Maximal extension of the Schwarzschild metric: From Painlevé-Gullstrand to Kruskal-Szekeres

Abstract

We find a specific coordinate system that goes from the Painlevé-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 $1\le E<\infty$ with the limit $E=\infty$ 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-Lemaître family parameterized also 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 $1\le E<\infty$ and being partial, not maximal, and moreover its $E=\infty$ limit evanescing in a Minkowski spacetime rather than ending in the Kruskal-Szekeres spacetime.

arXiv comment18 pages, 7 figures

References in corpus (11)

• Hawking Radiation as Tunneling
• Regular coordinate systems for Schwarzschild and other spherical spacetimes
• Simple excision of a black hole in 3+1 numerical relativity
• Static detectors and circular-geodesic detectors on the Schwarzschild black hole
• Simple Analytic Models of Gravitational Collapse
• Multi-horizon spherically symmetric spacetimes with several scales of vacuum energy
• Backreaction of accreting matter onto a black hole in the Eddington-Finkelstein coordinates
• Painleve-Gullstrand Coordinates for the Kerr Solution
• Ingoing Eddington-Finkelstein Metric of an Evaporating Black Hole
• Coordinate families for the Schwarzschild geometry based on radial timelike geodesics
• Inflying perspectives of reduced phase space

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• Beyond Schwarzschild: New Pulsating Coordinates for Spherically Symmetric Metrics
• Symmetry Transformations in Cosmological and Black Hole Analytical Solutions
• Novikov Coordinates and the Physical Description of Gravitational Collapse