Eddington–Finkelstein coordinates

Eddington–Finkelstein coordinates
L3erdnik · CC0 · source
Name	Eddington–Finkelstein coordinates
Field	General relativity
Introduced	1950s
Namedafter	Sir Arthur Eddington, David Finkelstein

Eddington–Finkelstein coordinates are a coordinate system used in General relativity to describe spacetime around a non-rotating Schwarzschild metric black hole, providing regular coordinates at the event horizon and clarifying causal structure near singularities and horizons. They are fundamental in analyses by researchers at institutions like Cambridge University, Princeton University, Harvard University, Caltech, and Institute for Advanced Study, and are applied in studies involving figures such as Albert Einstein, Subrahmanyan Chandrasekhar, Roger Penrose, Stephen Hawking, and John Archibald Wheeler. Their use appears in textbooks and reviews from publishers associated with Oxford University Press, Cambridge University Press, and Springer Science+Business Media.

Introduction

Eddington–Finkelstein coordinates were developed to resolve coordinate singularities in early solutions of Einstein field equations exemplified by the Schwarzschild solution, and to connect analytic work by Arthur Eddington and later contributions by David Finkelstein to understand black hole horizons, event horizons, and causal domains in contexts explored by Roy Kerr, Subrahmanyan Chandrasekhar, and Kip Thorne. They provide a bridge between classical treatments found in works by Karl Schwarzschild, Willem de Sitter, Hermann Weyl, and modern expositions by James B. Hartle, Sean Carroll, Carlo Rovelli, and Steven Weinberg.

Derivation and coordinate transformation

Starting from the Schwarzschild metric in coordinates introduced by Karl Schwarzschild and analyzed by Arthur Eddington and Geoffrey Burbidge, one performs a transformation using null coordinates inspired by methods attributed to Paul Dirac and techniques from Hermann Bondi and Felix Pirani. Define an advanced time coordinate by adding a logarithmic term involving the Schwarzschild radius r_s associated with Karl Schwarzschild and mass parameters treated in studies by Subrahmanyan Chandrasekhar and Lev Landau, which yields regularity at r = r_s similar to transformations used by Kruskal and Szekeres in the Kruskal–Szekeres coordinates construction. The transformation connects the original Schwarzschild t and r to null coordinates analogous to constructions in work by Roger Penrose on conformal diagrams and by Edward Witten on analytic continuations.

Ingoing and outgoing forms

There are two principal variants: the ingoing form associated with analyses in papers by David Finkelstein and the outgoing form used in scattering treatments by Richard Feynman and Gerard 't Hooft. The ingoing coordinates are suited to describing matter collapse studied by Wheeler and John Michell, while the outgoing form appears in treatments of Hawking radiation by Stephen Hawking and in semiclassical analyses by Don Page and Luis Alvarez-Gaumé. Both variants parallel null coordinate methods used in the Vaidya metric studied by Pravda and techniques in Raychaudhuri analyses.

Metric properties and horizons

In these coordinates the metric components are regular at the event horizon r = r_s, clarifying distinctions emphasized by Roger Penrose between coordinate singularities and true curvature singularities at r = 0 first discussed by Karl Schwarzschild and later by Roy Kerr for rotating cases. The feature that light cones tilt across the horizon underlies results in the Penrose diagram literature and is related to singularity theorems proved by Stephen Hawking and Roger Penrose. Studies by Israel, Arnowitt, Deser, and Misner on gravitational energy also reference horizon-regular coordinates in quasi-local analyses.

Geodesics and causal structure

Eddington–Finkelstein coordinates simplify integration of null and timelike geodesics used by Subrahmanyan Chandrasekhar in studies of orbital motion and by Kip Thorne in analyses of tidal forces, allowing explicit tracing of causal trajectories past the event horizon as in thought experiments by John Wheeler and Roger Penrose. The coordinates facilitate construction of global causal diagrams used by Stephen Hawking and Eric Poisson to demonstrate inextendibility results and to compare with maximal extensions like the Kruskal–Szekeres coordinates and analytic continuations studied by Felix Klein and Bernhard Riemann.

Extensions and applications

Beyond the non-rotating case the approach inspires coordinate choices for rotating Kerr metric investigations by Roy Kerr and charged Reissner–Nordström metric work tracing back to Hermann Reissner and Gunnar Nordström, and appears in numerical relativity codes developed at Max Planck Institute for Gravitational Physics, NASA, Jet Propulsion Laboratory, and European Space Agency collaborations. Applications include black hole perturbation theory used by Vishveshwara, gravitational-wave modeling advanced by Barry Barish and Rainer Weiss, and quantum gravitational contexts explored by John Preskill and Edward Witten.

Historical context and naming

The coordinates are named to acknowledge the pioneering observational and theoretical legacies of Arthur Eddington in solar eclipse tests and stellar structure work and David Finkelstein for his 1958 conceptual clarification of horizons, following historical threads involving Albert Einstein, Karl Schwarzschild, Hermann Weyl, Subrahmanyan Chandrasekhar, Roger Penrose, and institutional contexts at Cambridge University, University of Chicago, Yale University, and Bell Labs.

Category:General relativity