Cosmological Spacetimes and Observers

Relativistic cosmology begins by specifying a spacetime, a family of observers, and the measurements made along their worldlines. This chapter separates exact FLRW symmetry from its observational motivation, constructs the metric and its preferred congruence, and derives redshift and conformal null propagation without solving for the expansion history. Its final page relaxes isotropy through Bianchi cosmologies and uses selected Einstein constraint and shear-evolution equations as a bounded diagnostic. The central discipline is to keep geometry, coordinates, observer measurements, and the assumptions entering dynamics distinct.

Helpful background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian geometry; Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields background; this chapter uses its affine-null-geodesic, timelike-geodesic, and proper-time material for light rays and observer worldlines, not Jacobi-field theory. The Einstein Equation prepares the equations sampled on the Bianchi page and developed systematically in the next chapter; Relativistic Matter and Compact Objects supplies fluid stress-energy and anisotropic stress for the observer and Bianchi comparisons.

Prepare and choose a route

Before beginning, check that you can distinguish a timelike observer from a null-ray tangent, explain why proper time fixes a lapse but not a scale-factor history, and identify curvature as a geometric quantity rather than a coordinate label. If any distinction is unfamiliar, use the linked background pages above before the corresponding derivation.

Preparation check. An observer has a unit timelike four-velocity $U^aU_a=-1$, whereas a light ray has a null tangent $k^ak_a=0$ whose normalization depends on its affine parameter. Setting the lapse to one makes the chosen time coordinate equal to proper time along the fundamental congruence; it does not supply an equation for the scale factor. Curvature is read from tensors or invariants, not from whether a line element happens to look time dependent or spatially curved.

For the FLRW-and-light route, follow pages 1–5 in order. For the symmetry-and-anisotropy route, follow pages 1–3 and then page 6; return to the light pages when observer measurements are needed. The complete chapter map is:

1. The Cosmological Principle distinguishes exact homogeneity and isotropy from statistical homogeneity, sky isotropy about one observer, and a scale-dependent approximation supported by surveys.
2. FLRW Geometry derives the constant-curvature spatial metric, lapse, scale factor, curvature parameter, and the local-versus-global topology distinction.
3. Comoving Observers and Cosmic Time identifies the fundamental congruence and computes its expansion, acceleration, vorticity, and shear.
4. Cosmological Redshift starts from the invariant frequency $-U\mathbin{\cdot}k$ and derives $1+z=a_o/a_e$ for comoving FLRW endpoints.
5. Conformal Time and Comoving Coordinates makes radial null propagation and comoving-versus-physical scales transparent.
6. Bianchi Cosmologies and Anisotropy retains homogeneity but exposes the structure constants, spatial curvature, and shear hidden by exact isotropy.

This order matters. Redshift is an observer comparison along a null geodesic—not a coordinate definition. Conformal time is a change of coordinates—not a removal of curvature or expansion. A Bianchi class fixes the homogeneous Lie algebra—not a unique spacetime solution.

The FLRW geometric core

With normalized spatial curvature $\kappa=+1,0,-1$, use a dimensionful scale factor $a_{\rm dim}$ and dimensionless spatial coordinates. The cosmic-time line element is

$$ds^2=-dt^2+a_{\rm dim}^2(t) \left[d\chi^2+S_\kappa^2(\chi)d\Omega_2^2\right],$$

where $S_{+1}=\sin\chi$, $S_0=\chi$, and $S_{-1}=\sinh\chi$. The physical sectional curvature $\mathcal K(t)$ and spatial Ricci scalar are

$$\mathcal K(t)=\frac{\kappa}{a_{\rm dim}^2(t)}, \qquad {}^{(3)}R=6\mathcal K(t).$$

Choosing a reference epoch $t_0$ gives the equivalent package $A=a_{\rm dim}/a_{{\rm dim},0}$ and $K=\kappa/a_{{\rm dim},0}^{2}$. Here $A$ is dimensionless, while $K$ has dimensions of inverse length squared. The light and Fourier pages rename $A$ as $a$ after also rescaling the comoving coordinates; the FLRW-geometry page shows the complete translation.

Constant spatial curvature is a local geometric statement. It does not choose the global topology, determine the matter density, or decide whether the four-dimensional spacetime is curved. Milne coordinates, for example, have $\kappa=-1$ and $a_{\rm dim}=t$ but describe a region of flat Minkowski spacetime.

The fundamental observers have four-velocity $u^a=(\partial_t)^a$. Their proper time is $t$, and their kinematics satisfy

$$\nabla_a u^a=3H, \qquad \dot u^a=0, \qquad \omega_{ab}=0, \qquad \sigma_{ab}=0, \qquad H=\frac{\dot a_{\rm dim}}{a_{\rm dim}}.$$

These equations state what exact FLRW symmetry removes. They do not say that real galaxies have zero peculiar velocity or that a fluid with arbitrary stress must follow this congruence. Ellis and van Elst develop the covariant observer split and the FLRW specialization in Ellis and van Elst 1999, §§2.1–2.2, pp. 6–13, and §4 opening and §4.1, pp. 21–22, eqs. (103)–(105) (PDF).

Light, clocks, and comoving scales

A future-directed null tangent $k^a$ has frequency $\omega=-U_a k^a$ when measured by an observer with four-velocity $U^a$. Therefore redshift is always an endpoint comparison,

$$1+z =\frac{(-U\mathbin{\cdot}k)_e}{(-U\mathbin{\cdot}k)_o}.$$

For comoving FLRW endpoints, null-geodesic propagation gives $a_{\rm dim}\omega=\text{constant}$ and hence

$$1+z=\frac{a_{\rm dim}(t_o)}{a_{\rm dim}(t_e)}.$$

This result is kinematical: it needs the FLRW metric and observer choice but not a Friedmann equation. Peculiar velocities and gravitational potentials alter the endpoint contractions and the propagation; they are not failures of the invariant definition.

In this normalized-curvature package, define the dimensionless conformal coordinate $\eta_{\rm n}$ on an interval with $a_{\rm dim}\gt0$ by $d\eta_{\rm n}=dt/a_{\rm dim}(t)$. Then

$$ds^2=a_{\rm dim}^2(\eta_{\rm n}) \left[-d\eta_{\rm n}^2+d\chi^2 +S_\kappa^2(\chi)d\Omega_2^2\right].$$

Radial light rays obey $d\chi=\pm d\eta_{\rm n}$, so their dimensionless comoving travel interval is an integral of $dt/a_{\rm dim}$. The conformal factor preserves unparametrized null paths, not affine parameters, physical wavelengths, massive-particle proper times, or curvature. Hogg’s distance dictionary provides a useful later application of these distinctions in Hogg 2000, §§2–10, pp. 1–8 (PDF).

What isotropy was doing

Spatial homogeneity alone permits

$$ds^2=-dt^2+h_{ab}(t)\,\omega^a\omega^b, \qquad d\omega^a=-\frac12 C^a{}_{bc}\, \omega^b\wedge\omega^c .$$

The structure constants select a Bianchi algebra, while the trace-free part of the expansion defines the shear. In diagonal Bianchi I, with mean scale factor $a_{\rm B}=(a_1a_2a_3)^{1/3}$ and isotropic stress,

$$\dot{\sigma}^i{}_j+3H\sigma^i{}_j=0, \qquad \sigma^2\propto a_{\rm B}^{-6}.$$

These equations require a non-tilted orthogonal flow, vanishing anisotropic stress, and the Bianchi-I curvature specialization; see Ellis and van Elst 1999, §5.3, p. 38, eqs. (156)–(157) (PDF).

FLRW is recovered when the shear and all other direction-selecting quantities vanish and the spatial curvature becomes isotropic. The comparison reveals why a small present shear does not by itself prove exact FLRW history: anisotropy can scale rapidly, and isotropization must be stated using a dimensionless measure and a specified limit.

Where kinematics gives way to dynamics

The first five pages do not determine the FLRW scale-factor history. A positive differentiable $a_{\rm dim}(t)$ defines an FLRW geometry whether or not it satisfies a chosen field equation or matter model. The Bianchi page then uses the Hamiltonian constraint and a restricted shear equation to diagnose how anisotropy enters; it does not close the mean-scale-factor evolution for a general matter source. The next chapter inserts the FLRW metric and stress-energy tensor into Einstein’s equation to derive the Friedmann constraint, acceleration equation, and continuity law systematically.

This boundary prevents three common circular arguments. The cosmological principle does not prove a density history; the curvature sign does not by itself fix whether the universe recollapses; and $1+z=a_o/a_e$ does not select the function $H(z)$. Geometry supplies the stage and the measurement rules. Dynamics supplies the allowed histories.

Check the logical chain

Consider a spatially homogeneous model with flat slices and three unequal directional Hubble rates. A photon is emitted and received by observers orthogonal to the homogeneous slices. Which conclusions from the FLRW chain remain automatic, and which require additional assumptions?

Solution

Homogeneity makes invariant scalars functions of time and permits a common spatial foliation. Flat slices are compatible with Bianchi I, but they do not force the directional Hubble rates to agree. The orthogonal observers can use their proper time as a cosmic-time coordinate in this sector, yet their congruence has nonzero shear.

The invariant definition

$$1+z=\frac{(-U\mathbin{\cdot}k)_e}{(-U\mathbin{\cdot}k)_o}$$

always applies. The FLRW simplification $1+z=a_o/a_e$ is not automatic: in an anisotropic geometry the frequency evolution depends on the photon’s direction and the directional scale factors. Nor can a single FLRW conformal factor make the whole metric isotropic. Recovering the chapter’s FLRW formulas requires vanishing shear and the corresponding isotropic spatial geometry.

References

• Ellis, George F. R., and Henk van Elst. “Cosmological Models (Cargèse Lectures 1998).” In Marc Lachièze-Rey, ed., Theoretical and Observational Cosmology, pp. 1–116. Dordrecht: Springer, 1999. DOI. Open PDF, arXiv:gr-qc/9812046v5.
• Hogg, David W. “Distance Measures in Cosmology.” arXiv:astro-ph/9905116v4, 16 December 2000. Open PDF.