Comoving Coordinates

Comoving coordinates are a coordinate system attached to the expanding fabric of the universe rather than to physical rulers. An object that simply follows the cosmic expansion — neither pushed nor pulled by local forces — keeps the same comoving coordinates forever, even as the proper distance to it changes. A galaxy at comoving radius r = 5 stays at r = 5 for all time; what changes is the conversion factor between that label and a physical distance, namely the scale factor a(t).

The proper distance measured by laying physical rulers end to end is the comoving distance χ multiplied by the scale factor: d_proper(t) = a(t)·χ. Today, with a normalized to 1, proper and comoving distances coincide numerically. In the past they differed because a was smaller. Differentiating the relation reproduces the Hubble law, ȧχ = (ȧ/a)·d = H·d, showing that recession velocity is proportional to distance precisely because expanding space, not local motion, separates comoving objects.

Adopting comoving coordinates is what makes cosmology tractable and dissolves a famous confusion. Because comoving objects do not move through space, two galaxies can have a proper separation that grows faster than light without any object exceeding c — the speed limit of special relativity applies to motion through space, not to the expansion of space itself. The set of all comoving observers also defines a single universal cosmic time, the time read by clocks at rest in the local expansion, in which the universe is homogeneous.

The comoving framework emerged with the Robertson–Walker form of the metric in the 1930s, which made cosmic time and comoving radius the natural coordinates in which a homogeneous, isotropic universe is manifestly simple.