Using Singularity Theorems in Cosmology

A cosmological singularity theorem is an implication, not a slogan: a specified set of geometric and causal hypotheses forces geodesic incompleteness. Applying one responsibly means testing every hypothesis in the version actually cited, identifying the causal class and time direction of the incomplete geodesic, and refusing to replace that conclusion by curvature blow-up or a unique cosmic beginning.

Helpful background. Einstein Equations translate energy conditions into Ricci focusing, Relativistic Matter and Compact Objects supplies stress–energy examples, Causality and Global Structure develops the rigorous theorems and their hypotheses, Relativistic Cosmology supplies FLRW test models, and Mathematical Relativity clarifies completeness and extension.

The Hawking cosmological theorem used here

Assume a four-dimensional spacetime satisfies all four conditions below:

• the timelike convergence condition—called the geometric strong energy condition in Senovilla—$R_{ab}v^a v^b\ge0$ for every timelike vector $v^a$;
• strong causality;
• an edgeless, compact, achronal spacelike hypersurface $S$;
• positive initial expansion of the timelike geodesic congruence orthogonal to $S$.

Then the spacetime is past timelike geodesically incomplete. This is Senovilla’s Theorem 5.5, including its direction and causal class Senovilla 1998, Theorem 5.5 and proof, pp. 790–791 (PDF). The classic compact-slice formulation appears as Hawking–Ellis 1973, § 8.2, Theorem 4, p. 272; the corrected Senovilla review is used here because it exposes the exact causal and hypersurface hypotheses in the notation applied below.

No generic condition is part of this theorem version. It does not assume global hyperbolicity, call $S$ a Cauchy surface, or replace $S$ by a trapped surface. Related singularity theorems use those hypotheses, but importing them here would make it impossible to tell which result a model actually satisfies.

The geometric condition must not be replaced by a matter inequality without tracking the cosmological constant. With the site convention

$$G_{ab}+\Lambda g_{ab}=8\pi G T_{ab},$$

Einstein’s equation gives

$$R_{ab}v^a v^b =8\pi G\left(T_{ab}-\frac12Tg_{ab}\right)v^a v^b +\Lambda g_{ab}v^a v^b.$$

For the unit fluid velocity $u^a$, this becomes $R_{ab}u^a u^b=4\pi G(\rho+3p)-\Lambda$. More generally, for a unit timelike $v^a$ and $\gamma=-u_av^a\ge1$,

$$R_{ab}v^a v^b =8\pi G\left[(\rho+p)\gamma^2+\frac{p-\rho}{2}\right]-\Lambda.$$

Timelike convergence in every direction is therefore equivalent to $\rho+p\ge0$ and $4\pi G(\rho+3p)\ge\Lambda$. The matter inequality $\rho+3p\ge0$ is not by itself enough when a positive $\Lambda$ remains on the geometric side, and it does not control highly boosted directions without $\rho+p\ge0$. One may instead absorb $\Lambda$ into the total effective stress tensor and impose both corresponding inequalities on that total source.

Focusing and the finite proper-time bound

Let the future-directed orthogonal congruence have expansion $\theta_0\gt0$ on $S$. Compactness and continuity give a uniform lower bound $\theta_0\ge\theta_{\min}\gt0$. Reverse the tangent so that proper time $s$ increases into the past; its initial expansion is $\vartheta_0=-\theta_0\le-\theta_{\min}$. The congruence is hypersurface orthogonal, so its vorticity vanishes. Raychaudhuri’s equation and timelike convergence give

$$\frac{d\vartheta}{ds} =-\frac{\vartheta^2}{3}-\sigma_{ab}\sigma^{ab}-R_{ab}v^a v^b \le-\frac{\vartheta^2}{3}.$$

While $\vartheta$ is finite and negative,

$$\frac{d}{ds}\left(\frac{1}{\vartheta}\right)\ge\frac13.$$

For each geodesic, the equality solution diverges at $s=-3/\vartheta_0$. Since $\vartheta_0\le-\theta_{\min}$, this time is at most $3/\theta_{\min}$, and the inequality focuses at least as fast. Thus each past-directed orthogonal geodesic either is incomplete before $3/\theta_{\min}$ or, if extendible that far, develops a focal point to $S$ no later than $3/\theta_{\min}$.

The two expansion symbols prevent a frequent sign mistake. Positive $\theta_0$ describes the future-directed normals leaving an expanding slice, while $\vartheta_0=-\theta_0$ describes the same curves oriented into the past. Focusing occurs in that past direction; changing tangent orientation without changing the expansion sign would reverse the physical conclusion.

Focusing alone is local and does not yet prove incompleteness. Suppose, for contradiction, that the spacetime were past timelike complete. Follow every orthogonal geodesic through all proper times $0\le s\le B$, where $B=3/\theta_{\min}$. The normal exponential map sends the compact set $S\times[0,B]$ to a compact spacetime set. It is this whole family of segments, not just their endpoints at $B$, that can contain the relevant past domain. Maximizing normal geodesics cannot continue to maximize beyond a focal point; the causal-domain and horizon argument uses this fact to derive a contradiction with strong causality. The rigorous global step is the proof of Senovilla 1998, Theorem 5.5, pp. 790–791 (PDF), which invokes the supporting focusing, compactness, maximizing-geodesic, and causal-boundary results developed earlier in the review. Hawking Cosmological Singularity Theorems develops that proof architecture; this page applies it to cosmological models.

Testing FLRW models hypothesis by hypothesis

For an expanding closed FLRW model, a constant-$t$ slice has topology $S^3$ and is edgeless, compact, spacelike and achronal. The comoving normal congruence has

$$\theta=3H.$$

For a standard closed FLRW spacetime, cosmic time is a time function, so strong causality holds independently of the sign of $H$. On a chosen slice $t=t_0$ with $H(t_0)\gt0$, suppose the perfect-fluid matter, with $\Lambda$ retained on the geometric side, satisfies $\rho+p\ge0$ and $4\pi G(\rho+3p)\ge\Lambda$ throughout the spacetime. These two inequalities are equivalent to timelike convergence for every timelike vector. The theorem then implies past timelike incompleteness. Homogeneity gives $\theta_{\min}=3H(t_0)$, so the focusing estimate is $3/\theta_{\min}=H(t_0)^{-1}$; it is an upper bound, not an exact age.

Spatially flat FLRW with spatial topology $\mathbb R^3$ fails the compact-$S$ hypothesis of this theorem version. That fact does not establish completeness: direct geodesic integration or another theorem can still prove incompleteness. Changing the topology to a flat three-torus restores compactness without changing the local Friedmann equations.

For a concrete check, identify the spatial coordinates of flat dust FLRW periodically to make a three-torus, and take $a\propto t^{2/3}$ for $t\gt0$. Then $\Lambda=p=0$, $\rho\gt0$, and each cosmic-time slice is compact. The theorem applies. Direct integration gives a comoving past proper length $t_0=2/[3H(t_0)]$, smaller than the orthogonal-congruence focusing scale $H(t_0)^{-1}$. The bound and the exact solution agree without requiring equality. The unquotiented $\mathbb R^3$ model has the same local singularity but falls outside this compact-slice theorem’s hypotheses.

Global de Sitter supplies a discriminating counterexample. At $t\gt0$ its closed $S^3$ slices expand, are compact and achronal, and the spacetime is strongly causal and geodesically complete. For $R_{ab}=3H^2g_{ab}$,

$$R_{ab}v^a v^b=-3H^2\lt0$$

for a unit timelike vector. Timelike convergence fails, so there is no contradiction. A smooth bounce similarly has $\ddot a\gt0$ near its minimum and hence $R_{ab}u^a u^b=-3\ddot a/a\lt0$ for the comoving congruence in FLRW. Avoiding this theorem therefore has a precise cost in its hypotheses, not a refutation of its logic.

What the conclusion does and does not say

The conclusion guarantees at least one past-inextendible timelike geodesic with finite proper length. It does not imply that a scalar curvature diverges, that every observer is incomplete, that the boundary is spacelike, or that no regular extension exists. It also does not select a quantum-gravity resolution or establish that “time began.” Senovilla emphasizes these extension and interpretation limits in Senovilla 1998, § 6.5, pp. 806–807 (PDF).

This page applies one theorem to cosmological models. Proof-level mathematical relativity, low-regularity refinements and other trapped-set theorems remain separate. The disciplined workflow is: name the theorem version, list its exact hypotheses, test them independently, state its exact conclusion, and only then discuss physical interpretation.

Common pitfalls

Treating accelerated expansion as a theorem counterexample. Acceleration often violates the timelike convergence condition. A model outside a theorem’s hypotheses is not a counterexample to its conclusion.

Equating incompleteness with infinite curvature. The flat de Sitter patch is incomplete but has bounded curvature and a regular extension. Incompleteness, inextendibility and curvature pathology are different claims.

Exercises

Test Theorem 5.5 on global de Sitter at a slice $t_0\gt0$. Which hypotheses hold, which fails, and what conclusion follows?

Solution

The $t=t_0$ slice is an edgeless compact achronal $S^3$, its orthogonal congruence has positive expansion $3H\tanh(Ht_0)$, and global de Sitter is strongly causal. Timelike convergence fails because $R_{ab}v^a v^b=-3H^2$ for unit timelike $v^a$. The theorem therefore yields no incompleteness conclusion, consistently with global de Sitter’s geodesic completeness.

Why does a caustic in a congruence not by itself establish a spacetime singularity? Consider freely falling particles in Minkowski spacetime whose worldlines intersect.

Solution

The separation volume of the chosen particles can vanish at their intersection, so the congruence expansion becomes singular. Each straight worldline nevertheless continues smoothly, and spacetime curvature stays zero. The failure is in using that intersecting family as a smooth congruence, not in extending the geodesics. A singularity theorem needs its global causal argument in addition to the local focusing estimate.

References

• Hawking, Stephen W., and George F. R. Ellis. The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press, 1973. DOI. Chapter 8 DOI.
• Senovilla, José M. M. “Singularity Theorems and Their Consequences.” General Relativity and Gravitation 30 (1998): 701–848, corrected 2018 version. DOI. Open PDF.