The Schwarzschild 0riginal Solution and the Droste and Hilbert Solutions of Einstein Equations and Black-Holes. A Resume of a Long Story and the Many Misunderstandings it Produced.

I briey recall that the original Schwarzschild solution and the Droste and Hilbert solutions (and their maximal extension) to Einstein equations are in fact di¤erent solutions which de\u85ne manifolds with very di¤erent topologies. Moreover, whereas the original Schwarzschild solution does not imply in the existence of back-holes, the (maximal extension) of the Droste-Hilbert solution de\u85nes a manifold (Kruskal space-time) that con-tains a black-hole. All the above statements are pure mathematical ones and cannot be disputed. They have nothing to do with the possible ex-istence or not of black-holes in our physical world. Keeping this in mind put immediately an end to polemics that one \u85nd from time to time in the literature. 1. Schwarzschild [1] looked for a solution of Einstein equations supposing a priori that the space-time manifold where a point mass and the gravitational eld it generated live is M = R R3 where time takes values in R and R3 de