Showing ADM angular momentum to be well-defined and finite

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Hello! I am trying to show that the ADM definition of angular momentum is well-defined and finite. Here is the definition:

[itex]J^{i}[/itex] = -[itex]\frac{1}{2}[/itex][itex]lim_{r\rightarrow\infty}[/itex][itex]\int_{S_{r}}[/itex][itex]\epsilon_{ijm}[/itex] [itex]x^{j}[/itex] ([itex]k_{mn}[/itex] - [itex]\overline{g}_{mn}[/itex] [itex]tr k[/itex]) [itex]d[/itex][itex]S_{n}[/itex]

I'm working with an asymptotically flat, complete, Riemannian spacelike hypersurface along with the r-dependence for [itex]\overline{g}_{ij}[/itex] and [itex]k_{ij}[/itex].

Any insight on how I might go about showing that this is well-defined? Finite? Essentially I want to come up with a condition on the r-dependence that guarantees these properties.

Thanks for any help!
 

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