Showing ADM angular momentum to be well-defined and finite

[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!

 

[pat_connell]

To show that the ADM definition of angular momentum is well-defined and finite, you need to show that the integral over the surface at infinity converges. To do this, you must show that the integrand (i.e. the expression in the brackets) has a limit as r approaches infinity. Since \overline{g}_{mn} tr k is independent of r, it suffices to show that k_{mn} has a limit as r approaches infinity. To show that k_{mn} has a limit, first note that it can be written as a sum of terms of the form k_{mn} = \frac{1}{2}\partial_{a}\partial_{b}\overline{g}_{mn}. Now, since \overline{g}_{mn} is assumed to be asymptotically flat, all of its derivatives should become asymptotically constant as r approaches infinity. Thus, k_{mn} will be a sum of terms of the form k_{mn} = \frac{1}{2}\overline{C}_{ab}\overline{g}_{mn} where \overline{C}_{ab} is a constant tensor. Since the product of two constants is a constant, we have that k_{mn} is asymptotically constant as r approaches infinity and thus the integral over the surface at infinity will converge.