FLRW Metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is the line element describing a spatially homogeneous and isotropic universe. In reduced-circumference coordinates it reads ds² = −c²dt² + a(t)²[dr²/(1 − kr²) + r²(dθ² + sin²θ dφ²)], where a(t) is the scale factor, r is a comoving radial coordinate, and k ∈ {−1, 0, +1} is the constant sign of the spatial curvature. The time part is identical to flat spacetime; the entire space part is multiplied by a(t)² so that proper distances grow or shrink with the expansion.

The metric's form is not assumed but forced. The cosmological principle — that on large scales no place and no direction is special — restricts space to one of exactly three maximally symmetric geometries: flat (k = 0), positively curved and finite (k = +1, a 3-sphere), or negatively curved and infinite (k = −1, hyperbolic). All of the time evolution is then carried by the single function a(t), whose behaviour is determined by the Friedmann equations once the contents of the universe are specified.

The FLRW metric is the foundation of physical cosmology. From it follow the Hubble law (v = H·d), the cosmological redshift (1 + z = a_now/a_emit), the notion of a universal cosmic time, and the framework for tracing the universe back to a hot, dense early state. Measurements of the cosmic microwave background show the real universe is flat (k = 0) to within a fraction of a percent, making the simplest FLRW case the working model of the cosmos.

Built on Alexander Friedmann's dynamic solutions (1922, 1924) and Georges Lemaître's independent rediscovery and observational link (1927), then cast in its general, coordinate-independent form by Howard Robertson and Arthur Walker in the 1930s; all four names label the metric.