Painlevé–Gullstrand coordinates

Painlevé–Gullstrand coordinates
Name	Painlevé–Gullstrand coordinates

Painlevé–Gullstrand coordinates are a coordinate chart for the Schwarzschild spacetime introduced in the early 20th century that provides a regular description of a non-rotating black hole on the event horizon. Developed in parallel by Paul Painlevé and Allvar Gullstrand, these coordinates remove the coordinate singularity at the Schwarzschild radius and furnish an intuitive picture of infalling observers and the causal structure of Karl Schwarzschild's solution to Albert Einstein's field equations. They have been used in analyses by researchers connected to David Finkelstein, Subrahmanyan Chandrasekhar, Roger Penrose, and Kip Thorne.

Introduction

The chart offers a foliation of the Schwarzschild manifold by spatial slices that are flat in the sense used by Hermann Minkowski in the context of Special relativity, while retaining the global mass parameter introduced by Gustav Clemens in classical solutions. Conceived contemporaneously with works by Georges Lemaître and later interpreted in light of coordinate-regular descriptions by W. Rindler and John Archibald Wheeler, the coordinates are valuable to researchers studying horizon crossing, gravitational collapse analyzed by Oppenheimer–Snyder collapse approaches, and numerical relativity projects involving teams at institutions like Caltech and Max Planck Institute for Gravitational Physics.

Derivation

One derives the chart by performing a time coordinate transformation of the Schwarzschild metric originally written by Karl Schwarzschild in Schwarzschild coordinates, paralleling methods used by Georges Lemaître to remove coordinate singularities. Start from the Schwarzschild line element attributed to K. Schwarzschild and introduce a new time function tied to the proper time of radially infalling observers akin to trajectories studied by Arthur Eddington in his work on light propagation and by Albert Einstein in Einstein–Rosen bridge analyses. The transformation uses integrals similar to those found in calculations by Friedrich Bessel-type methods and was formalized by Paul Painlevé and Allvar Gullstrand to produce cross terms that eliminate the 1/(1-2M/r) divergence at the radius associated with Pierre-Simon Laplace's escape-velocity analogue.

Properties and interpretation

The metric in these coordinates displays manifest regularity at the event horizon, a feature also emphasized in studies by David Finkelstein and Roger Penrose. Spatial slices of constant new time are isometric to Euclidean space in the original derivation, resonating with interpretations by Hermann Minkowski and later by John Wheeler. The foliation corresponds to a family of geodesic observers analogous to the worldlines considered by Subrahmanyan Chandrasekhar in his treatment of test particle motion and by Kip Thorne in analyses of tidal forces. The coordinates make clear the one-way causal nature of the horizon emphasized in Stephen Hawking's work on black hole thermodynamics and illustrate trajectories relevant to Oppenheimer–Snyder models of gravitational collapse.

Extensions and variants

Researchers extended the Painlevé–Gullstrand framework to include rotating and charged solutions explored by Roy Kerr and Ernst Reissner respectively, producing variants analogous to Kerr–Schild coordinates used in studies by Roy Kerr and Alfred Schild. Generalizations informed by methods from Arnowitt Deser Misner (ADM) formalism and numerical relativity groups at Caltech and Max Planck Institute for Gravitational Physics led to horizon-penetrating slicings in spacetimes with cosmological constant as in work following Alexander Friedmann and Edwin Hubble. The approach also inspired coordinates for analogue gravity experiments pursued in laboratories associated with MIT, Stanford University, and University of Nottingham.

Applications in general relativity

Painlevé–Gullstrand coordinates have been applied in calculations of Hawking radiation building on analyses by Stephen Hawking and in studies of black hole accretion influenced by research at Princeton University and Institute for Advanced Study. They are used in pedagogical expositions by authors connected to Cambridge University Press and Oxford University Press in textbooks on Albert Einstein's theory, and by computational teams employing ADM and BSSN formalisms informed by Richard Arnowitt, Stanley Deser, and Charles Misner. Practical applications include modeling of infalling matter in accretion-disc studies associated with Shakura Sunyaev models, interpretations of gravitational redshift relevant to Joseph Taylor's pulsar timing, and horizon-penetrating numerical schemes developed by groups at Max Planck Institute for Gravitational Physics and Caltech.

Relationship to other coordinate systems

The coordinates relate closely to Lemaître coordinates introduced by Georges Lemaître and to Eddington–Finkelstein coordinates associated with Arthur Eddington and David Finkelstein; each removes the Schwarzschild coordinate singularity but differs in slicing and observer interpretation, a distinction examined by Roger Penrose and Kip Thorne. Transformations to Schwarzschild coordinates highlight the historical derivation by Karl Schwarzschild and comparisons with Kerr–Schild forms trace back to work by Roy Kerr and Alfred Schild. Connections to ADM slicing methods recall contributions by Richard Arnowitt, Stanley Deser, and Charles Misner, while modern numerical relativity implementations follow developments at institutions including Caltech and Max Planck Institute for Gravitational Physics.

Category:General relativity