Causality theory of spacetimes with continuous Lorentzian metrics revisited
aa r X i v : . [ g r- q c ] J a n Causality theory of spacetimes with continuous Lorentzianmetrics revisited
Leonardo Garc´ıa-Heveling ∗ Abstract
We revisit the causal structures J + and K + on a spacetime, and introduce a new one, called k + . The k + -relation can be used to characterize causal curves, and for smooth Lorentzian met-rics, it yields the same result as the standard J + -relation. If, on the other hand, the metricis merely continuous, then the different causal structures become inequivalent. We comparethem by investigating three properties, namely the validity of the push-up lemma, the open-ness of chronological futures, and the existence of limit causal curves. Depending on thedefinition of causal structure chosen, we show that at most two of these three properties holdfor continuous metrics. In particular, by using the new relation k + , the push-up lemma holdseven when the metric is continuous, while it generally does not for the standard J + -relation.Finally, we argue that, in general, no reasonable notion of causal structure can have all threeproperties. Key words: low regularity, causality theory, push-up lemma, causal bubbles.
The study of spacetimes with metrics of low regularity is a topic of rising importance in Lorentziangeometry. The main motivation stems from the strong cosmic censorship conjecture [7, 21] andthe occurrence of weak solutions to Einstein’s equations coupled to certain matter models [4, 10].It has hence become an important research question to establish which properties of the usual,smooth spacetimes are more “robust” or “fundamental”, in the sense that they continue to holdin lower regularity, and which, on the other hand, depend sensibly on the smoothness assumption.In trying to answer this question, the need arises to axiomatize the notion of spacetime, and inparticular, to treat the causal structure in an order-theoretic way. This, in turn, connects wellwith ideas in quantum gravity, such as causal set theory [3]. To make matters more concrete, inthe present paper we shall study spacetimes (
M, g ) where g is a continuous Lorentzian metric.However, since our approach is indeed of the order-theoretic type, it can easily be adapted to othersettings.Let us start by recalling the case of a classical spacetime ( M, g ) where g is smooth. Thechronological and causal relations I + and J + can then be defined using the notions of timelike andnon-spacelike curve respectively. The three following facts are well-known:1. The push-up lemma: if p ∈ I + ( q ) and q ∈ J + ( r ) then p ∈ I + ( r ).2. The limit curve theorem: the uniform limit of a converging sequence of causal curves is acausal curve.3. Openness of chronological pasts and futures: the sets I ± ( p ) are open, for any p ∈ M . ∗ Department of Mathematics, Radboud University, Nijmegen, The Netherlands.
Email: [email protected]
Acknowledgements:
I am very grateful to Annegret Burtscher for discussions and detailed comments on the draft. g or the manifold structure on M explicitly. This is confirmed by the “Lorentzian length spaces”approach of Kunzinger and S¨amann [14] and follow-up work [1, 9].In order to do causality theory on a spacetime with a C -metric, the first question is how thecausal structure should even be defined. The obvious answer is to define I + and J + throughtimelike and non-spacelike curves, just as in the smooth case. However, there are two potentialproblems:A. Points where the metric is not C do not admit normal neighborhoods.B. The regularity class where we define timelike curves becomes important.Chru´sciel and Grant [6] showed that because of A, the push-up lemma fails, while the limit curvetheorems are unaffected (points 1 and 2 above). As a consequence, spacetimes with continuousmetric exhibit so called “causal bubbles”, open regions contained in J + but not in I + . Regard-ing B, when the metric is at least C , it was shown by Chru´sciel [5] that one obtains the samechronological relation I + regardless of whether timelike curves are required to be Lipschitz orpiecewise-differentiable. In the case of continuous metrics, however, it was shown by Grant et al.[11] that this choice makes an important difference. In particular, they showed that the chrono-logical futures and pasts are open when using piecewise-differentiable curves, but not when usingLipschitz curves (point 3 above).A radically different, and in fact earlier, approach is that of Sorkin and Woolgar [22]. Theypropose to keep the definition of I + by piecewise-differentiable timelike curves, and then introduceanother relation K + as the smallest transitive, topologically closed relation containing I + . Therelation K + can then be used to replace J + . Even for smooth metrics, the two relations K + and J + do not coincide. Nonetheless, it is possible to define the usual causal curves (and hence J + ) interms of K + , without referring to the metric directly. The K + -relation has since found a varietyof applications, most notably Minguzzi’s works on stable causality [19] and time functions [18].However, there is no push-up lemma for the K + -relation.Following a similar philosophy, we propose a new causal relation k + . We define k + as thelargest relation such that the push-up lemma holds true. Just as Sorkin and Woolgar did with K + , we propose a definition of causal curve based on k + . When the metric is smooth, causal curvesdefined through k + coincide with those defined by the metric g , but when the metric is merelycontinuous, they do not. As a consequence, we show that while our new causal relation satisfies thepush up lemma, the limit curve theorems cease to hold. We then argue that, essentially becausewe chose k + to be maximal, there in fact do not exist any causal relations that can satisfy bothproperties at the same time. That is, at least, if we want to keep the usual definition of (piecewisecontinuously differentiable) timelike curve, the only one that guarantees open futures. We alsoexplore the possibility of alternative chronological relations, but we conclude that one runs intothe same problems.The goal of this paper is thus two-fold. On the one hand, we have shown that spacetimes withcontinuous metrics are unavoidably pathological. This strengthens the view, already present inthe literature, that one should focus on a special class of continuous metrics, the so-called causallyplain ones. These are the ones where the usual push-up lemma holds, and include, for example, theclass of Lipschitz metrics [6, Cor. 1.17]. On the other hand, we believe that the methods developedhere, mainly the k + -relation, can find further applications in (low regularity) Lorentzian geometry;for example, in the study of spacetimes with degenerate metrics [8], or with metrics that are noteven continuous [10]. Outline.
In Section 2 we provide more background, define the new causal relation k + andstudy its properties. In Section 3 we define causal curves in terms of ˜ k + , and show that these arejust the usual causal curves when the metric is smooth. In Section 4, we discuss other possiblechoices of causal structure. In Section 5, we summarize and discuss our results.2 The k + -relation Let M denote a Hausdorff, paracompact C -manifold, and g a C -Lorentzian metric. Assume that( M, g ) admits a C -vector field X such that g ( X, X ) <
0, called a time orientation. The pair (
M, g )together with a choice of time orientation is called a C -spacetime . Whenever we say that g is C (or smooth), or that ( M, g ) is a C (or smooth) spacetime, we mean that M admits a C (resp.smooth) subatlas such that g is C (resp. smooth) in this subatlas. By relation on M we will meana subset of M × M . The closure of a relation R , denoted by R , is the topological closure in theproduct topology on M × M . Likewise, we say that R is open if it is an open subset of M × M .There exist two different definitions for the notion of timelike curve , and thus two differentnotions of chronological relation . For C -metrics the two notions are equivalent [5, Cor. 2.4.11],but not for C -metrics [11]. One is based on the class L of locally Lipschitz curves, and the otherone on the class C of piecewise continuously differentiable curves: I + := { ( p, q ) ∈ M × M | there exists an L -curve γ : [ a, b ] → M such that γ ( a ) = p , γ ( b ) = q , g ( ˙ γ, ˙ γ ) < g ( ˙ γ, X ) < } , ˇ I + := { ( p, q ) ∈ M × M | there exists a C -curve γ : [ a, b ] → M such that γ ( a ) = p , γ ( b ) = q and g ( ˙ γ, ˙ γ ) < g ( ˙ γ, X ) < } . Recall that an L -curve is differentiable almost everywhere by Rademacher’s theorem. In the secondcase, when γ is C , the condition g ( ˙ γ, ˙ γ ) < I + is open,but not necessarily I + [11]. The standard g -causal relation is defined as J + := { ( p, q ) ∈ M × M | there exists an L -curve γ : [ a, b ] → M such that γ ( a ) = p , γ ( b ) = q and g ( ˙ γ, ˙ γ ) ≤ g ( ˙ γ, X ) < } , where we say that γ is a g -causal curve . We can analogously define the past relations I − , ˇ I − and J − by requiring g ( ˙ γ, X ) >
0, but since they are simply given by reversing the factors, there is noneed to treat them separately. Therefore we also shall not specify every time that our timelike andcausal curves are always future-directed. We do, however, sometimes use the notations q ∈ J + ( p ),and p ∈ J − ( q ), both meaning the same, namely ( p, q ) ∈ J + .We finish this subsection with a short digression about limit curve theorems. In the literature,there exist multiple statements with this name; a detailed review can be found in [16] for smoothspacetimes. In [6, Thm. 1.6] it is shown how the limit curve theorems form the smooth case alsocarry over to continuous spacetimes (see also [20, Thm. 1.5]). Rougly speaking, a limit curvetheorem is the combination of the following two statements:1. Under certain assumptions, a sequence of causal curves has a convergent (in some appropriatesense) subsequence.2. The limit of said subsequence is itself a causal curve.Here causal usually means g -causal, but we will also discuss alternative notions of causal curve.Regarding part 1, there exist many versions tailored to different applications. A common variationis to require the curves to be Lipschitz (as we did in our definition of J + ) and add some compactnessassumptions in order to apply the Arzel`a–Ascoli theorem. Part 2 is where the causal structurebecomes important; we discuss it in the context of our new k + -relation in Remark 3.4. k + We first introduce some nomenclature, the underlying concepts being fairly standard. By (
M, g )we continue to denote a C -spacetime, although some of the ideas make sense even if M is just a3et. The following definition gives a compatibility condition between the chronological and causalrelations, in this case denoted abstractly by R and S respectively. Definition 2.1.
Let
R, S ⊆ M × M be two relations. We say that S satisfies push-up relative to R if the following two properties hold:(i) ( x, y ) ∈ S, ( y, z ) ∈ R = ⇒ ( x, z ) ∈ R ,(ii) ( x, y ) ∈ R, ( y, z ) ∈ S = ⇒ ( x, z ) ∈ R .Let R, S, S ′ ⊆ M × M be relations. Then it is easy to see that:(a) If R is transitive, then R satisfies push-up relative to itself.(b) If S satisfies push-up relative to R , and S ′ ⊆ S , then also S ′ satisfies push-up relative to R .(c) If S and S ′ each satisfy push-up relative to R , then so does S ∪ S ′ .If ( M, g ) is C , then J + satisfies push-up relative to I + (and equivalently, ˇ I + ). This fact is knownas the push-up lemma [5, Lem. 2.4.14]. Those C -spacetimes where J + satisfies push-up relative ˇ I + are called causally plain . The term was coined by Chru´sciel and Grant [6, Def. 1.16], but bewarethat they used I + in place of ˇ I + . In any case, not all C -spacetimes are causally plain [6, Ex.1.11]. The failure of the push-up lemma on arbitraty C -spacetimes motivates our next, centraldefinition. Definition 2.2.
The k + -relation is the largest relation that satisfies push-up relative to ˇ I + . Proposition 2.3.
There exists a unique relation k + satisfying Definition 2.2. Moreover, k + istransitive and reflexive.Proof. Consider the set
S ⊆ P ( M × M ) of all relations satisfying push-up relative to ˇ I + . Weconstruct the maximal such relation by k + := [ S ∈S S, proving existence. The diagonal relation ∆ := { ( p, p ) | p ∈ M } is contained in S , since in fact itsatisfies push-up with respect to any relation. Hence ∆ ⊆ k + , meaning that k + is reflexive. Toshow transitivity, let S ′ := { ( x, y ) ∈ M | ∃ N ∈ N , ∃ p i ∈ M, i = 1 , ..., N such that p = x, p N = y and ( p i , p i +1 ) ∈ k + for all i = 1 , ..., N − } . Then k + ⊆ S ′ . It can be shown inductively that S ′ satisfies push-up relative to ˇ I + , hence also S ′ ⊆ k + . Therefore k + = S ′ , and S ′ is clearly transitive.The next lemma gives another important property of k + . Lemma 2.4.
It holds that k + ⊆ ˇ I + .Proof. Suppose ( p, q ) ∈ k + . Let γ : [0 , → M be any timelike curve with γ (0) = q . Then for all t ∈ (0 , q, γ ( t )) ∈ ˇ I + . Because ( p, q ) ∈ k + , the push-up property implies ( p, γ ( t )) ∈ ˇ I + . Since γ is continuous, ( p, γ ( t )) → ( p, q ) as t →
0, hence ( p, q ) ∈ ˇ I + .We end this subsection by showing how the k + -relation fits into the current literature. A C -spacetime ( M, g ) is said to be weakly distinguishing whenever, for all p, q ∈ M , ˇ I + ( p ) = ˇ I + ( q ) andˇ I − ( p ) = ˇ I − ( q ) together imply p = q [17, Def. 4.47]. Given two relations R, S on (
M, g ) (or anyset, for that matter), we say that the pair (
R, S ) is a causal structure in the sense of Kronheimerand Penrose [13, Def. 1.2] if:(i) S is transitive, reflexive and antisymmetric,(ii) R is contained in S and irreflexive, 4iii) S satisfies push-up relative to R . Proposition 2.5.
Let ( M, g ) be a C -spacetime. Then ( ˇ I + , k + ) is a causal structure in the senseof Kronheimer and Penrose if and only if ( M, g ) is weakly distinguishing.Proof. Point (iii) is satisfied by the very definition of k + . To see point (ii), recall that if ( M, g ) isweakly distinguishing, then (
M, g ) is chronological, meaning precisely that ˇ I + is irreflexive. Thatˇ I + is contained in k + is clear because ˇ I + , being transitive, must satisfy push-up with respect toitself (see property (b) right after Definition 2.1). Since k + is always transitive and reflexive byLemma 2.3, it only remains to show that k + is antisymmetric.Note that ( p, q ) ∈ k + if and only if ˇ I + ( q ) ⊆ ˇ I + ( p ) and ˇ I − ( p ) ⊆ ˇ I − ( q ). Hence, ( p, q ) ∈ k + and( q, p ) ∈ k + if and only if ˇ I + ( p ) = ˇ I + ( q ) and ˇ I − ( p ) = ˇ I − ( q ). Thus p = q for all such pairs ( p, q ) ifand only if ( M, g ) is weakly distinguishing.Another way of phrasing the last result in the usual language of causality theory is to say that“(
M, g ) is k + -causal if and only if it is weakly distinguishing”. k + Given a neighborhood U ⊆ M , we can define the localized relations ˇ I + U , J + U and k + U by applyingthe usual definitions to the spacetime ( U, g | U ). It is easy to see that ˇ I + U ⊆ ˇ I + and J + U ⊆ J + . Lemma 2.6.
Let U ⊆ M be an open neighborhood. Then k + U ⊆ k + .Proof. By definition, k + U satisfies push-up relative to ˇ I + U . We need to show that k + U also satisfiespush-up relative to ˇ I + , and then the claim follows from maximality of k + . Suppose that ( x, y ) ∈ k + U and ( y, z ) ∈ ˇ I + . Then there exists a timelike curve γ : [0 , → M from y to z . Since, by assumption, y ∈ U , we must have that for ǫ > γ | [0 ,ǫ ] ∈ U . Thus we have ( y, γ ( ǫ )) ∈ ˇ I + U , whichimplies ( x, γ ( ǫ )) ∈ ˇ I + U ⊆ ˇ I + by definition of k + U . Since also ( γ ( ǫ ) , z ) ∈ ˇ I + , transitivity of ˇ I + implies( x, z ) ∈ ˇ I + . Part (ii) of Definition 2.1 can be shown analogously.We want to investigate whether, for U small enough, k + U is closed. The motivation lies inthe limit curve theorems (see Remark 3.4 for the details). By Lemma 2.4, k + U ⊆ ˇ I + U . Since alsoˇ I + U ⊆ k + U , we conclude that k + U is closed if and only if k + U = ˇ I + U . By Definition 2.2, k + U = ˇ I + U if andonly if ˇ I + U satisfies push-up. Unfortunately, the next example [11, Ex. 3.1], shows that the latteris not necessarily the case. Example 2.7.
Let M = R with metric given by g α := − sin 2 θ ( x ) dt − θ ( x ) dxdt + sin 2 θ ( x ) dx where θ ( x ) := , x < − | x | α , − ≤ x ≤ π , x > , and 0 < α < g α is α -H¨older continuous, and in fact smooth outside of { x = − } ∪ { x = 0 } . This example was introduced by Grant et al. [11, Ex. 3.1], who showed thatˇ I + ( I + .Let p = (0 ,
0) and U ⊆ M any open neighborhood of p . Then the following hold:(i) I + U is not open,(ii) k + U is not closed.Point (i) is shown in [11, Ex. 3.1] (they in fact show that I + is not open, but their argument isalso valid on neighborhoods).In order to show point (ii), first note that the past ˇ I − ( p ) (blue region in Figure 1) is containedin { x > } . This is so because a timelike C -curve must have timelike tangent vector everywhere,5 γ ( − ǫ ′ ) q ˇ I − ( p ) γ xt Figure 1: The spacetime of Example 2.7, with the curve γ that lies outside of ˇ I − ( p ), while nonethe-less ( γ ( − ǫ ′ ) , p ) ∈ ˇ I + .which implies having a positive x -component when in { x ≥ } . When using L -curves, the past set I − ( p ) is in fact bigger [11, Ex. 3.1], but we will not discuss this further.Consider the curve γ : ( − ǫ, → M, s ( t ( s ) , x ( s )) given by t ( s ) := 11 − α A − α s, x ( s ) := − A | s | − α , where A > ǫ > γ is a timelike curvein the C -sense (but its extension to the endpoint s = 0 is not). Since γ ( s ) → p as s →
0, weconclude that ( γ ( − ǫ ′ ) , p ) ∈ ˇ I + for all ǫ ′ < ǫ .Next we show that ( γ ( − ǫ ′ ) , p ) k + . To see this, consider any point of the form q = ( x ( − ǫ ′ ) , t q )with t q < t ( − ǫ ′ ) (see Figure 1). Then ( q, γ ( − ǫ ′ )) ∈ ˇ I + , the connecting vertical segment beingan example of timelike C -curve between q and γ ( − ǫ ′ ). If we assume ( γ ( − ǫ ′ ) , p ) ∈ k + , then bypush-up it follows that ( q, p ) ∈ ˇ I + . However, ( q, p ) ˇ I + because x ( − ǫ ′ ) < I − ( p ) is containedin { x > } . Hence ( γ ( − ǫ ′ ) , p ) k + .Since we can pick ǫ ′ > t q smaller but arbitrarily close to t ( − ǫ ′ ), theprevious discussion applies to any neighborhood U of p . Thus k + U ( ˇ I + U , no matter how we choose U . Grant et al. showed that in the previous example also ˇ I + ( I + , and that I + is not open (recallthat ˇ I + is always open). If we were to define k + by requiring push-up with respect to I + insteadof ˇ I + , it may be that k + U is closed (for small enough U ). We do not explore this possibility here,and simply note that this would be at the cost of chronological futures not being open. Hence theconclusion is, either way, that one cannot have push-up, open futures and (local) closedness at thesame time. That is, at least, if one wants the chronological relation to be given by the usual I + orˇ I + . We explore alternatives to I + and ˇ I + in Section 4, but the conclusion there is also that oneof the three properties has to be sacrificed. k + -relation Throughout this section, F denotes an interval, meaning any connected subset of R . Definition 3.1.
A continuous curve γ : F → M is called k + -causal if for every t ∈ F and everyopen neighborhood U ⊆ M of γ ( t ), there exists an open neighborhood V ⊆ F of t such that s < s = ⇒ ( γ ( s ) , γ ( s )) ∈ k + U for all s , s ∈ V. emark 3.2. Similarly to Definition 3.1, one can define J + -causal curves, as was done already byHawking and Ellis in 1973 [12, Chap. 6.2], and K + -causal curves [22, Def. 17]. Any g -causal curveis automatically also J + -causal. However, the converse is not true, since a J + -causal curve maynot even have a well defined tangent vector. Nonetheless, if two points p, q ∈ M can be joined by a J + -causal curve γ , then ( p, q ) ∈ J + . In particular, there exists a g -causal curve σ , not necessarilyequal to γ , which joins p and q .Similarly to the previous remark, by transitivity and Lemma 2.6 it follows that if two points p, q can be joined by a k + -causal curve, then ( p, q ) ∈ k + . The next example motivates whyDefinition 3.1 has to be formulated in a local way, i.e. why we do not simply require s < s = ⇒ ( γ ( s ) , γ ( s )) ∈ k + for all s , s ∈ F . Example 3.3.
Let M = S × R with metric ds = − dt + dx . This spacetime is totally vicious,so ˇ I + = M × M and hence also k + = M × M . Therefore, any C -curve γ : F → M satisfies( γ ( s ) , γ ( s )) ∈ k + for all s , s ∈ F . However, M locally looks like Minkowski spacetime, where k + = J + , hence not all curves on M are k + -causal in the sense of Definition 3.1.Having defined k + -causal curves, we briefly return to Example 2.7 in order to better understandthe relationship between closedness of k + and limit curve theorems. Remark 3.4 (On limit curve theorems) . Suppose that ( γ n ) n is a sequence of k + -causal curvesconverging pointwise to a C -curve γ ∞ : F → M . Suppose that for every t ∈ F , there exists aneighborhood U ⊆ M of γ ∞ ( t ) such that k + U is closed. Then the curve γ ∞ is k + -causal, becauseany pair of points on γ ∞ can be written as a limit of pairs of points on γ n .In Example 2.7, we showed that the point p = (0 ,
0) does not admit any neighborhood U such that k + U is closed. Let γ be as in Example 2.7, and consider the sequence of curves givenby γ n = γ | ( − ǫ, /n ] . The sequence ( γ n ) n converges pointwise (even uniformly, after appropriatereparametrization) to a curve γ ∞ : ( − ǫ, → M which is just γ with p added as its endpoint.However, we showed in Example 2.7 that ( γ ( t ) , p ) k + for all − ǫ < t <
0, hence γ ∞ is not k + -causal.Moving on, we use k + -causal curves to define a new causal relation on M . Definition 3.5.
We define the ˜ k + -relation as follows: ( p, q ) ∈ ˜ k + if there exists a k + -causal curvefrom p to q .By Lemma 2.6 and transitivity of k + , ˜ k + ⊆ k + . In particular, ˜ k + satisfies push-up relative toˇ I + . It is also clear that the concatenation of two k + -causal curves is again k + -causal, hence ˜ k + istransitive. Example 3.3 shows that ˜ k + can be strictly smaller than k + . We finish this section withone of the main results of the paper, namely that ˜ k + = J + on smooth (and more generally, causallyplain) spacetimes. Recall that a C -spacetime is called causally plain if J + satisfies push-up relativeto ˇ I + . Lemma 3.6.
Let ( M, g ) be a C -spacetime, and γ : F → M a k + -causal curve. Then γ is also J + -causal.Proof. Let t ∈ F be arbitrary. By [6, Proposition 1.10], there exists a neighborhood U of p := γ ( t ),called a cylindrical neighborhood, such that ˇ I ± U ( p ) ⊆ J + U ( p ) (here we mean the closure of the setˇ I ± U ( p ) ⊆ U ). Let V ∈ F be a neighborhood of t as in Definition 3.1, and s ∈ V . Suppose s ≤ t , theother case being analogous. Because γ is k + -causal, we have ( p, γ ( s )) ∈ k + U . Let σ : [0 , ǫ ) → U beany timelike C -curve with σ (0) = γ ( s ). By push-up, Im( σ ) ⊆ ˇ I + ( p ). By continuity, σ ( u ) → p as u →
0. Hence, by the previous and our choice of U , γ ( s ) ∈ ˇ I + U ( p ) ⊆ J + U ( p ). In other words,( γ ( t ) , γ ( s )) ∈ J + U . Since t, s are arbitrary (as long as they are close enough), we conclude that γ is J + -causal. Theorem 3.7.
Let ( M, g ) be a causally plain C -spacetime. Then ˜ k + = J + .Proof. If (
M, g ) is causally plain, then J + satisfies push-up, hence J + ⊆ k + . In particular, ona subset U ⊂ M , we have J + U ⊆ k + U . Assume ( p, q ) ∈ J + . Then there exists a g -causal curve γ : [ a, b ] → M from p to q . By continuity, for every t ∈ [ a, b ] and every neighborhood U of γ ( t ),7here exists a neighborhood V ⊆ [ a, b ] of t small enough such that γ | V is contained in U . If s , s ∈ V and s < s , then ( γ ( s ) , γ ( s )) ∈ J + U ⊆ k + U . Thus γ is a k + -causal curve, and since γ is arbitrary, we conculde that J + ⊆ ˜ k + . The other inclusion follows from Lemma 3.6, by notingthat if two points p, q can be joined by a J + -causal curve, then ( p, q ) ∈ J + . It is possible to repeat the procedure of Section 3 for Sorkin and Woolgar’s K + , and define a relation˜ K + based on K + -causal curves (the latter class of curves is also studied in [22, Section 3]). Ona smooth spacetime, every point admits an arbitrarily small neighborhood U (a convex normalneighborhood) such that J + U = ˇ I + U . Thus we conclude that on smooth spacetimes, ˜ K + = J + .Unfortunately, Example 2.7 and Remark 3.4 tell us that ˜ K + cannot satisfy push-up with respectto ˇ I + on all C -spacetimes. That is because, if it did, then ˜ K + ⊆ k + . Recall that K + is closedand contains ˇ I + . It follows that if a curve γ is the limit of a sequence of C -timelike curves, then γ must be k + -causal. However, in Remark 3.4, we saw an example of such a γ where the endpointsare not k + -related to each other, a contradiction.A different approach is to consider J + to be the more fundamental relation, and then find anappropriate notion of chronological order, say ˆ I + . Ideally, we would like all of the following threeproperties to hold.(i) ˆ I + is open and contained in J + .(ii) J + satisfies push-up relative to ˆ I + .(iii) For every point p ∈ M and every neighborhood U of p , ˆ I ± ( p ) T U = ∅ .This leaves us with only one choice, namely ˆ I + = Int J + . To see this is the only option, note firstthat any relation not contained in Int J + would either not be open or not be contained in J + . Butif we choose ˆ I + strictly smaller than Int J + , then J + cannot satisfy push-up relative to it: For if( p, q ) ∈ Int J + , then we can find a point r ∈ ˆ I − ( q ) close enough to q such that ( p, r ) ∈ J + . Hence,push-up can only be satisfied if ( p, q ) ∈ ˆ I + . This being said, the following example shows that J + does not always satisfy push-up relative to Int J + . Example 4.1.
This example is adapted from [6, Ex. 1.11] and [15, Sec. 4.1]. Let M = ( − , × R with metric given by ds = − dt − − | t | / ) dtdx + | t | / (2 − | t | / ) dx . This metric is smooth everywhere except on the x -axis. A null curve starting at the point p =( − ,
0) can be parametrized as x γ ( x ) = ( t ( x ) , x ), and then˙ t = ( | t | / or | t | / − . By solving this equation, we obtain the boundary of J + ( p ) (light blue region in Figure 2). Considerthe first case of the equation, which is when the null curve γ moves upwards and to the right. Weare interested in finding the value x such that t ( x ) → x → x . It can easily be computed byseparation of variables: x = Z x dx = Z − dt | t | / = 2 . Let q = (0 , p, q ) ∈ Int J + . Consider a third point r = (0 , q, r ) ∈ J + , because the curve x (0 , x ) is null. However, ( p, r ) Int J + , since there are points ofthe form ( − ǫ,
3) arbitrarily close to r which cannot lie in J + ( p ) because they lie below the x -axisand their x -coordinate is larger than 2. Hence J + does not satisfy push-up relative to Int J + inthis example. 8 p rJ + ( q ) J + ( p ) xt Figure 2: The points p, q, r in Example 4.1, which satisfy ( p, q ) ∈ Int J + and ( q, r ) ∈ J + but( p, r ) Int J + .Finally, we point out yet another option, which is to define chronological futures via theLorentzian distance. Recall that in the smooth case, the Lorentzian distance d ( p, q ), given bymaximizing the length of g -causal curves between p and q , satisfies d ( p, q ) > ⇐⇒ ( p, q ) ∈ ˇ I + (see [2, Chap. 4] for details). On C -spacetimes, only the “ ⇐ = ” implication continues to hold. Tosee this, take the points p, q, r in Example 4.1 (depicted in Figure 2). We can connect p and q by avertical segment, which is timelike (hence causal) and has length equal to 1. We can connect q and r by a horizontal segment, which is also causal, and has length 0. Concatenating the two segments,we get a causal curve of length 1 from p to r , despite the fact that ( p, r ) ˇ I + . Further, we see that r ∈ ∂J + ( p ), and since { d > } ⊆ J + by definition, we conclude that { d > } is not open in thisexample. Nonetheless, another property of the Lorentzian distance, the inverse triangle inequality d ( p, r ) ≥ d ( p, q ) + d ( q, r ) if ( p, r ) , ( r, q ) ∈ J + , does hold for all C -metrics. We deduce from it that J + satisfies push-up relative to { d > } .Hence the combination of J + and { d > } gives us push-up and limit curve theorems, but at theprice of non-open futures. Table 1 summarizes the properties of different choices of causal and chronological relation on C -spacetimes. While each of the choices (rows) is distinct from the others for C -metrics, they allcoincide for smooth metrics. In particular, in the smooth case, the standard causal structure ticksall three boxes. For C -metrics, on the other hand, no combination of chronological and causalorder has all of the three properties that we considered. The newly introduced ( ˇ I + , ˜ k + ) is the onlycausal structure that has both push-up and an open chronological relation. Moreover, it defines acausal structure in the sense of Kronheimer and Penrose (see Proposition 2.5).It is fair to say that we have exhausted all reasonable possibilities. For if we want the chrono-logical relation to be given by timelike C -curves, then Example 2.7 and Remark 3.4 tell us thatno causal relation can satisfy push-up and at the same time admit a limit curve theorem. We donot know if this changes when using timelike L -curves, but even if so, we would loose the opennessof chronological futures instead [11]. If, on the other hand, we choose for the causal relation tobe given by g -causal curves, and try to define a compatible chronological relation, we run into thesame problems by the discussion in Section 4. 9able 1: Comparison of different causal structures on C -spacetimes by three important properties.Chronological Causal Push-up Open Limitorder order futures curvesˇ I + J + ✗ ✓ ✓ ˇ I + ˜ K + ✗ ✓ ✓ ˇ I + ˜ k + ✓ ✓ ✗ I + J + ✗ ✗ ✓ I + ˜ K + ? ✗ ✓ I + ˜ k + ✓ ✗ ? { d > } J + ✓ ✗ ✓ Int J + J + ✗ ✓ ✓ References [1] L. Ak´e Hau, A. J. Cabrera Pacheco, and D. A. Solis. On the causal hierarchy of Lorentzianlength spaces.
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