---
title: The FRW Metric and Cosmological Redshift
draft: false
module: Cosmic Expansion and Dynamics
moduleNumber: 11
lessonNumber: 2
order: 1102
summary: >
  The geometry of a homogeneous, isotropic universe is fixed by symmetry to the
  Robertson-Walker metric, with the entire freedom reduced to a scale factor and a
  single curvature constant selecting an open, flat, or closed space. From the metric
  the null geodesic of light gives comoving distance, the exact cosmological redshift,
  and the distinction between the proper distance we cannot measure and the redshift
  we can.
topics: [Cosmology]
sources:
  - book: Ryden
    ref: "Ch. 3 §3.3 The Robertson-Walker Metric; Ch. 4 — Cosmic Dynamics"
  - book: Carroll & Ostlie
    ref: "Ch. 29 — Cosmology; §29.2 The Robertson-Walker Metric"
---

The Newtonian sphere of the previous lesson gets the matter-dominated dynamics right
but cannot describe the geometry of space itself, the bending of light paths, or the
behaviour of radiation and vacuum energy. For those we need the spacetime metric. The
central result of this lesson is that the cosmological principle is so restrictive
that it fixes the metric almost completely: up to the time-dependent scale factor and
a single discrete choice of spatial curvature, there is exactly one metric consistent
with homogeneity and isotropy, the **Robertson-Walker metric**. From it we read off
the three possible spatial geometries, trace the null geodesic that light follows,
and derive the cosmological redshift rigorously, confirming the scale-factor formula
asserted earlier.

## Spacetime intervals and the metric

In special relativity the invariant interval between two nearby events separated by
$(c\,\d t, \d x, \d y, \d z)$ is

$$
\d s^{2} = -c^{2}\,\d t^{2} + \d x^{2} + \d y^{2} + \d z^{2} ,
$$

with the sign convention that timelike intervals have $\d s^2 < 0$. This interval is
the same in every inertial frame and encodes the entire geometry of flat spacetime. A
**metric** generalizes this: it is the rule that assigns a squared interval to any
infinitesimal displacement, written $\d s^{2} = g_{\mu\nu}\,\d x^{\mu}\,\d x^{\nu}$ with
the metric coefficients $g_{\mu\nu}$ possibly functions of position and time. Gravity,
in general relativity, is nothing but curvature of this metric. Cosmology asks: what is
the most general metric describing a spacetime that is spatially homogeneous and
isotropic at every instant?

A light ray always travels on a **null geodesic**, the path along which $\d s^{2} = 0$.
Setting the interval to zero is the statement that light traces the causal structure of
spacetime, and it is the equation we will integrate to connect emission and observation.
Massive particles at rest in the comoving frame follow **timelike geodesics** along which
only $\d t$ changes, so that the coordinate time $t$ is the proper time measured by a
comoving clock — cosmic time.

## The Robertson-Walker metric

Isotropy about every point forces the spatial part of the metric to be spherically
symmetric about every point, and homogeneity forces its curvature to be the same
everywhere. A space of constant curvature in three dimensions comes in exactly three
types, distinguished by the sign of the curvature. Writing the spatial line element in
comoving spherical coordinates $(r, \theta, \phi)$ and factoring the overall
time-dependent scale, the metric is the **Robertson-Walker** (or
Friedmann-Robertson-Walker, FRW) form:

$$
\d s^{2} = -c^{2}\,\d t^{2}
  + a(t)^{2}\left[\frac{\d r^{2}}{1 - k r^{2}} + r^{2}\big(\d\theta^{2} + \sin^{2}\theta\,\d\phi^{2}\big)\right].
$$

Everything about the expansion sits in $a(t)$; everything about the geometry sits in
the **curvature constant** $k$. By rescaling the comoving radial coordinate one can
always normalize $k$ to one of three values, $k \in \{+1, 0, -1\}$, in units where $r$
carries the dimension of length and $a$ is dimensionless. The three cases are the three
geometries:[^ry-rw]

- $k = +1$: **closed** (positively curved, spherical) — finite volume, no boundary.
- $k = 0$: **flat** (Euclidean) — the spatial slices are ordinary flat space.
- $k = -1$: **open** (negatively curved, hyperbolic) — infinite, saddle-like.

> **Definition (Robertson-Walker metric).** The unique spacetime metric consistent
> with spatial homogeneity and isotropy, carrying a single time-dependent scale factor
> $a(t)$ and a single spatial-curvature constant $k \in \{-1, 0, +1\}$.

It is often cleaner to use a rescaled radial coordinate $\chi$ defined by $\d\chi =
\d r / \sqrt{1 - k r^{2}}$, which turns the metric into

$$
\d s^{2} = -c^{2}\,\d t^{2}
  + a(t)^{2}\left[\d\chi^{2} + S_k(\chi)^{2}\big(\d\theta^{2} + \sin^{2}\theta\,\d\phi^{2}\big)\right],
$$

where the function $S_k(\chi)$ packages the curvature,

$$
S_k(\chi) = \begin{cases}
\sin\chi, & k = +1, \\
\chi, & k = 0, \\
\sinh\chi, & k = -1.
\end{cases}
$$

Here $\chi$ is the comoving radial distance and $S_k(\chi)$ is the comoving
transverse distance — the two coincide only in the flat case, and it is their divergence in
curved space that makes distant objects subtend anomalous angles.

$$
% caption: The three constant-curvature geometries as two-dimensional analogs: the
% saddle (open), the plane (flat), and the sphere (closed).
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% open: saddle
\begin{scope}[shift={(0,0)}]
\draw[very thick] (-1.0,0.5) .. controls (-0.3,-0.2) and (0.3,1.2) .. (1.0,0.5);
\draw[very thick] (-1.0,-0.5) .. controls (-0.3,0.2) and (0.3,-1.2) .. (1.0,-0.5);
\draw[black] (-1.0,0.5) -- (-1.0,-0.5);
\draw[black] (1.0,0.5) -- (1.0,-0.5);
\node[black, anchor=north] at (0,-1.3) {open: k = -1};
\end{scope}
% flat: plane
\begin{scope}[shift={(4.0,0)}]
\draw[very thick] (-1.0,-0.6) -- (1.0,-0.6) -- (1.4,0.4) -- (-0.6,0.4) -- cycle;
\node[black, anchor=north] at (0,-1.3) {f\/lat: k = 0};
\end{scope}
% closed: sphere
\begin{scope}[shift={(8.0,0)}]
\draw[very thick] (0,0) circle (0.9);
\draw[black] (0,0) ellipse (0.9 and 0.32);
\draw[black] (0,0.9) arc (90:270:0.32 and 0.9);
\node[black, anchor=north] at (0,-1.3) {closed: k = +1};
\end{scope}
\end{tikzpicture}
$$

## The three geometries

The sign of $k$ has concrete geometric consequences that are, in principle, directly
observable. In each geometry the familiar Euclidean theorems are modified. Consider the
sum of the interior angles of a triangle and the behaviour of initially parallel
geodesics:[^co-rw]

- In **flat** space ($k = 0$) the angle sum is exactly $180^{\circ}$, parallel lines
  stay parallel, and the circumference of a circle of radius $\chi$ is $2\pi\chi$.
- In **closed** space ($k = +1$) the angle sum **exceeds** $180^{\circ}$, initially
  parallel geodesics converge and eventually cross (like meridians on a globe), and a
  circle of comoving radius $\chi$ has circumference $2\pi\sin\chi < 2\pi\chi$. The
  total volume is finite.
- In **open** space ($k = -1$) the angle sum is **less than** $180^{\circ}$, parallel
  geodesics diverge, and a circle has circumference $2\pi\sinh\chi > 2\pi\chi$. The
  volume is infinite.

Which geometry describes our universe is an empirical question, settled by measuring
whether an object of known physical size subtends the angle predicted by flat geometry.
The position of the first acoustic peak in the cosmic microwave background performs
exactly this measurement and finds the universe flat to within about half a percent, so
$k = 0$ to current precision. The curvature constant is therefore not a free parameter
in practice, but the formalism must carry all three cases because flatness is a measured
result, not an assumption.

$$
% caption: A triangle drawn on each geometry. The interior angles sum to more than a
% straight angle when closed, exactly a straight angle when flat, and less when open.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% closed: fat triangle (bulging sides)
\begin{scope}[shift={(0,0)}]
\draw[very thick] (0,1.3) .. controls (-0.9,0.6) .. (-1.0,-0.7)
  .. controls (0,-1.0) .. (1.0,-0.7) .. controls (0.9,0.6) .. (0,1.3);
\node[black, anchor=north] at (0,-1.3) {closed: sum > 180};
\end{scope}
% flat: straight triangle
\begin{scope}[shift={(4.2,0)}]
\draw[very thick] (0,1.3) -- (-1.0,-0.7) -- (1.0,-0.7) -- cycle;
\node[black, anchor=north] at (0,-1.3) {f\/lat: sum = 180};
\end{scope}
% open: thin triangle (caved-in sides)
\begin{scope}[shift={(8.4,0)}]
\draw[very thick] (0,1.3) .. controls (-0.3,0.3) .. (-1.0,-0.7)
  .. controls (0,-0.4) .. (1.0,-0.7) .. controls (0.3,0.3) .. (0,1.3);
\node[black, anchor=north] at (0,-1.3) {open: sum < 180};
\end{scope}
\end{tikzpicture}
$$

## The null geodesic of light and comoving distance

To connect what we observe to where and when it was emitted, follow a light ray from a
distant source to us. Place ourselves at the spatial origin $\chi = 0$ and the source at
comoving radial coordinate $\chi_e$. Light travels radially along a null geodesic, so
$\d s^{2} = 0$ with $\d\theta = \d\phi = 0$. The Robertson-Walker metric then gives

$$
0 = -c^{2}\,\d t^{2} + a(t)^{2}\,\d\chi^{2}
\quad\Longrightarrow\quad
c\,\d t = -a(t)\,\d\chi ,
$$

taking the ray to move inward toward decreasing $\chi$ as $t$ increases. Rearranging
and integrating from emission at $t_e$ (radial coordinate $\chi_e$) to observation at
$t_0$ (origin),

$$
\chi_e = \int_0^{\chi_e} \d\chi = \int_{t_e}^{t_0} \frac{c\,\d t}{a(t)} .
$$

The left side is the **comoving distance** to the source, a fixed number independent of
time. The right side expresses it as an integral over cosmic history weighted by
$1/a(t)$: light emitted when $a$ was small covers more comoving distance per unit time.
The comoving distance is the fundamental distance in cosmology because it is constant in
the comoving frame; every observable distance measure is built from it and the curvature
function $S_k$.

$$
% caption: A light ray reaches the observer at the origin from a source at comoving
% radius chi. Emitted early, when a was small, each interval of time spans more chi.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {comoving distance};
\draw[->, black] (0,0) -- (0,4.0) node[above, black!70] {cosmic time};
% observer worldline at origin
\draw[black, thick] (0,0) -- (0,3.7);
\fill[black] (0,3.5) circle (0.09);
\node[black, anchor=east] at (-0.1,3.5) {observer now};
% source worldline
\draw[black, thick] (7.2,0) -- (7.2,3.7);
\fill[black] (7.2,1.0) circle (0.09);
\node[black, anchor=west] at (7.3,1.0) {emission};
% light path: curved because ds units of chi per dt vary; draw as smooth curve
\draw[acc, very thick] (7.2,1.0) .. controls (4.5,1.5) and (2.0,2.6) .. (0,3.5);
\node[acc, anchor=south] at (3.7,2.3) {null geodesic};
\end{tikzpicture}
$$

## The redshift derived from the metric

Now trace two successive wave crests, or two successive photons. The first crest is
emitted at $t_e$ and observed at $t_0$; the second, one period later, is emitted at
$t_e + \delta t_e$ and observed at $t_0 + \delta t_0$. Both traverse the same fixed
comoving distance $\chi_e$, because the source and observer are comoving. Applying the
null-geodesic integral to each crest,

$$
\int_{t_e}^{t_0} \frac{c\,\d t}{a(t)} = \chi_e
= \int_{t_e + \delta t_e}^{t_0 + \delta t_0} \frac{c\,\d t}{a(t)} .
$$

Subtracting the two integrals, the overlapping interior cancels and only the small
end pieces remain. To first order in the small periods, using that $a(t)$ is
essentially constant over one wave period,

$$
\frac{\delta t_0}{a(t_0)} = \frac{\delta t_e}{a(t_e)} .
$$

The emitted period is $\delta t_e$ and the observed period is $\delta t_0$; the
wavelengths are $\lambda_e = c\,\delta t_e$ and $\lambda_0 = c\,\delta t_0$. Therefore

$$
\frac{\lambda_0}{\lambda_e} = \frac{\delta t_0}{\delta t_e}
= \frac{a(t_0)}{a(t_e)} = \frac{a_0}{a(t_e)} ,
$$

and with the redshift defined by $1 + z = \lambda_0/\lambda_e$,

$$
1 + z = \frac{a_0}{a(t_e)} = \frac{1}{a(t_e)} .
$$

This is the rigorous derivation of the result stated in the previous lesson, now
following directly from the metric rather than from a heuristic. The redshift is a
measure of how much the universe has expanded between emission and observation — a pure
statement about the scale factor, containing no reference to velocity. The same
calculation shows that any periodic process at the source appears time-dilated by the
factor $1 + z$: distant supernova light curves, for example, are observed to evolve
more slowly by exactly $(1 + z)$, a direct confirmation that the redshift is cosmological
expansion and not, say, tired light losing energy en route.[^ry-redshift]

$$
% caption: Two wave crests emitted one period apart traverse the same comoving
% distance; the observed period is stretched by the ratio of scale factors.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {comoving distance};
\draw[->, black] (0,0) -- (0,3.6) node[above, black!70] {cosmic time};
% observer and source worldlines
\draw[black] (0,0) -- (0,3.4);
\draw[black] (6.8,0) -- (6.8,3.4);
\node[black, anchor=north] at (6.8,-0.15) {source};
\node[black, anchor=north] at (0,-0.15) {us};
% two crests
\draw[acc, very thick] (6.8,0.6) .. controls (4.0,1.0) and (1.8,1.8) .. (0,2.5);
\draw[acc, very thick, densely dashed] (6.8,0.95) .. controls (4.0,1.4) and (1.8,2.4) .. (0,3.15);
% emitted gap (small)
\draw[<->, black] (7.0,0.6) -- (7.0,0.95);
\node[black, anchor=west] at (7.1,0.78) {emitted gap};
% observed gap (large)
\draw[<->, black] (-0.25,2.5) -- (-0.25,3.15);
\node[black, anchor=east, align=right] at (-0.35,2.85) {observed\\gap larger};
\end{tikzpicture}
$$

## Proper distance versus the observed redshift

It is worth being precise about what we can and cannot measure. The **proper distance**
at cosmic time $t$ is the physical distance along a spatial slice of constant $t$,

$$
d_p(t) = a(t)\,\chi_e ,
$$

the length a chain of rulers laid end to end would read if we could freeze the
expansion. Differentiating at the present epoch reproduces Hubble's law, $\dot d_p =
H_0 d_p$. But $d_p$ is **not directly observable**: we cannot lay rulers across the
universe, and by the time light from a distant galaxy reaches us the galaxy has receded
further. What we actually measure is the redshift $z$, together with fluxes and angular
sizes. Redshift and proper distance are related only through the expansion history
$a(t)$: the same $z$ corresponds to different proper distances in universes with
different $a(t)$.

For small redshift the relation is simple. Expanding $a(t_e)$ about the present and using
$1 + z = 1/a(t_e)$ gives $z \approx H_0 d_p / c$ to leading order, the low-redshift Hubble
law. But at large $z$ the proper distance depends on the entire integrated history, and
several distinct distance measures — luminosity distance, angular-diameter distance,
comoving distance — diverge from one another. Those measures, and the way they encode the
contents of the universe, are the subject of the models lesson. The essential point here
is conceptual: the redshift is the primary observable and the scale factor is what it
reports; proper distance is a derived, model-dependent quantity.

$$
% caption: Comoving distance is fixed; proper distance is the comoving distance times
% the scale factor and grows with time. The redshift reports the scale factor at emission.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {cosmic time};
\draw[->, black] (0,0) -- (0,4.0) node[above, black!70] {distance};
% comoving distance: horizontal (constant)
\draw[black, very thick, densely dashed] (0.3,1.5) -- (8.0,1.5);
\node[black, anchor=south] at (5.4,1.55) {comoving distance: f\/ixed};
% proper distance: rising as a(t)
\draw[acc, very thick] (0.3,0.35) .. controls (3.0,0.9) and (5.5,2.2) .. (8.0,3.7);
\node[acc, anchor=south east] at (7.9,3.3) {proper distance = a chi};
% mark now
\draw[black, densely dotted] (8.0,0) -- (8.0,3.7);
\node[black, anchor=north] at (8.0,-0.15) {now};
\end{tikzpicture}
$$

## Summary

Homogeneity and isotropy force the spacetime metric into the Robertson-Walker form,
$\d s^{2} = -c^{2}\d t^{2} + a(t)^{2}[\d\chi^{2} + S_k(\chi)^{2}\d\Omega^{2}]$, with the
whole geometric freedom reduced to the scale factor $a(t)$ and a single curvature
constant $k \in \{-1, 0, +1\}$ selecting an open, flat, or closed space. The three
geometries differ in their triangle angle sums and in whether parallel geodesics
converge or diverge, and observation of the CMB finds the universe flat to sub-percent
precision. Light follows the null geodesic $c\,\d t = -a\,\d\chi$, whose integral defines
the comoving distance, and tracing successive wave crests gives the exact redshift
$1 + z = 1/a(t_e)$, confirming that the redshift measures the scale factor at emission
and dilates all source timescales by $(1 + z)$. Proper distance $d_p = a\chi$ follows
from the metric but is not directly measurable; the redshift is the true observable. The
next lesson supplies the dynamics that determine $a(t)$: the
[Friedmann
equations](/astrophysics-cosmology/cosmology-expansion-and-dynamics/the-friedmann-equations-and-cosmic-dynamics),
which fix how the scale factor evolves given the contents of the universe.

[^ry-rw]: Ryden, *Introduction to Cosmology*, Ch. 3 §3.3 — The Robertson-Walker Metric and the curvature constant $k$.
[^co-rw]: Carroll & Ostlie §29.2 — the three constant-curvature geometries and their observational signatures.
[^ry-redshift]: Ryden, Ch. 4 — the metric derivation of the cosmological redshift and the time dilation of distant sources by $(1+z)$.