Topology and the swartzschild solution - where is the mass?

[YangMills]

My professor and I were discussing the emergence of the Swartzschild solution from topological considerations, corresponding to the manipulations of a point singularity. He pointed out to me that mass nowhere enters into the considerations, and so classifying black holes according to mass is fallacious. I am not entirely sure how he arrived at this point, however, and have forgotten the general process. Could someone please explain this to me, and/or provide me with some other references?

Also, I recall reading somewhere that the Kerr solution emerges from considerations of a circle in a plane, using topology (as with the Swartzschild and points). Supposedly the killing field emerging from the isotropic nature of the space generates angular momentum. How would a charge arise if we were considering the Kerr-Neumman solution?

Thank you in advance

 

[grey_earl]

First, it's Schwarzschild, writing names correctly is (at least for me) a sign of respect to those great persons.
Second, there is no singularity. The Schwarzschild solution isn't defined on all of $$\mathbb{R}^4$$, but on $$\mathbb{R}^4 - {0}$$, which is topologically different, but I assume that's what you meant. Just to be mathematically sound.
Then on this space you just consider the most general static and asymptotically flat vacuum solution, which happens to be spherically symmetric and characterized by a parameter $$M$$, which up to now is completely arbitrary, can even be negative. (Israel theorem) In the case of $$M=0$$, you can actually continue the solution onto all of $$\mathbb{R}^4$$, but if $$M\neq 0$$, this isn't possible. Now asymptotical flatness allows you to give (at spatial infinity, by comparison with the Newtonian case) an interpretation of this parameter $$M$$, namely that of the total mass of the spacetime. And then you say: physical objects must have positive mass.
Not sure if that is what you meant, if not, just ask more.