The FRW Metric from Homogeneity and Isotropy

Derivation

The FRW Metric from Homogeneity and Isotropy

D-338 Home PU-308 Threads symmetry · fields Depends on maximally-symmetric-spaces, riemann-curvature-tensor

Statement

If spacetime is spatially homogeneous and isotropic about every point (the cosmological principle), then in comoving synchronous coordinates its line element is forced to the Friedmann–Robertson–Walker form \( ds^{2} = -c^{2}\,dt^{2} + a(t)^{2}\!\left[\dfrac{dr^{2}}{1-kr^{2}} + r^{2}\big(d\theta^{2}+\sin^{2}\!\theta\,d\varphi^{2}\big)\right] \), carrying exactly one free function of time — the scale factor \( a(t) \) — and one constant \( k \) fixing the sign of the spatial curvature.

Why it matters

This is the kinematic backbone of all of physical cosmology. Before any dynamics or matter content is specified, symmetry alone collapses the ten independent components of a general metric \( g_{\mu\nu}(x) \) down to a single time-dependent number \( a(t) \) and a discrete curvature choice. Every quantity the field uses — redshift, luminosity and angular-diameter distances, the horizon, the CMB — is read off this one line element.

It also cleanly separates geometry from dynamics: symmetry fixes the shape of the metric here, while the Einstein equations later fix only the evolution \( a(t) \) through the Friedmann equations. Understanding which features are symmetry-forced (and therefore not up for debate) is essential before interpreting any cosmological measurement.

Assumptions

Spatial homogeneity.Every point is equivalent: the intrinsic geometry of a spatial slice looks the same everywhere, so the curvature constant cannot depend on position. Drop it and the curvature becomes a field \( k(\vec x) \); one recovers inhomogeneous Lemaître–Tolman–Bondi models with a position-dependent expansion and no single \( a(t) \).

Spatial isotropy about every point.No spatial direction is preferred at any point. This forbids any \( g_{0i} \) cross term (a preferred spatial vector) and any direction-dependent expansion. Drop it and one gets anisotropic Bianchi cosmologies with shear \( \sigma_{ij}\neq 0 \) and a directionally dependent Hubble rate.

A smooth Lorentzian manifold with matter defining a comoving frame (Weyl's postulate).The dust of "typical" observers moves on non-intersecting timelike geodesics orthogonal to the spatial slices, giving a globally synchronizable cosmic time \( t \). Drop it and the slices need not be orthogonal to the flow, worldlines may cross, and no global \( t \) with \( g_{00}=-c^{2} \) exists.

Classical General Relativity holds on these scales.The metric is the fundamental field and curvature is finite. Drop it (near the singularity, at Planck curvature) and the classical FRW description is meaningless; a quantum-gravitational treatment is required.

Derivation

\[ \Sigma_{t}=\{\,t=\text{const}\,\},\qquad M \cong \mathbb{R}_t \times \Sigma_t \]

By Weyl's postulate the comoving matter worldlines are a congruence of timelike geodesics; the surfaces orthogonal to them foliate spacetime into spatial slices labelled by proper time \( t \) along the congruence. B

\[ g_{00}=-c^{2},\qquad g_{0i}=0 \]

Orthogonality of the slices to the geodesic congruence kills the cross terms \( g_{0i} \); a nonzero \( g_{0i} \) would be a preferred spatial vector at each point, violating isotropy. Reparametrizing \( t \) by the comoving proper time normalizes \( g_{00}=-c^{2} \) (synchronous/Gaussian-normal gauge). B

\[ ds^{2}=-c^{2}\,dt^{2}+g_{ij}(t,\vec x)\,dx^{i}dx^{j} \]

Assembling Steps 1–2, the only surviving metric data is the induced spatial metric \( g_{ij} \) on each slice. All remaining constraints act on \( g_{ij} \). A

\[ {}^{(3)}\!R_{ijkl}=\mathcal{K}\big(g_{ik}g_{jl}-g_{il}g_{jk}\big),\qquad \mathcal{K}=\text{const on }\Sigma_t \]

A space that is isotropic about every point is maximally symmetric (prior result: maximally-symmetric-spaces), so its Riemann tensor has this unique constant-curvature form; homogeneity forces the sectional curvature \( \mathcal{K} \) to be constant across the slice. C

\[ g_{ij}(t,\vec x)=a(t)^{2}\,\gamma_{ij}(\vec x) \]

A maximally symmetric space of fixed dimension and fixed curvature sign is unique up to an overall scale and isometries. As \( t \) varies continuously while each slice stays maximally symmetric, the metric can change only by an overall factor; write it as \( a(t)^{2} \) times a \( t \)-independent comoving metric \( \gamma_{ij} \). C

\[ \gamma_{ij}\,dx^{i}dx^{j}=f(r)\,dr^{2}+r^{2}\big(d\theta^{2}+\sin^{2}\!\theta\,d\varphi^{2}\big) \]

Isotropy about the origin makes the comoving 3-metric spherically symmetric; adapted spherical coordinates put it in this radial-plus-angular form, with the areal radius \( r \) chosen so the 2-spheres have area \( 4\pi r^{2} \). One unknown function \( f(r) \) remains. B

\[ {}^{(3)}\!R \;=\; \frac{2}{r^{2}}\!\left[\,1-\frac{d}{dr}\!\left(\frac{r}{f}\right)\right] \;=\; 6k \]

Compute the Ricci scalar of the ansatz in Step 6 and demand it equal the constant value \( 6k \) required for a maximally symmetric 3-space (for dimension \( n \), \( {}^{(3)}\!R=n(n-1)\mathcal{K}=6k \) with \( k \) the normalized curvature of \( \gamma \)). This is a first-order ODE for \( f \). C

\[ \frac{d}{dr}\!\left(\frac{r}{f}\right)=1-3kr^{2}\ \Longrightarrow\ \frac{r}{f}=r-kr^{3}+C \]

Rearranging Step 7 and integrating once. \( C \) is an integration constant to be fixed by a regularity condition at the origin. A

\[ C=0\ \Longrightarrow\ f(r)=\frac{1}{1-kr^{2}} \]

Regularity of the geometry at \( r=0 \) (a locally flat, non-conical origin) requires \( f(0)=1 \); a nonzero \( C \) would make \( r/f\to C\neq 0 \) as \( r\to 0 \), producing a conical/singular centre. Setting \( C=0 \) fixes \( f \) uniquely. B

\[ ds^{2}=-c^{2}\,dt^{2}+a(t)^{2}\!\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}\big(d\theta^{2}+\sin^{2}\!\theta\,d\varphi^{2}\big)\right] \]

Insert \( f \) from Step 9 into Step 6, then Step 6 into Step 5 into Step 3. Only the sign of \( k \) is invariant under the rescaling \( r\to\lambda r,\ a\to a/\lambda \), so \( k \) may be normalized to \( \{-1,0,+1\} \) or, keeping length in \( r \), to a curvature constant of dimension \( (\text{length})^{-2} \). B

Result

\[ ds^{2}=-c^{2}\,dt^{2}+a(t)^{2}\!\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}\big(d\theta^{2}+\sin^{2}\!\theta\,d\varphi^{2}\big)\right] \]

Reading. Homogeneity and isotropy leave exactly two pieces of freedom in the entire metric: a single positive function of time, the scale factor \( a(t) \), which stretches all comoving separations by the same factor; and one constant \( k \) whose sign selects a closed (\( k>0 \), 3-sphere), flat (\( k=0 \), Euclidean), or open (\( k<0 \), hyperbolic) spatial geometry. Comoving coordinates \( (r,\theta,\varphi) \) are frozen onto the matter; physical separations grow as \( a(t) \) grows. All the dynamics is deferred to \( a(t) \), which symmetry does not fix — the Einstein equations do.

Units check. Take \( a \) dimensionless with \( a(t_0)=1 \) today, \( r \) a comoving length, and \( k \) of dimension \( (\text{length})^{-2} \). Then \( c^{2}dt^{2} \) has units \( (\mathrm{m\,s^{-1}})^{2}(\mathrm{s})^{2}=\mathrm{m^{2}} \); \( kr^{2} \) is dimensionless so \( a^{2}dr^{2}/(1-kr^{2}) \) has units \( \mathrm{m^{2}} \); and \( a^{2}r^{2}d\Omega^{2} \) is \( \mathrm{m^{2}} \) since \( d\Omega^{2} \) is dimensionless. Every term is an area, as required for \( ds^{2} \).

Limiting cases

\( k=0 \) (flat): \( ds^{2}=-c^{2}dt^{2}+a(t)^{2}\big(dr^{2}+r^{2}d\Omega^{2}\big) \); the spatial slices are ordinary Euclidean 3-space, uniformly rescaled.
\( a(t)=\text{const} \): a static, non-expanding maximally symmetric universe (e.g. the spatial part of the Einstein static universe for \( k>0 \)); redshift vanishes.
\( r\ll 1/\sqrt{|k|} \) (small patch): \( 1-kr^{2}\to 1 \) and the metric is locally that of flat FRW; curvature is invisible on scales far below the curvature radius \( R_{\text{curv}}=1/\sqrt{|k|} \).
\( k>0 \) at \( r\to 1/\sqrt{k} \): \( g_{rr}\to\infty \); this is a coordinate artifact at the equator of the 3-sphere, not a physical singularity (use \( r=k^{-1/2}\sin\chi \)).

Breaks when

Small scales (galaxies, clusters, the Solar System). Homogeneity is badly violated by local overdensities; the correct metric is Schwarzschild/Kerr locally, and FRW applies only after averaging over \( \gtrsim 100\ \mathrm{Mpc} \).
Anisotropic or shearing cosmologies. If isotropy fails (primordial shear, magnetic fields, Bianchi models), off-diagonal or direction-dependent terms appear and the single-\( a(t) \) form is replaced by multiple directional scale factors \( a_i(t) \).
Near the initial singularity / Planck regime. As \( a\to 0 \) curvature invariants diverge; classical GR and hence the FRW metric cease to be valid and quantum gravity is required.
Isotropy about one point only. If space is isotropic about us but not about every point (a spherical LTB void), homogeneity fails and \( k \) becomes position dependent; FRW is not recovered.

Failure modes

Conflating comoving and proper distance. Writing the recession or separation as \( r \) rather than \( a(t)\,r \); the comoving \( r \) is frozen while the physical distance scales with \( a \).
Believing \( a(t) \) has an absolute value. Only ratios \( a(t_1)/a(t_2) \) are observable (through redshift); the overall normalization is pure gauge, conventionally \( a(t_0)=1 \).
Thinking \( k \) must be exactly \( \pm 1 \). The magnitude of \( k \) is coordinate-dependent (rescale \( r \)); only its sign is physical, and the curvature radius is \( 1/\sqrt{|k|} \).
Reading galaxy recession as motion through space. Comoving galaxies are at rest in these coordinates; it is the metric factor \( a(t) \) that grows, not peculiar velocity through a static space.
Assuming "isotropic about us" alone gives FRW. Isotropy about a single point plus the Copernican principle (no special location) is needed; isotropy about every point then implies homogeneity.
Treating the \( g_{rr} \) blow-up at \( r=k^{-1/2} \) as physical. It is a coordinate singularity at the 3-sphere equator, removable by \( r=k^{-1/2}\sin\chi \).

Discussion

The power of this result is that it is almost purely group-theoretic. "Homogeneous and isotropic about every point" is the statement that the spatial slices admit the maximal number of Killing vectors, \( n(n+1)/2 = 6 \) in three dimensions — three translations and three rotations. A theorem on maximally symmetric spaces then leaves only three geometries distinguished by the sign of \( k \), and time-dependence is confined to a single conformal factor. No field equation has been used; Einstein's equations enter only afterward, promoting \( a(t) \) from a free function to the solution of the Friedmann equations sourced by the energy content.

Physically, the comoving grid is welded to the "cosmic fluid": galaxies sit at fixed \( (r,\theta,\varphi) \) and the expansion of space is the growth of the metric coefficient \( a(t)^{2} \). The cosmic time \( t \) that labels the slices is the proper time of these comoving observers, which is why the CMB defines a natural rest frame — the frame in which the radiation is isotropic is precisely the comoving frame singled out by the derivation. This is the content of Weyl's postulate made geometric.

The name varies by convention: FRW (Friedmann–Robertson–Walker) or FLRW (adding Lemaître) all denote the same line element. Friedmann and Lemaître supplied the dynamics of \( a(t) \); Robertson and Walker proved, independently and rigorously, exactly the kinematic statement derived here — that homogeneity and isotropy force this metric regardless of the gravitational field equations.

A subtlety worth stating precisely: the foliation into homogeneous slices is not unique as a matter of pure geometry — one could slice a symmetric spacetime along tilted surfaces and spoil manifest homogeneity. What privileges the FRW slicing is the matter: the requirement that the slices be orthogonal to the comoving geodesic congruence (Weyl) picks out the surfaces of constant \( t \) uniquely. Equivalently, the FRW form is the statement that the isometry group \( G_6 \) (or \( G_3 \) acting transitively on 2-surfaces for the anisotropic Kantowski–Sachs exception) acts on the spatial slices; the maximal case \( G_6 \) is the one that yields the single-scale-factor metric. Recasting with conformal time \( d\eta=c\,dt/a \) turns the metric into \( a^{2}(\eta)\big[-d\eta^{2}+d\Sigma^{2}\big] \), manifestly conformal to a static maximally symmetric spacetime — the form that makes causal structure and horizons transparent.

Common misconceptions. The FRW metric does not assume the universe is flat (all three \( k \) are allowed), does not by itself predict expansion (that is dynamical, via \( a(t) \)), and does not require the universe to be finite (only \( k>0 \) with trivial topology is closed; \( k\le 0 \) slices can be infinite, and even \( k>0 \) admits multi-connected topologies). "The universe is expanding into something" is also wrong — nothing in the metric refers to an embedding space.

Worked examples

\[ D_{\text{prop}}(t)=a(t)\,r,\qquad v_{\text{rec}}=\frac{\dot a}{a}\,D_{\text{prop}}=H(t)\,D_{\text{prop}} \]

Flat proper distance and Hubble recession. Symbols first: in flat FRW (\( k=0 \)) the spatial line element along a radial direction is \( a(t)\,dr \), so the proper distance to a comoving galaxy at coordinate \( r \) is \( a(t)\,r \); differentiating at fixed \( r \) gives \( \dot D = \dot a\,r = (\dot a/a)D = HD \). A

\[ a(t_0)=1,\quad r=100\ \mathrm{Mpc}\ \Rightarrow\ D_0=100\ \mathrm{Mpc};\qquad a(t_1)=0.5\ \Rightarrow\ D_1=50\ \mathrm{Mpc} \]

Insert numbers. Today the proper distance equals the comoving value; when the universe was half its present size the same galaxy was 50 Mpc away. A

\[ v_{\text{rec}}=H_0 D_0=(70\ \mathrm{km\,s^{-1}\,Mpc^{-1}})(100\ \mathrm{Mpc})=7000\ \mathrm{km\,s^{-1}} \]

Evaluate the recession speed today with \( H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}} \). A

\[ D_1=50\ \mathrm{Mpc},\qquad v_{\text{rec},0}=7.0\times10^{3}\ \mathrm{km\,s^{-1}} \]

Reading. The comoving separation never changed; the physical distance and recession velocity are entirely encoded in \( a(t) \) and its rate \( H \). Units: \( \mathrm{Mpc}\times\mathrm{km\,s^{-1}\,Mpc^{-1}}=\mathrm{km\,s^{-1}} \), a speed.

\[ \chi(r)=\int_0^{r}\frac{dr'}{\sqrt{1-kr'^{2}}}=\frac{1}{\sqrt{k}}\,\arcsin\!\big(\sqrt{k}\,r\big),\qquad k>0 \]

Proper radial distance in a closed universe. Symbols first: the radial part of the spatial metric is \( dr/\sqrt{1-kr^{2}} \) (at fixed \( t \), with \( a=1 \)); integrating gives the proper radial distance \( \chi \), an arcsine because \( k>0 \). B

\[ R_{\text{curv}}=\frac{1}{\sqrt{k}}=10\ \mathrm{Gpc},\qquad r=5\ \mathrm{Gpc}=\tfrac{1}{2}R_{\text{curv}}\ \Rightarrow\ \sqrt{k}\,r=0.5 \]

Choose a curvature radius \( R_{\text{curv}}=1/\sqrt{k}=10\ \mathrm{Gpc} \) and a galaxy at half that coordinate. Then \( \sqrt{k}\,r=0.5 \). A

\[ \chi=R_{\text{curv}}\arcsin(0.5)=10\ \mathrm{Gpc}\times\frac{\pi}{6}=5.24\ \mathrm{Gpc} \]

Evaluate \( \arcsin(0.5)=\pi/6\approx0.5236 \). A

\[ \chi=5.24\ \mathrm{Gpc}\ >\ r=5.00\ \mathrm{Gpc} \]

Reading. In a positively curved space the true (proper) radial distance exceeds the areal-coordinate value \( r \): the geometry "bows outward," so 5 Gpc of coordinate corresponds to 5.24 Gpc of walked distance. For \( k=0 \) the two coincide; for \( k<0 \) (arcsinh) the proper distance is smaller than \( r \). Units: \( \mathrm{Gpc}\times(\text{dimensionless})=\mathrm{Gpc} \), a length.

Problems

Show that under the rescaling \( r\to\lambda r,\ a\to a/\lambda \) (with \( \lambda>0 \) constant) the FRW line element is invariant provided \( k\to k/\lambda^{2} \). Conclude that only the sign of \( k \) is physical and that \( |k| \) can always be set to \( 0 \) or \( 1 \).
Solution
Under \( r=\lambda\tilde r \), \( dr=\lambda\, d\tilde r \) and \( a=\tilde a/\lambda \). The angular term: \( a^{2}r^{2}d\Omega^{2}=(\tilde a/\lambda)^{2}(\lambda\tilde r)^{2}d\Omega^{2}=\tilde a^{2}\tilde r^{2}d\Omega^{2} \). The radial term: \( a^{2}\dfrac{dr^{2}}{1-kr^{2}}=(\tilde a/\lambda)^{2}\dfrac{\lambda^{2}d\tilde r^{2}}{1-k\lambda^{2}\tilde r^{2}}=\tilde a^{2}\dfrac{d\tilde r^{2}}{1-(k\lambda^{2})\tilde r^{2}} \). Defining \( \tilde k=k\lambda^{2} \) reproduces the identical form in \( (\tilde a,\tilde r,\tilde k) \). Since \( \tilde k=k\lambda^{2} \) has the same sign as \( k \) for any \( \lambda \), the sign is invariant, and choosing \( \lambda=1/\sqrt{|k|} \) (for \( k\neq0 \)) sets \( \tilde k=\pm1 \).
Count the independent Killing vectors of a spatial slice of the FRW metric and interpret them. What is the total for the case of maximal symmetry, and which physical symmetries do they represent?
Solution
A maximally symmetric \( n \)-dimensional space has \( n(n+1)/2 \) Killing vectors. For \( n=3 \) this is \( 3\cdot4/2=6 \): three correspond to homogeneity (spatial translations, moving any point to any other) and three to isotropy (rotations, mapping any direction to any other about a point). This is the maximal number; it is exactly the statement that the slice is homogeneous and isotropic. (The full 4D FRW spacetime generically has these same 6 spatial Killing vectors but is not maximally symmetric in 4D, which would require 10, because \( a(t) \) breaks time-translation invariance unless \( a=\text{const} \).)
In a flat (\( k=0 \)) FRW universe the scale factor during matter domination is \( a(t)=(t/t_0)^{2/3} \) with \( a(t_0)=1 \). Compute the proper distance today to a comoving galaxy at \( r=200\ \mathrm{Mpc} \), and its distance at \( t=t_0/8 \).
Solution
Proper distance \( D(t)=a(t)\,r \). Today \( a(t_0)=1 \Rightarrow D_0=200\ \mathrm{Mpc} \). At \( t=t_0/8 \): \( a=(1/8)^{2/3}=(2^{-3})^{2/3}=2^{-2}=1/4 \). Hence \( D=\tfrac14\times200\ \mathrm{Mpc}=50\ \mathrm{Mpc} \). The galaxy's comoving position is unchanged; only \( a \) rescales the physical separation.
Starting from the FRW metric, derive the redshift relation for a photon emitted at \( t_e \) and observed at \( t_o \) by showing that the comoving radial distance travelled by a radial light ray is \( \displaystyle\int_{t_e}^{t_o}\frac{c\,dt}{a(t)} \), and hence that successive wavecrests give \( 1+z=a(t_o)/a(t_e) \).
Solution
For a radial null ray \( ds^2=0,\ d\theta=d\varphi=0 \): \( 0=-c^2dt^2+a^2\,dr^2/(1-kr^2) \), so \( \dfrac{dr}{\sqrt{1-kr^2}}=\pm\dfrac{c\,dt}{a} \). Integrating from emitter to observer, the comoving distance \( \int_0^{r_e}dr/\sqrt{1-kr^2}=\int_{t_e}^{t_o}c\,dt/a(t) \) is fixed by the geometry. A crest emitted at \( t_e \) and the next at \( t_e+\delta t_e \) traverse the same comoving distance, so \( \int_{t_e}^{t_o}\frac{c\,dt}{a}=\int_{t_e+\delta t_e}^{t_o+\delta t_o}\frac{c\,dt}{a} \). Subtracting the common interval gives \( \dfrac{\delta t_e}{a(t_e)}=\dfrac{\delta t_o}{a(t_o)} \). Since \( \delta t\propto \) wave period \( \propto\lambda/c \), \( \dfrac{\lambda_o}{\lambda_e}=\dfrac{a(t_o)}{a(t_e)} \), i.e. \( 1+z\equiv\dfrac{\lambda_o}{\lambda_e}=\dfrac{a(t_o)}{a(t_e)} \).
The Ricci scalar of a spatial slice of the FRW metric is \( {}^{(3)}\!R=6k/a^{2} \). For a closed universe with present curvature radius \( R_{\text{curv},0}=1/\sqrt{k}=15\ \mathrm{Gpc} \) and \( a(t_0)=1 \), compute \( {}^{(3)}\!R \) today and at \( a=0.1 \). Comment on the trend.
Solution
With \( a_0=1 \): \( k=1/R_{\text{curv},0}^2=(15\ \mathrm{Gpc})^{-2}=4.44\times10^{-3}\ \mathrm{Gpc^{-2}} \). Today \( {}^{(3)}\!R=6k/1^2=2.67\times10^{-2}\ \mathrm{Gpc^{-2}} \) (equivalently curvature radius of the slice \( =\sqrt{6/{}^{(3)}\!R}=\sqrt{6}\,R_{\text{curv},0}/\sqrt{6}=15\ \mathrm{Gpc}\) up to the \( n(n-1) \) factor). At \( a=0.1 \): \( {}^{(3)}\!R=6k/a^2=2.67\times10^{-2}/(0.1)^2=2.67\ \mathrm{Gpc^{-2}} \), a factor \( 1/a^2=100 \) larger. The intrinsic spatial curvature scales as \( a^{-2} \): the early universe was far more sharply curved, and curvature dilutes as space expands (slower than matter, \( a^{-3} \), or radiation, \( a^{-4} \), which is why curvature becomes dynamically negligible at late times).