FLRW Geometry

Friedmann–Lemaître–Robertson–Walker geometry describes a spacetime whose spatial homogeneity and isotropy are shared by the full metric and its orthogonal fundamental-observer congruence. Symmetry leaves a lapse, one scale factor, and one constant-curvature reference metric. Choosing cosmic time fixes the lapse; it does not impose Einstein’s equation or determine the scale factor. This page constructs the geometry and its curvature before any matter model is introduced.

Required background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian metrics and causal cones; Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields background; this page uses its timelike-worldline, affine-null-ray, and proper-time material for the invariant interpretation of the adapted coordinates, not Jacobi-field theory.

Helpful background. The Einstein Equation supplies the Einstein equations that determine the scale factor once stress energy is specified, and Relativistic Matter and Compact Objects supplies the perfect-fluid interpretation used later.

Constant-curvature spatial slices

Let $\Sigma_t$ be a spatial slice with induced metric $h_{ij}(t)$. Isotropy about every point requires its Riemann tensor to be invariant under all spatial rotations. In three dimensions the only allowed form is

$${}^{(3)}R_{ijkl} =\mathcal K(t)\left(h_{ik}h_{jl}-h_{il}h_{jk}\right),$$

so ${}^{(3)}R_{ij}=2\mathcal K h_{ij}$ and ${}^{(3)}R=6\mathcal K$. Homogeneity makes $\mathcal K$ constant across each slice, although it can vary from one slice to another. Write

$$h_{ij}(t)=a_{\rm dim}^2(t)\gamma_{ij}, \qquad \mathcal K(t)=\frac{\kappa}{a_{\rm dim}^2(t)}, \qquad {}^{(3)}R[h]=\frac{6\kappa}{a_{\rm dim}^2(t)},$$

where the time-independent reference metric $\gamma_{ij}$ has normalized curvature $\kappa=+1,0,-1$. In this normalized-curvature form, $a_{\rm dim}$ carries length and sets the curvature-radius scale when $\kappa\ne0$. The dimensionless scale-factor convention used for redshift is translated explicitly below; the two conventions must not be mixed within one formula.

In geodesic polar coordinates the reference line element is

$$d\Sigma_\kappa^2=d\chi^2+S_\kappa^2(\chi)d\Omega_2^2, \qquad S_\kappa(\chi)= \begin{cases} \sin\chi,&\kappa=+1,\\ \chi,&\kappa=0,\\ \sinh\chi,&\kappa=-1. \end{cases}$$

The identities

$$S_\kappa''=-\kappa S_\kappa, \qquad (S_\kappa')^2=1-\kappa S_\kappa^2$$

give an independent curvature calculation. For the warped metric above, a radial two-plane has sectional curvature $-S_\kappa''/S_\kappa=\kappa$, while a tangential two-plane has curvature $[1-(S_\kappa')^2]/S_\kappa^2=\kappa$. Thus all sectional curvatures of $\gamma_{ij}$ equal $\kappa$, so $R[\gamma]=6\kappa$ and the constant rescaling by $a_{\rm dim}^2$ gives $R[h]=6\kappa/a_{\rm dim}^2$.

The same local geometry is often written

$$d\Sigma_\kappa^2=\frac{dr^2}{1-\kappa r^2}+r^2d\Omega_2^2,$$

with $r=S_\kappa(\chi)$. For $\kappa=+1$, $r=\sin\chi$ turns around at the equator, so this $r$ chart is not global. Tong constructs the flat, spherical, and hyperbolic coordinate forms using embeddings in Tong 2019, §1.1.1, pp. 7–10 (PDF).

A third useful chart is stereographic:

$$d\Sigma_\kappa^2 =\frac{d\rho^2+\rho^2d\Omega_2^2} {\left(1+\kappa\rho^2/4\right)^2}, \qquad r=\frac{\rho}{1+\kappa\rho^2/4}.$$

This form makes conformal flatness of each spatial slice explicit and supplies an independent coordinate check on the curvature. For $\kappa=+1$, $0\leq\rho\lt\infty$ covers the three-sphere except for the pole at $\rho=\infty$; for $\kappa=0$ the same range covers Euclidean space. For $\kappa=-1$, the complete hyperbolic slice is the ball $0\leq\rho\lt2$, whose conformal boundary lies at $\rho=2$.

Constant local curvature does not fix topology. Flat space may be $\mathbb R^3$ or a flat compact quotient, and spherical or hyperbolic spaces also admit quotients. Generic quotients need not preserve global isotropy or even global homogeneity. The local FLRW ansatz therefore fixes the spatial metric only up to global identifications; imposing the cosmological principle globally restricts those identifications further.

Lapse, scale factor, and time coordinates

The symmetry must constrain how the slices evolve as well as their intrinsic geometry. For example, every fixed-time slice of diagonal Bianchi I is Euclidean after rescaling its three coordinates, yet unequal directional expansion rates give nonzero shear. The rotations that make one slice isotropic then need not extend to symmetries of the spacetime. In FLRW, the same spatial symmetry acts throughout the evolution, so the expansion is isotropic and only one scale factor is needed. The Bianchi comparison makes this counterexample explicit.

Choose a coordinate chart adapted to the fundamental congruence and orthogonal spatial slices. This gauge choice sets the shift to zero; a nonzero shift can reappear in a different coordinate representation of the same FLRW geometry. Homogeneity makes the lapse spatially constant, $N=N(\lambda)$, while isotropy removes direction-dependent spatial deformation. Before choosing a clock, the line element is

$$ds^2=-N^2(\lambda)d\lambda^2 +a_{\rm dim}^2(\lambda)d\Sigma_\kappa^2.$$

The reparameterization $dt=N(\lambda)d\lambda$ defines cosmic time and gives

$$ds^2=-dt^2+a_{\rm dim}^2(t)d\Sigma_\kappa^2.$$

Worldlines of fixed $(\chi,\theta,\phi)$ have proper time $t$ and form the comoving congruence. The choice $N=1$ is a coordinate gauge choice, whereas the proper expansion $H=\dot a_{\rm dim}/a_{\rm dim}$ is a scalar measured by that congruence. Ellis and van Elst give the covariant symmetry and congruence construction in Ellis and van Elst 1999, §2.1, pp. 6–9, and §4.1, pp. 21–22 (PDF).

In this normalized-curvature package, conformal time is the dimensionless coordinate defined by $d\eta_{\rm n}=dt/a_{\rm dim}(t)$, so

$$ds^2=a_{\rm dim}^2(\eta_{\rm n}) \left[-d\eta_{\rm n}^2+d\Sigma_\kappa^2\right].$$

This form makes radial null paths simple, but the conformal factor remains physical: it controls proper times, areas, photon wavelengths, and curvature. To pass to the convention used for redshift and Fourier scales, choose a reference epoch $t_0$ and define

$$A(t)=\frac{a_{\rm dim}(t)}{a_{{\rm dim},0}}, \qquad K=\frac{\kappa}{a_{{\rm dim},0}^{2}}, \qquad d\Sigma_K^2=a_{{\rm dim},0}^{2}d\Sigma_\kappa^2,$$

where $a_{{\rm dim},0}=a_{\rm dim}(t_0)$. Rescale the conformal and comoving coordinates at the same time:

$$\eta=a_{{\rm dim},0}\eta_{\rm n}, \qquad \chi=a_{{\rm dim},0}\chi_{\rm n}, \qquad d\eta=\frac{dt}{A(t)}.$$

Then $A(t_0)=1$, $A$ is dimensionless, $K$ has dimensions of inverse length squared, and $\eta$ and the radial coordinate $\chi$ have dimensions of length; angular coordinates remain dimensionless. Flat Cartesian comoving coordinates are rescaled by the same constant as $\chi$. The unchanged physical metric can be written

$$ds^2=-dt^2+A^2(t)d\Sigma_K^2 =A^2(\eta)\left[-d\eta^2+d\Sigma_K^2\right].$$

Later pages rename $A$ as $a$ and use $K$ for spatial curvature. A Fourier wave number conjugate to $x^i$ is correspondingly dimensional. For $\kappa=0$, the constant split between scale factor and comoving coordinate is arbitrary.

Four-dimensional curvature checks

For the site curvature convention and cosmic time, direct calculation gives

$$R_{00}=-3(\dot H+H^2),$$ $$R_{ij}= \left(\dot H+3H^2+\frac{2\kappa}{a_{\rm dim}^2}\right)g_{ij}, \qquad R=6\left(\dot H+2H^2+\frac{\kappa}{a_{\rm dim}^2}\right).$$

These expressions provide four useful checks. First, a constant $a_{\rm dim}=a_*\gt0$ with $\kappa=0$ gives Minkowski spacetime after a constant spatial-coordinate rescaling. Second, $a_{\rm dim}=t$ and $\kappa=-1$ also gives $R_{ab}=0$: the Milne metric is merely an expanding coordinate patch of Minkowski spacetime. Thus nonzero spatial curvature need not mean nonzero spacetime curvature. Third, $\kappa=0$ and constant $H$ gives $R=12H^2$, the flat slicing of de Sitter spacetime. Spatial flatness therefore does not mean a flat four-geometry. Fourth, the Einstein-static geometry has $a_{\rm dim}=a_E$, $H=0$, and $\kappa=+1$:

$$R_{00}=0, \qquad R_{ij}=\frac{2}{a_E^2}g_{ij}, \qquad R=\frac{6}{a_E^2}.$$

The FLRW Weyl tensor vanishes. This expresses the absence of a local tidal field not already encoded in Ricci curvature, but conformal flatness alone is not a definition of FLRW: one must also identify the homogeneous, isotropic slicing and its observer congruence.

What the curvature parameter measures

With normalized $\kappa$, the physical curvature radius is $a_{\rm dim}(t)$ for $|\kappa|=1$; with the dimensionless scale factor it is $A/\sqrt{|K|}$. The sign of the curvature parameter labels spherical, flat, or hyperbolic intrinsic geometry. It does not by itself specify topology, total mass density, causal horizons, or the sign of four-dimensional curvature. Connections between curvature and density arise only after the Friedmann constraint is imposed.

This separation is the main geometric discipline for the rest of the volume: $a_{\rm dim}(t)$ describes the scale of the preferred slices, $\kappa/a_{\rm dim}^2=K/A^2$ their intrinsic sectional curvature, and Einstein’s equation supplies their dynamics. The next chapter derives that dynamics rather than building it into the metric.

Exercises

Starting from the $r$ form of $d\Sigma_\kappa^2$, show that $r=S_\kappa(\chi)$ produces the geodesic-polar form for all three normalized curvature signs.

Solution

For $\kappa=+1$, take $r=\sin\chi$. Then $dr^2/(1-r^2)=d\chi^2$ and $r^2d\Omega_2^2=\sin^2\chi\,d\Omega_2^2$. For $\kappa=0$, $r=\chi$ is immediate. For $\kappa=-1$, take $r=\sinh\chi$; since $1+r^2=\cosh^2\chi$, the radial term is again $d\chi^2$, while the angular term becomes $\sinh^2\chi\,d\Omega_2^2$. The positive-curvature substitution also reveals why one $r$ chart cannot distinguish the two hemispheres.

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.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas Term 2019. Official PDF.