Geodesic Completeness in Cosmology

Geodesic completeness is a statement about the parameter length of freely falling curves, not about whether a coordinate reaches an endpoint or whether one chosen curvature scalar diverges. A cosmological chart can end while the spacetime continues, and an incomplete spacetime can have perfectly bounded local curvature. The correct diagnosis therefore combines geodesics, invariants and extendibility.

Helpful background. Einstein Equations relate curvature to sources, Relativistic Matter and Compact Objects supplies matter regularity tests, Causality and Global Structure supplies completeness and extension language, Relativistic Cosmology supplies FLRW dynamics, and Mathematical Relativity clarifies the differentiability class of an extension.

Three logically separate diagnoses

A timelike geodesic is complete in one direction if its proper-time parameter can be extended without bound in that direction; a null geodesic uses an affine parameter instead. Timelike and null completeness are separate properties. “The coordinate $t$ stops” is not yet a geometric statement, because another chart may continue the same geodesic.

Curvature gives a second diagnostic. Divergence of a scalar polynomial invariant such as $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ prevents a sufficiently regular extension through that limit, but bounded scalars do not prove regularity: tidal components, topology or incomplete curves can still obstruct an extension. Extendibility is the third question and must name a regularity class, such as a $C^2$ Lorentzian metric. Senovilla’s FLRW and punctured-flat-space examples make these distinctions explicit Senovilla 1998, § 3, especially Example 3.1, pp. 750–753 (PDF).

FLRW affine and proper-time tests

For a spatially flat radial FLRW metric,

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

translation symmetry gives the conserved comoving momentum $P=a^2\,d\chi/d\lambda$. Along a null geodesic,

$$\frac{dt}{d\lambda}=\frac{\lvert P\rvert}{a(t)}, \qquad \Delta\lambda=\frac{1}{\lvert P\rvert}\int a(t)\,dt.$$

Thus conformal travel time $\int dt/a$ diagnoses causal reach, whereas null affine completeness uses $\int a\,dt$. Confusing these two integrals is a common error.

To see where the different powers come from, use the null condition $-(dt/d\lambda)^2+a^2(d\chi/d\lambda)^2=0$ and then insert $d\chi/d\lambda=P/a^2$. The affine parameter records the photon’s geodesic tangent, whose energy redshifts as $a^{-1}$; conformal time instead straightens its path in the coordinate diagram. A null curve has zero proper time, so proper time cannot replace its affine parameter.

For a timelike geodesic parametrized by proper time $\tau$,

$$\frac{dt}{d\tau}=\sqrt{1+\frac{P^2}{a^2(t)}}, \qquad \Delta\tau=\int\frac{dt}{\sqrt{1+P^2/a^2(t)}}.$$

A comoving observer has $P=0$, so cosmic time is proper time. If an FLRW model has $t\in(0,t_0]$, every comoving geodesic reaches the excluded endpoint $t=0$ in finite proper time: that specified patch is past incomplete. A larger spacetime may continue it, but that is a separate extension claim. Noncomoving timelike curves require the second integral, especially when $a\to0$.

For $a(t)=A t^p$ with $p\gt0$ and $K=0$, the Kretschmann scalar is

$$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} =12\left[\left(\frac{\ddot a}{a}\right)^2+H^4\right] =\frac{12\left[p^2(p-1)^2+p^4\right]}{t^4}.$$

It diverges at $t=0$. The Ricci scalar alone is insufficient: $R=6p(2p-1)/t^2$ vanishes for the radiation value $p=1/2$, even though the Kretschmann scalar diverges. This is a useful counterexample to any one-invariant criterion.

Flat and global de Sitter

The flat de Sitter chart has $a(t)=e^{Ht}$ and $t\in(-\infty,\infty)$. Its scalar curvature is the finite constant $R=12H^2$. Comoving geodesics have infinite proper time to $t=-\infty$, but a past-directed null geodesic has

$$\Delta\lambda\propto\int_{-\infty}^{t_0}e^{Ht}\,dt =\frac{e^{Ht_0}}{H},$$

which is finite. For $P\ne0$ the timelike integral can also be done exactly:

$$\Delta\tau =\int_{-\infty}^{t_0} \frac{e^{Ht}\,dt}{\sqrt{e^{2Ht}+P^2}} =\frac1H\operatorname{arsinh} \left(\frac{e^{Ht_0}}{\lvert P\rvert}\right).$$

It is finite for every nonzero $P$, but grows without bound as $P\to0$. This nonuniform limit explains why the comoving worldline is exceptional. The flat patch is therefore past null and noncomoving-timelike incomplete despite its regular curvature.

This is a chart boundary, not a de Sitter curvature singularity. Global coordinates use closed spatial slices. In the following line element the angular coordinates on the unit three-sphere are dimensionless, so its radius $\ell(T)$ is a length:

$$ds^2=-dT^2+\ell^2(T)d\Omega_3^2,\qquad \ell(T)=H^{-1}\cosh(HT).$$

The domain $-\infty\lt T\lt\infty$ covers the complete de Sitter spacetime. The constant $H=\sqrt{\Lambda/3}$ is its curvature scale; the global slice expansion is $\dot\ell/\ell=H\tanh(HT)$ and is not constant. Tong compares the flat, open and global closed slicings in Tong 2019, § 1.3.3, pp. 41–42 (PDF). The example establishes that incompleteness of a patch need not be inextendibility of the maximal spacetime.

A decision procedure

The following examples separate the three diagnostics:

Specified geometry	Past causal geodesics	Curvature at the endpoint	Extension
Flat radiation FLRW, $t\gt0$	Finite affine and proper lengths	$R=0$, but the Kretschmann scalar diverges	No $C^2$ extension through that divergent limit
Flat de Sitter patch	Null and noncomoving timelike curves are incomplete	All curvature invariants are constant	Extends to global de Sitter
Global de Sitter	Causal geodesics are complete in both directions	Regular and constant-curvature	No missing causal endpoint of this kind

First specify the manifold and coordinate domain. Next solve representative causal geodesics and test their proper or affine parameter at each endpoint. Then examine a sufficient set of curvature or tidal diagnostics. Finally ask whether a regular extension exists and whether the proposed extension is physically and globally admissible. These steps distinguish a removable chart edge, a quasiregular or topological defect, and a curvature pathology.

The procedure does not supply a universal definition of every singular boundary, nor does it replace proof-level mathematical-relativity results. It does prevent two overclaims: finite curvature is not sufficient for completeness, and geodesic incompleteness by itself does not identify curvature blow-up, a unique extension or a “beginning of time.”

Exercises

In the flat de Sitter patch, show that a radial null geodesic reaches $t=-\infty$ in finite affine parameter while a comoving timelike geodesic does not.

Solution

For the null geodesic, conservation of $P$ gives $d\lambda=a\,dt/\lvert P\rvert$. Hence the past affine length is $e^{Ht_0}/(H\lvert P\rvert)$, which is finite. A comoving timelike geodesic has $P=0$ and $d\tau=dt$, so the proper-time interval from $-\infty$ to $t_0$ diverges. Completeness must therefore be stated separately for different causal geodesics.

References

• Senovilla, José M. M. “Singularity Theorems and Their Consequences.” General Relativity and Gravitation 30 (1998): 701–848, corrected 2018 version. DOI. Open PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019. Official PDF.