Cosmological singularities from high matter density

We prove non-spacelike geodesic incompleteness for all non-static, chronological spacetimes satisfying the timelike convergence condition and a past null focusing condition. By a 'past null focusing condition' we mean any restriction on the spacetime Ricci tensor which forces null geodesics, when past complete, to become chronal. By the Einstein field equations, the condition is satisfied if the density of matter fields remains sufficiently high towards the past, as in certain cosmological scenarios. Unlike previous cosmological singularity theorems based on low causality conditions, we make no initial topological or geometric assumptions. We also obtain various corollaries pertaining to global topology, CMC slice existence, and to the type of geodesic incompleteness expected. Some information regarding certain cosmological models proposed by Minguzzi are also obtained.


Introduction
The classic splitting theorem of Cheeger and Gromoll demonstrates the rigidity associated with lines in the presence of non-negative Ricci curvature ; see chapter of 9 [13]. In this statement, a 'line' is an inextendible curve every closed segment of which realizes the Riemannian distance between the segment endpoints. Yau conjectured that there hold a Lorentzian analog to the above theorem. Roughly a decade later and through the work of a number of authors, the following splitting theorem was formulated.
Theorem 1.2 (Lorentzian splitting theorem). Let (M, g) be a spacetime that satisfies the timelike convergence condition, i.e., Ric(u, u) ≥ 0 for all timelike u. Suppose that (M, g) is either globally hyperbolic or timelike geodesically complete. Then, if (M, g) contains a complete timelike line, (M, g) splits as a globally static metric product (R × Σ, −dt 2 + h).
In this statement, a 'timelike line' is an inextendible timelike curve every closed segment of which realizes the Lorentzian distance between the segment endpoints.
The Lorentzian splitting theorem captures to some extent the rigidity of timelike geodesic completeness. Around the time this theorem was formulated, Bartnik [2] proposed a stronger form of rigidity in the context of cosmological spacetimes, i.e., spacetimes with compact Cauchy surface that satisfy the timelike convergence condition, Ric g (u, u) ≥ 0 for any timelike vector u. Conjecture 1.1 (Bartnik [2]). A cosmological spacetime, if timelike geodesically complete, is a globally static metric product (R × Σ, −dt 2 + h).
His conjecture has been settled under a number of different auxiliary hypothesis, with the weakest to date occurring in the recent article of Galloway and Vega [8].
Our first result here is to note that Bartnik's conjecture admits an equivalent formulation in terms of null rays, where by a 'null line' we mean an extendible achronal causal curve, and by a future (past) 'null ray' we mean a future (past) inextendible achronal causal curve. Proposition 1.3. A timelike geodesically complete cosmological spacetime splits as in Bartnik's conjecture if and only if it does not contain a null ray.
One of these directions is well known. Indeed, it is a standard consequence of Lorentzian geometry that a spacetime with compact Cauchy surface admits either a timelike or null line; in fact, the admission of such a line can be deduced from much weaker assumptions, as in theorem 3.1 below. If the spacetime is timelike geodesically complete and the line is timelike, then the splitting follows from the Lorentzian splitting theorem. Thus, if there are no null rays, there are no null lines and the line must be timelike.
Here we prove the other direction, namely that if a timelike geodesically complete cosmological spacetime splits as in the Bartnik conjecture, then there are no null rays.
Our second result, based on a local geometric condition prohibiting null rays, is the following new cosmological singularity theorem.
Theorem 1.4. Let (M, g) be a non-static, chronological spacetime that satisfies the timelike convergence condition and a past null focusing condition. Then (M, g) is past null geodesically incomplete, or, if not, then every timelike geodesic is past incomplete, (M, g) is globally hyperbolic, there is a past crushing singularity and a CMC foliation by Cauchy surfaces, which, moreover, have the topology of a spherical space.
By a 'static' spacetime we mean one in which there exists a hypersurface orthogonal timelike Killing vector field.
By a 'past null focusing condition' we mean any local geometric condition that forces past complete null geodesics to develop focal points. Such conditions may be expressed as restrictions on Ric g (n, n) where n is any null vector, and indeed they have been considered, for instance, in [14] and [10]. One particular explicit example in the future direction, Prop 1. in [14], is the following. For all future complete (affinely parametrized) null geodesics γ : [a, ∞) → M, require that where n is the tangent vector to γ at γ(s). In our terminology, this is an example of a future null focusing condition.
By a 'CMC foliation' we mean that there is a foliation of the spacetime (M, g) by Cauchy surfaces such that, for each Cauchy surface V , the mean curvature H V = tr h (K) is constant on V . Note that here we are writing the constaint equations of the Einstein field equations as where R h is the scalar curvature associated with the initial data set (V, h, K), h is the Riemannian metric induced on V by g, K is the second fundamental form of V in M, ρ = T (u, u) where u is the timelike vector field normal to V and T (·, ·) is the stress energy tensor appearing in the Einstein field equations. The interest in inferring the possibility of a CMC foliation stems from their various useful properties; see [4] for a recent introductory review of their significance and a list of open questions. By a 'crushing singularity' we mean that there is a sequence of slices V i approaching the future edge of the MGHD, with mean curvatures H i satisfying sup V i H i → −∞, where we are using the sign convention that tr h (K) = H < 0 means that the universe is contracting to the future. We recall, at this point, the following well known fact. If two slices have, respectively, mean curvature H 1 and H 2 , where the latter is the future of the former, then, for any mean curvature H satisfying sup H 1 ≤ H ≤ inf H 2 , there is a slice with mean curvature H in the spacetime, between the two slices. A common use for this result is that, if (M, g) has CMC slices of mean curvature H 1 ≤ H 2 , then the region between the slices is foliated by CMC slices, with mean curvatures monotonically increasing along the foliation from H 1 to H 2 . Thus, if there is a crushing singularity, we can pick any slice V as a one barrier, then use the crushing singularity to find a slice V ′ with sup V ′ H ′ ≤ inf V H. A CMC slice will then necessarily lie between them. Finally, following [6], by a 'spherical space' we mean that the Cauchy surface of (M, g) is topologically S 3 or a quotient S 3 \Γ where Γ is isomorphic to a subgroup of SO (4).
The interest in this theorem lies in some of the atypical features which it posesses. The first feature is that most if not all known singularity theorems based on weak causality conditions rely on the generic conditions 1 , and or some global topological assumption such as the existence of a compact spacelike slice. Here, we make no global topological assumption and we replace the generic condition with dynamics, in the form of the non-static condition, and an extra energy condition, in the form of the past null focusing condition.
The second interesting feature is that, in one of its possibilities, one obtains very specific global and causal properties. In particular: the spacetime must be globally hyperbolic, the singularity must be 'crushing', there is a CMC foliation, and even the topology of the Cauchy surfaces is restricted to that of a spherical space.
The causal geometric part of the result is due to recent results of Minguzzi [10]. The CMC foliability existence part stems from older results of Bartnik [2], Tipler [15], and Tipler-Marsden [16]. And finally, the spherical space conclusion stems from the results of Andersson-Galloway [1], and Galloway-Ling [6], which together combine various classic results of minimal surface theory, geometric measure theory, and the positive resolution of the elliptization conjecture, which includes the Poincaré conjecture.
It is likely that theorem 1.4 can be generalized to one that is based on an averaged form of the timelike convergence condition. This will hold if the Lorentzian splitting theorem admits such a generalization. We plan to address this in a further work. We think that this would be a conceptually preferable in view of recent doubts on the physicality of the timelike convergence condition. As for the past null focusing condition, this can be interpreted as an assumption that the energy density of matter fields remains high in the past. The extent to which this is expected in cosmological models with matter fields is a question for further investigation.

Preliminaries
Our conventions are basically as in [3]. A spacetime (M, g) is an n-dimensional Lorentzian manifold. A spacetime (M, g) is said to satisfy chronology (causality) if there are no closed timelike (causal) curves in M. An open set U ⊂ M is said to be causally convex is no non-spacelike curve intersects U in a disconnected set. Given a point p ∈ M, the spacetime (M, g) is said to be strongly causal at p if p has arbitrarily small causally convex neighborhoods. The spacetime is strongly causal if it is strongly causal at every point.
A spacetime (M, g) is globally hyperbolic if and only if it is causal and J + (p) ∩ J − (q) is compact for all p, q ∈ M. Standard causality theory (see chapter 3 of [3]) shows that this is equivalent to the existence of a Cauchy surface S, i.e., so that M = D(S) where D(·) denotes the domain of dependence defined in terms of causal curves.
We recall the following definition of an edge; as in chapter 14 of [3].
Definition 2.1. Let S ⊂ M be achronal. Then p ∈ S is an edge point of S provided every neighborhood U(p) of p contains a timelike curve γ from I − (p, U) to I + (p, U) that does not meet S. We denote by edge(S) the set of edge points of S.
From this follows the following standard result of causality theory.
Proposition 2.1. Let S be closed. Then each p ∈ ∂I + (S)\S lies on a null geodesic contained in ∂I + (S)\S, which either has a past endpoint on S, or else is past inextendible in M.
A spacetime (M, g) is said to be causally disconnected by a compact set V ⊂ M if there exists two infinite sequences of points {p i }, {q i } with q i ≤ p i , which diverges to infinity, such that for any i, all future directed non-spacelike curves from p i to q i intersect K. Here , an infinite sequence in a non-compact topological space is said to 'diverge to infinity' if given any compact subset C, only finitely many elements of the sequence are contained in C.
The Lorentzian distance function d(p, q) is defined as in chapter 4 of [3]. We note the following useful lemma, which appears as corollary 4.7 in [3].
Lemma 2.2. For any globally hyperbolic spacetime (M, g), the lorentzian distance function is continuous and d(p, q) for p, q ∈ M is finite.
We shall also briefly touch on the notion of the causal boundary of a spacetime; see chapter 6 of [3] for an introduction. The key notions we shall use stem from the work of [9]. The causal boundary of a spacetime was developed with the aim of describing some of the spacetime's global causal and geometric properties, by attaching to it a notion of a boundary representing the 'edge' of the spacetime. The important sets in this construction are terminally indecomposable past or future sets, i.e., TIP or TIF, which are constructed as follows.
A past (future) set A is a set in M such I −(+) (A) = A. An indecomposable past (future) set is an open past set that cannot be written as a union of two proper subsets both of which are open past (future) sets. A terminally indecomposable past (future) set is an indecomposable past (future) set which is not the chronological past (future) of any point in the spacetime. We note the following key lemma which will be used below. In the arguments to follow, we shall need a number of results from [6]. We refer our reader to [6] for the relevant definitions, to avoid repeating these here.

Proof of proposition 1.3
We take (M, g) to admit a compact Cauchy surface and to be timelike geodesically complete as well as satisfying the timelike convergence condition. We shall show that if (M, g) is without null rays then (M, g) splits as in the Bartnik conjecture, and, conversely, that if (M, g) splits as in the Bartnik conjecture then (M, g) has no null rays.
The first claim in fact follows immediately from the following standard theorem of Lorentzian geometry; which appears as theorem 8.13 in [3]. If (M, g) admits a compact Cauchy surface then it is clearly causally disconnected by a compact set, i.e., its Cauchy surface. Thus, (M, g) admits a line which is either timelike or null. If (M, g) admits no null rays then it has no null lines and thus the line is timelike, which by assumption is complete. The splitting then follows from the Lorentzian splitting theorem. Now we investigate the other direction.
When (M, g) splits as in the Bartnik conjecture, (M, g) is null geodesically complete and we also have that I +(−) (γ) = M for γ any past (future) inextendible timelike curve; see [8] for a justification of this fact. This implies that the future (past) causal boundary of M consists of a single element. Standard causality theory tells us that the boundary to any past (future) inextendible timelike curve is a set that is generated by null rays. It thus follows that any such set is empty. There can be no past (future) null rays arising as the boundary to a TIF (TIP). Thus, seeking a contradiction, we shall assume the presence of a future null ray and show that we can construct a TIP such that the future (past) null ray not contained in the TIP. This will be a contradiction for Bartnik splitting implies that there is only one TIP, which in particular is the whole of M. The same argument could of course be done in the past direction to rule out past null rays.
Suppose that (M, g) admits a future null ray η : [0, ∞) → M, where completeness follows from the fact that (M, g) is null geodesically complete. Consider an increasing sequence of parameter values n i < n i+1 with n i ∈ (0, ∞), i ∈ N. For each n i , there is a timelike maximizing curve δ i : [0, b i ] → M whose endpoints are given by The existence of such a curve is a standard consequence of global hyperbolicity. Denote the lorentzian length of δ i by d(p i , q i ). Consider a nearby timelike maximizer with endpoints also in V and on η such that d(p i+1 , q i+1 ) > d(p i , q i ). Continue this sequence of timelike maximizers to arbitrarily large values of n i . By construction and timelike geodesic completeness, b i → ∞ as i → ∞. By compactness of V , the sequence of points {p i } = {δ i (0)} ∈ V has an accumulation point p ∈ V . By Minguzzi's limit curve theorem [11], there passes a future inextendible non-spacelike ray µ : [0, ∞) → M with past endpoint µ(0) = p. By continuity of the Lorentzian length function in globally hyperbolic spacetimes, this ray is timelike. By lemma 2.2, the set I − (µ) thus defines a TIP. By construction, we have δ i ⊂ I − (η(n i )). By achronality of η, each curve δ i satisfies η ∩ I − (δ i ) = ∅. By our construction of µ, we have η ∩ I − (µ) = ∅, which is what we sought to show.

Proof of singularity theorem
If (M, g) is past null geodesically complete, then the past null focusing condition forces (M, g) to have no past null rays. Given that (M, g) is chronological, the following theorem of Minguzzi [10] implies that (M, g) is globally hyperbolic with a single TIF. Owing to an argument in Penrose's [12] singularity theorem, we can further argue that the Cauchy surface must be compact.
In particular, let Σ be a Cauchy surface for (M, g). Let V ⊂ Σ be any compact spacelike submanifold with codimension 2 with non-empty edge. By standard causality theory, ∂I − (V ) is a set which is generated by achronal null geodesics which are either future inextendible or with future endpoint intersecting edge(V ). Given that (M, g) is null geodesically complete and globally hyperbolic, the former possibility does not occur. Thus, all such generators intersect edge(V ). By the absence of past null rays, it cannot occur that these achronal null geodesics are past inextendible. Thus, the null hypersurface ∂I − (V ) must end. Since ∂I − (V ) is closed, it is also compact. By the standard homeomorphism constructed in the proof of the Penrose singularity theorem, eg., see [12] or chapter 9 of [17], it follows that Σ must be compact.
So (M, g) admits a compact Cauchy surface and by assumption satisfies the timelike convergence condition. The absence of past null rays then implies, by the above result, that (M, g) splits as in the Bartnik conjecture if (M, g) is timelike geodesically complete. But in that case (M, g) is static and we have a contradiction.
So (M, g) is either past null geodesically complete and timelike geodesically incomplete, or past null incomplete. In the former case, we can in fact show that every timelike geodesic must be past incomplete. This will follow from the following proposition, applied in the time reversed direction. Proposition 4.2. Let (M, g) be globally hyperbolic with a single TIP. Let γ 1 , γ 2 be two future inextendible timelike curves. If γ 1 is future incomplete then so is γ 2 .
Proof. Suppose otherwise. Let γ 1 and γ 2 be two future inextendible timelike curves such that γ 1 incomplete and γ 2 complete. Clearly these two curves are future inextendible. By lemma 2.2, each of these curves defines a TIP. Since there is a single TIP, we must have I − (γ 1 ) = I − (γ 2 ). We now show that this leads to a contradiction. To do this, we shall repeatedly use the standard fact that for a globally hyperbolic spacetime, the Lorentzian distance function d(p, q) is finite and continuous, and, furthermore, that for any two causally related points, there always exists a causal curve (in fact necessarily a geodesic) that realizes the lorentzian distance between the points; see chapter 3 and 4 of [3] for a review of the properties implied by global hyperbolicity. First, we construct a certain sequence along γ 1 and γ 2 .
Consider two sequences {p i } and {q j } of points lying on, respectively, γ 1 and γ 2 . Choose the sequence such that p i and q j lie, respectively, in the chronological past of p i+1 and q j+1 . By belonging to the same TIP, we may also choose this sequence to be such that for each triple p i , q j , p i+1 , there exists three future directed causal curves connecting these points as follows: η i connects p i to p i+1 , δ i connects p i q j , and β i connects q j to p i+1 . We may choose these points and these curves so that each curve realizes the Lorentzian distance d(p i , p i+1 ), which is either zero in the case of points connecting by an achronal causal curve or strictly positive otherwise.
By the reverse triangle inequality, for each i, j we have d(p i , p i+1 ) ≥ d(p i , q j ) + d(q j , p i+1 ). Since γ 1 is timelike incomplete, by considering p i further along γ 1 , we must have that d(p i , p i+1 ) → 0 as i → ∞. It thus follows that both d(p i , q j ) and d(q j , p i+1 ) approach 0 as i, j → ∞. Now consider the first point of our sequence p 0 . Using once more the reverse triangle inequality, we have d(p 0 , p i+2 ) ≥ d(p 0 , q j+1 ) + d(q j+1 , p i+2 ). Finiteness of the Lorentzian length implies d(p 0 , p i+2 ) ≤ k for some k > 0. From before, we know that d(q j+1 , p i+2 ) → 0. Putting this together, we have that as j → ∞, d(p 0 , q j+1 ) ≤ k ′ . We now show that the boundedness of d(p 0 , q j+1 ) as j → ∞ leads to a contradiction.
Since γ 2 is complete, as j → ∞, q j will lie on arbitrarily large parameter values of γ 2 . Thus d(q 0 , q j ) will grow unbounded as j → ∞. By the reverse triangle inequality, we have d(p 0 , q j ) ≥ d(p 0 , q 0 ) + d(q 0 , q j ). But this implies that d(p 0 , q j ) → ∞ as j → ∞ and we have a contradiction with the previous statement that d(p 0 , q j+1 ) is bounded as j → ∞.
So every past inextendible timelike geodesic is past incomplete. Given the absence of past null rays, we also know that the past causal boundary consists of a single element. We may now use the following result from [16], also recently used in [6], [4]. The further restriction that the Cauchy surfaces for (M, g) must be spherical spaces follows straightforwardly from results of Andersson and Galloway [1] and Galloway and Ling [7]. Since all the relevant details are contained in those papers, our presentation thereof will be brief.
Before justifying this, let us say that we do not have in mind the direct application of theorem 4.4, which is their main theorem; appearing as theorem 1 in [7]. The reason for this is that here, there is no a priori guarantee that the 'expanding in all directions' condition is satisfied. Indeed, as we shall see, we shall not need this condition to infer our conclusion.
The key idea employed in [1] and [7], is to derive null incompleteness of the spacetime by applying a refinement of Penrose's singularity theorem [12] in a suitable covering spacetime. Once an incomplete null geodesic is inferred in the covering spacetime, the original spacetime must also be null incomplete. For this purpose, our notion of a covering spacetime will stem from lemma 4 of [6]. If one can demonstrate the existence of an immersed past trapped surface in the initial data set of the spacetime, then this surface lifts to the covering space of the Cauchy surface. If one can construct a non-compact cover for the original Cauchy surface, one can then use a refinement of Penrose's singularity theorem (based on immersed trapped surfaces) in the covering spacetime.
Recall that in the proof of Penrose's singularity theorem, one uses the well known causal geometric result that, given any trapped surface Π, the achronal null geodesic emanating from this surface which generate ∂I − (Π) must, if complete, focus. Thus, if one can demonstrate the presence of a past immersed trapped surfaceΣ in the cover V of the initial data set V , then if the covering spacetime (M,g) is null geodesically complete, the null hypersurface, ∂I − (Σ) must be compact. This leads to a contradiction by considering a homeomorphism, guaranteed by global hyperbolicity of (M ,g), between ∂I − (Σ) andṼ which is non-compact; details for this appear in chapter 9 of [17].
The arguments in [1] and [7] can now be divided into topological and causal geometric components. The topological part of the argument is to infer, owing to certain topological properties, the presence of a minimal surface in the initial data set. Once the presence of a minimal surface in the initial data set is inferred, the positive definite condition on K ensures that this minimal surface is in fact past trapped.
The interplay between this causal theoretic and topological arguments is embodied in proposition 5, essentially proven in [1]. For that proposition, one appeals to a result of geometric measure theory to deduce the existence of a past trapped immersed surface. The rest of the argument in [1], [7] is then as described: construct a suitable non-compact cover for the initial data set, and use a refinement of Penrose's theorem.
It is now possible to see that the same ideas can be used here, where, in particular, the 'topological' part of the argument in [7] proceeds completely unaltered. The main difference is in the causal theoretic argument, which is different here since in our case we posit no restriction on K. Here, it is the past null focusing condition which plays this role. This condition ensures that, under the assumption of past null completeness, the hypersurface ∂I − (Σ) is compact where Σ is now simply a minimal, rather than trapped, surface. This is because, as in a time reverse of proposition 2.1, the hypersurface ∂I − (Σ) is generated by achronal null geodesics which have future endpoint on S. If the spacetime is past null geodesically complete and there are no past null rays, then these generators cannot persist arbitrarily far into the past, oth-erwise they would be past null rays. So ∂I − (Σ) ends, and, thus, must be compact.
The key is that this will still be true in the covering spacetime, where now a contradiction can be deduced from the non-compactness of the Cauchy surface. This follows because the past null focusing condition is also obeyed in the covering spacetime. This is easily seen from the fact that the condition is a local geometric condition on Ric g , and thus lifts to the covering spacetime of lemma 4.5. Then, arguing as above, ∂I − (Σ) is compact, whereΣ is the lift of Σ. The standard contradiction with the non-compactness ofṼ proceeding by homeomorphism can now be applied. This completes the proof.
5 Appendix -an application of theorem 1.4 We note the following singularity theorem by Minguzzi [10].
Theorem 5.1 (Minguzzi [10]). Let (M, g) be a spacetime of dimension greater than 2. If (M, g) is null geodesically complete, chronological, contains a future trapped surface, satisfies the timelike convergence condition, the generic condition, together with a past null focusing condition, then (M, g) is globally hyperbolic with compact Cauchy surface V and has an incomplete timelike line.
Minguzzi writes, on the above theorem, the following remark.
"Since the existence of trapped surfaces is a quite natural consequence of general relativity if matter concentrated enough, [this] theorem ... supports the global hyperbolicity of the spacetime (and a closed space) provided it is null geodesically complete. Since the conditions are quite reasonable one concludes that the spacetime is either null geodesically incomplete or timelike geodesically incomplete (or both).
Finally I would like to stress that the assumption of null geodesic completeness does not lead to a spacetime picture which contradicts observations. Thus [certain] theorems ... may have a positive role in proving the good causal property of spacetime rather than being used only to prove its singularity. As a matter of fact they can be used to do both ... [as above]." Pending Minguzzi's remarks, we note the following corollary of theorem 1.5 and theorem 5.1, where in particular the non-static condition is replaced by the assumption of the generic condition, since then Bartnik's splitting possibility in the proof of theorem 1.4 cannot occur.
Theorem 5.2. Let (M, g) be a 4-dimensional spacetime satisfying the conditions of theorem 5.1. Then, in addition to the consequences derived in theorem 5.1, the compact Cauchy surfaces of M have the topology of a spherical space, every timelike geodesic is past incomplete, there is past crushing singularity and a CMC foliation.
Theorem 5.2 shows that we can in fact obtain relatively specific restrictions on the universe models suggested by Minguzzi's theorem 5.1.