Maximal extension of the Schwarzschild metric: From Painlevé-Gullstrand to Kruskal-Szekeres
Abstract
It is shown that one can link in a continuous manner the Painlevé-Gullstrand partial extension to the Kruskal-Szekeres maximal analytical extension, and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture. This is done by finding a coordinate system that adopts 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 parametrized 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 E family of extensions. This family is different from the Novikov-Lemaître family also parametrized by the energy E of timelike geodesics, with the Novikov extension holding for 0 < E < 1 and being maximal, and the Lemaître extension holding for 1 ≤ E < ∞ and being partial, not maximal, and moreover its E = ∞ limit evanescing in a Minkowski spacetime or inflating into a Kasner spacetime, rather than ending in the Kruskal-Szekeres spacetime.
- Publication:
-
Annals of Physics
- Pub Date:
- July 2021
- DOI:
- 10.1016/j.aop.2021.168497
- arXiv:
- arXiv:2005.14211
- Bibcode:
- 2021AnPhy.43068497L
- Keywords:
-
- General Relativity and Quantum Cosmology;
- Astrophysics - High Energy Astrophysical Phenomena;
- High Energy Physics - Theory;
- Mathematical Physics
- E-Print:
- 18 pages, 7 figures