Causality and skies: is non-refocussing necessary?
CCAUSALITY AND SKIES: IS NON-REFOCUSSING NECESSARY?
A. BAUTISTA, A. IBORT AND J. LAFUENTE
Abstract.
It is shown that if M is a strongly causal free of naked singularities space-time,then its causal structure is completely characterized by a partial order in the space of skiesdefined by means of a class non-negative Legendrian isotopies. It is also proved that suchpartial order is determined by the class of future causal celestial curves, that is, curves inthe space of light rays which are tangent to skies and such that they determine non-negativesky-Legendrian isotopies.It will also be proved that the space of skies Σ equipped with Low’s (or reconstructive)topology is homeomorphic and diffeomorphic to M under the only additional assumption that M separates skies, that is, that different points determine different skies. The sky-separatingproperty of M being weaker than the “non-refocussing” property encountered in the previousliterature is sharp and the previous result provides the answer to the question of what isthe class of space-times whose causal structure, topology and differentiable structure can bereconstructed from their spaces of light rays and skies.Finally, the previous results allow a formulation of Malament-Hawking theorem in termsof the partial order defined on the space of skies. Contents
1. Introduction 12. The space of light rays and the space of skies 32.1. The space of light rays 32.2. The smooth structure of N T N and the contact structure of N
43. Reconstruction of the causal structure 53.1. The space of skies and its topology 53.2. The partial order in the space of skies 53.3. Celestial curves and twisted null curves 63.4. Celestial curves and the partial order in the space of skies 94. The smooth structure of the space of skies and the non-refocussing property 124.1. Regular sets 124.2. The topology of the space of skies and regular sets 135. Conclusions and discussion 16References 161.
Introduction
In a recent paper by Bautista et al [Ba14] it was shown, following a suggestion by Low[Lo01, Lo06], that if M is a strongly causal, free of naked singularities, non-refocussing space-time M , then the sky map is a diffeomorphism, that is, the topology and differentiable structure This work has been partially supported by the Spanish MICIN grant MTM 2010-21186-C02-02 andQUITEMAD P2009 ESP-1594. a r X i v : . [ g r- q c ] N ov A. BAUTISTA, A. IBORT AND J. LAFUENTE of M can be recovered from the natural topology and differentiable structure on its space of skiesΣ and light rays N . Moreover if M and M are two strongly causal space-times (where M is nullnon-conjugate), and φ : N → N is a diffeomorphism mapping causal celestial curves into causalcelestial curves then there exists a conformal immersion Φ : M → M such that φ ( γ ) = Φ ◦ γ for any light ray γ ∈ N and conversely ([Ba14, Thm. 4]). Let us recall that a celestial curveΓ is a differentiable curve in the space of light rays which is tangent everywhere to a sky andit is called past (future) causal if it defines a non-negative (non-positive) Legendrian isotopyof skies. The class of causal celestial curves emerges thus as the relevant geometrical structureon N characterizing the original conformal class of the Lorentzian metric on M . Moreover theprevious conditions cannot be weakened as the examples discussed in [Ba14] show.Let us recall that strongly causal space-times are natural candidates for a reconstructiontheorem because they constitute a class of space-times whose spaces of unparametrized nullgeodesics are smooth manifolds (see for instance [Lo89, Prop. 2.1. and ff.]). We will assume inwhat follows that the space-time M is strongly causal, hence time-orientable, and free of nakedsingularities (the later condition guaranteeing that its space of light rays is a Hausdorff space).It is important to recall that the space of light rays carries a canonical contact structure thatprovides an additional piece of geometry relevant in the analysis that follows.Each event x ∈ M determines the congruence of light rays S ( x ) passing through it and calledthe sky of x . There is a natural map from M into the set of skies Σ called the sky map S : M → Σ, x (cid:55)→ S ( x ). The reconstruction theorem for the topological (respect., differentiable) structure of M will consists in determining under what conditions the sky map S is a homeomorphism(respect., a diffeomorphism), that is, under what conditions the space of skies Σ inherits anatural topology (respect., a smooth manifold structure) from N such that the sky map is ahomeomorphism (a diffeomorphism).Simple examples, like the Einstein cylinder R × S m − equipped with the standard productmetric g = − dt ⊕ h , where h is the induced Euclidean metric on S m − , show that even globallyhyperbolic spaces could have many-to-one sky maps. Thus a natural condition that has to beimposed on M is that the sky map is injective or, in other words, that skies separate events, i.e.,given two different events x (cid:54) = y their skies are different S ( x ) (cid:54) = S ( y ). Thus for a strongly causalspace-time with the property that skies separate events, the sky map S is invertible with inverse P (called the ‘parachute map’). Now equipping Σ with the natural topology induced from N itis easy to show that S is continuous.In order to guarantee that the sky map is open Low introduced in [Lo93], [Lo01] and [Lo06]an apparently weaker property called non-refocusing : a space-time M is refocusing at x ∈ M if there exists an open neighbourhood U of x such that for every open neighbourhood V ⊂ U of x there exists y / ∈ U such that every null geodesic through y enters V . This property hasbeen studied in depth in [Ki11] and plays an important role in the proofs given in [Ba14, Prop.3] and [Ki11, Prop. 4.1], that the sky map S : M → Σ is a homeomorphism. We claim thatthe hypothesis of non-refocusing is not necessary if the space-time M is sky-separating, i.e., anyspace-time with skies separating events is homeomorphic and diffeomorphic to its space of skies.This conclusion will be reached as a consequence of Thm. 4.4 at the end of Section 4.Moreover, and this will constitute the first objective of this project, it is expected that notonly the topological structure, but the causal structure of M could be characterized in terms ofthe topological structure of N and Σ too. Again, it was conjectured by R. Low that two events ina space-time are causally related iff their corresponding skies, which are Legendrian knots withrespect to the canonical contact structure in the space of null geodesics, are linked. Recentlyit was shown by Chernov and Rudyak [Ch08] and Chernov and Nemirovski [Ch10] that Low’sconjecture is actually true in a globally hyperbolic space with a Cauchy surface whose universalcovering is diffeomorphic to an open domain in R n . A fundamental role in such analysis is AUSALITY AND SKIES 3 played by the partial relation defined by non-negative Legendrian isotopies. Recently Chernovand Nemirovski [Ch14] had extended the previous ideas to show that the causal structure ofa simply connected globally hyperbolic space-time M can be reconstructed from the partialordering in the universal covering of Legendrian isotopy class of the fibres of the sphere bundleof a smooth Cauchy surface.In Sect. 3.4 it will be shown that the causal structure of M can be recovered from a partialordering introduced in the space of skies by a restricted class of non-negative Legendrian isotopiescalled sky isotopies. Without entering in the analysis of Low’s Legendrian conjecture here, itwill be shown that the analysis of the causal structure of M in terms of Σ is deeply related tothe study of celestial curves. It will be shown that celestial curves are in correspondence with aclass of null curves that will be called twisted null curves. The causal structure of the originalspace-time will be characterized completely at the end of Sect. 3 in terms of the partial orderrelation induced in the space of skies by future (past) causal twisted null curves.Finally, the proof that the space of skies of a sky-separating space-time is homeomorphic(and diffeomorphic) to the original space proceeds by constructing a basis for the reconstructive(or Low’s) topology by means of regular open subsets of Σ, where ‘regular’ here means thatthe corresponding tangent spaces to the skies elements of the open set ‘pile up’ nicely defininga regular submanifold in the tangent space to N . The definition and discussion of the mainproperties of regular sets constitutes the core of Section 4, where again the properties of twistedcausal null curves will be used in a critical way.Thus we offer an answer to the question of characterizing a large class of space-times M such that the pair ( N , Σ) is capable of reconstructing the causal, topological and differentiablestructures of M . However the question of what is the largest class of space-times such that twodifferent skies which are related by a future causal celestial curve are topologically linked asstated in Low’s conjecture is still open.2. The space of light rays and the space of skies
The space of light rays.
Let M be a second countable paracompact m -dimensionalsmooth manifold and C a conformal class of Lorentzian metrics of signature ( − + · · · +) suchthat M becomes a time-orientable strongly causal space-time. We will denote by g a represen-tative metric on C and a time-like vector field T determining a time-orientation on M will befixed in what follows.Let N denote the space of unparametrized inextensible future-oriented null geodesics, calledin what follows light rays, i.e., N is the space of equivalence classes of inextensible smooth nullcurves γ : I → M , with I an interval in R , such that ∇ γ (cid:48) γ (cid:48) = 0, g ( γ (cid:48) , T ) <
0, and two such curvesare equivalent if they are related by an affine reparametrization for the chosen representative g of the conformal class C .We will consider in what follows the fibre bundle N over M consisting of nonzero null vectors,and the corresponding components of future (past) null vectors N ± . If we denote N + x = { v ∈ N x | v (cid:54) = 0 , g x ( v, T ( x )) < } and N − x = { v ∈ N x | v (cid:54) = 0 , g x ( v, T ( x )) > } , we have N ± = (cid:83) x ∈ M N ± x and N = N + ∪ N − . We will denote by π : N → M the restriction of the canonical tangent bundleprojection T M → M to N (and N ± ).We will denote again the canonical projection π : PN + → M , where PN + denotes the quotientspace of N + by the action of the multiplicative group of positive real numbers R + by scalarmultiplication. Notice that there is a canonical surjection σ : PN + → N , given by σ ([ u ]) = γ [ u ] ,where γ [ u ] (or [ γ u ] as it will be used in what follows too) denotes the unparametrized geodesiccontaining γ u and γ u ( t ) indicates the unique future parametrized geodesic such that γ u (0) = π ( u ),and γ (cid:48) u (0) = u . Moreover, because γ λu ( t ) = γ u ( λt ), u ∈ N + and the previous notation isconsistent. A. BAUTISTA, A. IBORT AND J. LAFUENTE
The smooth structure of N . The space of light rays N can be equipped with the struc-ture of a second countable paracompact smooth manifold of dimension 2 m −
3, if dim M = m , andsuch that the map σ becomes a submersion, in two different ways. We will succinctly describethem in the following paragraphs.First, we can use the local structure of M , i.e., because M is strongly causal, given any event x ∈ M , there exists a globally hyperbolic neighbourhood U x of x and a local smooth Cauchyhypersurface C x ∈ U x [Mi08]. We can take U x small enough such that it is contained in a localchart of M . Hence we can define an atlas for N as follows, select for any event x ∈ M a globallyhyperbolic open neighbourhood U x as before with Cauchy hypersurface C x . We consider therestriction of the projective bundle PN + to C x and we denote it by PN + ( C x ). There is a naturalembedding i x : PN + ( C x ) → PN + . Then the composition σ ◦ i x : PN + ( C x ) → N will provide thecharts of the atlas we are looking for and the open sets U x = σ ◦ i x ( PN + ( C x )) ⊂ N will be thedomains of the corresponding charts (see [Ba14, Sect. 2.3] for more details).Alternatively, we can induce a smooth structure on N from the smooth structure of the bundle N + by considering the foliation defined by the leaves of the integrable distribution generated bythe vector fields X g and ∆, where X g denotes the geodesic spray of a fixed representative metric inthe conformal class C and ∆ is the dilation or Euler field. Because [ X g , ∆] = X g , the distribution D = span { ∆ , X g } is integrable and denoting by D the corresponding foliation, we have that thespace of leaves N + / D ∼ = N . If M is strongly causal it can be shown that D is a regular foliationand the space of leaves inherits a smooth structure from N + . Again, it is not hard to show thatboth smooth structures coincide.2.3. The tangent bundle T N and the contact structure of N . Let Γ : ( − (cid:15), (cid:15) ) → N be adifferentiable curve such that Γ(0) = γ and let χ ( s, t ) : ( − (cid:15), (cid:15) ) × I → M be a geodesic variationby null geodesics of a parametrization γ ( t ) of the null geodesic γ , that is, χ is a smooth functionsuch that χ ( s, t ) = γ s ( t ) are null geodesics, γ ( t ) is a parametrization of γ , and [ γ s ] = Γ( s ) where[ γ s ] denotes the unparametrized geodesic containing γ s . Then the vector field along γ definedby J = ∂χ/∂s | s =0 is a Jacobi field. The set of Jacobi fields along γ ( t ) will be denoted by J ( γ )and they satisfy the second order differential equation: J (cid:48)(cid:48) = R ( γ (cid:48) , J ) γ (cid:48) , where J (cid:48) denotes the covariant derivative of J along γ (cid:48) ( t ). Notice that since the geodesic variation χ is by null geodesics, we have (cid:104) J, γ (cid:48) (cid:105) = constant and we denote by L ( γ ) the linear space of Jacobifields satisfying this property.Equivalence classes of curves Γ( s ) possessing a first order contact define tangent vectors to N at γ , hence tangent vectors at γ correspond to equivalence classes of Jacobi fields with re-spect to the equivalence relation defined by reparametrization of the geodesic variation χ . Suchreparametrizations will correspond to Jacobi fields of the form ( at + b ) γ (cid:48) ( t ), then there is a canon-ical projection L ( γ ) → T γ N , mapping each Jacobi field J into a tangent vector [ J ] = J mod γ (cid:48) whose kernel is given by Jacobi fields proportional to γ (cid:48) . In what follows the tangent vectors [ J ]will be denoted again as J unless there is risk of confusion.There is a canonical contact structure on N defined by the maximally non-integrable hyper-plane distribution H γ ⊂ T γ N formed by the vectors orthogonal to their supporting light ray,i.e.,(2.1) H γ = { J ∈ T γ N | (cid:104)
J, γ (cid:48) (cid:105) = 0 } . It is easy to show that H does not depend on the representative metric used to define, therepresentative J chosen for the tangent vector, or the parametrization γ ( t ) we chose for the lightray γ . AUSALITY AND SKIES 5
Let us recall that if X is a contact manifold with contact distribution a maximally non-integrable codimension one distribution H , the contact structure is said to be exact or co-orientable if there exists a globally defined 1-form α , such that H = ker α and such 1-formis called a contact 1-form for the contact structure H .It is obvious that the canonical contact structure H on N , Eq. (2.1), can be locally definedby the family of 1-forms α x defined on the open sets U x of the atlas described in Sect. 2.2 aboveand given by the explicit formula: α xγ : J (cid:55)→ (cid:104) J, γ (cid:48) (cid:105) , where the parametrization γ ( t ) of the light ray γ is determined by the Cauchy surface C x ⊂ U x and (cid:104) T ( x ) , γ (cid:48) (0) (cid:105) = −
1. The local 1-forms α x do not define a global 1-form, however because N is paracompact we can use a partition of the unity subordinated to a locally finite refinementof the open covering {U x } of N defined by family of globally hyperbolic open neighbourhoods { U x | x ∈ M } , and paste the local 1-forms to define a globally defined 1-form whose kernel is H .Notice however the space of not oriented unparametrized null geodesics still carries a canonicalcontact structure (defined by the same formula above, Eq. (2.1)s ) which is not co-oriented.3. Reconstruction of the causal structure
The space of skies and its topology.
As it was explained in the introduction, the skyof an event is the congruence of light rays passing through it. Thus if x ∈ M denotes an event,the corresponding sky will be denoted either by S ( x ) or X . Then S ( x ) = { γ ∈ N | x ∈ γ } .Notice that there is a canonical map σ x : PN + x → S ( x ), σ x ([ u ]) = γ [ u ] . Clearly the sky S ( x ) as asubmanifold of N is diffeomorphic to the sphere of dimension m −
2. The family of all skies willbe denoted by Σ, that is, Σ = { X = S ( x ) | x ∈ M } , and the canonical map S : M → Σ, x (cid:55)→ S ( x ), is called the sky map. The sky map is clearlysurjective, however it doesn’t have to be injective as indicated in the introduction. Hence wewill say that M separates skies if S is injective, that is, if x (cid:54) = y , then S ( x ) (cid:54) = S ( y ). If M separates skies, the map P : Σ → M , inverse to the sky map, is well defined and will be calledthe parachute map.The space of skies Σ carries a canonical topology called the reconstructive topology defined asfollows. Let U ⊂ N be an open set, then consider the set of all skies X such that X ⊂ U . We willdenote this set by Σ( U ). It is clear that the family of sets Σ( U ) satisfies Σ( U ) ∩ Σ( V ) = Σ( U ∩ V ),then they constitute a basis for a topology on Σ called the reconstructive topology.It is easy to prove that the sky map S is continuous with respect to the reconstructive topology.However it is not obvious if it is open or not. As it was discussed in the introduction it is one ofthe objectives of this paper to determine under what conditions S is open, i.e., P continuous, ornot.We will end these remarks by observing that if X = S ( x ) is a sky, then given γ ∈ X , atangent vector J to X at γ is determined by a geodesic variation such that all their geodesicspass through the point x at time 0, then J (0) = 0. This implies that (cid:104) J, γ (cid:48) (cid:105) = 0 for all J ∈ T γ X and T X ⊂ H . Thus skies are Legendrian spheres because, in addition, 2 m − H γ =2 dim T γ X = 2( m − The partial order in the space of skies.
The canonical contact structure on N allowsto define a natural partial ordering in the space of skies.Let us recall first that if X is a co-oriented contact manifold with contact distribution H = ker α where α is a contact 1-form, a differentiable family Λ s , s ∈ [0 , F : Λ × [0 , → X verifying F (Λ , s ) = Λ s ⊂ X . The map A. BAUTISTA, A. IBORT AND J. LAFUENTE F s : Λ → Λ s , given by F s ( λ ) = F ( λ, s ) is a diffeomorphism for all s ∈ [0 , s is said to be non-negative (non-positive) if ( F ∗ α )( ∂/∂s ) ≥ F ∗ α )( ∂/∂s ) ≤ F a parametrization of Λ s . It is easy to check that the previous definition does not dependon the chosen parametrization.If we consider now the class S of Legendrian spheres on the contact manifold X , we can definea partial order on S by saying that S ≺ S , S , S ∈ S , if there exists a non-negative Legendrianisotopy F : S × [0 , → X , joining S and S , i.e., such that F ( S ) = S , F ( S ) = S .We will consider the previous ideas in the contact manifold N of light rays of a given space-time M . The class S of Legendrian spheres in M contains the space of skies Σ. Then the partialorder ≺ described before induces a partial order in Σ. However we would like to restrict theprevious partial order because it could happen that two skies X = S ( x ) and X = S ( x ) wouldbe related, X ≺ X , but the non-negative Legendrian isotopy X s joining X and X will fallout of Σ, that is, not all Legendrian spheres X s will be the sky of a point x s ∈ M .Thus we will weaken the partial order ≺ by restricting the class of Legendrian isotopies to thoseconsisting of skies. Hence let F : X × [0 , → N be a Legendrian isotopy such that X s = F s ( X )is the sky of x s ∈ M , i.e., X s = S ( x s ) and it defines a differentiable curve µ : [0 , → M , givenby µ ( s ) = x s . Conversely, let x ∈ M be an event and X = S ( x ) its sky which is a Legendriansphere, then any differentiable curve µ : [0 , → M with µ (0) = x defines a Legendrian isotopyparametrized by the function F µ : X × [0 , → N given by F µ ( γ [ u ] , s ) = γ [ u s ] , and u s ∈ N + µ ( s ) isthe parallel transport of u ∈ N + x along µ . Notice that then F µ is a Legendrian isotopy of skiesand F s ( X ) = S ( µ ( s )), s ∈ [0 , ≺ Σ .On the other hand there is a natural partial order relation in M defined by the conformalclass of the Lorentzian metric. Thus given two events x, y ∈ M , we say that y is in the causalfuture of x and it will be denoted by x ≺ y , if y ∈ J + ( x ), i.e., y can be reached by a futureoriented causal curve starting at x .Now it is simple to show that the curve µ : [0 , → M is causal past (future) iff F µ is anon-negative (respect. non-positive) sky isotopy. Hence we have the following characterizationof causality in terms of definite sky isotopies ([Ba14, Prop. 4]). Proposition 3.1. x ≺ y iff X ≺ Σ Y . The previous observations and results lead naturally to the following:
Definition 3.2.
A continuous curve χ : [0 , → Σ will be causal past (future) if it defines anon-negative (respect. non-positive) Legendrian isotopy in N . Two skies X, Y ∈ Σ are said tobe past (future) causally related if there is a causal past (future) curve χ such that χ (0) = X and χ (1) = Y , and it will be denoted by X ≺ c Y ( Y ≺ c X ). As a consequence of the “Twisted Curve Theorem”, Thm. 3.9, the “ µ -Lemma”, Lemma 3.8,and Cor. 4.5 below it follows that the space-time M is diffeomorphic and order isomorphic tothe space of skies Σ equipped with the partial order ≺ c and the natural differentiable structureinduced from the space of light rays N . Corollary 3.3.
Let M be a strongly causal free of naked singularities and sky-separating space-time, then M is diffeomorphic and order isomorphic to its space of skies Σ3.3.
Celestial curves and twisted null curves.
As stated in the introduction, the recon-struction theorem in [Ba14] asserts that the conformal structure of M is captured by the classof causal celestial curves, that is by curves in N that are everywhere tangent to skies. Moreformally: AUSALITY AND SKIES 7
Definition 3.4.
A non-zero tangent vector J ∈ (cid:98) T γ N , (with (cid:98) T γ N = T γ N − { } ), will be called acelestial vector if there exists a sky S ∈ Σ such that J ∈ (cid:98) T γ S . A differentiable curve Γ : I → N is called a celestial curve if Γ (cid:48) ( s ) is a celestial vector for all s ∈ I . We will analyze in this section the relation existing between celestial curves and the causalityproperties of M and of the space of skies Σ. To do that we will introduce first the notion oftwisted causal null curve that will prove to be useful in the arguments to follow. Definition 3.5.
A continuous curve µ : [ a, b ] → M will be called a piecewise twisted null curve if there exists a partition a = s < s < . . . < s k = b such that for every i = 1 , . . . , k : i. µ | ( s i − ,s i ) is differentiable. ii. g ( µ (cid:48) ( s ) , µ (cid:48) ( s )) = 0 for all s ∈ ( s i − , s i ) . iii. µ (cid:48) ( s ) and Dµ (cid:48) ds ( s ) are linearly independent for all s ∈ ( s i − , s i ) .We say that µ is causal if µ | ( s i − ,s i ) is causal future (respect. causal past) for all i = 1 , . . . , k .If k = 1 then µ will be simply called twisted null curve . Now it is clear that if we are given a parametrized null geodesic γ : [0 , → M , a curve λ : ( − (cid:15), (cid:15) ) → M verifying that λ (0) = γ (0), and W ( s ) a null vector field along λ such that W (0) = γ (cid:48) (0), the family of curves:(3.1) f ( s, t ) = exp λ ( s ) ( tW ( s ))is a geodesic variation of γ ( t ) formed by null geodesics with f (0 , t ) = γ ( t ) and J ( t ) = ∂ f ∂s (0 , t ).If µ is a null curve then we may use W ( s ) = µ (cid:48) ( s ) and obtain a geodesic variation of γ that,in addition, defines a celestial curve in N . Actually more is true as it is shown by the following: Proposition 3.6. [Ba14]
If the curve
Γ : [0 , → N with Γ ( s ) = γ s ∈ N is celestial thenthere exists a differentiable null curve µ : [0 , → M such that γ s ( τ ) = exp µ ( s ) ( τ σ ( s )) where σ ( s ) ∈ N + µ ( s ) is a differentiable curve proportional to µ (cid:48) ( s ) wherever µ is regular. In fact, by construction, the curve µ in Prop. 3.6 runs the points in M such that the celestialcurve Γ is tangent to their skies, in other words, Γ (cid:48) ( s ) ∈ (cid:98) T S ( µ ( s )) for all s ∈ [0 , Corollary 3.7.
Given a celestial curve
Γ : [0 , → N such that Γ (cid:48) ( s ) ∈ (cid:98) T S ( p ) , ≤ s ≤ ,then the curve µ : [0 , → M of the previous proposition 3.6 is unique verifying µ ( s ) = p ∈ M .Proof. Consider that there exists µ , µ : [0 , → M associated to Γ in the sense of propo-sition 3.6 and verifying µ ( s ) = µ ( s ) = p for s ∈ [0 , A = { s ∈ [0 ,
1] : µ ( s ) = µ ( s ) } . Clearly, A is not empty and closed in [0 , U ⊂ M of p . Since U is open, then there exist δ > µ i (( s − δ, s + δ )) ⊂ U for i = 1 , s = 0 then we consider µ i ([0 , δ )) ⊂ U andanalogously for s = 1). Let us suppose that for s ∈ ( s − δ, s + δ ) we have that µ ( s ) (cid:54) = µ ( s )and since U is causally convex, then the segment of the light ray Γ ( s ) = γ s ∈ N connecting µ ( s ) and µ ( s ) is totally contained in U and, moreover since Γ (cid:48) ( s ) ∈ (cid:98) T S ( µ ( s )) ∩ (cid:98) T S ( µ ( s )),then the points µ ( s ) and µ ( s ) are mutually conjugated along γ s but, in virtue of [On83, Prop.10.10], this is not possible in a normal neighbourhood contradicting U is normal. Then we havethat µ ( s ) = µ ( s ) and hence the set A is also open in [0 , A is open, closed and notempty in [0 ,
1] then A = [0 ,
1] and we conclude that µ = µ . (cid:3) Given a celestial curve Γ the unique curve µ associated to it in the sense of Prop. 3.6 passingby p ∈ S − ( X ) will be called the “dust” of Γ by X and denoted by µ Γ X . The previousarguments can be made more precise by proving that the dust of a celestial curve is a twistednull curve. This is the content of the next Lemma. A. BAUTISTA, A. IBORT AND J. LAFUENTE
Lemma 3.8 ( µ -Lemma) . Let
Γ : [0 , → N be a celestial curve such that Γ (cid:48) (0) ∈ (cid:98) T X with X ∈ Σ . Then there exists a unique curve χ Γ X : [0 , → Σ such that it is continuous in Low’stopology and verifies χ Γ X (0) = X and Γ (cid:48) ( s ) ∈ (cid:98) T χ Γ X ( s ) . Moreover, the dust curve µ Γ X is apiecewise twisted null curve in M running along the image of S − ◦ χ Γ X .Conversely, given a regular twisted null curve µ : [0 , → M such that µ (0) = x = S − ( X ) , µ (cid:48) (0) (cid:54) = 0 (cid:54) = µ (cid:48) (1) , then the curve Γ µ : [0 , → N defined by the variation of null geodesics x : [0 , × I → M such that x ( s, t ) = exp µ ( s ) ( tµ (cid:48) ( s )) = Γ µ ( s ) | t is celestial with Γ (cid:48) (0) ∈ (cid:98) T X and χ Γ X ( s ) = S ( µ ( s )) .Proof. Let Γ : [0 , → N be a celestial curve such that Γ ( s ) = γ s ∈ N and Γ (cid:48) (0) ∈ (cid:98) T X with X = S ( x ) ∈ Σ. By corollary 3.7, there exists a unique differentiable curve µ : [0 , → M anda partition { a ≤ b < a ≤ b < · · · < a n − ≤ b n − < a n ≤ b n = 1 } ⊂ [0 , γ s ( τ ) = exp µ ( s ) ( tσ ( s ))where σ : [0 , → N is a differentiable curve verifying σ ( s ) = λ k ( s ) µ (cid:48) ( s ) for s ∈ ( b k , a k +1 )and λ k differentiable with k = 1 , . . . , n −
1. This curve µ also verifies µ ( s ) = p k ∈ M for all s ∈ [ a k , b k ].Now, we can define the curve χ Γ X = S ◦ µ : [0 , → Σ. Recall that for an open set
U ⊂ N containing a sky X ∈ Σ, the set of all skies contained in U is denoted as Σ ( U ). By the definitionof the Low’s topology, the set Σ ( U ) is open in Σ and these collection of open sets forms a basisat X .In order to show that χ Γ X is continuous, we will show that, given any U ⊂ N containing a sky S ( µ ( s )) ∈ Σ then (cid:0) χ Γ X (cid:1) − (Σ ( U )) is open in [0 ,
1] is verified. So, take any s ∈ [0 ,
1] and consideran open set
U ⊂ N such that χ Γ X ( s ) ⊂ U and then χ Γ X ( s ) ∈ Σ ( U ). Choose a collection ofnested intervals I sn ⊂ R such that { s } = (cid:84) n I sn . Let us suppose that there exists s n ∈ I sn suchthat χ Γ X ( s n ) / ∈ Σ ( U ). Then there is a light ray γ n ∈ χ Γ X ( s n ) ∈ Σ such that γ n / ∈ U . Recallthat a light ray is fully determined by a point p ∈ M and a direction [ v ] ∈ PN + p , so γ n can bedefined by µ ( s n ) ∈ γ n ⊂ M and a null direction [ v n ] ∈ PN + µ ( s n ) . Since lim µ ( s n ) = µ ( s ) and dueto the compactness of the fibres PN + µ ( s n ) , then with no lack of generality taking a subsequenceof [ v n ] if necessary, there exists a direction [ v ] ∈ PN + µ ( s ) defining, together with µ ( s ), the lightray γ such that lim γ n = γ ∈ χ Γ X ( s ) ⊂ U .But since U is open, there exists an integer K such that for every n > K we have that γ n ∈ U contradicting that χ Γ X ( s n ) / ∈ Σ ( U ). Therefore there exist I sn such that χ Γ X ( s n ) ∈ Σ ( U ) andhence (cid:0) χ Γ X (cid:1) − (Σ ( U )) is open in [0 , µ Γ X , we will cut off the segments µ | ( a k ,b k ) from µ and glue togetherthe segments µ | [ b k ,a k +1 ] . We call c = 0 and for every k = 1 , . . . , n −
1, let us define c k +1 = a k +1 − (cid:80) ki =1 ( b i − a i ) ∈ [0 ,
1] and consider the change of parameter h k : [ c k , c k +1 ] → [ b k , a k +1 ]defined by h k ( τ ) = τ + a k +1 − c k +1 . Since µ is differentiable and h k is a diffeomorphism forevery k = 1 , . . . , n − µ k ( τ ) = µ ◦ h k ( τ ) is differentiable for τ ∈ ( c k , c k +1 ). Moreover, since µ (cid:48) k ( τ ) = µ (cid:48) ( h k ( τ )) then g ( µ (cid:48) k ( τ ) , µ (cid:48) k ( τ )) = g ( µ (cid:48) k ( h k ( τ )) , µ (cid:48) k ( h k ( τ ))) = 0 AUSALITY AND SKIES 9 for τ ∈ ( c k , c k +1 ). Also, the covariant derivatives verify Dµ (cid:48) k ( τ ) dτ = h (cid:48)(cid:48) k ( τ ) µ (cid:48) ( h k ( τ )) + ( h (cid:48) k ( τ )) Dµ (cid:48) ( h k ( τ )) ds = Dµ (cid:48) ( h k ( τ )) ds then denoting J s as the Jacobi field along γ s defined by the variation 3.2, we have J s (0) = µ (cid:48) ( s )and J (cid:48) s (0) = Dσ ( s ) ds = D ( λ k ( s ) µ (cid:48) ( s )) ds = λ (cid:48) k ( s ) µ (cid:48) ( s ) + λ k ( s ) Dµ (cid:48) ( s ) ds for s ∈ ( b k , a k +1 ). Since Γ is celestial, then J s (cid:54) = 0 (mod γ (cid:48) s ) and so, Dµ (cid:48) ( s ) ds is not proportional to µ (cid:48) ( s ) for s ∈ ( b k , a k +1 ), therefore Dµ (cid:48) k ( τ ) dτ and µ (cid:48) k ( τ ) are linearly independent for τ ∈ ( c k , c k +1 ).We have shown that for any k = 1 , . . . , n − µ k are twisted null curves. Since h − k ( a k +1 ) = h − k +1 ( b k +1 ) then all the segments µ k glue together continuously. Therefore we candefine, with no ambiguity, the curve µ Γ X : [0 , a ] → M such that µ Γ X ( τ ) = µ k ( τ ) if τ ∈ [ c k , c k +1 ]for k = 1 , . . . , n − , a ] = ∪ n − k =1 [ c k , c k +1 ]. This curve µ Γ X is then a piecewise twisted nullcurve associated to the partition { c < c < · · · < c n = a } ⊂ [0 , a ] and it is unique except byreparametrization.Conversely, let us consider a twisted null curve µ : [0 , → M such that µ (0) = x = S − ( X ).Then, we can define the variation of null geodesics x : [0 , × I → M such that x ( s, t ) = exp µ ( s ) ( tµ (cid:48) ( s )) = γ s ( t )which verifies γ (cid:48) s (0) = µ (cid:48) ( s ). Now, define the curve Γ µ ( s ) = γ s ∈ N for every s ∈ [0 , J s of the variation x along γ s verifies J s (0) = µ (cid:48) ( s ) = γ (cid:48) s (0) and J (cid:48) s (0) = Dµ (cid:48) ds ( s ) and,since µ is twisted null then Dµ (cid:48) ds is not proportional to γ (cid:48) s . Therefore (Γ µ ) (cid:48) ( s ) = J s (mod γ (cid:48) s ) (cid:54) =0 (mod γ (cid:48) s ) and hence (Γ µ ) (cid:48) ( s ) ∈ (cid:98) T S ( γ s (0)) = (cid:98) T S ( µ ( s ))then Γ µ is celestial. (cid:3) Celestial curves and the partial order in the space of skies.
We have already pointedit out that if x ≺ y , then their corresponding skies are related S ( x ) ≺ c S ( y ). The discussion tofollow will show that such relation can actually be refined by proving that in case of y ∈ I + ( x ) ,there exists a causal piecewise twisted null curve joining x and y , hence relating the causalproperties of Σ to the existence of appropriate celestial curves. Theorem 3.9 (Twisted null curve theorem) . Let p, q ∈ M such that q ∈ I + ( p ) , then there existsa future piecewise twisted null curve µ joining p to q . To prove the previous Theorem we will need some lemmas.
Lemma 3.10.
Let M be a 3–dimensional space-time and γ : I → M be a future time-like geo-desic. Then there exists δ > such that for any t ∈ ( t , t + δ ] , there exists a future twisted nullcurve µ joining γ ( t ) to γ ( t ) .Proof. Given the future time-like geodesic γ : I → M and t ∈ I , it is known, e. g. by [La03, §
97] and [Pe72, def. 7.13], that there exists a synchronous coordinate system (
U, φ = ( t, x, y ))with γ ( t ) ⊂ U in which the metric g of M can be written as( g ij ) = − g g g g Recall that y ∈ I + ( x ) means that there exists a future time-like curve from x to y . where g ij ≡ g ij ( t, x, y ) for i, j = 1 , U is causally convex and the expression of the geodesic γ in these coordinates is φ ( γ ( s )) = ( s, , ∈ R . For a point γ (cid:0) t (cid:1) ∈ U , it is possible to find R > U = (cid:8) ( t, x, y ) : x + y ≤ R, t ≤ t ≤ t (cid:9) is contained in U .As candidates for the required twisted null curve, we will study curves µ r such that φ ( µ r ( s )) = ( f r ( s ) , r (1 − cos s ) , r sin s )where 0 ≤ r ≤ R/ f r = f r ( s ) is a function. If µ r is a null curve, then g ( µ (cid:48) r , µ (cid:48) r ) = 0 andtherefore − ( f (cid:48) r ( s )) + r g sin s + 2 r g sin s cos s + r g cos s = 0where g ij = g ij ( φ ( µ r ( s ))). Thus, we have a first order ordinary differential equation whichdescribes a null curve passing through γ ( t )(3.3) (cid:26) f (cid:48) r ( s ) = r (cid:112) g sin s + 2 g sin s cos s + g cos sf r (0) = t Since the metric in the hypersurfaces { t = c } with t ≤ c ≤ t is positive definite, then the termunder the square root in 3.3 is always positive. Moreover, since f (cid:48) r > µ r is future.Let us show that we can find r > µ r is twisted. A simple calculation gives( dφ ) µ r ( s ) (cid:18) Dµ (cid:48) r ds ( s ) (cid:19) = (cid:0) f (cid:48)(cid:48) r + r ϕ ( r, s ) , r cos s + r ϕ ( r, s ) , − r sin s + r ϕ ( r, s ) (cid:1) where ϕ i = ϕ i ( r, s ) with i = 0 , , U depending on the Christoffelsymbols and the components of µ (cid:48) r . In order to show that Dµ (cid:48) r ds and µ (cid:48) r are linearly independent,it is enough to see that the determinant of their components x , y does not cancel out, so (cid:12)(cid:12)(cid:12)(cid:12) r cos s + r ϕ ( r, s ) r sin s − r sin s + r ϕ ( r, s ) r cos s (cid:12)(cid:12)(cid:12)(cid:12) = r (1 + r ( ϕ ( r, s ) cos s + ϕ ( r, s ) sin s ))hence, since ϕ and ϕ are continuous in U , they are also bounded in the compact set U andthere exists r ≤ R/ r ( ϕ ( r, s ) cos s + ϕ ( r, s ) sin s ) (cid:54) = 0for all r ∈ (0 , r ], and in this case, Dµ (cid:48) r ds and µ (cid:48) r are linearly independent.At this moment, we have seen that µ r is a twisted null curve passing through γ ( t ) for0 < r ≤ r , and it remains to show that there exists δ > µ r also passes through γ ( t )for every t ∈ ( t , t + δ ].Now, we want to prove that for every r ∈ (0 , r ] there exists s r > f r ( s r ) = t .Given r ∈ (0 , r ], we define ω r = sup { s : f r ( s ) exists } . Let us assume that lim s (cid:55)→ ω r f r ( s ) = c ≤ t .In case of ω r < + ∞ , the solution f r of equation 3.3 verifying the initial condition f r ( ω r ) = c would coincide with f r = f r ( s ) for s < ω r contradicting the maximality of f r up to ω r because inthat case f r could be extended beyond s = ω r . On the other hand, if ω r = + ∞ , the derivabilityof f r would imply that lim s (cid:55)→ + ∞ f (cid:48) r ( s ) = 0 and hence the curve solution µ r would approximate tothe curve β r verifying β r ( s ) = ( c, r (1 − cos s ) , r sin s ) ∈ U in T M , i.e. for every s ∈ R the sequence { s n = s + 2 πn } n ∈ N would verifylim s (cid:55)→ + ∞ µ r ( s n ) = β r ( s ) and lim s (cid:55)→ + ∞ µ (cid:48) r ( s n ) = β (cid:48) r ( s ) AUSALITY AND SKIES 11
By the continuity of the metric g then we havelim s (cid:55)→ + ∞ g ( µ (cid:48) r ( s n ) , µ (cid:48) r ( s n )) = g ( β (cid:48) r ( s ) , β (cid:48) r ( s )) (cid:54) = 0since β r is contained in the space-like hypersurface { t = c } , but this contradicts that g ( µ (cid:48) r , µ (cid:48) r ) =0. Therefore, independently from ω r , for every r ∈ (0 , r ] we have that lim s (cid:55)→ ω r f r ( s ) > t and hence,for every r ∈ (0 , r ] there exists s r ∈ (0 , ω r ) such that f r ( s r ) = t .Since the functions g ij are continuous in U for i, j = 1 ,
2, then their restrictions to the compactset U reach their maximum, therefore there exists M ij > | g ij ( t, x, y ) | ≤ M ij for( t, x, y ) ∈ U . Then,0 < f (cid:48) r ( s ) = r (cid:113) g sin s + 2 g sin s cos s + g cos s ≤≤ r (cid:113)(cid:12)(cid:12) g sin s (cid:12)(cid:12) + 2 | g sin s cos s | + | g cos s | ≤≤ r (cid:112) M + 2 M + M = rM where M = √ M + 2 M + M ∈ R is independent from r and s . So integrating, we have that t ≤ f r ( s ) ≤ rM s + t and therefore t = f r ( s r ) ≤ rM s r + t ⇒ t − t rM ≤ s r then there exists ρ ∈ (0 , r ] small enough such that s r ≥ π for all r ∈ (0 , ρ ] and hence theparameter s of f r can be extended beyond s = 2 π . Since f (cid:48) ρ ( s ) > f ρ ( s ) > t forall s >
0, therefore there exists δ > f ρ (2 π ) = t + δ . So, by the inequality t ≤ f r (2 π ) ≤ πrM + t we have that lim r (cid:55)→ f r (2 π ) = t and for every t ∈ ( t , t + δ ] thereexists r ∈ (0 , ρ ] such that µ r (0) = ( t , ,
0) = φ ( γ ( t )) µ r (2 π ) = ( f r (2 π ) , ,
0) = ( t, ,
0) = φ ( γ ( t ))therefore we have shown that there exists δ > t ∈ ( t , t + δ ] the points γ ( t )and γ ( t ) can be connected by some future twisted null curve µ r . Analogously, this constructioncan be done to obtain a future twisted null curve joining γ ( t ) to γ ( t ) for all t ∈ [ t − δ, t ). (cid:3) Lemma 3.11.
The statement of Lemma 3.10 is true in a m –dimensional spacetime M .Proof. We can find a synchronous coordinate system (
U, φ ) with φ = ( t, x , . . . , x m − ) (asdone previously) such that the expression of the geodesic γ in these coordinates is φ ( γ ( s )) =( s, , . . . , ∈ R m , so this chart is adapted to γ . Consider the restriction V = { ( t, x , . . . , x m − ) : x i = 0 , i = 3 , . . . , m − } ⊂ φ ( U )then N = φ − ( V ) ⊂ M is a 3–dimensional manifold embedded in M . Moreover, by [On83,Lemma 4.3] we have that Levi-Civita connection in N coincides with the orthogonal projectionover N of the Levi-Civita connection in M , hence we have D N ds = tan (cid:0) Dds (cid:1) where D N ds and Dds denote the covariant derivatives in N and M respectively. So the geodesics in M contained in N are also geodesics in N and the restriction ( N, φ | N = ( t, x , x )) of the synchronous coordinatesystem is still a synchronous coordinate system for N . Then, since γ is a geodesic contained in N , by step 3.10, there exists δ > µ ⊂ N such that µ joins γ ( t )to γ ( t + δ ). Since the metric in N is the restriction of the metric in M , then µ as curve in M is also null. Finally, since µ (cid:48) and D N µ (cid:48) ds = tan (cid:16) Dµ (cid:48) ds (cid:17) are lineally independent in T µ ( s ) N thenis an immediate consequence that µ (cid:48) and Dµ (cid:48) ds are lineally independent in T µ ( s ) M . Therefore,we have shown that there exists δ > µ a future twisted null curve in M joining γ ( t ) to γ ( t + δ ). (cid:3) We can prove now as a direct consequence of the previous lemmas, Lemma 3.10 and 3.11, thefollowing:
Proposition 3.12.
Let γ : I → M be a future timelike geodesic. Then, for any t , t ∈ I , thereexists a future piecewise twisted null curve µ joining γ ( t ) to γ ( t ) .Proof. By Lemma 3.11, for all t ∈ [ t , t ] there exists an open interval I t = [ t − δ t , t + δ t ] ⊂ [ t , t ]relative to [ t , t ] such that γ ( t ) can be joined to γ ( u ) with u ∈ I t by means of a piecewise twistednull curve. By the compactness of [ t , t ], we can extract a finite covering { I n } n =1 ,...,N such that,with no lack of generality, verifies I i ∩ I k (cid:54) = ∅ ⇔ k = i ±
1. We can choose a partition { t = a < b < · · · < a N − < b N − < a N = t } such that a i ∈ I i and b i ∈ I i ∩ I i +1 and therefore there exists future twisted null curves joining γ ( a i ) to γ ( b i ) and γ ( b i ) to γ ( a i +1 ) for i = 1 , . . . , N −
1. The union of these curves forms afuture piecewise twisted null curve connecting γ ( t ) to γ ( t ). (cid:3) Now we can proceed with the proof of Theorem 3.9.
Proof.
Theorem 3.9: Consider p, q ∈ M such that q ∈ I + ( p ), then there exists a continuousfuture time-like curve λ connecting p and q . By compactness of λ between p and q , there existsa finite covering { W k } k =1 ,...,K of globally hyperbolic and causally convex open sets, then it ispossible to built a continuous curve γ joining p and q formed by segments γ k ⊂ W k of futuretime-like geodesics with endpoints at λ . So γ becomes a future piecewise time-like geodesicBy Prop. 3.12, the endpoints of the time-like geodesic segments γ k of γ can be connected bya future piecewise twisted null curve µ k . Since γ is continuous, we can glue together all µ k toobtain another piecewise twisted null curve µ joining p and q . (cid:3) The smooth structure of the space of skies and the non-refocussing property
Regular sets.
The smooth structure on the space of skies will be obtained by selecting afamily of neighbourhoods possessing the properties that will make obvious the construction of anatlas on Σ. We will call such neighbourhoods regular neighbourhoods and they refine the notionof regular set already introduced in [Ba14, Def. 3].Let W ⊂ Σ be a non-empty set satisfying the conditions:(1) (cid:98)
T X ∩ (cid:98) T Y = ∅ for all X (cid:54) = Y ∈ W .(2) The union (cid:99) W = (cid:91) X ∈ W (cid:98) T X ⊂ (cid:98) T N is a regular (3 m − (cid:98) T N .(3) Let (cid:98) D be the distribution in (cid:99) W whose leaves are (cid:101) X = (cid:98) T X . Then the space of leaves (cid:102) W = (cid:110) (cid:101) X : X ∈ W (cid:111) = (cid:99) W / (cid:98) D is a differentiable quotient manifold.It is clear that in this case, (cid:102) W can be identified to W via the bijective map(4.1) Θ : W → (cid:102) WX (cid:55)→ (cid:101) X and hence W inherits the quotient topology such that U ⊂ W is open ⇔ (cid:98) U = (cid:91) X ∈ U (cid:98) T X ⊂ (cid:99) W is open,and also a differentiable structure from (cid:102) W . So, we will denote W equipped with the previousstructure as W ( ∼ ) (cid:39) (cid:102) W . AUSALITY AND SKIES 13 (4) For every X ∈ W and every celestial curve Γ : I (cid:15) → N such that Γ (cid:48) (0) ∈ (cid:98) T X ,(a) there exists 0 < δ ∈ I (cid:15) such that Γ (cid:48) : I δ → (cid:99) W with I δ = ( − δ, δ ).(b) the curve χ Γ X : I δ → W ( ∼ ) defined in Lemma 3.8 is differentiable.(5) Given (cid:101) X, (cid:101) Y ∈ (cid:102) W , for any causal curve χ : [ a, b ] → Σ, joining X and Y , then χ ( s ) ∈ W for all s ∈ [ a, b ].Now we are ready to state the next definition: Definition 4.1.
A not–empty subset W ⊂ Σ is said to be a regular subset, and denoted as W ⊂ reg Σ , if it verifies conditions (1) to (5) above. Observe that both the definition of regular subset and the differentiable structure of W ( ∼ ) (cid:39) (cid:102) W depend only on N and Σ.4.2. The topology of the space of skies and regular sets.
We will show next that the classof regular subsets is not empty.We will say that V ⊂ M is an open normal set is V is globally hyperbolic, causally convex,relatively compact, open set of M . A classical theorem due to Whitehead guarantees the existenceof convex normal neighbourhoods V at any point x ∈ M , (see [On83, chapter 5] and [Mi08,theorem 2.1 and definition 3.22] for a treatment of this result in Lorentz manifolds). Thus fora strongly causal space-time M there exists a basis of neighbourhoods at any p ∈ M formed bynormal open sets. Proposition 4.2.
Let V ⊂ M be a normal open set, then U = S ( V ) ⊂ reg Σ is regular. Moreover, S : V → U ( ∼ ) is a diffeomorphism.Proof. Let V ⊂ M be a normal open set, then condition (1) is verified since V is causally convex.By [Ba14, Thm. 1], condition (2) is verified. The condition (3) and the fact of S : V → U ( ∼ ) being a diffeomorphism are consequences of [Ba14, Thm. 2]. Lemma 3.8 trivially implies (4a)and permits to construct the curve χ Γ X as the following composition of differentiable mapsΓ π Θ − I δ −→ (cid:98) U −→ (cid:101) U −→ U ( ∼ ) s (cid:55)→ Γ (cid:48) ( s ) (cid:55)→ (cid:98) T χ Γ X ( s ) (cid:55)→ χ Γ X ( s )then (4b) is verified. Finally, in order to verify (5), we know that Γ (cid:48) ( a ) ∈ (cid:98) T X , Γ (cid:48) ( b ) ∈ (cid:98) T Y and
X, Y ∈ U , by lemma 3.8, there exists a piecewise twisted null curve µ : [ a, b ] → M such that µ ( a ) = x ∈ V and µ ( b ) = y ∈ V . Since V is causally convex, then µ is fully contained in V andtherefore χ = S ◦ µ is fully contained in U = S ( V ). So, we conclude that U ⊂ reg Σ. (cid:3) We may call the regular sets U = S ( V ) with V open normal, elementary regular sets in Σ.Using now the technical lemma: Lemma 4.3.
Given W ⊂ reg Σ a regular set and X = S ( x ) ∈ W , then for any twisted nullcurve µ : I (cid:15) → M such that µ (0) = x there exists δ > verifying that µ (( − δ, δ )) ⊂ S − ( W ) .Proof. Consider X = S ( x ) ∈ W ⊂ reg Σ, then by lemma 3.8, there exists a celestial curveΓ : I (cid:15) → N and a continuous curve χ Γ X : I (cid:15) → Σ such that χ Γ X = S ◦ µ . Since W is regular,then there exists δ > χ Γ X : ( − δ, δ ) ⊂ I (cid:15) → W ( ∼ ) is differentiable. Then we have µ (( − δ, δ )) = S − ◦ χ Γ X (( − δ, δ )) ⊂ S − (cid:16) W ( ∼ ) (cid:17) = S − ( W ) . (cid:3) It is easy to prove the following:
Theorem 4.4.
Let W ⊂ reg Σ be a regular set, then S − ( W ) is open in M .Proof. Given W ⊂ reg Σ and consider X ∈ W such that x = S − ( X ) ∈ M . Take a futuretwisted null curve µ : I (cid:15) → M with µ (0) = x , then by lemma 4.3, there exists δ > µ (( − δ, δ )) ⊂ S − ( W ). Without any lack of generality, we can assume that δ is small enoughfor V = I + ( µ ( − δ )) ∩ I − ( µ ( δ )) being globally hyperbolic and causally convex. Observe that x ∈ V and for any p ∈ V , we have that p ∈ I + ( µ ( − δ )), then by theorem 3.9, for any p ∈ V there exists a future piecewise twisted null curve µ p connecting µ ( − δ ) and µ ( δ ) passing through p (see Figure 1). Now, since W is regular, then by property (5), the curve χ p = S ◦ µ p is fullycontained in W , therefore p ∈ S − ( W ) and hence V ⊂ S − ( W ) and S − ( W ) is open in M . (cid:3) In virtue of Proposition 4.2 and Theorem 4.4, since the sky map S is an homeomorphism withthe Low’s topology in Σ, it is clear that this topology coincides with the topology generated byregular sets in Σ. So, by [Ba14, Cor. 1, Thm. 2 and Cor. 2], we get the following corollary. Corollary 4.5.
The family of regular sets { W | W ⊂ reg Σ } is a basis for the Low’s topology of Σ . Moreover, there exists a unique differentiable structure in Σ compatible with the manifolds W ( ∼ ) ⊂ Σ that makes of S : M → Σ a diffeomorphism. In the previous construction of the topology of Σ by mean of regular sets, the hypothesis ofnon–refocusing in M has not been used, but we have obtained, with no further hypotheses, thatthe resultant topology coincides with Low’s topology in Σ.The following Lemma corroborates the relation between neighbourhood basis of M and itsspace of skies Σ and will be used to establish the conclusion that sky-separating implies non-refocussing. Lemma 4.6.
Let B ( x ) be a neighbourhood basis consisting on globally hyperbolic, normal andcausally convex open sets.For any U ∈ B ( x ) , denote by U = { γ ∈ N : γ ∩ U (cid:54) = ∅ } . Then { Σ ( U ) : U ∈ B ( x ) } is a neighbourhood basis of S ( x ) ∈ Σ . µ ( t ) µ (0) = x µ ( ) µ ( ) S ( W ) J + ( µ ( )) J ( µ ( )) p V piecewise twisted null curve Figure 1.
AUSALITY AND SKIES 15
Proof.
Because the bundle PN ( M ) → M is locally trivial, let us take a neighbourhood V ⊂ M of x ∈ M such that there is a diffeomorphism ϕ : V × S m − → PN ( V ) with ϕ (cid:0) { y } × S m − (cid:1) = PN y for all y ∈ V .Consider the map σ : PN ( V ) → V ⊂ N defined by σ ([ v ]) = γ [ v ] . It is clear that σ is continuousand hence σ = σ ◦ ϕ : V × S m − → V is also so. Observe that S ( x ) = σ (cid:0) { x } × S m − (cid:1) , and σ ( V × S m − ) = V .Now, take any open W ⊂ V containing the sky S ( x ), then { x } × S m − ⊂ σ − ( S ( x )) ⊂ σ − ( W )Since σ is continuous then σ − ( W ) is open in V × S m − .For any ( y, q ) ∈ V × S m − there exists a neighbourhood basis whose elements are U ( y,q ) = K y × H q where K y ⊂ V and H q ⊂ S m − are open neighbourhoods of y ∈ V and q ∈ S m − respectively. Then for any ( x, q ) ∈ { x } × S m − , there exist U ( y,q ) with ( x, q ) ∈ U ( y,q ) ⊂ σ − ( W ).Since { x } × S m − is compact, then there exists a finite sub-covering { U j = K j × H j } j =1 ,...,n ⊂ σ − ( W ). Then { x } × S m − ⊂ n (cid:91) j =1 U j ⊂ σ − ( W )Observe that K = (cid:84) nj =1 K j is an open neighbourhood of x and (cid:83) nj =1 H j = S m − .Since B ( x ) is a neighbourhood basis of x ∈ M , there exists U ∈ B ( x ) such that U ⊂ K .For any ( y, q ) ∈ U × S m − , we have that( y, q ) ∈ U × n (cid:91) j =1 H j therefore there exists j such that q ∈ H j and since y ∈ K ⊂ K j , then ( y, q ) ∈ U j ⊂ σ − ( W ).This implies that { x } × S m − ⊂ U × S m − ⊂ σ − ( W ) . and hence S ( x ) ⊂ σ (cid:0) U × S m − (cid:1) ⊂ W and since U = σ (cid:0) U × S m − (cid:1) then S ( x ) ∈ Σ ( U ) ⊂ Σ ( W )is verified. Then { Σ ( U ) : U ∈ B ( x ) } is a neighbourhood basis of S ( x ) ∈ Σ as we claimed. (cid:3)
A direct consequence of the previous results is the following:
Theorem 4.7.
Let M be a space-time separating skies such that it is refocussing at x , then thesky map S : M → Σ is not open.Proof. We will show that there exists a sequence { x n } in M that does not converge to x andsuch that S ( x n ) converges to S ( x ) in Σ does contradicting the statement that S is open.Because M is refocussing at x there exists an open neighbourhood W ⊂ M of x such that forevery open neighbourhood V ⊂ W of x there is y / ∈ W such that every light ray passing through y enters V . Let us choose a sequence of globally hyperbolic neighbourhoods V xn ⊂ W of x suchthat ∩ n V xn = { x } . More specifically, let σ ( t ) be a time-like curve contained on a causally convex,globally hyperbolic neighbourhood U ⊂ W of x and let a n (respect. b n ) be a sequence of pointson σ , in the past (future) of x , such that a n → x (respect. b n → x ). Now we choose the sequenceof open neighbourhoods as V xn = I + ( a n ) ∩ I − ( b n ). Then for any V xn in the previous sequence there exists x n / ∈ W such that γ ∩ V xn (cid:54) = ∅ and x n ∈ γ ∈ N . Hence, since x n / ∈ W for all n , then x n cannot converge to x .On the other hand, considering the open subsets U n = { γ ∈ N | γ ∩ V xn (cid:54) = ∅} , and becauseof Lemma 4.6, it is clear that Σ( U n ) define a neighbourhood basis at S ( x ) in Σ, and because S ( x n ) ∈ Σ( U n ) then we conclude that S ( x n ) → S ( x ). (cid:3) Then we get as a corollary of Thm. 4.7:
Corollary 4.8.
If the skies of M separate events then M is non–refocusing. Conclusions and discussion
We have reached the main conclusion that the topological, differentiable and causal structuresof sky-separating strongly causal space-times can be reconstructed from the corresponding onesin their spaces of light rays and skies. It is also important to point out that because of Lemma 4.6any strongly causal space-time is locally sky-separating, thus the property of being sky-separatinghas a global character.The possibility of describing the causal structure of a space-time in terms of the partial orderinduced in the space of skies by non-negative Legendrian isotopies in the space of light rays,provides a new interpretation to the Malament-Hawking theorem, [Ma77], [Ha76], in the sensethat the partial order relation defined on the space of skies characterise the conformal structure ofthe original space-time. Actually, suppose that Φ : N → N is a sky preserving diffeomorphismbetween the spaces of light rays of two strongly causal sky-separating space-times M and M .If the map Φ preserves the partial orders ≺ a , a = 1 , and Σ,i.e., if X ≺ Y then Φ( X ) ≺ Φ( Y ), for any X, Y ∈ Σ , then because of Cor. 3.3, we have thatΦ induces a causal diffeomorphism ϕ : M → M , hence a conformal diffeomorphism.The characterisation of causal relations in terms of sky isotopies opens a new direction inthe foundations of the causal sets programme to quantum gravity [Br91], [Ri00], as it showsthat causal structures need for their description the additional structure provided by the contactstructure in the space of light rays.It is also worth to point out here that the causal completion of a given spacetime is justcontinuous and often fails to be smooth (as in the case of Minkowski) space. According to thereconstruction theorems discussed in this paper a similar analysis could be performed directlyon the space of light rays and skies. In this setting a concrete proposal of a new causal boundaryconstruction was proposed by R. Low [Lo06] but has not been discussed in detail so far.A particularly interesting situation happens for three dimensional space-times that will bediscussed in a forthcoming paper. In such case the space of light rays happens to be threedimensional again as well as the space of skies. In such case Low’s causal boundary can beconstructed explicitly and their topology can then be compared with that of the original space-time. References [Ab88] R. Abraham, J. Marsden, T. Ratiu,
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E-mail address : [email protected], [email protected] Depto. de Geometr´ıa y Topolog´ıa, Univ. Complutense de Madrid, Avda. Complutense s/n, 28040Madrid, Spain.
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