ABSTRACT
Among the solutions of space-times with compact sources are the Black Hole solutions. The very first Schwarzschild solution provided the initial model of space-time near the Sun with which general relativity passed its first tests. Its mathematical extension (decreasing the radial coordinate, first below the physical radius of the star and then below the Schwarzschild radius) already revealed the exotic nature of the space-time of a point mass. We have seen the Kruskal extension of the Schwarzschild solution and indicated the similar one for the Reissner-Nordstrom solution. These and the Kerr-Newmann family of solutions are all asymptotically flat. The portion of space-time connected to the asymptotic region is the exterior region. The mathematical extension refers to extension away from the asymptotic region, in the interior region. The different regions are signalled in terms of coordinates where some of the metric components vanish or diverge and are demarcated by the zeros of the function ∆(r) = (r − r+)(r − r−) where r± are constants. As noted before, although some of the metric components vanish or diverge, the Riemann tensor - which encodes the physical effects of gravity - is perfectly well behaved. The geodesics across the r = r± surfaces are well behaved too and in fact signal how an extension may be sought. Fundamentally, an extension across a chart boundary is sought by changing the local coordinates, obtaining the corresponding metric and continuing the same metric form to a larger neighborhood till the next chart boundary where the extended metric or the curvature may develop singularities.
