Comoving Observers and Cosmic Time

Comoving FLRW observers remain at fixed spatial coordinates and measure the same proper-time coordinate $t$. Their four-velocity defines the preferred cosmological rest frame, while its expansion scalar gives $3H$. In exact FLRW the congruence is geodesic, hypersurface orthogonal, shear free, and vorticity free. These are invariant statements about a particular family of worldlines, not claims that every galaxy is exactly comoving or that all spacetimes possess a universal cosmic clock.

Required background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian metrics, causal cones, normalization, and spatial projection; Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields background; this page uses its timelike-geodesic and proper-time material for freely falling worldlines, not Jacobi-field theory.

Helpful background. The Einstein Equation supplies the Einstein equations connecting measured expansion to stress energy, while Relativistic Matter and Compact Objects supplies the perfect-fluid frame.

The fundamental observer congruence

Use the dimensionless-scale-factor package introduced on the FLRW-geometry page:

$$ds^2=-dt^2+a^2(t)d\Sigma_K^2, \qquad a(t_0)=1,$$

where $d\Sigma_K^2=\gamma_{ij}dx^i dx^j$ is time independent, carries units of length squared, and has constant curvature $K$ with units of inverse length squared. Dimensional Cartesian-like or radial coordinates are used; angular coordinates remain dimensionless. In these cosmic-time coordinates, the curves $x^i=\text{constant}$ have unit tangent

$$u^a=(\partial_t)^a, \qquad u^a u_a=-1.$$

Along each curve $ds^2=-dt^2$, so $t$ is its proper time up to an additive constant. The spatial projector in the observer’s instantaneous rest space is

$$h_{ab}=g_{ab}+u_a u_b.$$

Because $u_a=-\nabla_a t$, the congruence is orthogonal to the $t=$ constant slices. Frobenius’ condition is automatically satisfied and the vorticity vanishes. Choosing the additive constants consistently gives one cosmic-time label across the FLRW region. This construction depends on the preferred congruence and foliation; it is not an absolute simultaneity structure for an arbitrary spacetime.

The word comoving means fixed coordinates in this adapted chart. It is physically useful because the homogeneous perfect-fluid stress tensor has $u^a$ as its timelike eigenvector. When $\rho+p\ne0$, this singles out the fluid rest frame. Pure vacuum stress, $T_{ab}=-\rho_\Lambda g_{ab}$, instead has every timelike vector as an eigenvector: it does not select a unique comoving congruence without a specified slicing or other matter. A real source can instead have

$$U^a=\gamma\left(u^a+v^a\right), \qquad u_a v^a=0,$$

where $v^a$ is its peculiar velocity, $v^2=h_{ab}v^av^b\lt1$, and $\gamma=(1-v^2)^{-1/2}$. A coordinate choice does not make $v^a$ vanish. Ellis and van Elst formulate this observer split covariantly in Ellis and van Elst 1999, §§2.1.1–2.1.3, pp. 6–8 (PDF).

Expansion, shear, rotation, and acceleration

For any unit timelike congruence, decompose its covariant derivative as

$$\nabla_bu_a =\frac{1}{3}\theta h_{ab}+\sigma_{ab}+\omega_{ab}-a_a u_b.$$

Here $\theta=\nabla_a u^a$ is the expansion, $a^a=u^b\nabla_bu^a$ the four-acceleration, $\sigma_{ab}$ the projected symmetric trace-free shear, and $\omega_{ab}$ the projected antisymmetric vorticity. These four pieces have direct geometric meanings: volume change, failure of free fall, shape distortion at fixed volume, and local rotation.

For the FLRW metric, $\sqrt{-g}=a^3\sqrt{\gamma}$, hence

$$\theta=\frac{1}{\sqrt{-g}} \partial_t\!\left(\sqrt{-g}\right)=3\frac{\dot a}{a}=3H.$$

Direct projection gives

$$a^a=0, \qquad \sigma_{ab}=0, \qquad \omega_{ab}=0.$$

The first equality can also be read from $\Gamma^i{}_{00}=0$: fixed-spatial- coordinate worldlines are timelike geodesics. The other two follow from isotropy, because a spatial antisymmetric tensor or trace-free symmetric tensor would select a direction. Tong derives the same expansion from neighboring comoving separations in Tong 2019, §1.1.2, pp. 10–13 (PDF).

If a small comoving cell has coordinate volume $d^3x$, its physical volume is $dV=a^3\sqrt{\gamma}\,d^3x$. Therefore

$$\frac{1}{dV}\frac{d(dV)}{dt}=3H=\theta.$$

This local volume law is the invariant content behind the phrase “space expands.” It does not require imagining a substance flowing into space or an external dimension into which the universe grows.

Cosmic time and measured separation

Cosmic time synchronizes ideal comoving clocks, but a finite distance between two worldlines still requires a simultaneity convention. On a $t=$ constant slice, a fixed dimensional comoving interval $d\ell_{\rm com}$ has physical length

$$d\ell_{\rm phys}(t)=a(t)d\ell_{\rm com},$$

so neighboring comoving observers satisfy

$$\frac{d\,d\ell_{\rm phys}}{dt}=H\,d\ell_{\rm phys}.$$

This is a relation between nearby observers on the preferred slice. For widely separated events there is no unique relative velocity without specifying how tangent vectors are compared. Consequently $H D$ may exceed the speed of light for a large slice distance without any local observer measuring a material object to pass at superluminal speed.

An observation is local to a tetrad. A comoving observer measures a photon frequency $\omega=-u_a k^a$ and decomposes any stress tensor into energy density, pressure, flux, and anisotropic stress relative to $u^a$. A moving observer uses $U^a$ instead and sees Doppler and aberration corrections. The next page derives cosmological redshift from precisely this measured-frequency definition.

Where the idealization fails

The four vanishing or isotropic kinematical quantities diagnose departures from FLRW. A Bianchi cosmology retains homogeneity but generally has $\sigma_{ab}\neq0$. A rotating model has $\omega_{ab}\neq0$ and need not admit global hypersurfaces orthogonal to its matter flow. In an inhomogeneous fluid, a spatial pressure gradient can give $a^a\neq0$. Multiple components may also be tilted relative to one another, so no single congruence makes every energy flux vanish.

Cosmic time should therefore be used with its domain stated: it is the proper time of the exact-background comoving congruence. Perturbation theory can retain that background clock while individual sources, gauges, and observed-redshift slices differ from it. Confusing those levels creates fictitious forces, double-counted velocities, or claims of simultaneity that no observable requires.

Exercises

Use $\theta=(1/\sqrt{-g})\partial_a(\sqrt{-g}\,u^a)$ to compute the expansion of comoving observers in an $(n+1)$-dimensional FLRW spacetime.

Solution

For $n$ spatial dimensions, $\sqrt{-g}=a^n\sqrt{\gamma}$ and $u^a$ has only $u^t=1$. Thus

$$\theta=\frac{1}{a^n\sqrt{\gamma}} \partial_t(a^n\sqrt{\gamma})=n\frac{\dot a}{a}=nH.$$

The result also follows because each of the $n$ orthogonal linear dimensions of a comoving volume scales by $a$. It reduces to $\theta=3H$ in three spatial dimensions and vanishes when the scale factor is constant.

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.