Maximal extension of the Schwarzschild metric: From Painlev\'e-Gullstrand to Kruskal-Szekeres
Abstract
We find a specific coordinate system that goes from the Painlev\'e-Gullstrand partial extension to the Kruskal-Szekeres maximal extension and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture. We do this by adopting two time coordinates, one being the proper time of a congruence of outgoing timelike geodesics, the other being the proper time of a congruence of ingoing timelike geodesics, both parameterized by the same energy per unit mass E. E is in the range 1≤ E<∞ with the limit E=∞ yielding the Kruskal-Szekeres maximal extension. So, through such an integrated description one sees that the Kruskal-Szekeres solution belongs to this family of extensions parameterized by E. Our family of extensions is different from the Novikov-Lema\itre family parameterized also by the energy E of timelike geodesics, with the Novikov extension holding for 0<E<1 and being maximal, and the Lema\itre extension holding for 1≤ E<∞ and being partial, not maximal, and moreover its E=∞ limit evanescing in a Minkowski spacetime rather than ending in the Kruskal-Szekeres spacetime.