The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity and which describes a homogeneous, isotropic expanding/contracting universe. Depending on geographical/historical preferences, this may be referred to under the names of a preferred subset of the four scientists Alexander Friedmann, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker, e.g. Friedmann-Robertson-Walker (FRW) or Robertson-Walker (RW).

The FLRW metric is used as a first approximation for the standard big bang cosmological model of the universe. Because the FLRW assumes homogeneity, some popular accounts mistakenly assert that the big bang model cannot account for the observed lumpiness of the universe. In actuality, the FLRW is used as a first approximation for the evolution of the universe because it is simple to calculate, and models which calculate the lumpiness in the universe are added onto FLRW as extensions. As of 2003, the theoretical implications of the various extensions to FLRW appear to be well understood, and the goal is to make these consistent with observations from COBE and WMAP.

The metric can be written as

$ds^2\; =\; c^2\; dt^2-a(t)^2d\backslash Omega^2$

where:

$a(t)\; =$ the scale factor of the universe at time t

$\backslash bar\{r\}\; =\backslash begin\{cases\}$

R_C \sinh(r/R_C), &\mbox{for negative curvature} \\ r, &\mbox{for zero curvature} \\R_C \sin(r/R_C), &\mbox{for positive curvature} \end{cases}

where $R\_C\; =$the absolute value of the radius of curvature

$d\backslash Omega^2\; =\; d\backslash theta^2+\backslash sin^2\backslash theta\; d\backslash phi^2$

In this formulation of the metric,

$r$ gives the comoving distance from the observer

$\backslash bar\{r\}$ gives the proper motion distance.

Under the assumptions of homogeneity and isotropy, together with an appropriate energy-momentum tensor, the Einstein field equations reduce to the Friedmann equations. The solution of the Friedmann equations is the FLRW metric. Most cosmologists agree that the observable universe is well approximated by an almost FLRW model, that is, a model which follows the FLRW metric apart from primordial density fluctuations. In a strictly FLRW model, there are no clusters of galaxies, stars or people, since these are objects much denser than a typical part of the universe.

However, for brevity, the almost FLRW model is often referred to simply as the FLRW model (or the FRW model).

External links

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References

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• . See chapter 23 for a particularly clear and concise introduction to the FLRW models.

Coordinate charts in general relativity

Mètrica FLRW | Friedman-Lemaître-Robertson-Walker | Friedmann-Lemaître-Robertson-Walker | Metrica di Friedmann - Lemaître - Robertson - Walker | Metryka Friedmana-Lemaître'a-Robertsona-Walkera