The FRW Metric and Cosmological Redshift

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 $(c\mathbf{d}t,\mathbf{d}x,\mathbf{d}y,\mathbf{d}z)$ is

with the sign convention that timelike intervals have $\mathbf{d}{s}^2<0$. 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 $\mathbf{d}{s}^2=g_{\mu \nu }\mathbf{d}{x}^{\mu }\mathbf{d}{x}^{\nu }$ 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 $\mathbf{d}{s}^2=0$. 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 $\mathbf{d}t$ 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 $(r,\theta ,\varphi )$ and factoring the overall time-dependent scale, the metric is the Robertson-Walker (or Friedmann-Robertson-Walker, FRW) form:

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:¹

• $k=+1$: 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.

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

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

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.

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:²

• 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.

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 $\mathbf{d}{s}^2=0$ with $\mathbf{d}\theta =\mathbf{d}\varphi =0$. The Robertson-Walker metric then gives

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),

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 $S_k$.

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,

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,

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

and with the redshift defined by $1+z={\lambda }_0/{\lambda }_e$,

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.³

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 $t$,

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.

Summary

Homogeneity and isotropy force the spacetime metric into the Robertson-Walker form, $\mathbf{d}{s}^2=-c^2\mathbf{d}{t}^2+a(t{)}^2[\mathbf{d}{\chi }^2+S_k(\chi {)}^2\mathbf{d}{\Omega }^2]$, 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 $c\mathbf{d}t=-a\mathbf{d}\chi$, 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, which fix how the scale factor evolves given the contents of the universe.