Curvature and expansion — the Robertson–Walker metric

How the radius of curvature of a closed universe is tied to its scale factor, why curvature is a relativistic effect, and the Robertson–Walker metric that describes the expanding universe.

Follows MIT 8.286 lecture 11MIT 8.286 Lecture 11: Non-Euclidean Spaces: Closed UniversesOn MIT OpenCourseWare: the lecture and its readings

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Full reading path (17 pages)

Vectors (8.01)
Position and velocity in one dimension (8.01)
Acceleration in one dimension (8.01)
Motion in two dimensions and projectiles (8.01)
Newton's laws of motion (8.01)
The universal law of gravitation (8.01)
The homogeneous expanding universe (8.286)
Why a static universe cannot work (8.286)
The expanding sphere and the equation for a(t) (8.286)
The Friedmann equation (8.286)
Open, closed and flat universes (8.286)
The closed universe as a cycloid (8.286)
The age of the universe and the Big Bang singularity (8.286)
The age of a closed universe from H and Ω (8.286)
The open universe and the age of any matter universe (8.286)
Non-Euclidean geometry (8.286)
The metric of a sphere and of a closed universe (8.286)

On this page · 4 sections

The radius grows with the scale factor
Curvature is a relativistic effect
The Robertson–Walker metric
Summary

The 3-sphere describes the space of a closed universe at one moment. But the universe expands. In the last part of Lecture 11 Guth connects the radius of the 3-sphere to the scale factor $a(t)$ — the one place in the course where we meet, without deriving it, how matter curves space.

The radius grows with the scale factor#

From the Newtonian model, physical distances are $\ell_p(t) = a(t)\,\ell_c$ , with comoving objects at fixed coordinates. Every distance in the 3-sphere metric is proportional to its radius $R$ , so for the geometry to expand with the universe, $R(t)$ must be proportional to $a(t)$ .

The units fix the rest. $R$ is in metres, $a$ in metres per notch and $k$ in 1/notch², so the only possibility is

Radius from units

R^2(t) = (\text{dimensionless constant}) \times \frac{a^2(t)}{k}

In words

R squared (t) equals ( dimensionless constant ) times a squared (t) over k

Source: 8.286 Lecture 11 transcript, pp. 21–22

The constant says how strongly a given amount of matter curves space, and finding it needs Einstein's field equations. Guth simply states the answer — the constant is 1:

Radius of curvature of a closed universe

R^2(t) = \frac{a^2(t)}{k}

In words

R squared (t) equals a squared (t) over k

Source: 8.286 Lecture 11 transcript, p. 23

This answers a question left open in Lecture 9: $a/\sqrt k$ is the radius of curvature of space. Guth describes $1/R$ as the curvature; in general relativity courses the (Gaussian) curvature of a sphere is usually defined as $1/R^2$ .

Curvature is a relativistic effect#

Recall $k = -2E/c^2$ , with $E < 0$ for a closed universe. Then $R^2 = a^2c^2/(2|E|)$ — the radius contains $c$ . In terms of quantities astronomers measure, using $\rho = \Omega\rho_c$ and $\rho_c = 3H^2/(8\pi G)$ :

Radius of curvature from H and Ω

R = \frac{c\,H^{-1}}{\sqrt{\Omega - 1}}

In words

R equals c H to the power minus 1 over the square root of omega minus 1

Source: 8.286 Lecture 11 transcript, p. 24

Two limits are revealing. If $c$ were infinite (at fixed density and expansion rate), $R$ would be infinite: space would be flat. Curvature is a relativistic effect. And as $\Omega \to 1$ , $R \to \infty$ : a universe at exactly the critical density is flat, which is why the critical-density case is called "flat".

Loading graph…

Radius of curvature of a closed universe, computed from R = cH⁻¹/√(Ω − 1). As Ω approaches 1 the radius grows without limit and space becomes flat.

Worked example

How curved could our universe be?

In Lecture 7 Guth quoted measurements putting $\Omega_\text{total}$ within about half a percent of 1. If the universe were closed with $\Omega - 1 = 0.005$ and $H = 67.3$ km/s/Mpc, how large would its radius of curvature be?

Hubble length

$c/H = 299{,}792 / 67.3 = 4450$ Mpc.

Radius

$R = 4450/\sqrt{0.005} = 4450 \times 14.1 = 63{,}000$ Mpc, about 63 Gpc.

Meaning

That is several times larger than the horizon distance, so over the part of the universe we can see, space would be only very slightly curved. (The relation $R = cH^{-1}/\sqrt{\Omega - 1}$ also holds with dark energy included, as long as $\Omega$ is the total density parameter.)

The Robertson–Walker metric#

Put $R^2 = a^2/k$ into the 3-sphere metric written with $u = \sin\psi$ , and define a new radial coordinate $r \equiv u/\sqrt k$ , measured in notches. The metric of space becomes

Robertson–Walker metric (space, closed universe)

ds^2 = a^2(t)\left[\frac{dr^2}{1 - k r^2} + r^2\left(d\theta^2 + \sin^2\theta\,d\phi^2\right)\right]

In words

ds squared equals a squared (t) [ dr squared over 1 minus k r squared plus r squared (d theta squared plus sine squared theta d phi squared ) ]

Source: 8.286 Lecture 11 transcript, p. 25

This is the Robertson–Walker metric, the standard description of a homogeneous, isotropic, expanding universe. The scale factor multiplies everything, so every distance grows with $a(t)$ , and the constant $k$ from the Friedmann equation now appears directly in the geometry. Guth derives it here for the closed case; open universes follow in Lecture 12, "Non-Euclidean Spaces: Open Universes and the Spacetime Metric".

Summary#

Every distance in a closed universe scales with $a(t)$ , so its radius of curvature does too: $R^2 = (\text{const})\,a^2/k$ , and general relativity fixes the constant to 1.
$a/\sqrt k$ is the radius of curvature; in observable terms $R = cH^{-1}/\sqrt{\Omega - 1}$ .
Curvature disappears as $c \to \infty$ (it is relativistic) and as $\Omega \to 1$ (the flat universe).
Substituting into the 3-sphere metric gives the Robertson–Walker metric $ds^2 = a^2(t)[dr^2/(1 - kr^2) + r^2(d\theta^2 + \sin^2\theta\,d\phi^2)]$ .

Symbols on this page

Show the 21 symbols used here

$\theta$ theta: Polar angle: measured from the north pole (0) to the south pole (π). Say it: “theta” (sounds like “thay-ta or thee-ta”).
$\pi$ pi: The ratio of a circle's circumference to its diameter, π ≈ 3.14159. Say it: “pi”.
$\rho$ rho: Density: mass (or energy divided by c²) per unit volume. ρ_c is the critical density, ρ̄ the mean density. Say it: “rho”.
$\phi$ phi: Azimuthal angle (longitude), running from 0 to 2π around the axis. Say it: “phi” (sounds like “fye”).
$\psi$ psi: The extra angle of the 3-sphere: how far round the closed universe a point is. Say it: “psi” (sounds like “sigh”).
$\Omega$ capital omega: The density parameter Ω = ρ/ρ_c. Subscripts pick one ingredient: Ω_m matter, Ω_vac (or Ω_Λ) vacuum energy, Ω_r radiation, Ω_k curvature. Say it: “omega”.
$\ell$ script l: A length or distance; ℓ_p is a physical distance and ℓ_c a comoving one. Say it: “ell”.
$\infty$ infinity: Infinity: larger than any number. Say it: “infinity”.
$\propto$ proportional to: Is proportional to: doubling one doubles the other. Say it: “is proportional to”.
$\equiv$ identical to: Is defined as (or is identically equal to). Say it: “is defined as”.
$\to$ arrow: Goes to (tends to), or becomes. Say it: “goes to”.
$\times$ times: Times (multiplication). Between two vectors: the cross product. Say it: “times”.
$\sin$ sine: Sine of an angle. Say it: “sine”.
$a$ a: The scale factor a(t): how much the universe has expanded. Physical distance = a × comoving distance (Guth's units: metres per notch). Say it: “a”.
$H$ H: The Hubble parameter H = ȧ/a: the expansion rate of the universe at a given time. Say it: “H”.
$G$ G: Newton's gravitational constant, G ≈ 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻². Say it: “G”.
$c$ c: The speed of light, c ≈ 2.998 × 10⁸ m/s. Say it: “c”.
$k$ k: The curvature constant: k > 0 closed, k = 0 flat, k < 0 open (Guth's units: 1/notch²). Say it: “k”.
$\rho_{c}$ critical density: The critical density, 3H²/(8πG): the density of a flat universe (about 9 × 10⁻²⁷ kg/m³ today). Say it: “rho c”.
$\ell_{p}$ physical distance: A physical (proper) distance: what a ruler would measure at one moment, ℓ_p = a(t) ℓ_c. Say it: “ell p”.
$\ell_{c}$ comoving distance: A comoving distance: fixed as the universe expands, measured in coordinate units (Guth's notches). Say it: “ell c”.

Tap or click a symbol in a formula to see the same explanation. All symbols: symbols and notation.

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How is the radius of curvature $R$ of a closed universe related to the scale factor?
If the speed of light were infinite (at fixed density and expansion), what would the radius of curvature be?
What happens to $R$ as $\Omega \to 1$ ?

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