The FLRW Metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is the load-bearing wall of modern cosmology: every distance, every redshift, and every age we quote for the universe is computed inside it. Its content is a theorem about symmetry. Ask only that the universe look the same at every point (homogeneity) and the same in every direction (isotropy) on large scales — the Cosmological Principle — and Robertson and Walker proved in the 1930s that these two demands alone fix the spacetime line element up to a single function of time, the scale factor $a(t)$ , and a single discrete constant, the curvature index $k \in \{-1,0,+1\}$ . Ten free functions of a general metric collapse to one function and one sign. The dynamics of $a(t)$ are then handed off to the Friedmann equations, which come from Einstein's field equations plus an assumed matter content; the metric supplies the stage, the field equations supply the play. Physically the metric splits spacetime into a universal cosmic time $t$ — the proper time of comoving observers who see the microwave sky as isotropic — and spatial slices of constant $t$ that are maximally symmetric three-dimensional spaces. Maximal symmetry forces constant curvature, so there are exactly three spatial geometries: flat Euclidean space ( $k=0$ ), the positively curved 3-sphere ( $k=+1$ ), and negatively curved hyperbolic 3-space ( $k=-1$ ). The scale factor multiplies the entire spatial part, so cosmic expansion is nothing more exotic than the growth of $a(t)$ , stretching every comoving separation in lockstep. Galaxies riding the Hubble flow keep fixed comoving coordinates; their physical separations grow because $a(t)$ grows. That single fact delivers the Hubble–Lemaître law $v = H d$ , explains cosmological redshift as photon wavelengths stretching with $a(t)$ , and makes "the age of the universe" a well-defined number. This article derives the metric from symmetry, extracts the redshift from null geodesics, and connects it to the Friedmann framework.

Key takeaways

The FLRW metric is the unique line element compatible with a spatially homogeneous and isotropic universe; all its freedom lives in one time-dependent scale factor $a(t)$ and one curvature sign $k \in \{-1,0,+1\}$ .
In comoving coordinates $\mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a(t)^2\left[\mathrm{d}r^2/(1-kr^2) + r^2\mathrm{d}\Omega^2\right]$ , with $k = -1, 0, +1$ for open, flat, and closed spatial sections.
Comoving observers keep fixed $(r,\theta,\phi)$ and their proper time is the cosmic time $t$ ; physical (proper) distances scale as $a(t)$ , giving the Hubble law $v = H d$ with $H \equiv \dot a/a$ .
Cosmological redshift follows directly from the metric: $1+z = a(t_0)/a(t_e)$ — wavelengths stretch in exact proportion to the scale factor, so $z=3$ means the universe was a quarter of its present size.
Curvature lives only in the spatial slices; a spatially flat ( $k=0$ ) universe still has a strongly curved 4D spacetime whenever $a(t)$ evolves.
Feeding the metric into Einstein's equations with a perfect fluid yields the Friedmann equation $(\dot a/a)^2 = \tfrac{8\pi G}{3}\rho - kc^2/a^2 + \Lambda c^2/3$ .
Key idea: homogeneity plus isotropy force spacetime into $\mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a(t)^2\,\mathrm{d}\Sigma^2$ , where $\mathrm{d}\Sigma^2$ is a maximally symmetric 3-space of constant curvature $k$ , so all of cosmology reduces to the single function $a(t)$ .

The core insight

Homogeneity plus isotropy force spacetime into the form $\mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a(t)^2\,\mathrm{d}\Sigma^2$ , where $\mathrm{d}\Sigma^2$ is the line element of a maximally symmetric 3-space of constant curvature $k$ ; every cosmic observable then reduces to the single function $a(t)$ and the sign of $k$ .

Where this lives today

The FLRW metric is not a historical relic; it is the coordinate system in which every precision-cosmology result is stated. Planck's 2018 analysis of the microwave background pins the curvature to $\Omega_k = 0.0007 \pm 0.0019$ , consistent with the flat $k=0$ metric to a fraction of a percent — the measurement behind Worked example 3. Baryon acoustic oscillations imprint a $\sim150\ \mathrm{Mpc}$ standard ruler on the galaxy distribution, and DESI and Euclid measure its apparent size versus redshift to map $S_k(\chi)$ and $H(z)$ directly. Type Ia supernovae — the Pantheon+ compilation now reaching $z \sim 2$ — trace the luminosity distance $d_L(z) = (1+z)S_k(\chi)$ whose deviation from a decelerating model revealed dark energy in 1998. And the metric frames cosmology's sharpest current puzzle: the Hubble tension, the persistent $\sim5\sigma$ gap between the early-universe $H_0 \approx 67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ inferred from Planck and the late-universe $H_0 \approx 73\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ from the distance ladder — a disagreement over the present slope of $a(t)$ that gravitational-wave standard sirens are now being enlisted to arbitrate.

The physical picture

Paint galaxies onto a rubber grid and stretch the grid uniformly. Each galaxy keeps its grid address — its comoving coordinate — forever, but the physical spacing between grid lines grows in proportion to $a(t)$ . The FLRW metric is precisely the rule that converts grid units to metres: the same conversion everywhere, growing uniformly with cosmic time. Nothing moves through the grid; the grid itself inflates. A photon launched across it is a wave painted on the same rubber, so as space stretches the wave stretches too — its wavelength grows by exactly the factor the grid grows, and that is cosmological redshift. The first figure makes "space stretches, wavelength stretches with it" one picture, and attaches a real number to it.

Figure

A photon crosses expanding space from a galaxy at scale factor a = 0.5 to an observer at a = 1: comoving grid cells widen, the wavelength stretches in lockstep from λₑ = 500 nm to λ₀ = 1000 nm, so 1 + z = a(t₀)/a(tₑ) = 2 and z = 1

Isotropy about every point is a ferociously strong constraint. If space looks the same in all directions from every location, its intrinsic curvature cannot vary from place to place or single out a direction — so the curvature must be one constant number, and only its sign is free. That is the trichotomy $k = +1$ (a 3-sphere), $k = 0$ (flat), $k = -1$ (a hyperbolic saddle). The sign is not a cosmetic label: it decides how a fixed transverse ruler at comoving distance $\chi$ maps onto an angle in your sky, through the transverse comoving distance $S_k(\chi)$ . In a closed universe light rays reconverge like great circles and a standard rod looks larger than Euclidean intuition expects; in an open universe they diverge and it looks smaller. The second figure is the working punchline — the three $S_k(\chi)$ curves and the real numbers at $\chi = 1$ that a geometer of the cosmos actually reaches for.

Figure

Transverse comoving distance Sₖ(χ) for the three FLRW geometries: closed sin χ for k = +1, flat χ for k = 0, open sinh χ for k = −1. At χ = 1 the same radial distance gives Sₖ = 0.841, 1.000, 1.175 — so a fixed rod subtends a larger angle in a closed universe and a smaller one in an open universe

The core equations

Everything below is a translation of "same everywhere, same in every direction" into geometry, and then a reading-off of observables. We start from the line element the symmetry forces, rewrite it in the coordinate that makes angular sizes transparent, then extract the expansion rate, the redshift, the curvature scale, and finally the source term that hands the scale factor to the Friedmann equations. The reduced-circumference form is the workhorse. It puts the curvature explicitly into the radial term, where the coefficient $1/(1-kr^2)$ carries the entire geometry of the spatial slice.

\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: $\mathrm{d}s^2$ — invariant spacetime interval; $c$ — speed of light; $t$ — cosmic (proper) time of comoving observers; $a(t)$ — scale factor (here $a$ carries the length scale, so for $k=\pm1$ it is the curvature radius); $r,\theta,\phi$ — dimensionless comoving spherical coordinates; $k\in\{-1,0,+1\}$ — curvature index. Reading: the whole metric is one factor $a(t)^2$ multiplying a fixed spatial shape — expansion is a uniform rescaling, nothing more. Trading $r$ for the comoving radial distance $\chi$ absorbs the awkward $1/(1-kr^2)$ into a clean $\mathrm{d}\chi^2$ and packages all the curvature into a single transverse function $S_k(\chi)$ . This is the form to use whenever angles and distances are the question.

\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: $\chi$ — dimensionless comoving radial distance; $S_k(\chi)$ — transverse comoving distance, the "radius" that governs transverse separations and angular sizes; $\mathrm{d}\Omega^2 = \mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2$ — solid-angle element. Reading: radial and transverse comoving distances are equal only when space is flat; curvature makes them differ, and that difference is a measurable geometry test. The three curvatures give three closed forms for $S_k$ . The flat case is the familiar Euclidean $S_k = \chi$ ; the closed and open cases replace it with a sine and a hyperbolic sine, plotted in the second figure.

S_k(\chi) = \begin{cases} \sin\chi & k=+1 \\ \chi & k=0 \\ \sinh\chi & k=-1 \end{cases} \qquad (3)

where: $S_k$ — maps a radial comoving distance onto the transverse comoving radius. Reading: for small $\chi$ all three agree ( $\sin\chi \approx \chi \approx \sinh\chi$ ), so near-field cosmology cannot see curvature — you must reach large $\chi$ , i.e. high redshift, to tell the geometries apart. Because comoving observers keep fixed coordinates while $a(t)$ grows, the proper distance to a galaxy is $a(t)\chi$ , and differentiating that at fixed cosmic time produces the Hubble–Lemaître law with the single rate $H \equiv \dot a/a$ .

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

where: $H(t)$ — Hubble parameter (fractional expansion rate), present value $H_0 \approx 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ ; $d_{\text{prop}}$ — proper (physical) distance to a comoving object; $v_{\text{prop}}$ — proper recession velocity. Reading: recession speed rises linearly with distance not because galaxies fly apart but because more grid sits between you — and this is the only velocity field homogeneity permits. The cosmological redshift is the metric's most-used single line. It equates the observed-to-emitted wavelength ratio with the ratio of scale factors, and follows purely from the null geodesic of a photon (Derivation B).

1 + z = \frac{\lambda_0}{\lambda_e} = \frac{a(t_0)}{a(t_e)} \qquad (5)

where: $z$ — redshift; $t_e, t_0$ — cosmic times of emission and observation; $\lambda_e, \lambda_0$ — emitted and observed wavelengths. Reading: redshift is a direct scale-factor odometer — measure $z$ and you know how much smaller the universe was when the light left, $a(t_e)/a(t_0) = 1/(1+z)$ , with no dynamics assumed. The curvature enters observations through the density parameter $\Omega_k$ , which packages $k$ , the present scale factor, and $H_0$ into one dimensionless number, and fixes the physical radius of curvature of space today.

\Omega_k \equiv -\frac{kc^2}{a_0^2 H_0^2}, \qquad R_{\text{curv},0} = \frac{a_0}{\sqrt{|k|}} = \frac{c/H_0}{\sqrt{|\Omega_k|}} \qquad (6)

where: $\Omega_k$ — curvature density parameter ( $<0$ closed, $=0$ flat, $>0$ open); $a_0$ — present scale factor; $R_{\text{curv},0}$ — present radius of spatial curvature; $c/H_0$ — Hubble radius. Reading: measuring $\Omega_k$ close to zero does not prove exact flatness — it pushes the curvature radius far beyond the observable horizon, so space is locally Euclidean to superb precision (Worked example 3). Finally, the metric becomes physics when it meets Einstein's equations with a perfect-fluid source. The comoving fluid's density and pressure are the only matter data the symmetry allows, and they drive $a(t)$ .

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, \qquad T_{\mu\nu} = \left(\rho + \frac{p}{c^2}\right)u_\mu u_\nu + p\,g_{\mu\nu} \qquad (7)

where: $G_{\mu\nu}$ — Einstein tensor built from the FLRW metric; $\Lambda$ — cosmological constant; $T_{\mu\nu}$ — perfect-fluid stress-energy; $\rho, p$ — mass density and pressure of the cosmic fluid; $u_\mu$ — comoving four-velocity, $u^\mu \propto (1,0,0,0)$ . Reading: homogeneity kills every off-diagonal source, so the ten Einstein equations collapse to just two independent scalar equations — the Friedmann and acceleration equations (Derivation C gives the first).

Derivation AHow maximal symmetry fixes the line element and the comoving radial coordinate

The goal is to show why the metric of equations (1) and (2) is forced rather than assumed, and to build the coordinate $\chi$ that turns the ungainly $1/(1-kr^2)$ into a clean transverse function $S_k(\chi)$ . We start from the symmetry demand, invoke the theorem that a maximally symmetric 3-space has constant curvature, write the resulting spatial metric, and then perform the single integral that converts the radial coordinate — arriving at exactly the sine, linear, and hyperbolic-sine forms that distinguish the three universes and that the second figure plots.

Step 1. Impose homogeneity and isotropy on each constant-time slice. A 3-space with these symmetries carries the maximal number of independent isometries, $n(n+1)/2 = 6$ for $n=3$ , and a theorem of differential geometry says such a space has constant curvature: its Riemann tensor is $R_{ijkl} = K(g_{ik}g_{jl} - g_{il}g_{jk})$ with $K$ the same at every point. Why: constant curvature is not an extra assumption but the unique consequence of "no point and no direction is special," which is precisely the Cosmological Principle applied to space.

Step 2. Solving for the metric of a constant-curvature 3-space in spherical coordinates gives $\mathrm{d}\Sigma^2 = \mathrm{d}r^2/(1-kr^2) + r^2\,\mathrm{d}\Omega^2$ , and multiplying by the time-dependent factor $a(t)^2$ and appending the universal cosmic-time term $-c^2\mathrm{d}t^2$ yields the FLRW line element (1). Why: the factor $a(t)^2$ is pulled out front because homogeneity forbids the spatial shape from varying with position — only its overall scale may change with time, which is what "the scale factor" means.

Step 3. The radial term $\mathrm{d}r^2/(1-kr^2)$ is not a clean proper interval, so define $\chi$ by $\mathrm{d}\chi = \mathrm{d}r/\sqrt{1-kr^2}$ , i.e. $\chi = \int_0^r \mathrm{d}r'/\sqrt{1-kr'^2}$ . Why: $\chi$ is the comoving distance actually travelled along a radial line, so photons and rulers see $\chi$ directly, whereas $r$ is just a convenient label.

Step 4. Do the integral for each curvature: $k=+1$ gives $\chi = \arcsin r$ , $k=0$ gives $\chi = r$ , and $k=-1$ gives $\chi = \operatorname{arcsinh} r$ . Inverting, $r = S_k(\chi)$ with $S_k = \sin\chi,\ \chi,\ \sinh\chi$ , so the angular term $r^2\mathrm{d}\Omega^2$ becomes $S_k(\chi)^2\mathrm{d}\Omega^2$ and the metric takes the form (2). Why: this is the whole reason curvature shows up as a bent transverse distance — the same radial trek $\chi$ subtends a different transverse arc depending on whether the sines curve rays back together or the hyperbolic sines fan them apart. *Note: the small- $\chi$ limit $S_k \to \chi$ recovers Euclidean space for all $k$ .*

Go deeper · Derivation B · 1 minCosmological redshift from a radial null geodesic

Derivation BCosmological redshift from a radial null geodesic

The goal is to derive $1+z = a(t_0)/a(t_e)$ from the metric alone, with no appeal to Doppler shifts or velocities, and thereby show that redshift is a statement about the geometry a photon traverses rather than about anyone's motion. We send a photon along a radial null path, use the constancy of the comoving distance between a fixed source and a fixed observer to relate two successive wavecrests, and read off that the wavelength ratio is the scale-factor ratio — the single most important observable the FLRW metric produces.

Step 1. A radial photon has $\mathrm{d}s^2 = 0$ and $\mathrm{d}\theta = \mathrm{d}\phi = 0$ , so from equation (2) the null condition is $0 = -c^2\mathrm{d}t^2 + a(t)^2\mathrm{d}\chi^2$ , giving $c\,\mathrm{d}t/a(t) = \mathrm{d}\chi$ . Why: setting the interval to zero is what "travelling at the speed of light" means in curved spacetime, and it converts a time integral into a fixed comoving distance.

Step 2. Integrate from emission to observation: $\displaystyle\int_{t_e}^{t_0} \frac{c\,\mathrm{d}t}{a(t)} = \int_0^{\chi} \mathrm{d}\chi' = \chi$ , a constant, because the source and observer keep fixed comoving coordinates. Why: the comoving separation between two Hubble-flow objects never changes — only the physical separation does — so this integral is the same for every crest the source emits.

Step 3. Emit a second crest one period $\delta t_e$ later, received a period $\delta t_0$ later: $\displaystyle\int_{t_e + \delta t_e}^{t_0 + \delta t_0} \frac{c\,\mathrm{d}t}{a(t)} = \chi$ as well. Subtracting the two integrals, and noting $a$ is essentially constant over one optical period, leaves $\delta t_0/a(t_0) = \delta t_e/a(t_e)$ . Why: the shared endpoints cancel and only the two short period-slivers survive, directly tying the received period to the emitted period through the scale factors.

Step 4. A period is a wavelength, $\lambda = c\,\delta t$ , so $\lambda_0/\lambda_e = \delta t_0/\delta t_e = a(t_0)/a(t_e)$ , and by definition $1 + z \equiv \lambda_0/\lambda_e = a(t_0)/a(t_e)$ . Why: this is equation (5) — a pure geometric stretching, which is why it is exact for any expansion history and why a "recession velocity" interpretation only survives as an approximation at small $z$ . *Note: for nearby sources this reduces to the special-relativistic Doppler formula $z \approx v/c$ .*

Go deeper · Derivation C · 1 minFrom the FLRW metric to the first Friedmann equation

Derivation CFrom the FLRW metric to the first Friedmann equation

The goal is to feed the FLRW metric into Einstein's field equations and extract the equation that governs $a(t)$ — the first Friedmann equation. The full computation of the curvature tensors is a page of Christoffel algebra, so we work in units with $c = 1$ to keep the bookkeeping clean, quote the standard nonzero Christoffel symbols and the time-time Einstein component, contract against the comoving perfect fluid, and restore $c$ in the boxed result. The payoff is the master equation of cosmic dynamics, whose $k=0$ case defines the critical density.

Step 1. With $c=1$ the metric is $\mathrm{d}s^2 = -\mathrm{d}t^2 + a^2 \tilde g_{ij}\,\mathrm{d}x^i\mathrm{d}x^j$ , and the nonzero Christoffel symbols are $\Gamma^0_{ij} = a\dot a\,\tilde g_{ij}$ and $\Gamma^i_{0j} = (\dot a/a)\,\delta^i_j$ , where $\tilde g_{ij}$ is the constant-curvature spatial metric. Why: these are the only connection coefficients the diagonal, time-dependent metric produces, and they encode exactly how the expansion couples the time and space directions.

Step 2. Contracting the Riemann tensor built from these gives the time-time Ricci component $R_{00} = -3\,\ddot a/a$ , and assembling the spatial parts gives the time-time Einstein component $G_{00} = 3\left[(\dot a/a)^2 + k/a^2\right]$ . Why: the curvature constant $k$ enters $G_{00}$ through the spatial Ricci scalar of the constant-curvature slice, which is why spatial geometry and expansion rate appear together in one equation.

Step 3. For the comoving perfect fluid the time-time stress-energy is the energy density, $T_{00} = \rho$ (with $c=1$ ), so the time-time Einstein equation $G_{00} + \Lambda g_{00} = 8\pi G\,T_{00}$ with $g_{00} = -1$ becomes $3(\dot a/a)^2 + 3k/a^2 - \Lambda = 8\pi G\,\rho$ . Why: homogeneity forces the fluid to be at rest in comoving coordinates, so only the energy density fills the time-time slot and all momentum-flux terms vanish.

Step 4. Solve for the expansion rate and restore $c$ by dimensional analysis (the curvature and $\Lambda$ terms each carry a $c^2$ ): $\left(\dfrac{\dot a}{a}\right)^2 = \dfrac{8\pi G}{3}\rho - \dfrac{kc^2}{a^2} + \dfrac{\Lambda c^2}{3}$ . Why: this is the first Friedmann equation — the expansion rate set by density, curvature, and the cosmological constant — and setting $k = \Lambda = 0$ defines the critical density $\rho_c = 3H^2/8\pi G$ .

\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}

Note: the flat, matter-only case gives $a \propto t^{2/3}$ .

Limiting cases

The FLRW geometry specializes cleanly with the curvature sign and the behaviour of the scale factor.

Case	Result	Meaning
Flat ( $k=0$ )	$\mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a^2(\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2)$	Euclidean spatial slices; $S_k(\chi)=\chi$ , so transverse and radial comoving distances coincide. Favoured by the CMB, $\|\Omega_k\|\lesssim 0.001$ .
Closed ( $k=+1$ )	$\mathrm{d}\Sigma^2 = \mathrm{d}\chi^2 + \sin^2\chi\,\mathrm{d}\Omega^2$	3-sphere of finite volume; geodesics reconverge and $\chi \in [0,\pi]$ . A standard rod looks larger than in flat space.
Open ( $k=-1$ )	$\mathrm{d}\Sigma^2 = \mathrm{d}\chi^2 + \sinh^2\chi\,\mathrm{d}\Omega^2$	Hyperbolic 3-space of infinite volume; geodesics diverge and $\chi \in [0,\infty)$ . A standard rod looks smaller.
Small $\chi$ (near field)	$S_k(\chi) \approx \chi$ for all $k$	Curvature is invisible nearby; the three geometries agree to first order, so tests of $k$ need high redshift.
Static Minkowski ( $a=\text{const},\,k=0$ )	$\mathrm{d}s^2 \to \eta_{\mu\nu}$	Constant scale factor and flat slices give flat spacetime — no expansion, no redshift, $H=0$ .

Common trapk = 0 means the whole universe is flat

It does not. The index $k$ describes only the curvature of the spatial slices at fixed cosmic time. A spatially flat ( $k=0$ ) FLRW universe still has a strongly curved four-dimensional spacetime whenever $a(t)$ changes with time — the time-dependence of the metric produces nonzero components of the spacetime Riemann tensor (for instance $R_{tt} = -3\ddot a/a$ ), which is exactly what drives the redshift and the Friedmann dynamics. "Flat" in cosmology is a statement about the geometry of a snapshot of space, never about spacetime as a whole. A second, quieter trap lives in the units: whether the scale factor $a$ is dimensionless with $a(t_0)=1$ (so comoving coordinates carry length and $S_k$ hides a curvature radius) or carries length itself (so $\chi$ and $S_k$ are dimensionless, the convention used here) differs between textbooks. The physics is identical; always check which convention a formula assumes before plugging in numbers.

Worked examples

1 · Reading the scale factor off a quasar's redshift (fully worked)

A distant quasar's Lyman-α line, emitted at rest wavelength $\lambda_e = 121.6\ \mathrm{nm}$ , is observed at $\lambda_0 = 486.4\ \mathrm{nm}$ . Find the redshift, the scale factor at emission relative to today, and how much smaller the universe was when the light left.

Step 1. The cosmological redshift is the wavelength ratio (equation 5): $1 + z = \lambda_0/\lambda_e = 486.4/121.6 = \mathbf{4.00}$ , so $z = \mathbf{3.00}$ . No dynamics or distance assumed — the redshift is measured directly from two wavelengths.

Step 2. Invert the redshift for the scale factor: $a(t_e)/a(t_0) = 1/(1+z) = 1/4 = \mathbf{0.25}$ . The linear size of the universe at emission was one quarter of today's, so every comoving separation was four times more compressed.

Step 3. Volumes go as the cube of the linear scale, so the universe was $(1/4)^3 = 1/64$ of its present volume and the mean density of any conserved matter was $64\times$ higher. The quasar's light left when the universe was a quarter of its present linear size ( $a = 0.25$ , $z = 3$ ) and 64 times denser — a fact read straight off the FLRW metric with zero cosmological model input.

2 · Recession velocity and the Hubble sphere (fully worked)

Take $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Find the Hubble radius, the recession velocity of a galaxy at present proper distance $d = 6000\ \mathrm{Mpc}$ , and decide whether that velocity contradicts relativity.

Step 1. The Hubble radius is the proper distance at which the recession speed equals $c$ : $D_H = c/H_0 = (2.998\times10^5\ \mathrm{km\,s^{-1}})/(70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}) = \mathbf{4283\ Mpc}$ . Galaxies inside this sphere recede subluminally, those beyond it superluminally.

Step 2. Apply the Hubble–Lemaître law (equation 4): $v = H_0\,d = (70)(6000) = 4.2\times10^{5}\ \mathrm{km\,s^{-1}} = \mathbf{1.40\,c}$ . The galaxy at $6000\ \mathrm{Mpc}$ sits well outside the Hubble sphere and recedes at $1.40$ times the speed of light.

Step 3. This is not a violation of relativity. The recession speed is the rate of growth of a proper distance summed across intervening space, not a velocity measured in any single local inertial frame — and special relativity only forbids the latter from exceeding $c$ . No galaxy ever passes a light beam locally; light emitted toward us from beyond the Hubble sphere can still reach us later because the sphere itself grows. A galaxy at $6000\ \mathrm{Mpc}$ recedes at $1.40\,c$ , entirely consistent with relativity — superluminal recession is a property of the expanding metric, not of motion through space.

3 · The curvature radius from the Planck flatness bound (fully worked)

Planck 2018 constrains the curvature density to $|\Omega_k| \lesssim 0.001$ . With $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , what lower bound does this place on the present radius of curvature of space?

Step 1. The radius of curvature follows from equation (6): $R_{\text{curv},0} = (c/H_0)/\sqrt{|\Omega_k|} = D_H/\sqrt{|\Omega_k|}$ , with the Hubble radius $D_H = 4283\ \mathrm{Mpc}$ from Worked example 2.

Step 2. Insert the bound: $\sqrt{|\Omega_k|} \lesssim \sqrt{0.001} = 0.0316$ , so $R_{\text{curv},0} \gtrsim 4283/0.0316 = \mathbf{1.35\times10^{5}\ Mpc}$ , about $\mathbf{32}$ Hubble radii.

Step 3. The observable universe extends only a few Hubble radii, so a curvature radius exceeding $\sim32$ of them means any curvature of space is undetectable within everything we can see — space is Euclidean to superb local precision even though the metric permits it to close. Flatness data do not prove $k=0$ exactly; they push the curvature radius past $10^5\ \mathrm{Mpc}$ , far beyond the horizon, which is why the $k=0$ FLRW metric is such an excellent working model.

Physics of the portrait

The FLRW metric carries four names, but two of them — Robertson and Walker — belong to the theorem that made it inevitable. Friedmann and Lemaître had already found expanding solutions of Einstein's equations; it was Howard Percy Robertson who proved, rigorously, that homogeneity and isotropy leave no other possibility for the line element, turning a family of models into a uniqueness statement.

Origins · 1 minHoward Percy Robertson and how this was discovered

Howard Percy Robertson

1903–1961 · the uniqueness theorem for the isotropic, homogeneous line element

Howard Percy Robertson was an American mathematical physicist, trained at Caltech and Princeton, whose fingerprints are on a surprising range of twentieth-century physics. In a series of papers around 1935–1936 ("Kinematics and World-Structure", Astrophysical Journal) he proved the theorem that anchors this article: any spacetime that is spatially homogeneous and isotropic must have the line element now written with his and Walker's names, with all freedom confined to the scale factor $a(t)$ and the curvature sign $k$ . Arthur Geoffrey Walker, a British mathematician, gave an independent and more general proof in 1937, and the joint result — the "Robertson–Walker" metric, or FLRW once Friedmann's and Lemaître's dynamical solutions are credited — became the geometric skeleton of cosmology. Robertson's reach went well beyond this: he worked out the Poynting–Robertson effect, the radiation drag that spirals interplanetary dust into the Sun, and sharpened the Heisenberg uncertainty relation into the general Robertson–Schrödinger inequality. His cosmological legacy is methodological as much as technical: he insisted that the smoothness of the universe be treated as a stated symmetry with provable consequences, so that "why this metric and not another?" has a theorem for an answer rather than a shrug.

History

1917

— Einstein builds the first relativistic cosmological model, assuming homogeneity and isotropy

1922

— Alexander Friedmann derives the expanding and contracting solutions of the field equations

1927

— Georges Lemaître links an expanding metric to galaxy redshifts and predicts a linear velocity–distance law

1935

— Howard Percy Robertson proves the line element is the unique homogeneous, isotropic form

1937

— Arthur Geoffrey Walker gives an independent proof, completing the Robertson–Walker theorem

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 3–5 (the metric, redshift, distances) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 29 · Dodelson, Modern Cosmology, 2nd ed., ch. 2–3 · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 · Weinberg, Cosmology, 2008, ch. 1 · Misner, Thorne & Wheeler, Gravitation, ch. 27. Key results used throughout this article: the coordinate integral $\chi = \int \mathrm{d}r/\sqrt{1-kr^2}$ yields $S_k = \sin\chi,\,\chi,\,\sinh\chi$ ; the redshift relation $1+z = a(t_0)/a(t_e)$ follows from a radial null geodesic; the first Friedmann equation comes from the time-time Einstein component. The anchor numbers are: $\lambda_0/\lambda_e = 4 \Rightarrow z=3,\ a=0.25$ ; $D_H = c/H_0 = 4283\ \mathrm{Mpc}$ and $v(6000\ \mathrm{Mpc}) = 1.40c$ ; $R_{\text{curv}} \gtrsim 1.35\times10^5\ \mathrm{Mpc}$ from $|\Omega_k|\lesssim0.001$ ; $\sinh 1 = 1.175$ , $\sin 1 = 0.841$ . Historical milestones: Einstein 1917, Friedmann 1922, Lemaître 1927, Robertson 1935, Walker 1937; $H_0$ and $\Omega_k$ values from Planck 2018.

Prerequisites. The Cosmological Principle · Homogeneity and Isotropy in Cosmology · Comoving and Proper Coordinates · The Expansion of the Universe and the Scale Factor. Leads to. The Friedmann Equations · The Fluid and Acceleration Equations · Cosmological Redshift · Spatial Curvature of the Universe · Critical Density and Density Parameters.

Ask the tutor: Ask me to walk through how the FLRW metric turns the observed redshift $z$ into the three cosmological distances — comoving, angular-diameter, and luminosity — that supernova and BAO surveys actually measure.

If the universe is expanding, what is it expanding into?

Into nothing — the question presumes an outside that FLRW does not contain. Homogeneity means there is no edge and no centre, so expansion is not matter flying outward through pre-existing space from a special point. It is the metric distance between comoving points growing everywhere at once, encoded in $d_{\text{prop}} = a(t)\chi$ . Every observer sees all others receding by the same Hubble law and each equally regards itself as at rest. A closed ( $k=+1$ ) universe is finite yet boundaryless, like the surface of a balloon seen from within; a flat or open one is infinite and was infinite at every past moment. There is no exterior region for space to invade, and the metric never refers to one.

Does k = 0 mean spacetime is flat?

No — it means only that the three-dimensional space at a fixed cosmic time is flat (Euclidean). The full four-dimensional spacetime is curved whenever the scale factor evolves, because the time dependence of the metric produces nonzero components of the spacetime Riemann tensor, such as $R_{tt} = -3\ddot a/a$ . That curvature is not a technicality: it is what makes light redshift and what the Friedmann equations describe. So a universe can be spatially flat and still be a strongly curved spacetime — indeed our own universe is measured to be spatially flat while expanding and accelerating, which is a thoroughly non-flat spacetime.

Is cosmological redshift just a Doppler shift?

Not in the usual sense. A Doppler shift comes from relative motion through space measured in one frame; cosmological redshift comes from the scale factor growing while the photon is in flight, stretching its wavelength by $a(t_0)/a(t_e)$ (Derivation B). The source and observer can both be perfectly at rest in comoving coordinates and the redshift still appears — it is a property of the geometry traversed, not of any velocity. For nearby galaxies the two pictures agree, because a small expansion looks like a small recession and $z \approx v/c$ . But at high redshift the Doppler formula fails and only the scale-factor relation is exact, which is why $z$ can exceed 1 (and reaches $\sim1100$ for the CMB) without anything moving faster than light.

How can galaxies recede faster than light without breaking relativity?

Because a recession velocity $v = H\,d$ is the growth rate of a proper distance summed across all the space in between, not a speed measured in a single local inertial frame. Special relativity forbids only the latter from exceeding $c$ , and it never is: no galaxy overtakes a passing light beam locally. Any galaxy beyond the Hubble radius $D_H = c/H_0 \approx 4283\ \mathrm{Mpc}$ recedes superluminally (Worked example 2), yet we still see light from far beyond it, because the Hubble sphere itself expands and photons can cross from regions that were once receding faster than light. Superluminal recession is a feature of the expanding metric, fully allowed by general relativity.

What is the difference between comoving and proper distance?

Comoving distance $\chi$ is the fixed "grid coordinate" separation between two Hubble-flow objects — it does not change as the universe expands, by construction. Proper distance $d_{\text{prop}} = a(t)\chi$ is the physical separation you would measure with a chain of rulers laid end to end at one instant of cosmic time, and it grows with the scale factor. Today, with $a(t_0)$ normalising the two, they numerically coincide for nearby objects; in the past the proper distance was smaller by the factor $a(t_e)/a(t_0) = 1/(1+z)$ . Comoving distance is the natural bookkeeping variable because it removes the expansion, which is exactly why redshift, distance integrals, and $S_k(\chi)$ are all written in terms of $\chi$ .

Why are there no cross terms between time and space in the metric?

Because isotropy forbids them. A cross term $g_{ti}\,\mathrm{d}t\,\mathrm{d}x^i$ would be a spatial vector built into the geometry, and a vector points somewhere — it would single out a preferred direction, contradicting the assumption that space looks the same in every direction. Their absence has a physical payoff: the constant- $t$ surfaces are orthogonal to the comoving worldlines, so the cosmic time coordinate $t$ coincides with the proper time ticked by comoving clocks. That is what makes "the age of the universe" a single, observer-independent number rather than a frame-dependent quantity.