# Showing ADM angular momentum to be well-defined and finite

1. Apr 25, 2013

### Albereo

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

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

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

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!