Friedmann–Lemaître–Robertson–Walker metric

Friedmann–Lemaître–Robertson–Walker metric
NASA / WMAP Science Team · Public domain · source
Name	Friedmann–Lemaître–Robertson–Walker metric
Field	Cosmology, General relativity
Introduced	1922–1935
Notable people	Alexander Friedmann, Georges Lemaître, Howard P. Robertson, Arthur Geoffrey Walker

Friedmann–Lemaître–Robertson–Walker metric The Friedmann–Lemaître–Robertson–Walker metric is a family of exact solutions in Albert Einstein's General relativity that model homogeneous, isotropic expanding or contracting universes such as those described in Friedmann equations, Lambda-CDM model, and early Big Bang scenarios. It underpins modern observational programs like Hubble Space Telescope, Planck (spacecraft), and Dark Energy Survey while connecting to theoretical work by Alexander Friedmann, Georges Lemaître, Howard P. Robertson, and Arthur Geoffrey Walker.

Overview and definition

The metric defines a spacetime geometry with maximal spatial symmetry used in models such as Einstein–de Sitter universe, Lambda-CDM model, inflationary cosmology, and analyses by George Gamow and Vesto Slipher; it is specified by a scale factor a(t), a curvature parameter k, and a spatial metric of constant curvature relevant to observations from Hubble (telescope), WMAP, and missions like Planck (spacecraft). The metric is central to theoretical frameworks developed by Alexander Friedmann, Georges Lemaître, and later formalized by Howard P. Robertson and Arthur Geoffrey Walker in works connected to Royal Society and institutions like Princeton University and Cambridge University.

Mathematical form and properties

In comoving coordinates the line element is ds^2 = -c^2 dt^2 + a(t)^2 [dr^2/(1 - kr^2) + r^2 dΩ^2], a form used in calculations by Albert Einstein, Edwin Hubble, Robertson and applied in analyses by Stephen Hawking, Roger Penrose, and Kip Thorne. The scale factor a(t) enters the Friedmann equations derived from Einstein field equations with stress–energy tensors as in works by Lev Landau, Evgeny Lifshitz, and Subrahmanyan Chandrasekhar. Key properties include maximal spatial symmetry under isometry groups related to Felix Klein's Erlangen program and classification theorems studied at University of Cambridge and ETH Zurich.

Cosmological solutions and dynamics

Solutions for a(t) include matter-dominated, radiation-dominated, and vacuum-dominated cases employed in Big Bang cosmology, Cosmic microwave background, and structure formation studies involving James Peebles, Alan Guth, Andrei Linde, and Guth–Linde inflationary models. The metric yields the Friedmann equations that relate a(t), energy density ρ, pressure p, and the cosmological constant Λ appearing in debates involving Albert Einstein, Willem de Sitter, and Paul Dirac. Specific solutions like the Einstein static universe, de Sitter space, and anti-de Sitter space are limits used in analyses by Arthur Eddington, Willem de Sitter, and researchers at Institute for Advanced Study.

Spatial curvature and coordinate systems

Spatial sections are constant-curvature three-manifolds classified as spherical, flat, or hyperbolic—linked to theories by Bernhard Riemann, Carl Friedrich Gauss, and handled with coordinates introduced by Arthur Geoffrey Walker, Howard P. Robertson, and in textbooks from Princeton University Press and Cambridge University Press. Coordinate choices include comoving, conformal, and proper distance coordinates employed in observational pipelines at European Space Agency, National Aeronautics and Space Administration, and in data analyses by Planck Collaboration and Sloan Digital Sky Survey. The curvature parameter k connects to topology studies related to William Thurston and global analyses by Stephen Hawking.

Relation to observational cosmology

Predictions from the metric feed directly into observables such as redshift–distance relations measured by Edwin Hubble, luminosity distances used in Supernova Cosmology Project and High-Z Supernova Search Team results that involved Saul Perlmutter, Brian Schmidt, and Adam Riess, and angular-diameter relations probed by Planck (spacecraft) and Atacama Cosmology Telescope. Constraints on parameters a(t), Λ, and curvature k derive from datasets assembled by Sloan Digital Sky Survey, Dark Energy Survey, WMAP, and gravitational-wave standard-siren observations by collaborations like LIGO Scientific Collaboration and Virgo (detector).

Derivation from symmetry principles

The metric follows from imposing spatial homogeneity and isotropy—symmetries analyzed by Élie Cartan, Hermann Weyl, and formal group-theoretic methods inspired by Felix Klein and applied in relativity by Weyl and Robertson. Demanding a maximally symmetric three-space yields the Robertson–Walker form via Killing vector analyses that appear in treatments by Roger Penrose, Kip Thorne, and in lectures at Princeton University and Cambridge University.

Extensions and generalizations

Generalizations include anisotropic models like Bianchi classification solutions studied by L. Bianchi, George Ellis, and Menahem Yaakov, inhomogeneous models such as Lemaître–Tolman metric used by Georges Lemaître and Richard Tolman, and perturbation theory developed by James Peebles, Yakov Zel'dovich, and John M. Bardeen for structure formation. Other extensions connect to modified gravity proposals by researchers at Perimeter Institute, Institut des Hautes Études Scientifiques, and groups working on f(R) gravity, Brans–Dicke theory, and quantum cosmology programs linked to Stephen Hawking, Martin Rees, and Carlo Rovelli.

Category:Cosmology