Painlev\'e-Gullstrand synchronizations in spherical symmetry

TL;DR

This paper investigates the conditions under which Painlevé-Gullstrand synchronizations exist in spherically symmetric spacetimes, linking geometric properties to energy conditions, and demonstrates how Schwarzschild solutions can be derived in these coordinates.

Contribution

It provides a detailed analysis of Painlevé-Gullstrand synchronizations in spherical symmetry, including existence conditions, energy interpretations, and a method to derive Schwarzschild solutions directly in these coordinates.

Findings

Synchronization exists where $(dr)^2 \\leq 1$

Energy densities vanish in these slices

Schwarzschild solution obtained in these coordinates

Abstract

A Painlev\'e-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 Painlev\'e-Gullstrand synchronization only exists in the region where $(dr)^2 \leq 1$, $r$ 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 Painlev\'e-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 Painlev\'e-Gullstrand synchronization, the Schwarzschild solution is directly obtained in a whole coordinate domain that includes the horizon and both its interior and exterior regions.