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| FLRW metric | |
|---|---|
| Name | FLRW metric |
| Introduced | 1922–1927 |
| Authors | Alexander Friedmann, Georges Lemaître, Howard P. Robertson, Arthur G. Walker |
| Field | General relativity |
FLRW metric The FLRW metric is the class of homogeneous and isotropic cosmological solutions used in General relativity to model expanding or contracting universes on large scales. Developed through contributions by Alexander Friedmann, Georges Lemaître, Howard P. Robertson, and Arthur G. Walker, it underpins the standard Big Bang cosmology and provides the geometric foundation for interpreting observations from projects such as the Hubble Space Telescope and the Planck mission. The metric enters the theoretical framework that connects Albert Einstein’s field equations to the Friedmann equations and empirical probes including the Cosmic microwave background and Type Ia supernova surveys.
The FLRW metric arises as the most general spatially homogeneous and isotropic line element consistent with Albert Einstein’s field equations under the assumptions of the cosmological principle. Historical work by Alexander Friedmann produced time-dependent solutions; later synthesis by Georges Lemaître, Howard P. Robertson, and Arthur G. Walker formalized the symmetry classification used in modern cosmology. This metric provides the stage on which phenomena studied by collaborations such as the Sloan Digital Sky Survey, Supernova Cosmology Project, and Baryon Oscillation Spectroscopic Survey are interpreted.
In comoving coordinates the FLRW line element is written using a scale factor a(t) and a spatial curvature parameter k, connecting to Albert Einstein’s field equations and the stress–energy content described by Andrei Sakharov-motivated fluids and fields. The form is commonly expressed in terms of temporal coordinate t, radial coordinate r, and angular coordinates θ and φ; alternative coordinate choices relate to metrics used by Kruskal–Szekeres coordinates and coordinate charts developed in David Hilbert-inspired formulations. The metric components feed directly into the Christoffel symbols and Ricci curvature tensors needed to derive the Friedmann equations as shown in treatments by Lev Landau, Evgeny Lifshitz, and expositions in texts from Cambridge University Press and Princeton University Press.
Spatial slices of constant cosmological time are maximally symmetric 3-manifolds with curvature sign determined by k, paralleling geometries classified by Bernhard Riemann and exemplified in constant-curvature spaces studied by Carl Friedrich Gauss and Nikolai Lobachevsky. Isotropy about every point is implemented by the action of the rotation group and the homogeneous nature ties to Lie groups used in classification work by Élie Cartan. Killing vectors and the associated isometry groups simplify the analysis used in Mathematical Reviews-level treatments and in the symmetry arguments employed by Noether theorem-based conservation considerations found in the literature of Yakov Zel'dovich and Igor Novikov.
The dynamics come from substituting the FLRW ansatz into Albert Einstein’s field equations with a stress–energy tensor usually modeled as a perfect fluid characterized by energy density ρ(t) and pressure p(t), concepts developed in the context of research by Lev Landau and Evgeny Lifshitz. This yields the Friedmann equations—first and second Friedmann relations—relating a(t), ρ(t), p(t), and k, and introducing the Hubble parameter H(t). Solutions and parameter constraints are central to investigations by the Planck collaboration, the WMAP team, and analyses such as those by the Supernova Cosmology Project that inferred acceleration attributed to Dark energy and the Cosmological constant Λ originally proposed by Albert Einstein.
Specific equation-of-state choices produce standard cosmological epochs and models: radiation-dominated solutions tied to early-universe physics studied by George Gamow and Ralph Alpher; matter-dominated solutions relevant to structure-formation work by P. J. E. Peebles and J. Richard Gott; and Λ-dominated accelerating solutions consistent with work by the Supernova Cosmology Project and the High-Z Supernova Search Team. Inflationary extensions built on FLRW backgrounds were developed in models by Alan Guth, Andrei Linde, and Alexei Starobinsky, and are tested by datasets analyzed by Planck and ground-based arrays like the Atacama Cosmology Telescope.
Predictions of the FLRW framework determine angular-diameter and luminosity-distance relations used in analyses by the Hubble Space Telescope Key Project, the Sloan Digital Sky Survey, and the Baryon Oscillation Spectroscopic Survey. The metric underlies interpretation of the Cosmic microwave background anisotropy power spectrum measured by WMAP and Planck, baryon acoustic oscillation features cataloged by 2dF Galaxy Redshift Survey and Sloan Digital Sky Survey, and cosmological parameter estimation efforts coordinated by institutions such as NASA and the European Space Agency.
Generalizations relax homogeneity or isotropy: anisotropic cosmologies like the Bianchi classification and inhomogeneous solutions such as the Lemaître–Tolman metric extend FLRW to model effects studied in gravitational-collapse work by Roger Penrose and Stephen Hawking. Alternative-gravity theories replacing General relativity with frameworks like f(R) gravity, Brans–Dicke theory, and other scalar–tensor models produce modified cosmological dynamics explored by research groups at institutions including Caltech, Harvard University, and Princeton University. Numerical-relativity studies connecting FLRW backgrounds to nonlinear structure formation are pursued by collaborations associated with Max Planck Society and large computing centers.