The –Lemaître–Robertson–Friedmann–Lemaître–Robertson–Walker metric (FLRW; ) is a metric based on the exact solution of the Einstein field equations of general relativity. The metric describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. 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). This model is sometimes called the Standard Model of modern cosmology, although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 1930s.
General metric
The FLRW metric starts with the assumption of homogeneity and isotropy of space. It also assumes that the spatial component of the metric can be time-dependent. The generic metric which meets these conditions is
where ranges over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below. does not depend on t – all of the time dependence is in the function a(t), known as the "scale factor".
Reduced-circumference polar coordinates
In reduced-circumference polar coordinates the spatial metric has the form
k is a constant representing the curvature of the space. There are two common unit conventions: k may be taken to have units of length−2, in which case r has units of length and a(t) is unitless. k is then the Gaussian curvature of the space at the time when a(t) = 1. r is sometimes called the reduced circumference because it is equal to the measured circumference of a circle (at that value of r), centered at the origin, divided by 2 (like the r of Schwarzschild coordinates). Where appropriate, a(t) is often chosen to equal 1 in the present cosmological era, so that measures comoving distance. Alternatively, k may be taken to belong to the set (for negative, zero, and positive curvature respectively). Then r is unitless and a(t) has units of length. When , a(t) is the radius of curvature of the space, and may also be written R(t).
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.)
Hyperspherical coordinates
In hyperspherical or curvature-normalized coordinates the coordinate r is proportional to radial distance; this gives
where is as before and
As before, there are two common unit conventions: k may be taken to have units of length−2, in which case r has units of length and a(t) is unitless. k is then the Gaussian curvature of the space at the time when a(t) = 1. Where appropriate, a(t) is often chosen to equal 1 in the present cosmological era, so that measures comoving distance. Alternatively, as before, k may be taken to belong to the set (for negative, zero, and positive curvature respectively). Then r is unitless and a(t) has units of length. When , a(t) is the radius of curvature of the space, and may also be written R(t). Note that when , r is essentially a third angle along with θ and φ. The letter χ may be used instead of r.
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
or as
where sinc is the unnormalized sinc function and is one of the imaginary, zero or real square roots of k. These definitions are valid for all k.
Cartesian coordinates
When k = 0 one may write simply
This can be extended to by defining
, , and , where r is one of the radial coordinates defined above, but this is rare.
Curvature
Cartesian coordinates
In flat FLRW space using Cartesian coordinates, the surviving components of the Ricci tensor are
and the Ricci scalar is
Spherical coordinates
In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are