FLRW Spacetime and Cosmological Coordinates

An FLRW spacetime is specified geometrically by a homogeneous-isotropic spatial metric, a scale factor, and a choice of time coordinate. This record fixes the normalized-curvature convention $k\in\{-1,0,+1\}$ and compares cosmic time, conformal time, geodesic-polar radius, and curvature radius. It states only stable geometry and redshift relations: matter dynamics and fitted cosmological parameters are separate questions.

Helpful background. Lorentzian Metrics and Causal Cones supplies causal classification and conformal rescaling; Exact Solutions of Einstein’s Equations supplies the chart-domain and invariant-check method.

Normalized-curvature spatial charts

Let $t$ range over an interval $I$ on which $a(t)\gt0$. In the normalized-curvature package, $a$ has dimensions of length and

$$ds^2=-dt^2+a^2(t)d\Sigma_k^2, \qquad d\Sigma_k^2=d\chi^2+S_k^2(\chi)d\Omega_2^2,$$

where

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

For the simply connected spatial models, $0\leq\chi\leq\pi$ when $k=+1$ and $0\leq\chi\lt\infty$ when $k=0$ or $-1$. The angular coordinates have their usual overlaps, and the two-sphere collapses at every regular polar origin. Homogeneity and isotropy fix the local constant-curvature geometry, not the global topology: suitable quotients can have the same local curvature while changing identifications and global symmetries.

The substitution

$$r=S_k(\chi), \qquad \left(S_k'(\chi)\right)^2=1-kS_k^2(\chi)$$

gives the alternative form

$$d\Sigma_k^2= \frac{dr^2}{1-kr^2}+r^2d\Omega_2^2.$$

$k$	$S_k(\chi)$	Geodesic-polar range	Curvature-radius chart
$+1$	$\sin\chi$	$0\leq\chi\leq\pi$ on the round three-sphere	$0\leq r\lt1$ covers one open hemisphere per monotone branch; $r\to1$ approaches the equator, where this radial chart degenerates
$0$	$\chi$	$0\leq\chi\lt\infty$ on the simply connected flat model	$0\leq r\lt\infty$
$-1$	$\sinh\chi$	$0\leq\chi\lt\infty$ on the simply connected hyperbolic model	$0\leq r\lt\infty$

The intrinsic scalar curvature of a cosmic-time slice is

$${}^{(3)}R=\frac{6k}{a^2}.$$

Thus $k$ labels spatial curvature in this normalization; it does not by itself determine four-dimensional curvature. For $k=+1$, use the $\chi$ chart or a second curvature-radius branch to cross the equator. The complete exact-spacetime treatment is FLRW Spacetimes. The symmetry construction and coordinate forms are developed in Ellis and van Elst 1999, §4.1, printed pp. 21–22, PDF.

Cosmic time, conformal time, and redshift

Worldlines of fixed $(\chi,\theta,\phi)$ are comoving, and $t$ is their proper time. Define conformal time by

$$d\eta=\frac{dt}{a(t)}.$$

Then

$$ds^2=a^2(\eta) \left(-d\eta^2+d\Sigma_k^2\right).$$

With a dot denoting $d/dt$ and a prime denoting $d/d\eta$,

$$H=\frac{\dot a}{a}, \qquad \mathcal H=\frac{a'}{a}=aH.$$

In this package $\eta$ and $\chi$ are dimensionless. A radial proper distance measured within one cosmic-time slice is $a(t)$ times the corresponding $\chi$ interval, while the area radius of a symmetry two-sphere is $a(t)S_k(\chi)$. Neither quantity should be called simply “the distance” without naming the slice and operational definition.

For a future-directed null ray with tangent $p^a$, a comoving observer $u^a=(\partial_t)^a$ measures angular frequency $\omega=-u_ap^a$. The null geodesic equation gives $a\omega=\text{constant}$, so comoving emission and observation events obey

$$1+z=\frac{\omega_e}{\omega_o} =\frac{a(t_o)}{a(t_e)}.$$

No Einstein equation is needed for this kinematic relation. Peculiar velocities, metric perturbations, and noncomoving endpoint frames require the invariant observer-frequency formula and are treated in Cosmological Redshift.

A constant rescaling can be moved between the scale factor and comoving coordinates without changing the physical metric. Normalizing nonzero curvature to $k=\pm1$ fixes that freedom by making $a$ the curvature radius; in the flat case an arbitrary constant spatial rescaling remains. An alternative convention can instead make the scale factor dimensionless and the curvature parameter dimensionful, but the two packages must not be mixed within one formula.

Choosing the coordinates

Choice	What it makes simple	Domain or interpretation caveat
Cosmic time $t$	Comoving proper time, expansion $H$, and spatial geometry on orthogonal slices	Its interval is the chosen branch of the spacetime, not automatically all real numbers
Conformal time $\eta$	Radial null propagation and conformal structure	Its range follows from $\int dt/a$ and need not match the range of $t$
Geodesic-polar radius $\chi$	Proper radial distance and complete simply connected $k=+1$ slices	Angular coordinates still require overlaps at their axes and origins
Curvature radius $r$	The familiar factor $(1-kr^2)^{-1}$	For $k=+1$, separate northern and southern branches are required

The scale factor is not determined by choosing one of these coordinates. Once stress-energy and a gravitational theory are supplied, its dynamics belong to The Friedmann Equations, not to this chart card.

Invariant and limiting checks

For the site curvature convention, the four-dimensional Ricci scalar is

$$R=6\left(\dot H+2H^2+\frac{k}{a^2}\right),$$

and the Weyl tensor vanishes. These statements provide checks of the geometry, but conformal flatness alone does not imply FLRW without the homogeneity, isotropy, and foliation hypotheses.

If $k=0$ and $a$ is constant, a constant spatial rescaling gives inertial Minkowski form, $H=0$, and zero comoving redshift. Milne coordinates instead use $k=-1$ and $a(t)=t$ on $t\gt0$. The map

$$T=t\cosh\chi, \qquad R=t\sinh\chi$$

identifies that metric with the future light cone of Minkowski spacetime. Hence $a=0$ at the Milne chart tip is not a curvature singularity. Curvature invariants, geodesic behavior, and an overlapping chart—not the scale factor alone—decide whether such an endpoint is singular.

Reall develops the FLRW geometry and causal checks in Reall 2013, §§13.3–13.4, printed pp. 158–168, PDF.

References

• Ellis, George F. R., and Henk van Elst. “Cosmological Models.” In Theoretical and Observational Cosmology, edited by Marc Lachièze-Rey, 1–116. Dordrecht: Springer, 1999. DOI. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. Open PDF.