Painlevé-Gullstrand synchronizations in spherical symmetry

Abstract

A Painlevé-Gullstrand synchronization is a slicing of the space-time by a family of flat spacelike 3-surfaces. For spherically symmetric space-times, we show that a Painlevé-Gullstrand synchronization only exists in the region where $(dr{)}^2\le 1$, $r$ being the curvature radius of the isometry group orbits ($2$-spheres). This condition says that the Misner-Sharp gravitational energy of these 2-spheres is not negative and has an intrinsic meaning in terms of the norm of the mean extrinsic curvature vector. It also provides an algebraic inequality involving the Weyl curvature scalar and the Ricci eigenvalues. We prove that the energy and momentum densities associated with the Weinberg complex of a Painlevé-Gullstrand slice vanish in these curvature coordinates, and we give a new interpretation of these slices by using semi-metric Newtonian connections. It is also outlined that, by solving the vacuum Einstein's equations in a coordinate system adapted to a Painlevé-Gullstrand synchronization, the Schwarzschild solution is directly obtained in a whole coordinate domain that includes the horizon and both its interior and exterior regions.

16 pages

References in corpus (12)

• 3+1 Formalism and Bases of Numerical Relativity
• Present status of the Penrose inequality
• Generalized Lemaitre-Tolman-Bondi Solutions with Pressure
• Classification of spacelike surfaces in spacetime
• Gravitational collapse of spherically symmetric plasmas in Einstein-Maxwell spacetimes
• Spherically Symmetric Black Hole Formation in Painlevé-Gullstrand Coordinates
• Spherically Symmetric Gravitational Collapse of General Fluids
• Generalized Painlevé-Gullstrand metrics
• Uniqueness of Flat Spherically Symmetric Spacelike Hypersurfaces Admitted by Spherically Symmetric Static Spactimes
• Involute, minimal, outer and increasingly trapped spheres
• Coordinates and frames from the causal point of view
• Some notes on the Kruskal - Szekeres completion

Cited by in corpus (7)

• Separable geodesic action slicing in stationary spacetimes
• Maximal slicings in spherical symmetry: local existence and construction
• On the uniqueness of the space-time energy in General Relativity. The illuminating case of the Schwarzschild metric
• Spacetime characterizations of $\Lambda$-vacuum metrics with a null Killing 2-form
• On the meaning of Painlevé--Gullstrand synchronization
• Slicing black hole spacetimes
• Nonsingular spherically symmetric black-hole model with holonomy corrections