Comoving and Proper Coordinates

Every distance in cosmology wears two labels at once, and confusing them is the single richest source of paradox in the subject. The first label is comoving: a permanent address, glued to the cosmic fluid, that a galaxy riding the mean expansion keeps unchanged for all cosmic time. The second is proper: the physical separation you would read off right now with rulers laid end to end on a surface of constant cosmic time. The cosmological principle — homogeneity and isotropy — singles out a privileged family of observers, those at rest with respect to the matter and the cosmic microwave background, and it is natural to freeze a coordinate grid onto them. In the Friedmann–Lemaître–Robertson–Walker (FLRW) line element this choice is stark: the time-dependence of the whole spatial geometry is squeezed into one overall factor, the scale factor $a(t)$ , so that proper distance is nothing but comoving distance multiplied by $a(t)$ . That one relation, $d_{\rm prop}(t)=a(t)\,\chi$ , is the master key. Differentiate it and Hubble's law $v=H\,d$ falls out exactly — not as a small-velocity approximation but as the definition of recession. Track a light wave across the fixed comoving grid and the redshift $1+z=a_0/a_{\rm emit}$ appears with no Doppler bookkeeping. Ask whether two patches of sky could ever have exchanged a signal and the answer reduces to comparing comoving intervals, because in conformal time light travels on tidy $45°$ lines. Comoving coordinates make the dynamics and causal structure simple; proper quantities are what telescopes measure. Fluency in translating between them is the foundation of redshift, horizons, and the whole cosmic distance ladder.

Key takeaways

Comoving coordinates are spatial labels carried along by the cosmic fluid: a galaxy in the Hubble flow keeps fixed comoving coordinates $(\chi,\theta,\phi)$ forever.
Proper distance is the instantaneously measured physical separation; it equals the comoving distance times the scale factor, $d_{\rm prop}(t)=a(t)\,\chi$ , so it grows with expansion even while the labels stay frozen.
The FLRW metric puts cosmic time $t$ (the proper time of comoving clocks) outside, and hides all time dependence of space inside a single factor $a(t)^2$ .
Differentiating $d_{\rm prop}=a\chi$ gives Hubble's law $v=\dot a\chi=H\,d_{\rm prop}$ exactly; it exceeds $c$ beyond the Hubble radius $c/H$ without violating relativity.
Light on the fixed comoving grid stretches with $a$ , giving $1+z=a_0/a_{\rm emit}$ ; in conformal time $\mathrm{d}\eta=\mathrm{d}t/a$ null rays run on $45°$ lines, so horizons become simple comoving intervals.
Key idea: write every physical separation as $d_{\rm prop}(t)=a(t)\,\chi$ with $\chi$ a frozen comoving label and $a(t)$ the universal scale factor — redshift, recession, and horizons all follow by differentiating or integrating this one relation.

The core insight

Write physical separations as $d_{\rm prop}(t)=a(t)\,\chi$ , with $\chi$ a fixed comoving coordinate and $a(t)$ the universal scale factor; the whole kinematics of an expanding universe follows from differentiating or integrating this single relation.

Where this lives today

The comoving–proper split is the working language of every quantitative measurement in cosmology. Simulations of structure formation — from the Millennium run to today's billion-particle boxes — are integrated in comoving coordinates precisely so the mean Hubble expansion factors out, leaving only peculiar displacements relative to the grid as genuine dynamics. The cosmic microwave background fixes the comoving rest frame observationally: our motion relative to it appears as a dipole anisotropy $\Delta T\approx3.36\times10^{-3}\ \mathrm{K}$ (Planck 2018), a Solar-System velocity of about $370\ \mathrm{km\,s^{-1}}$ ; subtract it and the sky is isotropic to one part in $10^5$ — a direct validation of the cosmological principle that underwrites comoving coordinates. Baryon acoustic oscillations, the frozen $\sim150\ \mathrm{Mpc}$ comoving sound-horizon ruler, serve as a standard comoving length measured by DESI and Euclid to pin down $H(z)$ and the dark-energy equation of state. And the "Hubble tension" — the $\sim5\sigma$ gap between locally measured $H_0\approx73$ and CMB-inferred $H_0\approx67\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ — is at bottom an argument about how to translate a fixed comoving geometry into the proper distances and velocities that rulers and clocks report.

The physical picture

Paint galaxies as dots on the surface of a balloon. The latitude and longitude of each dot are its comoving coordinates and never change as the balloon inflates. What grows is the rubber-surface distance between any two dots — the proper distance — in exact proportion to the balloon's radius, which plays the role of $a(t)$ . Nothing crawls across the rubber; the rubber itself stretches. This is the precise content of the slogan "space expands": the labels stay put, the physical separations grow. Equivalently, comoving coordinates are stretchable graph paper glued to the matter — a galaxy's grid label is fixed, but a physical ruler has fixed proper length, so the number of rulers needed to reach that label grows with time. The first figure freezes this at two epochs.

Figure

Comoving grid at two epochs — galaxies keep the same comoving labels χ while every cell stretches with the scale factor a(t). A pair 150 Mpc apart in comoving distance sits 75 Mpc apart when a = 0.5 and 150 Mpc apart today at a₀ = 1, since d_prop = a χ

The second picture is where the label-confusion actually bites. Differentiating the master relation for a galaxy pinned to the grid gives a recession speed strictly proportional to proper distance, $v=H\,d_{\rm prop}$ . Plot that line and it crosses the speed of light at a finite distance — the Hubble radius $c/H$ — beyond which comoving galaxies recede faster than light. This is no paradox: $v$ is the time-derivative of an integrated proper distance stitched across many local frames, not a velocity measured at one event. Remarkably, many galaxies now receding faster than $c$ are still visible, because the comoving Hubble radius evolves and light emitted long ago from beyond today's Hubble sphere can still be crossing into it.

Figure

Recession velocity of the Hubble flow v = H₀ d is exactly linear in proper distance. With H₀ = 70 km s⁻¹ Mpc⁻¹ it reaches c at the Hubble radius c/H₀ = 4.28 Gpc ≈ 14.0 Gly; beyond that galaxies recede faster than light, yet many remain observable because the comoving Hubble radius grows

A third idea completes the toolkit: conformal time $\eta$ , defined by $\mathrm{d}\eta=\mathrm{d}t/a$ . In the pair $(\eta,\chi)$ the FLRW metric is a scale factor times flat Minkowski space, so radial light rays satisfy $\Delta\chi=\pm c\,\Delta\eta$ — they run on $45°$ lines exactly as in special relativity, and every causal question collapses into flat-space geometry on the fixed comoving grid. Comoving = "who", proper = "how far right now", conformal time = "the clock that makes light straight".

The core equations

The starting point is the FLRW line element written in comoving coordinates. Homogeneity and isotropy force the spatial slices to be maximally symmetric, and the only freedom left is an overall time-dependent size — the scale factor. Cosmic time $t$ is chosen to be the proper time of the fundamental observers.

\mathrm{d}s^2=-c^2\,\mathrm{d}t^2+a(t)^2\left[\frac{\mathrm{d}r^2}{1-kr^2}+r^2\left(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2\right)\right] \qquad (1)

where: $t$ — cosmic time, the proper time of comoving observers; $a(t)$ — dimensionless scale factor (conventionally $a_0\equiv1$ today); $r,\theta,\phi$ — comoving spatial coordinates, fixed for each fluid element; $k$ — spatial curvature constant, rescaled to $+1,0,-1$ . Read it as: all the time dependence of space is quarantined inside the single factor $a(t)^2$ . For radial work it is cleaner to trade $r$ for the comoving radial coordinate $\chi$ that measures proper radial distance divided by $a$ . The curvature is then carried by a single function $S_k(\chi)$ multiplying the angular part.

\mathrm{d}s^2=-c^2\,\mathrm{d}t^2+a(t)^2\left[\mathrm{d}\chi^2+S_k(\chi)^2\,\mathrm{d}\Omega^2\right] \qquad (2)

where: $\chi$ — comoving radial distance, $\mathrm{d}\chi=\mathrm{d}r/\sqrt{1-kr^2}$ ; $S_k(\chi)=\sin\chi,\ \chi,\ \sinh\chi$ for $k=+1,0,-1$ ; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ . Read it as: along a radial line the metric is simply $\mathrm{d}s=a\,\mathrm{d}\chi$ , which makes the next equation immediate. The proper distance to a comoving object is the integral of that radial element on a slice of constant cosmic time. Because $a(t)$ is the same everywhere on the slice, it pulls straight out of the integral.

d_{\rm prop}(t)=\int_0^{\chi}a(t)\,\mathrm{d}\chi'=a(t)\,\chi \qquad (3)

where: $d_{\rm prop}$ — instantaneous proper (physical) distance at time $t$ ; $\chi$ — the fixed comoving distance to the object. Read it as the master relation of the whole article: physical separation equals a frozen label times a universal clock-driven scale. Differentiating the master relation for a galaxy at fixed comoving $\chi$ gives Hubble's law. It is exact — the linearity in distance is built into the geometry, not an expansion for small $v$ .

v_{\rm rec}=\dot d_{\rm prop}=\dot a\,\chi=\frac{\dot a}{a}\,d_{\rm prop}=H(t)\,d_{\rm prop} \qquad (4)

where: $v_{\rm rec}$ — recession velocity, the rate of change of proper distance; $H=\dot a/a$ — Hubble parameter. Read it as: $v_{\rm rec}=c$ at the Hubble radius $d_{\rm prop}=c/H$ , and $v_{\rm rec}>c$ beyond it, with no local frame ever seeing a superluminal passage. Conformal time rescales the clock so that expansion multiplies the entire flat metric. It is the coordinate in which light behaves as in special relativity.

\mathrm{d}\eta=\frac{\mathrm{d}t}{a(t)},\qquad \mathrm{d}s^2=a(\eta)^2\left[-c^2\,\mathrm{d}\eta^2+\mathrm{d}\chi^2+S_k^2\,\mathrm{d}\Omega^2\right] \qquad (5)

where: $\eta$ — conformal time; $a(\eta)$ — scale factor as a function of it. Read it as: radial null rays obey $\Delta\chi=\pm c\,\Delta\eta$ , so causal diagrams are drawn with straight $45°$ light cones. Tracking a light wave across the fixed comoving grid yields the cosmological redshift, tying the observed stretch directly to how much the universe has grown since emission.

1+z=\frac{\lambda_{\rm obs}}{\lambda_{\rm emit}}=\frac{a(t_0)}{a(t_{\rm emit})} \qquad (6)

where: $z$ — observed cosmological redshift; $t_0,t_{\rm emit}$ — times of observation and emission. Read it as: redshift is a direct ruler of the scale factor at emission, not a velocity in disguise. Finally, the comoving distance to a source is the total comoving increment light covers between emission and now — equivalently, $c$ times the elapsed conformal time.

d_C\equiv c\int_{t_{\rm emit}}^{t_0}\frac{\mathrm{d}t}{a(t)}=c\,(\eta_0-\eta_{\rm emit}) \qquad (7)

where: $d_C$ — comoving distance (with $a_0=1$ it is also the present proper distance); $\eta_0-\eta_{\rm emit}$ — conformal lookback time. Read it as: every operational distance in cosmology is built from this integral, then dressed with factors of $(1+z)$ .

Derivation AProper distance and the exactness of Hubble's law

The goal is to show, from the FLRW metric alone, that the proper distance to a comoving object is $a(t)\chi$ and its rate of change is exactly $H\,d_{\rm prop}$ — a strict identity, not a low-velocity approximation — and then to read off why the recession speed passes through $c$ at the Hubble radius without any local observer measuring a superluminal motion. We work on a constant-time slice with $k=0$ ; nothing essential changes for curved slices.

Step 1. Restrict the metric to a constant-time radial path: set $\mathrm{d}t=0$ and $\mathrm{d}\theta=\mathrm{d}\phi=0$ , so the line element collapses to $\mathrm{d}s=a(t)\,\mathrm{d}\chi$ . Why: proper distance is by definition measured simultaneously in cosmic time along a spatial direction, which kills the time and angular terms and leaves a pure radial ruler.

Step 2. Integrate from the observer at $\chi=0$ to the object at $\chi$ : because $a(t)$ is position-independent on the slice, $d_{\rm prop}=\int_0^\chi a\,\mathrm{d}\chi'=a(t)\,\chi$ . Why: homogeneity guarantees a single scale factor everywhere, so the geometry of the slice is Euclidean up to the overall stretch $a$ , and the integral is trivial.

Step 3. Differentiate at fixed comoving label, $\dot\chi=0$ : the product rule gives $\dot d_{\rm prop}=\dot a\,\chi=(\dot a/a)(a\chi)=H\,d_{\rm prop}$ . Why: a galaxy in the Hubble flow does not move in comoving coordinates, so the entire time dependence of its proper distance comes from $a(t)$ — making $v=Hd$ an identity rather than a fit.

Step 4. Set $v_{\rm rec}=c$ : this happens at $d_{\rm prop}=c/H$ , the Hubble radius, and for larger distances $v_{\rm rec}>c$ . Yet $v_{\rm rec}$ is $\mathrm{d}/\mathrm{d}t$ of an integrated proper distance summed across many local inertial frames, not a relative velocity at one event, so local Lorentz invariance ( $|v_{\rm local}|<c$ ) is untouched. Why: the speed limit of relativity constrains what a single observer sees passing by, and no such observer ever clocks a comoving neighbour moving faster than light; the superluminal number is a property of the global slicing, and photons from beyond the Hubble radius can still reach us because the comoving Hubble radius $c/(aH)$ evolves in time.

Go deeper · Derivation B · 1 minCosmological redshift from light on the comoving grid

Derivation BCosmological redshift from light on the comoving grid

The goal is to derive the exact relation $1+z=a_0/a_{\rm emit}$ by following two successive wavecrests along a radial null geodesic between a comoving source and observer, using only that both endpoints keep fixed comoving coordinates. The payoff is conceptual: the redshift is not a Doppler shift from a recession velocity but a direct imprint of how much the scale factor has grown, which is why the formula holds even when the recession speed exceeds $c$ .

Step 1. Impose the radial null condition $\mathrm{d}s^2=0$ with $\mathrm{d}\theta=\mathrm{d}\phi=0$ , giving $c\,\mathrm{d}t=a(t)\,\mathrm{d}\chi$ for an incoming ray. Why: light travels on null worldlines, and on the comoving grid this reads as a fixed comoving increment per unit conformal time, decoupling the geometry of propagation from the expansion history.

Step 2. Integrate to get the comoving distance the light covers, $\chi=\int_{t_{\rm emit}}^{t_0} c\,\mathrm{d}t/a(t)$ , and note this same $\chi$ applies to every crest because source and observer are comoving. Why: neither endpoint moves in comoving coordinates, so successive crests traverse the identical comoving span, which is the constraint that fixes the arrival cadence.

Step 3. Apply the equal- $\chi$ condition to crest $n$ and crest $n{+}1$ emitted a time $\delta t_{\rm e}$ apart and received $\delta t_0$ apart; subtracting the two integrals cancels the common bulk and leaves the boundary terms $\delta t_0/a(t_0)=\delta t_{\rm e}/a(t_{\rm emit})$ . Why: over one wave period the scale factor is essentially constant, so only the endpoints of the integration interval contribute, converting a statement about paths into one about clock rates.

Step 4. Convert intervals to wavelengths, $\lambda=c\,\delta t$ , to obtain $\lambda_0/\lambda_{\rm e}=a_0/a_{\rm emit}=1+z$ . Why: wavelength stretches in exact lockstep with proper distance, so the measured redshift is a direct readout of the expansion factor between emission and observation — a ruler for $a$ , not a speedometer.

Go deeper · Derivation C · 1 minWhy comoving observers are the freely-falling clocks that read cosmic time

Derivation CWhy comoving observers are the freely-falling clocks that read cosmic time

The goal is to justify the two properties that make comoving coordinates physically privileged rather than a bookkeeping choice: a comoving clock's proper time equals cosmic time $t$ , and a comoving worldline is a timelike geodesic — the observer is in free fall despite the surrounding expansion. Both follow from a short inspection of the FLRW metric and its Christoffel symbols.

Step 1. For a worldline with $\mathrm{d}x^i=0$ the metric gives $\mathrm{d}\tau^2=-\mathrm{d}s^2/c^2=\mathrm{d}t^2$ , so $\tau=t$ . Why: the time–time component is $g_{tt}=-c^2$ , a constant, so a clock that stays at fixed spatial coordinates simply ticks off cosmic time, which is what lets us speak of "the age of the universe" unambiguously.

Step 2. Compute the relevant connection coefficient: $\Gamma^i_{tt}=\tfrac12 g^{ij}(2\partial_t g_{jt}-\partial_j g_{tt})=0$ , since the metric is diagonal ( $g_{jt}=0$ ) and $g_{tt}$ is spatially constant. Why: this particular Christoffel is the "force" term that would push a stationary observer around, and its vanishing is the technical heart of the free-fall claim.

Step 3. Insert into the geodesic equation for a comoving four-velocity $u^\mu=(c,0,0,0)$ : $\mathrm{d}u^i/\mathrm{d}\tau+\Gamma^i_{tt}(u^t)^2=0$ reduces to $\mathrm{d}u^i/\mathrm{d}\tau=0$ , so $u^i=0$ is consistent — the observer stays comoving forever without thrust. Why: a geodesic is the worldline of an unforced particle, so comoving galaxies are not "held in place" against expansion; they are freely falling, and their apparent recession is pure geometry.

Step 4. Note the isotropy: about any comoving point the spatial metric is rotationally symmetric, so a comoving observer sees no preferred direction in the mean flow. Why: this is the cosmological principle made operational — the frame in which the universe looks isotropic is precisely the comoving frame, which is why the CMB rest frame and the comoving frame coincide.

Limiting cases

How the comoving–proper relation specializes in the standard cosmological regimes:

Case	Result	Meaning
Static limit	$a=\text{const}\Rightarrow d_{\rm prop}=a\chi=\text{const}$	No expansion: $H=0$ , proper and comoving distances differ only by a fixed conventional scale, no recession.
Matter-dominated (Einstein–de Sitter)	$a\propto t^{2/3}\Rightarrow H=2/(3t)$	Decelerating expansion; proper distance to a fixed comoving object grows sublinearly, $d_{\rm prop}\propto t^{2/3}\chi$ .
Radiation-dominated	$a\propto t^{1/2}\Rightarrow H=1/(2t)$	Early universe; proper distances grow slowly but the comoving horizon opens fast, $\chi_{\rm ph}\propto t^{1/2}$ .
de Sitter / dark-energy-dominated	$a\propto e^{Ht}\Rightarrow d_{\rm prop}=a\chi\propto e^{Ht}$	Exponential growth at fixed comoving $\chi$ ; a cosmic event horizon sits at fixed proper distance $c/H$ .

Common trapRecession velocity is a velocity through space that cannot exceed c

This is the central misconception the two-label picture exists to dissolve. The recession velocity $v=H\,d_{\rm prop}$ is not a local relative velocity through space; it is the time-derivative of an integrated proper distance stitched across many local inertial frames on a constant-time slice. Special relativity limits only relative velocities measured at a single event, where two worldlines actually cross — and no comoving observer ever measures a neighbour passing faster than $c$ . So $v>c$ beyond the Hubble radius violates nothing. The companion trap is thinking such galaxies are forever unreachable by their light: whether we see them depends on the comoving Hubble radius $c/(aH)$ and its evolution, not on today's recession speed — which is why we routinely observe galaxies (all sources with $z\gtrsim1.5$ in the real universe) whose present recession velocity exceeds $c$ .

Worked examples

1 · Superluminal recession at twice the Hubble radius (fully worked)

Take $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Find the Hubble radius, then the recession velocity of a galaxy whose present proper distance is twice that, and reconcile the result with relativity.

Step 1. The Hubble radius is $c/H_0=(2.998\times10^5\ \mathrm{km\,s^{-1}})/(70\ \mathrm{km\,s^{-1}\,Mpc^{-1}})=\mathbf{4283\ Mpc=4.28\ Gpc}$ , which in light-travel terms is $4283\times3.262\ \mathrm{Mly}=\mathbf{14.0\ Gly}$ . At exactly this distance a comoving galaxy recedes at $v=H_0\,(c/H_0)=c$ .

Step 2. For a galaxy at present proper distance $d_{\rm prop}=2\times4283=8566\ \mathrm{Mpc}$ , Hubble's law gives $v_{\rm rec}=H_0\,d_{\rm prop}=70\times8566=5.996\times10^5\ \mathrm{km\,s^{-1}}$ , i.e. $v_{\rm rec}/c=5.996\times10^5/2.998\times10^5=\mathbf{2.00}$ . The galaxy recedes at twice the speed of light.

Step 3. Nothing breaks. No local observer stationed along the way ever clocks a neighbour passing at $2c$ ; each measures the local relative speed to be small. The number $2c$ is the rate of growth of a proper distance summed across $\sim$ thousands of such frames. Hubble radius = 4.28 Gpc (14.0 Gly); a galaxy at 8.57 Gpc recedes at $v_{\rm rec}=2.00\,c$ , fully consistent with relativity.

2 · Proper distances and redshift in an Einstein–de Sitter universe (fully worked)

In a flat matter-dominated universe, $a\propto t^{2/3}$ , the comoving distance to redshift $z$ is $d_C=(2c/H_0)\left[1-(1+z)^{-1/2}\right]$ . For $z=1$ and $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , find the present proper distance, the proper distance at emission, the present recession velocity, and the observed wavelength of the Lyman-α line emitted at 121.6 nm.

Step 1. With $2c/H_0=2\times4283=8566\ \mathrm{Mpc}$ and $(1+z)^{-1/2}=1/\sqrt{2}=0.7071$ , the comoving distance is $d_C=8566\times(1-0.7071)=8566\times0.2929=\mathbf{2509\ Mpc}$ . Since $a_0=1$ , this is also the present proper distance, $d_{\rm prop}(t_0)=2509\ \mathrm{Mpc}$ .

Step 2. At emission the scale factor was $a_{\rm emit}=a_0/(1+z)=1/2$ , so the proper distance then was $d_{\rm prop}(t_{\rm emit})=a_{\rm emit}\,\chi=\tfrac12\times2509=\mathbf{1254\ Mpc}$ . The object was half as far in physical units when it emitted the light we now receive.

Step 3. The present recession velocity of that comoving source is $v_{\rm rec}=H_0\,d_{\rm prop}(t_0)=70\times2509=1.756\times10^5\ \mathrm{km\,s^{-1}}$ , i.e. $v_{\rm rec}/c=\mathbf{0.586}$ — subluminal, because $z=1$ sits inside the Hubble radius in this model.

Step 4. The wavelength stretches by exactly $1+z=2$ : the Lyman-α line at $\lambda_{\rm emit}=121.6\ \mathrm{nm}$ arrives at $\lambda_{\rm obs}=(1+z)\,\lambda_{\rm emit}=2\times121.6=\mathbf{243.2\ nm}$ . As a check on causal reach in this model, the particle horizon is a pure comoving interval $\chi_{\rm ph}=c\int_0^{t_0}\mathrm{d}t/a=2c/H_0=8566\ \mathrm{Mpc}$ (present proper size $\approx27.9$ Gly), exactly twice the Hubble radius. At $z=1$ (EdS): present proper distance 2509 Mpc, emission proper distance 1254 Mpc, $v_{\rm rec}=0.586\,c$ , and Lyα shifts $121.6\to243.2$ nm.

Physics of the portrait

The distinction between comoving and proper coordinates is inseparable from the man who first grasped that the universe expands, wrote the recession law two years before Hubble measured it, and understood galaxies as fixed markers carried apart by growing space.

Origins · 1 minGeorges Lemaître and how this was discovered

Georges Lemaître

1894–1966 · the expanding universe and the comoving description

A Belgian Catholic priest and mathematical physicist, Georges Lemaître is the father of modern cosmology. In his 1927 paper Un univers homogène de masse constante et de rayon croissant, he solved Einstein's equations for a homogeneous isotropic universe, found it must expand, and derived the linear relation $v=H\,d$ — complete with an estimate of the coefficient — two years before Edwin Hubble's observational paper of 1929. Crucially, Lemaître read the redshifts not as motion through space but as the stretching of space: galaxies sit at fixed comoving positions while the geometry between them grows, exactly the picture this article formalizes. He introduced the scale factor as the single dynamical size of the cosmos and saw that the natural coordinates ride with the matter. Einstein, who had told him "your calculations are correct, but your physics is abominable," came around after Hubble's data. Lemaître went further still, proposing in 1931 that expansion run backward implies a dense beginning — the "primeval atom," the seed of the Big Bang. The rigorous proof that the comoving-coordinate metric is the unique form consistent with homogeneity and isotropy came from Howard Robertson and Arthur Walker, building on Alexander Friedmann's 1922 solutions, which is why the metric now carries all four names.

History

1917

— Einstein and de Sitter build the first relativistic cosmological models, static by construction

1922

— Alexander Friedmann derives the expanding and contracting solutions of Einstein's equations

1927

— Georges Lemaître introduces comoving coordinates and the linear velocity–distance law v = H d

1929

— Edwin Hubble publishes the empirical redshift–distance relation from Cepheid distances

1935

— Robertson and Walker prove the FLRW metric is the unique homogeneous isotropic form

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §27 · Wald, General Relativity, Ch. 5 · Carroll, Spacetime and Geometry, Ch. 8 · Hartle, Gravity, Ch. 18 · Schutz, A First Course in General Relativity, Ch. 12 · Weinberg, Cosmology, Ch. 1 (for the redshift and distance measures). Key numerical anchors used above: Hubble radius $c/H_0=4.28\ \mathrm{Gpc}=14.0\ \mathrm{Gly}$ and Hubble time $1/H_0=14.0\ \mathrm{Gyr}$ at $H_0=70$ ; the Einstein–de Sitter comoving distance $d_C=(2c/H_0)[1-(1+z)^{-1/2}]$ giving $2509\ \mathrm{Mpc}$ at $z=1$ ; the CMB dipole $\Delta T\approx3.36\times10^{-3}\ \mathrm{K}$ and Solar-System velocity $\approx370\ \mathrm{km\,s^{-1}}$ (Planck 2018); the BAO comoving sound horizon $\approx150\ \mathrm{Mpc}$ ; and the CMB isotropy level $\sim10^{-5}$ .

Prerequisites. The Cosmological Principle · The FLRW Metric · The Scale Factor and Hubble Parameter. Leads to. Cosmological Redshift · Cosmological Horizons · The Friedmann Equations.

Ask the tutor: Ask me to walk you through why a galaxy receding at twice the speed of light is still shining in our telescopes — or to compute the proper distance and redshift for any expansion history you choose.

What is the difference between comoving and proper distance in one sentence?

Comoving distance is a permanent coordinate label, fixed to the cosmic fluid, that does not change as the universe expands; proper distance is the physical separation you would measure right now with rulers on a constant-time slice, and it equals comoving distance times the current scale factor, $d_{\rm prop}=a(t)\,\chi$ . Two galaxies keep the same comoving distance forever while their proper distance grows with $a(t)$ . Comoving answers "which galaxy?"; proper answers "how far is it at this instant?". Almost every apparent paradox in expanding-universe kinematics dissolves once you track which of the two a statement is about.

How can a galaxy recede faster than light without breaking relativity?

Because the recession velocity $v=H\,d_{\rm prop}$ is not a local velocity through space. It is the rate of change of a proper distance built by adding up ruler-lengths across many local inertial frames on a slice of constant cosmic time. Special relativity forbids one object from passing another faster than $c$ at a single event, and that never happens here: every local observer along the line of sight measures small relative speeds. The superluminal figure is a global, coordinate-dependent statement about how fast an integrated distance grows. Photons from beyond today's Hubble radius can still reach us because the comoving Hubble radius itself evolves, so "receding faster than $c$ " and "unobservable" are not the same thing.

Why is the cosmological redshift not just a Doppler shift?

A Doppler shift is computed from a relative velocity at the moment of emission or reception. The cosmological redshift instead depends on the whole expansion history between emission and observation, through $1+z=a_0/a_{\rm emit}$ : the wavelength stretches in exact proportion to the scale factor while the light travels. For nearby galaxies the two viewpoints agree numerically, so the low- $z$ relation looks Doppler-like. But at large $z$ they diverge, and the naive special-relativistic Doppler formula gives the wrong answer and wrongly caps the velocity at $c$ . The clean statement: redshift measures how much the universe has grown since the light set out — a ruler for $a(t)$ , not a speedometer.

Do comoving coordinates mean galaxies are not really moving?

On average, yes — a galaxy perfectly in the Hubble flow sits at fixed comoving coordinates and is in free fall; its apparent recession is pure geometry, the stretching of space between fixed markers. Real galaxies also carry a peculiar velocity, a genuine motion relative to the local comoving frame from the gravitational tugs of neighbours and clusters (ours is a few hundred $\mathrm{km\,s^{-1}}$ ). The observed redshift combines the cosmological part from expansion and a peculiar-velocity Doppler part. Comoving coordinates cleanly separate the two: subtract the smooth Hubble flow and only peculiar motions remain as real dynamics — exactly how $N$ -body simulations and peculiar-velocity surveys are organized.

Why does cosmic time equal the proper time of comoving observers?

Because the FLRW metric has a constant time–time component, $g_{tt}=-c^2$ , with no cross terms between time and space. For a worldline at fixed spatial coordinates ( $\mathrm{d}x^i=0$ ), the line element reduces to $\mathrm{d}\tau^2=-\mathrm{d}s^2/c^2=\mathrm{d}t^2$ , so the observer's own clock ticks off cosmic time directly. This is what makes "the universe is 13.8 billion years old" well-defined — it is the proper time elapsed on a comoving clock since the Big Bang. The same features that give $\tau=t$ also make $\Gamma^i_{tt}$ vanish, so comoving observers follow geodesics.

What makes conformal time so useful, and how does it differ from cosmic time?

Cosmic time $t$ is what a comoving clock reads. Conformal time $\eta$ , defined by $\mathrm{d}\eta=\mathrm{d}t/a$ , is a rescaled clock that runs faster when the universe is small and slower when it is large. Its magic is that it turns the FLRW metric into an overall factor $a^2$ times flat Minkowski space, so radial light rays obey $\Delta\chi=\pm c\,\Delta\eta$ — they move on straight $45°$ lines exactly as in special relativity. This makes causal questions trivial to draw: the particle horizon is simply how much comoving distance light has covered in the elapsed conformal time, and whether two regions could ever have communicated reduces to comparing comoving intervals. Cosmic time is best for dynamics and ages; conformal time is best for causality and horizons.