Chronological spacetimes without lightlike lines are stably causal
aa r X i v : . [ g r- q c ] J un Chronological spacetimes without lightlike linesare stably causal
E. Minguzzi
Dipartimento di Matematica Applicata, Universit`a degli Studi di Firenze, Via S. Marta 3,I-50139 Firenze, ItalyE-mail: ettore.minguzzi@unifi.it
Abstract:
The statement of the title is proved. It implies that under physicallyreasonable conditions, spacetimes which are free from singularities are necessarilystably causal and hence admit a time function. Read as a singularity theorem itstates that if there is some form of causality violation on spacetime then eitherit is the worst possible, namely violation of chronology, or there is a singularity.The analogous result: “Non-totally vicious spacetimes without lightlike rays areglobally hyperbolic” is also proved, and its physical consequences are explored.
1. Introduction
While the local structure of spacetime is fairly simple to describe, there are stilla number of open problems concerning the causal behavior of the spacetimemanifold in the large. About three decades ago Geroch and Horowitz in theconclusions of their review “Global structure of spacetimes” [8] identified theproblem of giving good physical reasons for assuming stable causality as one ofthe most important questions concerning the global aspects of general relativitytogether with the proof of the cosmic censorship conjecture. Indeed, if stablecausality holds then the spacetime does not suffer any pathological behaviorconnected with the presence of almost closed causal curves, and, more impor-tantly, it admits a (non-unique) time function [9], that is a function which iscontinuous and increases on every causal curve.In order to understand the role of stable causality it is useful to recall thatmost conformal invariant properties can be ordered in the so called causal ladderof spacetimes (see figure 1). If the real Universe were represented by a globallyhyperbolic manifold (the top of the ladder) then a number of mathematicaland physical nice properties would hold. The problem is that, though there isevidence that the spacetime manifold evolves according to the Einstein equa-tions, it is not clear whether the evolution from physically reasonable Cauchy
E. Minguzzi data would introduce naked singularities and would eventually produce a non-globally hyperbolic spacetime. If so, the Cauchy data would be insufficient forthe determination of the spacetime geometry and one would have to take intoaccount the information coming from infinity. However, Penrose gave argumentswhich support the view that the so developed manifold would actually be glob-ally hyperbolic [23] (strong cosmic censorship).Some other authors claim that one should only expect that the non-predictablebehavior due to naked singularities be confined behind horizons (weak cosmiccensorship). Other authors note that there is not even compelling reasons forexcluding chronologically violating regions, in fact in some cases they allow tokeep the spacetime non-singular even in presence of trapped surfaces [21]. Fromthis point of view chronology violating sets should not be discarded a priori,instead they should be considered in the same footing as naked singularities, aphysical possibility which hopefully remains hidden behind an horizon. Theseconsiderations show that the class of mathematically reasonable spacetimes isconsiderably large, and therefore physicists look for physical arguments whichallow to get as close as possible to global hyperbolicity. In short physicists lookfor results which allow to climb the causal ladder.The first step would be to justify the chronology property. Actually this as-sumption is philosophically satisfactory because its violation would arise issuesrelated to the free will of the generic observer. However, the notion of free will isnot modeled in general relativity, therefore it becomes reasonable to search forother physical mechanisms, perhaps based on quantum mechanics, which pre-vent the formation or stability of chronology violating sets. The idea that sucha mechanism should indeed exist and that starting from well behaved initialconditions closed timelike curves can not form has been referred by Hawkingas the chronology protection conjecture [10]. As I commented above there is nogeneral consensus on its validity and the evidence coming from classical generalrelativity is under investigation [29,28,32,13].It is natural to separate the remainder of the causal ladder in two parts.That going from chronology up to stable causality (causality, distinction, strongcausality belong to it), and that going from stable causality up to global hy-perbolicity (passing through causal continuity and causal simplicity). While theformer part deals with each time more demanding conditions conceived to avoidalmost closed causal curves, the latter part presents each time more demandingconditions in order to reduce the effects of points at infinity on spacetime.The problem of climbing the causal ladder from chronology up to stablecausality will be considered and solved in this work. It has received less at-tention than the latter problem, that is, that of going from stable causality upto global hyperbolicity which is indeed more closely related to the strong cosmiccensorship conjecture [23].I am going to prove that chronology plus the absence of lightlike lines impliesstable causality (theorem 6). The theorem is formulated so that every mentionedproperty is conformally invariant. It is therefore a theorem on the causal struc-ture of spacetime. In this respect it is important to use the weaker assumption of absence of lightlike lines instead of the more common null convergence, genericityand completeness conditions, though these have a more direct physical mean-ing. In any case the requirement of absence of lightlike lines can be regarded asa null completeness assumption, that is, it follows from demanding absence of hronological spacetimes without lightlike lines are stably causal 3 singularities. I shall say more on this correspondence in the first section. Thusthe theorem physically can be interpreted by saying that under chronology, theabsence of singularities implies stable causality and hence the existence of a timefunction . It is the first result of this form which reduces the existence of a timefunction to considerable less demanding properties. Moreover, note that in theprevious statement the required absence of singularities is more precisely onlya null completeness requirement: the spacetime manifold could still be timelikeincomplete in a way compatible with the singularity theorems (I shall say moreon that in sections 4 and 6).Recall that stable causality is the best possible constraint in order to removealmost closed causal curves and hence causality violations. The theorem can thenbe regarded as a singularity theorem, indeed, rewritten in the form non-stablycausal spacetimes either are non-chronological or admit lightlike lines receivesthe following physical interpretation if there is a form of causality violation onspacetime then either it is the worst possible, namely violation of chronology, orthe spacetime is singular . Regarded in this way the theorem clarifies the influenceof causality violations on singularities. In fact, if the violation of chronology isregarded as a sort of singularity then the theorem states that if there is no timefunction then the spacetime is singular in this broader sense.I refer the reader to [20,16] for most of the conventions used in this work.In particular, I denote with (
M, g ) a C r spacetime (connected, time-orientedLorentzian manifold), r ∈ { , . . . , ∞} of arbitrary dimension n ≥ − , + , . . . , +). On M × M the usual product topology is defined. For convenienceand generality I often use the causal relations on M × M in place of the morewidespread point based relations I + ( x ), J + ( x ), E + ( x ) (and past versions). Allthe causal curves that we shall consider are future directed (thus also the pastrays). The subset symbol ⊂ is reflexive, X ⊂ X . Several versions of the limitcurve theorem will be repeatedly used, particularly those referring to sequencesof g n -causal curves, where the metrics in the sequence g n may differ. The readeris referred to [15] for a sufficiently strong formulation.
2. Absence of lightlike lines
In this section I consider the property of absence of lightlike lines and commenton its physical meaning.Two spacetimes belonging to the same conformal class ( M, g ) share the samelightlike geodesics up to reparametrizations, and the condition of maximality forthe lightlike geodesic γ reads “there is no pair of events x, z ∈ γ , ( x, z ) ∈ I + ”,which makes no mention to the full metric structure and hence is independentof the representative of the conformal class. Thus, it is convenient to give thefollowing conformal invariant definition, Definition 1.
A lightlike line is an achronal inextendible causal curve.
The definition implies, by achronality, that the causal curve is a lightlikegeodesic and that it maximizes the Lorentzian length between any of its points.It is well known that [22, Chap. 10, Prop. 48]
Proposition 1.
If a inextendible lightlike geodesic admits a pair of conjugateevents then it is not a lightlike line.
E. Minguzzi
It can be proved that the notion of conjugate points along a lightlike geodesicis conformally invariant [20], thus the previous proposition relates two confor-mal invariant properties. In particular note that the requirement every lightlikegeodesic has a pair of conjugate points is stronger than absence of lightlike lines ,e.g. 1+1 Minkowski spacetime with x = 0 and x = 1 identified. From the pointof view of Lorentzian geometry any statement should be formulated so as tomake its conformal invariance clear. For physical reasons some authors prefer tomention physically motivated but non-conformal invariant conditions. The con-sequence, however, is that several results have been formulated in an unnecessaryweak form as the assumptions of the theorems are not really used. Definition 2.
An inextendible lightlike geodesic γ of the spacetime ( M, g ) sat-isfies the generic condition if at some x ∈ γ the tangent vector n to the curveis a generic vector , that is, n c n d n [ a R b ] cd [ e n f ] = 0 . A spacetime satisfies the null generic condition if every inextendible lightlike geodesic satisfies the genericcondition. A spacetime can be generic only if n ≥ null generic condition is generic is clarified by [3, Prop. 2.15].It is usually assumed on the physical ground that if a lightlike geodesic does notsatisfy it then arbitrarily small metric perturbation in the geodesic path wouldmake it true. Definition 3.
The spacetime ( M, g ) satisfies the timelike convergence conditionif R ( v, v ) ≥ for all timelike, and hence also for all lightlike, vectors v . Thespacetime ( M, g ) satisfies the null convergence condition if R ( v, v ) ≥ for alllightlike vectors v (cf. [11, p.95] [3, Def. 12.8]). Thus the null convergence condition is a consequence of the positivity of theenergy density.
Definition 4.
A spacetime ( M, g ) is null geodesically complete if every inex-tendible lightlike geodesic is complete. Proposition 2.
In a spacetime ( M, g ) of dimension dim M ≥ , which satis-fies the null convergence condition, the null generic condition and that is nullgeodesically complete every inextendible lightlike geodesic admits a pair of con-jugate events. In particular ( M, g ) does not have lightlike lines.Proof. It follows from the existence of some pair of conjugate points in thelightlike geodesics accordingly to [11, Prop. 4.4.5] [3, Prop. 12.17].This proposition has been improved by Tipler [30,31] and Chicone and Ehrlich [6](see also Borde [5]) by weakening the null convergence condition to the averagednull convergence condition. This possibility is important because many quantumfields on spacetime determine a stress-energy tensor and hence a Ricci tensorwhich does not comply with the null convergence condition while it satisfies theaveraged null convergence condition.Proposition 2 implies that the condition of absence of lightlike lines is quitereasonable from a physical point of view at least if the spacetime is assumed tobe non-singular (see also the discussion in [11, Sect. 4.4]) or just null geodesicallycomplete. hronological spacetimes without lightlike lines are stably causal 5
In the next sections I will prove that the assumption of absence of light-like lines has the effect of identifying the levels of the causal ladder betweenchronology and stable causality. In this respect the hard part will come withthe inclusion of stable causality. A key role will be played by the property of K -causality introduced by Sorkin and Woolgar [27], and for the last step by anew property which I study in the next section. PSfrag replacementsAbsence of Absence oflightlike lines lightlike raysGlobal hyperbolicity ⇓ Causal simplicity ⇓ Causal continuity ⇓ Stable causality ↓ ↑ ? K -causality ⇓ A ∞ -causality ⇓ Compact stable causality ⇓ A ∞ -causality ⇓ A -causality ⇓ Strong causality ⇓ Non-partial imprisonment ⇓ Distinction ⇓ Non-total imprisonment ⇓ Causality ⇓ Chronology ⇓ Non-total viciousness
Fig. 1.
The causal ladder displaying the new levels considered in section 3. Penrose’s infiniteladder between A -causality and A ∞ -causality is omitted [16], as well as the levels of weakdistinction and feeble distinction [19]. For the placement of the non-imprisonment propertiesthe reader is referred to [18]. The arrow C ⇒ D means that C implies D and there are exampleswhich show that C differs from D . Stable causality implies K -causality but it is not knownif they coincide. The implications climbing the ladder express the geometrical content of thetheorems proved in this work. E. Minguzzi
3. Compact stable causality
Recall that a non-total imprisoning spacetime is a spacetime for which thereis no future-inextendible causal curve totally imprisoned in a compact (futurenon-total imprisonment is equivalent to past non-total imprisonment [2,18]).It is known that every relatively compact open set in a non-total imprisoningspacetime [18] is stably causal when regarded as a spacetime with the inducedmetric [2]. Actually, this property characterizes non-total imprisonment, indeedwe have
Theorem 1.
A spacetime ( M, g ) is non-total imprisoning iff for every relativelycompact open set B , ( B, g | B ) is stably causal.Proof. The implication to the right was proved by Beem [2]. To the left, assume(
M, g ) has a compact C in which some curve γ is future imprisoned. In [18] Iproved that there is a lightlike line η contained in C such that η ⊂ Ω f ( η ) where Ω f ( η ) is the set of accumulation points in the future of η (in analogy with theset of ω -limit points of dynamical systems). Let B be a relatively compact openset such that C ⊂ B . Take q ∈ η and, given a convex neighborhood U ∋ q , U ⊂ B , take p ∈ η ∩ J − ( U,g | U ) ( q ). Take g ′ > g in B ( g ′ need not be defined on B C ) then p ∈ I − ( U,g ′ ) ( q ), but recall that p ∈ Ω f ( η ) is an accumulation point forthe future-inextendible g ′ -timelike curve given by the portion of η which startsfrom q . Thus since I − ( U,g ′ | U ) is open it is possible to construct a closed g ′ -timelikecurve contained in B . The argument holds for any choice of g ′ thus it is not truethat for every relatively compact open set B , ( B, g | B ) is stably causal.Note that non-total imprisonment is a quite weak property (it is impliedby weak distinction [18]). A related problem is that of establishing if, givenan arbitrary compact on spacetime, the metric can be widened in it withoutintroducing closed causal curves in the whole spacetime. If this is possible thespacetime satisfies a condition which is stronger than non-total imprisonment.We can define a new property Definition 5.
A spacetime ( M, g ) is compactly stably causal if for every rel-atively compact open set B there is a metric g B ≥ g such that g B > g on B , g B = g on B C and ( M, g B ) is causal.Remark 1. There are some equivalent definitions, for instance: (
M, g ) is com-pactly stably causal if for every compact set C there is g C ≥ g such that g C > g on C and ( M, g C ) is causal. In order to prove the equivalence one has to takeappropriate convex combinations of metrics with smooth coefficients.Some natural questions arise, among them the placement of compact stablecausality in the causal ladder of spacetimes. Before considering this questionlet me recall some notation and terminology [16]. Following Woodhouse [33,1] I denote with A + the closure of the causal relation, that is A + = ¯ J + . Aspacetime is A ∞ -causal if there is no finite cyclic chain of distinct A + -relatedevents. This property is equivalent to the antisymmetry of the relation A + ∞ = ∪ + ∞ i =1 ( A + ) i , which is the smallest transitive relation containing A + . Analogously,a spacetime is A ∞ -causal if the relation A + ∞ is antisymmetric. The relation K + is the smallest closed and transitive relation containing J + , and the spacetime hronological spacetimes without lightlike lines are stably causal 7 is K -causal if the relation K + is antisymmetric [27]. It is known that stablecausality implies K -causality, although it is not known if these two conditionscoincide [17]. We have Theorem 2. K -causality implies A ∞ -causality.Proof. Since J + ⊂ K + , any causal relation obtained from J + by taking closuresor by making the relation transitive through the replacement R + → ∪ + ∞ i =0 ( R + ) i ,is still contained in K + . Since A + ∞ has this form A + ∞ ⊂ K + , thus K -causalityimplies A ∞ -causality. Remark 2.
Given a relation R + the two involutive operations given by (a) clo-sure: R + → ¯ R + , and (b) transitivization: R + → R + ∞ = ∪ + ∞ i =1 ( R + ) i , oncealternatively applied to J + generate a chain of relations all contained in K + whose first members are J + , A + , A + ∞ , A + ∞ , · · · . By demanding the antisym-metry one obtains a ladder of causal properties whose first members are causal-ity, A -causality, A ∞ -causality and A ∞ -causality, all necessarily weaker than K -causality. If at a certain point two adjacent relations coincide then they coincidewith K + as they are both closed an transitive and they are certainly the small-est relations with this property. In this case the mentioned ladder of relationsfinishes there where this coincidence occurs. As we shall see, the mentioned firstlevels are all different but it is not known if from some point on the levels wouldstart to coincide, that is, if after a finite number of operations of closure andtransitivization one would get K + and K -causality. Examples support the viewthat this coincidence occurs at a level which increases with the dimensionalityof the spacetime. Lemma 1.
Let ◦ denote the composition of relations, then J + ◦ A + ⊂ A + and A + ◦ J + ⊂ A + .Proof. Let us consider the former case, the latter being analogous. Let ( x, y ) ∈ J + and ( y, z ) ∈ A + , and let γ n be a sequence of causal curves of endpoints( y n , z n ) → ( y, z ). Take x k ∈ I − ( x ), x k → x , so that x k ≪ y and for sufficientlylarge n , x k ≪ y n ≤ z n , thus ( x k , z n ( k ) ) ∈ I + and in the limit ( x, z ) ∈ A + . Theorem 3. A ∞ -causality implies compact stable causality.Proof. Suppose (
M, g ) is A ∞ -causal but non-compactly stably causal, then thereis a relatively compact open set B such that for every g ′ ≥ g , g ′ > g on B , g ′ = g on B C , ( M, g ′ ) is not causal. Let g n be a sequence of metrics g n ≥ g , g n > g on B , g n = g on B C , g n +1 ≤ g n , and g n → g pointwisely on the appropriatetensor bundle. For every choice of n , ( M, g n ) is not causal, and since ( M, g ) iscausal there must be a closed g n -causal curve γ n intersecting B (see figure 2). Let p n ∈ γ n ∩ B and parametrize the curves with respect to a complete Riemannianmetric h so that p n = γ n (0).Assume an infinite number of γ n is entirely contained in ¯ B . Beem [2] hasshown that there would be a inextendible g -causal limit curve contained in ¯ B in contradiction with the non-total imprisoning property of the spacetime (re-call that A -causality implies distinction which implies the non-total imprisoningproperty). Thus without loss of generality we can assume that none of the γ n is E. Minguzzi entirely contained in ¯ B . We conclude that γ n intersects ˙ B at least once to enter B C . Without loss of generality we can also assume that p n → p ∈ ¯ B .Using again the limit curve argument, through p there passes a future inex-tendible (hence its h -length parameter has domain ( −∞ , + ∞ )) g -causal curve γ which can’t pass through p twice as it would imply a violation of causality for( M, g ). In particular since (
M, g ) is non-partial imprisoning it escapes ¯ B at a lastpoint q ∈ ˙ B to never reenter ¯ B . Let γ n be a subsequence of γ n which convergesto γ uniformly on compact subsets and let s be the value of the parametersuch that q = γ ( s ). Since γ n ( s + 2) → γ ( s + 2) / ∈ ¯ B pass to a subsequencedenoted in the same way so that γ n ( s + 2) / ∈ ¯ B . Let (¯ s n , t n ) ∋ s + 2 be thelargest open connected interval so that γ n ((¯ s n , t n )) ⊂ ( ¯ B ) C . Define ¯ q n , p n ∈ ˙ B as ¯ q n = γ n (¯ s n ) and p n = γ n ( t n ). Let p ∈ ˙ B be an accumulation point for p n , without loss of generality we can assume p n → p . Note that the segment γ n | [¯ s n ,t n ] is entirely contained in B C and hence it is g -causal. Since ¯ s n ∈ [0 , s +2],without loss of generality we can assume ¯ s n → ¯ s for some ¯ s . Now, ¯ s ≤ s indeed if ¯ s > s then ¯ q n ∈ ¯ B converges to γ (¯ s ) a point that does not be-long to ¯ B which is impossible. In particular, it is possible to find a sequence s n ,¯ s n < s n < s + 2, such that s n → s . Then q n = γ n ( s n ) / ∈ ¯ B converges to q andthe g -causal sequence of curves γ n | [ s n ,t n ] has endpoints ( q n , p n ) ∈ J + such that( q n , p n ) → ( q , p ), i.e. ( q , p ) ∈ A + . Note that ( p , q ) ∈ J + as both pointsbelong to γ , hence ( p , p ) ∈ A + . PSfrag replacements ¯ p n p = γ (0) γ n (0)= p n γ n γ q = γ ( s ) γ n (¯ s n )= ¯ q n γ n ( s n )= q n γ ( s + 2)¯ p n γ p n = γ n (0) p q ¯ q n q n p n p g n -causalclosed g -causalinextendibleinextendible g -causal B Fig. 2.
The argument of the proof that A ∞ -causality implies compact stable causality. The limit curve theorem states that t n → + ∞ otherwise p would belong tothe prolongation of γ which is impossible since q is the last point of γ in ¯ B . Thesegments γ n | [¯ s n ,t n ] are not all contained in a compact because γ escapes every hronological spacetimes without lightlike lines are stably causal 9 compact to never return and for every k > γ n ( s n + k ) → γ ( s + k ) because t n → + ∞ . As a consequence the pair ( p , p ) ∈ A + can be regarded as the limitof the pairs of endpoints of g -causal segments which are not all contained in acompact. (In order to construct these segments take ¯ p k ∈ I − ( p ), ¯ p k → p sothat q ∈ I + (¯ p k ) and hence since I + is open q n ( k ) ∈ I + (¯ p k ) for a sufficientlylarge n ( k ). Next follow the g -causal segment γ n | [ s n ,t n ] which are not all containedin a compact, finally redefine the parametrization of the sequence ¯ p k and passif necessary to a subsequence so that (¯ p n , q n ) ∈ I + and hence (¯ p n , p n ) ∈ J + with (¯ p n , p n ) → ( p , p ).) In particular, p = p since the spacetime is stronglycausal.Now, translate all the parametrizations of γ n so that t n gets replaced by0. Repeat the previous steps where now p plays the role of p and the foundsequence γ n is a reparametrized subsequence of γ n .Continue in this way, defining for at each step analogous subsequences andevents so that p k ∈ ¯ B , ( p k , p k +1 ) ∈ A + , p k = p k +1 , and for each k there isa sequence of g -causal curves, not all contained in a compact, so that the end-points of the sequence converge to ( p k , p k +1 ). Note that for every pair of positiveintegers a ≤ b , ( p a , p b ) ∈ A + ∞ .Since ¯ B × ¯ B is compact, there is a subsequence denoted ( p k s , p k s +1 ) suchthat ( p k s , p k s +1 ) → ( x, z ) as s → + ∞ . Moreover, x = z because otherwise forevery relatively compact causally convex neighborhood U ∋ x , for sufficientlylarge s , ( p k s , p k s +1 ) ∈ U , and the sequence of g -causal curves not all containedin a compact, whose endpoints converge to ( p k s , p k s +1 ) would contradict thecausal convexity of U . Since A + is closed, ( x, z ) ∈ A + and x = z . Since p k s isa subsequence of p k , for every s , k s + 1 ≤ k s +1 , thus ( p k s +1 , p k s +1 ) ∈ A + ∞ andin the limit s → + ∞ , ( z, x ) ∈ A + ∞ . As a consequence ( M, g ) is not A ∞ -causalwhich is the searched contradiction. Theorem 4.
Compact stable causality implies A ∞ -causality.Proof. Assume the spacetime is compactly stably causal, and suppose it is not A ∞ -causal then there is a finite closed chain of A + -related events ( x i , x i +1 ) ∈ A + , i = 1 , . . . , n , x n +1 = x .Consider a relatively compact open set B which contains all x i , i = 1 , . . . , n ,and let g B ≥ g , g B > g on B , g B = g on B C . We want to prove that A + ∩ ( B × B ) ⊂ J +( M,g B ) , from which it follows that ( M, g B ) is not causal whateverthe choice of g B and hence ( M, g ) is not compactly stably causal, the searchedcontradiction. Let ( y, z ) ∈ A + , y, z ∈ B , then by the limit curve theorem either( y, z ) ∈ J + ⊂ J +( M,g B ) or there are a future inextendible g -causal curve σ y starting from y , and a past inextendible g -causal curve σ z ending at z such thatfor every y ′ ∈ σ y \{ y } and z ′ ∈ σ z \{ z } , ( y ′ , z ′ ) ∈ A + . At least a segment of σ y near y is timelike for ( M, g B ) and analogously for σ z , thus ( y, y ′ ) ∈ I +( M,g B ) , and( z ′ , z ) ∈ I +( M,g B ) finally since ( y ′ , z ′ ) ∈ A + ⊂ J +( M,g B ) , it is ( y, z ) ∈ I +( M,g B ) . Remark 3.
All the properties of the previous theorems differ. In [16] I gave anexample of non- K -causal A ∞ -causal spacetime. A closer inspection proves thatit is actually non- A ∞ -causal but compactly stably causal. Moreover, it is possibleto construct an example, similar to that of [16] which is A ∞ -causal but non- K -causal (simply repeat the figure of [16] three times vertically, and then identify the holes cyclically). The properties A ∞ -causality and compact stable causalitydiffer because of the spacetime example of figure 3. A consequence of theseexamples is the perhaps surprising fact that compact stable causality differsfrom stable causality (see again the example of [16]). This fact means that thebehavior of the light cones near infinity is important in order to determine if aspacetime is properly compactly stably causal or not. Remove I d e n t i f y RemoveRemove { PSfrag replacements
K xx yy
Fig. 3. A A ∞ -causal but non-compactly stably causal spacetime. In order to constructthe spacetime start from R × S × R of coordinates ( t, θ, z ), θ ∈ [0 , g = − d t + d θ + d z , remove two spacelike surfaces and identify, after a translation by an ir-rational number, two spacelike surfaces as done in the figure . The coordinates ( x, y ) havebeen introduced on the identified surfaces so as to make the identification clear. The space-time is non-orientable but this feature is not essential. The spacetime is non-compactly stablycausal since any enlargement of the metric on K gives closed causal curves. Thanks to thetranslation by an irrational number there cannot be closed chains of A + related events.
4. The proof and some physical considerations
I start with a result due to Hawking [12] [11, Prop. 6.4.6] (he proved it with thestronger but inessential assumption that every lightlike geodesic admits a pairof conjugate points)
Lemma 2.
A chronological spacetime without lightlike lines is strongly causal.Proof.
Recall that in a strongly causal spacetime, given any neighborhood U of x ∈ M there exist a neighborhood V ⊂ U , x ∈ V , such that any future-directedcausal curve with endpoints at V is entirely contained in U (see for instance[20, Lemma 3.22]). Thus if ( M, g ) were not strongly causal there would be apoint x , a neighborhood U ∋ x , and a sequence of causal curves γ n of startingevent x n , ending event z n such that x n → x , z n → x , and the curves γ n arenot entirely contained in U . Hence there are the conditions required by the limitcurve theorem [15, theorem 3.1] case (2) which implies the existence of a lightlikeline passing through x , a contradiction.A fundamental step in the proof is Theorem 5.
If a spacetime does not have lightlike lines then the relation A + =¯ J + is transitive, that is K + = A + . Moreover, if the spacetime is also chronolog-ical then the spacetime is K -causal. hronological spacetimes without lightlike lines are stably causal 11 Proof.
Let us prove the transitivity of A + . Take two pairs ( x, y ) ∈ A + and( y, z ) ∈ A + and two sequences of causal curves σ n of endpoints ( x n , y n ) → ( x, y ),and γ n of endpoints ( y ′ n , z n ) → ( y, z ). Apply the limit curve theorem [15] to bothsequences, and consider first the case in which the limit curve in both cases doesnot connect the limit points. By the limit curve theorem, σ n has a limit curve σ which is a past inextendible causal curve ending at y . Analogously γ n has alimit curve γ which is a future inextendible causal curve starting from y . Theinextendible curve γ ◦ σ cannot be a lightlike line thus there are points x ′ ∈ σ \{ y } , z ′ ∈ γ \{ y } such that ( x ′ , z ′ ) ∈ I + and (pass to a subsequence) points x ′ n ∈ σ n , x ′ n → x ′ and z ′ n ∈ γ n , z ′ n → z ′ , thus, since I + is open, for sufficiently large n ,( x n , z n ) ∈ I + and finally ( x, z ) ∈ ¯ I + = A + .If both limit curves join the limit points then clearly ( x, z ) ∈ J + ⊂ A + . If,say, σ joins x to y but γ does not join y to z , take x ′ n ∈ I − ( x ), x ′ n → x , sothat x ′ n ≪ y and for large n , x ′ n ≪ y ′ n ≤ z n , thus in the limit ( x, z ) ∈ A + . Theremaining case is analogous. Thus A + is closed and transitive hence A + = K + .Assume ( M, g ) is chronological then by lemma 2 (
M, g ) is strongly causal.The relation A + is antisymmetric indeed let ( x, y ) ∈ A + and ( y, x ) ∈ A + , x = y ,and let σ n of endpoints ( x n , y n ) and γ n of endpoints ( y ′ n , z n ) be sequences ofcausal curves whose endpoints converge to the initial pairs ( x n , y n ) → ( x, y ),( y ′ n , z n ) → ( y, x ). Then we repeat the argument used above, that is we apply thelimit curve theorem to the accumulation point y . Call σ the limit causal curvefor σ n and analogously let γ be the limit causal curve for γ n . If σ connects x to y and γ connects y to x then there is a closed causal curve on spacetime acontradiction. Let U ∋ x , V ∋ y be two disjoint causally convex neighborhoods.If σ connects x to y but γ does not connect y to x , then it is possible to argueas above, i.e. take x ′ k ∈ I − ( x ), x ′ k → x , then for sufficiently large n , whichwe can choose so that n ( k ) > k , y ′ n ( k ) ∈ I + ( x ′ k ) ∩ V , from which it followsthat there is a sequence of causal curves of endpoints x ′ k , z n ( k ) , intersecting V . But ( x ′ k , z n ( k ) ) → ( x, x ) thus strong causality is violated at x . The case inwhich γ connects y to x is analogous. The remaining case is that in which σ is past-inextendible and γ is future-inextendible. Then γ ◦ σ is a inextendiblecausal curve which by assumption is not a lightlike line. Moreover, since strongcausality holds, this curve is not partially imprisoned in any compact, thus usingthe same argument as above (i.e. taking advantage of the chronality of γ ◦ σ )it follows that there is a sequence of causal curves of endpoints x n , z n not allcontained in a compact. Again there is a contradiction with the strong casualityat x .Clearly, if we could prove that K -causality is equivalent to stable causalitythen the main theorem would follow. Unfortunately, though there is evidence forthis coincidence [17] no proof has yet been given. In fact Seifert [24], even beforethe introduction of K -causality, gave an argument which would have impliedthe equivalence. Unfortunately, he only sketched the proof and a recent moredetailed study [17] has shown that those arguments were inconclusive. If the twocausal properties are indeed equivalent it is probable that the proof would berather involved because the K + relation is not as easy to handle as the othercausal relations. Fortunately, however, it is possible to circumvent this difficulty,and avoid a direct proof of the equivalence between stable causality and K -causality, by working on compact stable causality. Indeed, the previous resultwill be used in the following weaker form Corollary 1.
A chronological spacetime without lightlike lines is compactly sta-bly causal.
Now, the idea is to consider the property “(
M, g ) is compactly stably causaland does not admit lightlike lines” to show that it is inductive (see lemma 4),that is, invariant under enlargement of the light cones over compact sets. Thenit is possible to enlarge the light cones in a sequence of compact sets that cover M so as to obtain a causal spacetime with strictly larger light cones (theorem6). Lemma 3. On ( M, g ) let B be a relatively compact open set, let g n be a sequenceof metrics g n ≥ g , g n > g on B , g n = g on B C , g n +1 ≤ g n , and g n → g pointwisely on the appropriate tensor bundle. If ( M, g ) does not have lightlikelines then all but a finite number of ( M, g n ) do not have lightlike lines.Proof. If not we can, passing to a subsequence, assume that all (
M, g n ) havelightlike lines. Denote γ n a respective sequence of lightlike lines and assumethere is one, say γ ¯ n , which does not intersect B . Since g ¯ n and g coincide outside B , γ ¯ n is a g -causal curve. Also it is g -achronal because if there are two points p, q ∈ γ ¯ n such that ( p, q ) ∈ I + g then as g ≤ g ¯ n , ( p, q ) ∈ I + g n which is impossiblebecause γ ¯ n is a lightlike line on ( M, ¯ g n ). But γ ¯ n cannot be g -achronal as itwould be a lightlike line of ( M, g ), thus the overall contradiction proves that all γ n intersect B . Without loss of generality we can assume (pass to a subsequenceif necessary) that there are x n ∈ B ∩ γ n , and x ∈ ¯ B such that x n → x . By thelimit curve theorem there is a inextendible g -causal curve η passing through x .If η is not g -achronal there are y, z ∈ η such that ( y, z ) ∈ I + g ⊂ I + g n for every n . But since y and z are limit points of the sequence γ n and I + g ( ⊂ I + g n ) is opensome of the curves γ n are not lightlike lines. The contradiction proves that η is not only g -causal but also g -achronal thus it is a lightlike line. Again this isimpossible thus the assumption that an infinite number of ( M, g n ) does admitlightlike lines has lead to a contradiction. Lemma 4. If ( M, g ) is compactly stably causal and without lightlike lines thenfor every open set of compact closure B it is possible to find a metric g B ≥ g such that g B > g on B , g B = g outside B , and ( M, g B ) is compactly stablycausal and without lightlike lines.Proof. Since (
M, g ) is compactly stably causal we can find ˜ g B such that ˜ g B > g on B , ˜ g B = g outside B and ( M, ˜ g B ) is causal. Define g n = (1 − n ) g + n ˜ g B sothat g ≤ g n ≤ ˜ g B satisfies the assumptions of the previous lemma. Thus there isa certain element of the sequence, denote it g B , such that ( M, g B ) does not havelightlike lines and since g B ≤ ˜ g B , ( M, g B ) is causal. But every causal spacetimewithout lightlike lines is compactly stably causal thus the thesis. Theorem 6. If ( M, g ) is chronological and without lightlike lines then it is stablycausal.Proof. Let h be an auxiliary complete Riemannian metric, x ∈ M , and let B k = B ( x , k ) be the open balls of radius k centered at x . Define g = g . Bythe previous lemma it is possible to find a metric g > g on B , g = g outside B , such that ( M, g ) is compactly stably causal and without lightlike lines. Next hronological spacetimes without lightlike lines are stably causal 13 repeat the argument for the relatively compact open set B with respect to thespacetime ( M, g ): there is a metric g > g on B , g = g (= g ) outside B , suchthat ( M, g ) is compactly stably causal and without lightlike lines. Continue inthis way and find a sequence of metrics g k +1 ≥ g k ≥ g , g k +1 > g k on B k +1 .The open sets A = B , A k = B k +1 \ ¯ B k − for k ≥
2, cover M . Let { χ k } bea partition of unity so that the support of χ k is contained in A k , and define˜ g = P + ∞ k =1 χ k g k +2 (the sum has at most two non vanishing terms at each point)then ˜ g > g , moreover at x ∈ B k , ˜ g ( x ) ≤ g k +2 ( x ), because for n > k , χ n ( x ) = 0(see figure 4). But ( M, ˜ g ) is causal because otherwise there is a closed ˜ g -causalcurve σ , which being a closed set, is entirely contained in B s for some s . Since˜ g ≤ g s +2 on B s , this curve is g s +2 -causal which contradicts the (compact stable)causality of ( M, g s +2 ). Thus since ( M, ˜ g ) is causal and ˜ g > g , ( M, g ) is stablycausal. {{ {{{
PSfrag replacements gg g g g g g B B B B B B A A A A A M ˜ g Fig. 4.
The construction of the metric ˜ g > g and of the causal spacetime ( M, ˜ g ) in the proofof theorem 6. Remark 4.
This result is sharp in the sense that causal continuity can not replacestable causality in the statement of the theorem. Indeed, the 1+1 spacetime R × S of coordinates ( t, θ ), θ ∈ [0 , s = − d t + d θ with the timelikesegment θ = 1, 0 ≤ t ≤
1, removed does not have lightlike lines, is chronological,and thus stably causal ( t is a time function) but it is not reflective and henceit is not causally continuous. Analogously, chronology can not be weakened tonon-total viciousness indeed, for instance, the spacetime of figure 5 is non-totallyvicious, does not have lightlike lines but is not even chronological. Nevertheless,it is possible to relax slightly the chronology condition by asking, for instance,that the chronology violating set be confined in a compact or even more weaklyto have a compact boundary (see the next section).Recall that a time function t : M → R is a continuous function which increaseson every causal curves, that is, if γ : B → M is a causal curve, b < b implies t ( γ ( b )) < t ( γ ( b )). Hawking proved, improving previous results by Geroch [7], that stable causality holds if and only if the spacetime admits a time function(for the direction, time function ⇒ stable causality, see [9], for the other directionsee [11]). Actually the time function can be chosen smooth with timelike gradient[4] (see also [25]). Thus a corollary of theorem 6 is Theorem 7. If ( M, g ) is chronological and without lightlike lines then it admitsa time function (which can be chosen smooth with timelike gradient). Recall also that if t is a time function then F a = { p : t ( p ) > a } is an openfuture set and ˙ F a = { p : t ( p ) = a } . In particular, S a = ˙ F a is an acausal boundary(hence edgeless), that is, S a is a partial Cauchy hypersurface [11].The great advantage of theorem 7, is that it allows to considerably weakenthe causality and boundary conditions underlying most singularity theorems.Indeed, most of them assume some of the following: (a) global hyperbolicity,(b) a partial Cauchy hypersurface (c) a compact achronal edgeless set (d) atrapped set. Often these global assumptions are made without any further justi-fication, in fact Senovilla in his review [26, p. 803-8] expressed the opinion thatthese boundary assumptions may represent the main weak point of singularitytheorems. Fortunately, theorem 6 justifies the presence of a foliation of partialCauchy hypersurfaces and hence may be used to weaken the global assumptionsmade in singularity theorems. In this section I am going to consider the im-plications of the absence of lightlike rays. Recall that a future ray is a future-inextendible causal curve which is achronal. Past rays are defined analogously.Chosen a point c ∈ ( a, b ) in a lightlike line γ : ( a, b ) → M , the portion γ | [ c,b ) isa lightlike future ray while γ | ( a,c ] is a lightlike past ray, thus Lemma 5.
The absence of lightlike future (or past) rays implies the absence oflightlike lines.
Thus, assuming the absence of lightlike future rays one expects to obtain astronger property than stable causality. Indeed, we have (see also the relatedresult [29, Prop. 4])
Theorem 8. If ( M, g ) is chronological and without future lightlike rays then itis globally hyperbolic (and the only TIP is M ). An analogous past version alsoholds.Proof. Since there are no future rays then there are no lightlike lines and thespacetime is stably causal and admits a time function t . Let p ≤ q , we have toprove that C = J − ( q ) ∩ J + ( p ) is compact. Take r ∈ I + ( q ) so that a = t ( r ) > t ( q ),and consider the partial Cauchy surface S a . Since C ⊂ I − ( r ), all the points in C stay in the past set P a = { x : t ( x ) < a } . The set H − ( S a ) is generated byfuture lightlike rays (as S a is edgeless) and since by assumption there is no futurelightlike ray, H − ( S a ) is empty. Thus C ⊂ P a ⊂ D − ( S a ) ⊂ D ( S a ), the last setbeing globally hyperbolic. Note that no causal curve from p can escape D ( S a ) andhence P a to return to q , as t is a time function. Hence C = J − D ( S a ) ( q ) ∩ J + D ( S a ) ( p )is compact. Finally, ( M, g ) has no TIP but M because the boundary of any TIPis generated by future lightlike rays. hronological spacetimes without lightlike lines are stably causal 15PSfrag replacements M \ ¯ C C
RemoveIdentify
Fig. 5.
The figure displays 1+1 Minkowski spacetime with two spacelike slices identified anda triangle removed. If the angle at the top of the triangle is small enough there are no pastlightlike rays.
Note that in theorem 8 chronology can not be weakened to non-total vicious-ness, i.e. to the condition
C 6 = M where C is the chronology violating set. Indeed,figure 5 gives a counterexample. Nevertheless, if one replaces the absence of fu-ture lightlike rays with the absence of lightlike rays then the proof of theorem12 will show that a non-totally vicious spacetime is chronological (by showingthat ˙ C if non-empty, contains a lightlike ray), and thus one has: Theorem 9. If ( M, g ) is non-totally vicious and without lightlike rays then it isglobally hyperbolic (and there are no TIP or TIF but M ).4.2. Physical considerations. Theorem 8 can be used as a singularity theoremthough the null convergence condition is not enough to guarantee that a future-complete future-inextendible (affinely parametrized) lightlike geodesic γ : [ a, + ∞ ) → M admits a pair of conjugate points. A sufficient condition is Tipler’s [29, Prop.1] lim s → + ∞ [( s − a ) Z + ∞ s R cd n c n d d s ′ ] > , (1)where n c is the tangent vector to γ at γ ( s ). Weaker conditions were also con-sidered by Borde [5]. These conditions physically state that the energy densityshould not drop off too sharply. The assumption is reasonable in those caseswhere the universe is contracting (or taking the past version, expanding) as onewould expect the energy density to increase rather than decrease.Thus we get the following singularity theorem (past version) Theorem 10.
The following conditions cannot all hold(i) ( M, g ) is past null geodesically complete,(ii) ( M, g ) is chronological(iii) ( M, g ) is non-globally hyperbolic,(iv) Some energy condition which implies the presence of conjugate points inpast-complete past-inextendible lightlike geodesics (e.g. lim s →−∞ [( b − s ) Z s −∞ R cd n c n d d s ′ ] > , holds on any past-inextendible lightlike geodesic γ : ( −∞ , b ) → M ). The nice feature of this theorem is that there is essentially no boundary as-sumption and the causality conditions are quite weak. There is no assumption onthe existence of partial Cauchy surfaces or trapped sets. Of course, the strongestassumption which must be physically justified is made in (iv) but the local ex-pansion of the Universe together with the cosmic background radiation, seem tosupport it. Then the theorem states, that under the said energy conditions thespacetime is either globally hyperbolic or has singularities. Used in conjunctionwith Penrose’s (1965), and Hawking and Penrose’s (1970) singularity theorems[11] it allows to characterize quite precisely what a spacetime looks like if itcontains trapped surfaces and it is still null geodesically complete.We have
Theorem 11.
Let ( M, g ) be a spacetime of dimension greater than 2. If(i) ( M, g ) is null geodesically complete,(ii) ( M, g ) is chronological,(ii) There is a future trapped surface,(iv) The timelike convergence, the generic condition, together with some energycondition which implies the presence of conjugate points in past-complete past-inextendible lightlike geodesics (e.g. lim s →−∞ [( b − s ) Z s −∞ R cd n c n d d s ′ ] > , holds on any past-inextendible lightlike geodesic γ : ( −∞ , b ) → M ).then the spacetime is globally hyperbolic with compact space slices and has aincomplete timelike line.Proof. The conditions (i), (ii) and (iv) imply (v): the spacetime is globally hy-perbolic (theorem 10). The Cauchy hypersurfaces are either compact or non-compact. In the latter case (iii) and (v) imply, by the Penrose singularity theo-rem, that the spacetime is null geodesically incomplete. Thus (vi): the Cauchyhypersurfaces are compact. The proof of the Hawking-Penrose theorem impliesthat (i), (ii) or (vi), and (iv) imply that there is a incomplete timelike line.Since the existence of trapped surfaces is a quite natural consequence of gen-eral relativity if matter concentrate enough, theorem 8 supports the global hy-perbolicity of the spacetime (and a closed space) provided it is null geodesicallycomplete. Since the conditions are quite reasonable one concludes that the space-time is either null geodesically incomplete or timelike geodesically incomplete (orboth).Finally I would like to stress that the assumption of null geodesic complete-ness does not lead to a spacetime picture which contradicts observations. Thustheorems 8 and 6 may have a “positive” role in proving the good causal propertyof spacetime rather than being used only to prove its singularity. As a matter offact they can be used to do both (theorem 11). hronological spacetimes without lightlike lines are stably causal 17
5. The non-chronological case
So far we have studied the consequence of the absence of lightlike lines underthe assumption of chronology. Let us consider the other possibility, namely non-chronological spacetimes. Denote with C the chronology violating set, with C α , C = S α C α , its (open) components and with B αk the (closed) components of therespective boundaries ˙ C α = S k B αk .The next result joins two theorems, one by Kriele [14, Theorem 4] who im-proved previous results by Tipler [29] and the other by the author [15]. Theorem 12.
A non-chronological spacetime without lightlike lines is either to-tally vicious (i.e. C = M ) or it has a non-empty chronology violating set C , theboundaries ˙ C α of the components C α , are disjoint and the components B αk ofthose boundaries are all non-compact. In particular non-totally vicious space-times without lightlike lines are non-compact. For the proof that the sets ˙ C α are disjoint I refer the reader to [15]. Instead,I elaborate on Kriele’s argument by giving a slightly different proof that theboundaries B αk are non-compact. Indeed, I can give a shorter proof thanks tothe limit curve theorem contained in [15] and to the results on totally imprisonedcurves contained in [18].Recall that in the chronology violating set C , Carter’s equivalence relation p ∼ q iff p ≪ q ≪ p gives rise to open equivalence classes, moreover, since C is open, if x ∈ ˙ C it cannot be x ∈ C . Recall also that with Ω f ( η ) it is denotedthe set Ω f ( η ) = T t ∈ R η [ t, + ∞ ) of accumulation points in the future of the causalcurve η , and analogously in the past case. This set is always closed, moreover,it is non-empty iff the curve is partially imprisoned in a compact [18]. Proof.
Assume that B αk ⊂ ˙ C α is compact and let x ∈ B αk . Let x n ∈ C α suchthat x n → x , and let U ∋ x be a convex set. There are closed timelike curves σ n ⊂ C α of starting and ending point x n , which are necessarily not entirelycontained in U (every convex set is causal). Let z = x , then by the limit curvetheorem [15] (point 2) there are two cases (corresponding to 0 < b < + ∞ , or b = + ∞ in that reference).In the first case there is a closed continuous causal curve γ ∈ ¯ C α passingthrough x . It must be achronal since if p, q ∈ γ , p ≪ q , then x ≤ p ≪ q ≤ x andhence x ≪ x which implies x ∈ C a contradiction. Thus γ is a geodesic with nodiscontinuity in the tangent vectors at x . It can be extended to a lightlike line γ by making infinite rounds over γ (note that in this case Ω f ( γ ) = Ω p ( γ ) = γ ).In the second case there are a future inextendible continuous causal curve γ x ⊂ ¯ C α starting at x and a past inextendible continuous causal curve γ z ⊂ ¯ C α ending at x . If γ x ∩ I + ( x ) = ∅ and γ z ∩ I − ( x ) = ∅ then for sufficiently large n , since I + is open, it would be possible to complete a segment of γ n to aclosed timelike curve passing through x hence x ∈ C , a contradiction. Thus γ x or γ z , say γ x , is a lightlike ray. In particular γ x being a lightlike ray is achronaland hence can not enter C α , thus γ x ⊂ B αk . Now, since B αk is compact and B αk ∩ C = ∅ , results on totally imprisoned causal curves can be applied [18,theorem 3.6]. In particular there is a minimal non-empty closed achronal set Ω ⊂ Ω f ( γ x ) ⊂ B αk such that through each point of Ω there passes one and onlyone lightlike line, this line is entirely contained in Ω and for every line α ⊂ Ω , Ω f ( α ) = Ω p ( α ) = Ω . Just the existence of a lightlike line suffices to concludethe proof that the boundaries B αk are non-compact.The last statement in a slightly weaker form has been first obtained by Tipler[29, theorem 7]. It follows from the observation that a compact spacetime has anon-empty chronology violating set C (see [11, Prop. 6.4.2]) thus either C = M or ˙ C is non-empty and compact in contradiction with the absence of lightlikelines.These results restrict the possible chronology violation in spacetimes withoutlightlike lines, for instance they state that the chronology violation must extendto infinity. In principle this fact does not mean that a chronology violatingregion can not develop from regular data. For this to be the case stronger globalassumptions than the only absence of lightlike lines should be assumed [29,13].Instead of trying to remove chronology violating sets altogether from thespacetime, it is natural to consider what theorem 6 may say in the cases ofchronology violation. The idea is that if ( M, g ) has a non-empty chronologyviolating set but M = ¯ C then the spacetime ( N, g | N ), where N is any connectedcomponents of M \ ¯ C , has empty chronology violating set. PSfrag replacements Nγ γ RemoveRemoveIdentify
Fig. 6.
If (
M, g ) has a non-empty chronology violating set and has no lightlike line, (
N, g | N ),with N any component of the shaded region M \ ¯ C , may admit lightlike lines (e.g. the causalcurves γ or γ ). However, even if (
M, g ) does not have lightlike lines, (
N, g | N ) may have light-like lines (see figure 6). This may happen because a lightlike line γ for ( N, g | N )is not inextendible in M , and thus once extended it may enter the chronologyviolating set (the geodesic γ in the figure). Another possibility is that while γ is also inextendible in M , the enlargement of the spacetime enlarges the set oftimelike curves and hence the possibilities that γ is not a line (the geodesic γ inthe figure). Thus it is not possible to infer from the absence of lightlike lines for( M, g ) the same property for (
N, g | N ). Actually, neither the converse is true, theMisner spacetime (with region I= N , see figure 32 of [11]) does not have lightlikelines but its analytic extension (I+II) where II is the chronology violating setfor I+II, does admit a lightlike line given by the Misner boundary.There is therefore no immediate way to apply theorem 6 to the non-chronologicalcase apart from that of motivating on physical grounds that some component N does not have lightlike lines. hronological spacetimes without lightlike lines are stably causal 19
6. Conclusions
A proof has been given that chronological spacetimes without lightlike lines arestably causal, and that non-totally vicious spacetimes without lightlike rays areglobally hyperbolic (together with some other variations). The properties: (i)chronology, (ii) null convergence condition and (iii) null generic condition, arequite reasonable from a physical point of view, moreover, for our purposes (ii)can be weakened to the averaged null convergence condition. Assuming (i), (ii)and (iii) the result of the title of this work translates into the physical statementthat null geodesically complete spacetimes are stably causal and therefore admita time function. Since the existence of some partial Cauchy surface is assumedin most singularity theorems, this result can be used to weaken the assumptionsof those theorems. This result may also prove important when applied to thestudy of the real Universe. Indeed, let us recall that Hawking’s and Hawkingand Penrose’s theorems [11] suggest the existence of an incomplete causal curvewhich however could well be timelike. In other words our Universe may perhapsbe geodesically null complete but timelike incomplete, in which case the maintheorem could be applied in the “positive” way to infer the existence of a timefunction for the Universe. In fact theorem 11 shows that the assumption of nullgeodesic completeness leads to consequences that do not contradict physicalobservations.The Penrose’s singularity theorem seems to go against this conclusion as itpredicts null incompleteness in those cases in which trapped surfaces form. Itmust be remarked, however, that Penrose’s theorem assumes the existence of anon-compact Cauchy hypersurface thus (i) it assumes the existence of a timefunction and hence it cannot be used to dismiss the conclusion that a timefunction exists and (ii) for spacetimes with compact slices its conclusions donot hold. However, even if the space slices are compact, one can still extractinformation from the proof of Penrose’s theorem [22, theorem 14.61]. The resultis that, roughly speaking, black holes do not exist. Trapped surfaces may formand locally they may resemble black holes but the global behavior would bequite different. Indeed, their horizons would finally join and swallow the wholespacetime. Thus, without an “exterior”, the “interior” could not be distinguishedfrom a usual spacetime.In conclusion the theorems of this work can be used physically, either in the“negative” way, to prove the existence of singularities or of chronology violatingregions, or in the “positive” way to argue for the existence of a time function orof global hyperbolicity. In either case they shade new light on the existence androle of time at cosmological scales.
Acknowledgments
This work has been partially supported by GNFM of INDAM and by MIURunder project PRIN 2005 from Universit`a di Camerino.
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