Asymtotically flat spacetime applicable for spacetimes which are not diffeomorphic to $\mathbb{R}^4$

Question

I wanted to investigate changes on a compact 4-manifold $M$. More specifically it is the K3-surface. I follow a paper by Asselmeyer-Maluga from 2012.

The idea there was to make sure that the manifold is Ricci-flat first, then Asselmyer-Maluga applies a mathematical technique that changes the differential structure and cannot be Ricci-flat anymore. Topologically, though, nothing changes. The technique is based on replacing a suitably embedded torus by a complement of a knot. Asselmeyer-Maluga moves on to relate the used knots to possible particles.

My idea was to make the change of spacetime and its physical relevance more clear by showing that some suitable definition of mass was $0$ first, but cannot be afterwards.

To make it possible to investigate a more physical spacetime, i.e. to introduce a Lorentzian metric that does not contain closed time-like curves, Asselmeyer-Maluga removed a fourball $B^4$. Now I am new to the definitions of mass in General Relativity and do not really know what is possible to apply and calculate when.

Are all the masses like ADM and Bondi that need asymtotical flatness impossible to evaluate as they need "infinite space"? Or is this exactly the typical setting of "manifold minus ball" that I have seen in the coordinate free description of asymtotical flatness? My feeling so far is that it is necessary to use semi-local concepts of mass like the one of Brown-York.

Kind regards

This post imported from StackExchange Physics at 2014-06-29 09:39 (UCT), posted by SE-user Zonan