[1][2][3] The general form of the metric follows from the geometric properties of homogeneity and isotropy.
Depending on geographical or historical preferences, the set of the four scientists – Alexander Friedmann, Georges Lemaître, Howard P. Robertson and Arthur Geoffrey Walker – are variously grouped as Friedmann, Friedmann–Robertson–Walker (FRW), Robertson–Walker (RW), or Friedmann–Lemaître (FL).
When combined with Einstein's field equations the metric gives the Friedmann equation which as been developed in to the Standard Model of modern cosmology,[4] and the further developed Lambda-CDM model.
The metric is a consequence of assuming that the mass in the universe has constant density – homogeneity – and is the same in all directions – isotropy.
Assuming isotropy alone is sufficient to reduce the possible motions of mass in the universe to radial velocity variations.
[5]: 408 Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.
The equivalence principle applied to each galaxy means distance measurements can be made using special relativity locally.
, which plays a critical role in cosmology, has an analog in the radius of a sphere.
The radius of a sphere lives in the third dimension: it is not part of the 2 dimensional surface.
However, the value of this radius affects distances measure on the two dimensional surface.
[7]: 147 To apply the metric to cosmology and predict its time evolution requires Einstein's field equations together with a way of calculating the density,
This process allows an approximate analytic solution Einstein's field equations
giving the Friedmann equations when the energy–momentum tensor is similarly assumed to be isotropic and homogeneous.
[6]: 73 Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.
[8]: 25.1.3 The FLRW metric assume homogeneity and isotropy of space.
In reduced-circumference polar coordinates the spatial metric has the form[10][11]
There are two common unit conventions: A disadvantage of reduced circumference coordinates is that they cover only half of the 3-sphere in the case of positive curvature—circumferences beyond that point begin to decrease, leading to degeneracy.
(This is not a problem if space is elliptical, i.e. a 3-sphere with opposite points identified.)
As before, there are two common unit conventions: Though it is usually defined piecewise as above, S is an analytic function of both k and r. It can also be written as a power series
is one of the imaginary, zero or real square roots of k. These definitions are valid for all k. When k = 0 one may write simply
FLRW space using Cartesian coordinates, the surviving components of the Ricci tensor are[12]
In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are[13][failed verification]
In 1922 and 1924 the Soviet mathematician Alexander Friedmann[14][15] and in 1927, Georges Lemaître, a Belgian priest, astronomer and periodic professor of physics at the Catholic University of Leuven, arrived independently at results[16][17] that relied on the metric.
Howard P. Robertson from the US and Arthur Geoffrey Walker from the UK explored the problem further during the 1930s.
[18][19][20][21] In 1935 Robertson and Walker rigorously proved that the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic (as noted above, this is a geometric result and is not tied specifically to the equations of general relativity, which were always assumed by Friedmann and Lemaître).
This solution, often called the Robertson–Walker metric since they proved its generic properties, is different from the dynamical "Friedmann–Lemaître" models, which are specific solutions for a(t) that assume that the only contributions to stress–energy are cold matter ("dust"), radiation, and a cosmological constant.
By combining the observation data from some experiments such as WMAP and Planck with theoretical results of Ehlers–Geren–Sachs theorem and its generalization,[26] astrophysicists now agree that the early universe is almost homogeneous and isotropic (when averaged over a very large scale) and thus nearly a FLRW spacetime.
That being said, attempts to confirm the purely kinematic interpretation of the Cosmic Microwave Background (CMB) dipole through studies of radio galaxies [27] and quasars [28] show disagreement in the magnitude.
Taken at face value, these observations are at odds with the Universe being described by the FLRW metric.
Moreover, one can argue that there is a maximum value to the Hubble constant within an FLRW cosmology tolerated by current observations,