\int_{\Sigma }\partial ^{i}t_{i0}d^{3}x = lim_{r\rightarrow \infty }\int_{\bar{B_{r}(x)} }\partial ^{i}t_{i0}d^{3}x$. We would like to use the divergence theorem on this new integral next to the limit but to do that we would need to know that the closed balls of ever increasing radii are orientable and I have no idea how/if we can conclude that.