Observations and predictions from past lightcones
aa r X i v : . [ g r- q c ] J a n OBSERVATIONS AND PREDICTIONS FROM PAST LIGHTCONES
MARTIN LESOURD
Abstract.
In a general Lorentzian manifold M , the past lightcone of a point is aproper subset of M that does not carry enough information to determine the rest of M . That said, if M is a globally hyperbolic Cauchy development of vacuum initialdata on a Cauchy surface S and there is a point whose past lightcone contains S ,then the contents of such a lightcone determines all of M (up to isometry). We showsome results that describe what properties of M guarantee that past lightcones doindeed determine all or at least significant portions of M . Null lines and observerhorizons, which are well known features of the de-Sitter spacetime, play a prominentrole. Introduction
In Lorentzian geometry, an observer at a given time in a spacetime (
M, g ) isrepresented by a timelike curve with future endpoint p ∈ M , and the past lightcone J − ( p ) ⊂ M of p represents all signals in M that can reach the observer at p . One canthen ask the following. (A) Can an observer at p ∈ M know the global structure of M on the basis of J − ( p )? (B) Can an observer at p ∈ M make predictions about M \ J − ( p ) on the basis of J − ( p )?In this short note, we describe some of what is known about (A) and (B) , we provevarious further results, and we list some natural further questions.Part of the appeal in (A) and (B) is that they are subject to somewhat surprisingexamples. As described below, two (inextendible and globally hyperbolic) spacetimes( M ′ , g ′ ) and ( M, g ) can be non-isometric, in spite of the fact that each member of thecountable collection of past lightcones { I − ( p i ) } that covers M can be isometricallyembedded into ( M ′ , g ′ ) and likewise with M and M ′ interchanged. Here we will showwhen this cannot happen.We now recall some basic definitions of causal theory, cf. [1], [14], [16] for some classicreferences and [15] for a more recent authorative survey.A spacetime ( M, g ) is a connected C ∞ Hausdorff manifold M of dimension two orgreater with a Lorentzian metric g of signature ( − , + , + , ... ), and we will assume a timeand space orientation. Since regularity is not the issue here, for simplicity of expressionwe take g to be smooth, but many of the arguments can be extended to lower regularity.The lightcone structure inherited on the tangent space T p M at each p leads to thenotion of a causal curve, which in turn leads to defining J − ( p ) (or I − ( p )) as the col-lection of all points q ∈ M from which there exists a causal (or timelike) curve withfuture endpoint p and past endpoint q . A causal curve between p and q ∈ J + ( p ) is achronal iff q / ∈ I + ( p ). A null line is an inextendible achronal causal curve.A set is achronal ( acausal ) iff no two members of it can be connected by a time-like (causal) curve. The domain of dependence D ( S ) of a set S ⊂ M is given by D ( S ) = D + ( S ) ∪ D − ( S ) and D + ( S ) is defined as the collection of all points q in M such that any inextendible ( C ) past directed causal curve passing through q inter-sects S . ˜ D ( S ) is defined identically except that curves are timelike, rather than causal.From the perspective of the Cauchy problem of general relativity, D + ( S ) representsthe maximal portion of M that could be determined by initial data on S . If S is closedas a subset of M then D + ( S ) = ˜ D + ( S ). The future Cauchy horizon is defined as H + ( S ) ≡ D + ( S ) \ I − ( D + ( S )) and H ( S ) = H + ( S ) ∪ H − ( S ). S is a partial Cauchy hypersurface if it an edgeless acausal set, cf. Definition 14.27of [1] for the definition of edge( S ) for an achronal set S . A spacetime ( M, g ) is globallyhyperbolic iff there exists a partial Cauchy hypersurface S such that M = D ( S ). By [2],a smooth globally hyperbolic spacetime ( M, g ) is isometric to ( R × S, − f ( t ) dt + h ( t ))where f ( t ) is smooth and h ( t ) a Riemannian metric on S . From the perspective ofcausal structure, global hyperbolicity is equivalent to the spacetime being causal and J + ( p ) ∩ J − ( q ) being compact for all p, q ∈ M .If S is acausal and S ∩ edge( S ) = ∅ (eg., S is a partial Cauchy hypersurface), then D ( S ) is non-empty, open, and D ( S ) ∩ H ( S ) = ∅ . Note that the openness of D ( S ) maybe ruined if we take S to be achronal rather than acausal.A spacetime ( M, g ) is causally simple iff it is causal and J +( − ) ( p ) is closed for all p ∈ M . Global hyperbolicity is strictly stronger than causal simplicity.An isometric embedding of ( M, g ) into ( M ′ , g ′ ) is an injective (but not necessarilysurjective) map φ : M ֒ → M ′ such that φ is a diffeomorphism onto its image φ ( M ) and φ ∗ ( g ′ ) = g . If φ maps M surjectively onto M ′ then we write φ : M → M ′ and we saythat ( M, g ) and ( M ′ , g ′ ) are isometric . A spacetime ( M, g ) is inextendible when thereexists no isometric embedding φ of M into M ′ such that M ′ \ φ ( M ) = ∅ .A spacetime is future holed if there is a partial Cauchy hypersurface S ′ ⊂ M ′ andan isometric embedding φ : ˜ D ( S ′ ) ֒ → M such that φ ( S ′ ) is acausal in ( M, g ) where(
M, g ) is any spacetime, and φ ( H + ( S ′ )) ∩ D + ( φ ( S ′ )) = ∅ . A spacetime is hole-free ifit lacks future and past holes. Minguzzi [12] shows that causally simple, inextendiblespacetimes are hole free. Acknowledgements . We thank the Gordon Betty Moore Foundation and JohnTempleton Foundation for their support of Harvard’s Black Hole Initiative. We thankProfessors Erik Curiel, JB Manchak, Ettore Minguzzi, Chris Timpson, and JamesWeatherall for valuable comments that improved the paper.2.
Previous Work (A) . A natural definition coming from [8] (whose terminology is slightly different)is the following. No closed causal curves.
BSERVATIONS AND PREDICTIONS FROM PAST LIGHTCONES 3
Definition 2.1.
A spacetime (
M, g ) is weakly observationally indistinguishable from( M ′ , g ′ ) just in case there is an isometric embedding φ : I − ( p ) ֒ → M ′ for every point p ∈ M . If this is also true with M and M ′ interchanged, then the spacetimes are strongly observationally indistinguishable .One could replace I − ( p ) with J − ( p ), and although this makes no real difference toany of the results in [8] and [9] or indeed to what follows, that would capture a somewhatmore honest sense of distinguishability because observers can certainly receive signalsfrom J − ( p ) \ I − ( p ).One can also take observers to be inextendible timelike curves σ (as is also done in[8]), but in that case J − ( σ ) ⊂ M would represent “all possible observations that couldbe made supposing that the observer lasts forever with respect to M ”, which, thoughinteresting, is stronger than what happens in practice.Malament pg. 65-6 [8] gives examples of spacetimes that are strongly observationallyindistinguishable but non-isometric. His examples are globally hyperbolic, inextendible,and exploit the presence of observer horizons in the de-Sitter spacetime.Based on an explicit cut and paste argument, Manchak [9] shows the following. Proposition 2.2 (Manchak [9]) . Given any non-causally bizarre spacetime ( M, g ) ,there exists a spacetime ( M ′ , g ′ ) that is weakly observationally indistinguishable from ( M, g ) but not isometric to ( M, g ) . Although Manchak’s construction of ( M ′ , g ′ ) works for any (non-causally bizarre)spacetime ( M, g ), it relies on introducing a countably infinite collection of holes in( M ′ , g ′ ). It is unknown to us whether Proposition 2.2 holds for strong observationalindistinguishability.Viewed together, [8] and [9] lead one to the following. Question.
Find conditions { A, B, ... } satisfied by ( M, g ) and ( M ′ , g ′ ) such that:‘Weakly (or strongly) observationally indistinguishable + A + B + ... ’ ⇔ ‘( M, g ) and ( M ′ , g ′ ) are isometric’Proposition 3.6 and Corollary 3.11 below are in this direction. (B) . Geroch [7] defines prediction in general relativity as follows. Definition 2.3. p ∈ M is predictable from q , written as p ∈ P ( q ), iff p ∈ D + ( S )for some closed, achronal set S ⊂ J − ( q ). p ∈ M is verifiable iff p ∈ P ( q ) and p ∈ I + ( q ) \ J − ( q ).Manchak observes the following. Proposition 2.4 (Manchak [10]) . If P ( q ) ∩ ( I + ( q ) \ J − ( q )) = ∅ , then ( M, g ) admits anedgless compact achronal set. A spacetime is causally bizarre if there is a point p ∈ M such that M ⊂ I − ( p ). MARTIN LESOURD
A slightly different notion of prediction considered in [10] is as follows.
Definition 2.5. p ∈ M is genuinely predictable from q , written P ( q ), iff p ∈ P ( q )and for all inextendible spacetimes ( M ′ , g ′ ), if there is an isometric embedding φ : J − ( q ) ֒ → M ′ , then there is an isometric embedding φ ′ : J − ( q ) ∪ J − ( p ) ֒ → M ′ such that φ = φ ′| J − ( q ) .The idea here is that genuine predictions guarantee that observers with the samepast make the same predictions. By a short cut-and-paste argument, Manchak observesthe following. Proposition 2.6 (Manchak [10]) . Let ( M, g ) be any spacetime and q a point in M .Then P ( q ) ⊆ ∂J − ( q ) . Thus the domain of genuine predictions from q , if non-empty, is on the verge of beinga retrodiction. 3. Some Observations
We assume that spacetimes satisfy the field equations(3.1) G g ≡ Ric g − g Scal g = T φ,... − Λ g where Λ ∈ R is a constant, and with T φ,... the stress-energy tensor associated withpossible matter fields { φ, ... } in M . The system (3.1) leads to the formulation of aCauchy problem on a spacelike initial data set S . In the vacuum setting T = 0, Λ = 0,the Cauchy problem was shown [3] to be well posed in the sense that there exists aunique, up to isometry, maximal globally hyperbolic development of S obtained byCauchy evolution of the initial data on S according to (3.1) with T = 0, Λ = 0.This well posedness has been extended to more general settings and we take it as apre-condition for the spacetimes we consider. Definition 3.1.
Fix the constant Λ and fix an expression for T φ,... . Given a spacetime( M, g ) and an acausal edgeless connected set S ⊂ M , we say that ( ˜ D ( S ) , g | ˜ D ( S ) ) is a faithful development if it is uniquely determined, up to isometry, by Cauchy evolutionof the initial data on S according to (3.1). If ( M, g ) admits a connected acausaledgeless set, we say that it is locally Cauchy if, for any connected acausal edgeless set S , ( ˜ D ( S ) , g | ˜ D ( S ) ) is a faithful development. Remark . Note that this is slightly unorthodox in the sense that it is usually D ( S ),rather than ˜ D ( S ), which we think of as being determined by S . Given that S is closedfor the definition of locally Cauchy, we have D ( S ) = ˜ D ( S ), and so being locally Cauchyis only slightly stronger than asking for D ( S ) to be determined up to isometry. Remark . Since we want to guarantee isometric embeddings, we want to rule out ex-amples of regions that are globally hyperbolic but not determined by Cauchy evolution. Here we think of the Cauchy problem from the perspective of ‘initial data’, as opposed to ‘initialand boundary data’, though the latter is more natural for Λ < BSERVATIONS AND PREDICTIONS FROM PAST LIGHTCONES 5
To see a trivial example smooth functions transformations Ω , start with Minkowskispacetime ( R ,n , η ), identify some open set O ⊂ R ,n lying above t = 0, and modifyit by a conformal transformation η → Ω η . In that case, in spite of M \ O being vac-uum, O will in general have a non-vanishing Einstein tensor G g . But since there areCauchy hypersurfaces (in M \ O ) for ( R ,n , Ω η ) which are exactly flat , ( R ,n , Ω η ) isnot locally Cauchy if it is not isometric to ( R ,n , η ).We also make use of the following. Definition 3.4.
Given a partial Cauchy hypersurface S in a spacetime M we say thatan open subset of M is an ǫ -development of S , denoted D ǫ ( S )( ⊃ S ) if there is existsan ǫ > ∈ R ) such that D ǫ ( S ) admits a Cauchy surface S ǫ every point of which lies atdistance ≥ ǫ > S , as measured by a normalized timelike vector fieldorthogonal to S . Remark . The non-empty interior of a causal diamond J + ( p ) ∩ J − ( q ) = ∅ with p, q ∈ ( R ,n , η ) is not an ǫ -development because its Cauchy surfaces are anchored at ∂J + ( p ) ∩ ∂J + ( q ).We now observe the following. Proposition 3.6.
Let ( M, g ) and ( M ′ , g ′ ) be inextendible and locally Cauchy. Supposethat(i) ( M, g ) has a compact Cauchy surface and no null lines,(ii) ( M ′ , g ′ ) is causal and hole-free.Then ( M, g ) and ( M ′ , g ′ ) are isometric iff they are weakly observationally indistinguish-able. Note that by the examples in [8], Proposition 3.6 is false without the assumptionthat (
M, g ) lacks null lines.In either case of weakly o.i. or strongly o.i., it would be interesting to settle whetherthe compactness in (i) is necessary, cf. Proposition 3.13 below.
Proof.
The proof of Proposition 3.6 starts by strengthening Theorem 1 of [6]. Lemma 3.7.
Let ( M, g ) be a spacetime without null lines. Then given any compactregion K , there exists another compact K ′ ⊃ K such that if p, q / ∈ K ′ and q ∈ J + ( p ) − I + ( p ) , then any causal curve γ connecting p to q cannot intersect K .Proof. Suppose otherwise for some K and K ⊃ K . Then consider a sequence ofever bigger compact sets K i +1 ⊂ K ′ i . By assumption, each K i will have horismos Examples like this suggest the following problem. Given a Lorentzian manifold (
M, g ) with anarbitrary geodesically complete Lorentzian metric g , what conditions on M and g make it possible tosolve for a function Ω : M → R such that ( M, Ω g ) is complete and vacuum? In the language of the constraints, initial data sets of the form ( R n , g E , In [6], the authors assume that (
M, g ) is null geodesically complete, satisfies the null energy con-dition, and the null generic condition. These assumptions implies the absence of null lines. It wasthen observed by Galloway that one can prove the theorem by instead assuming that (
M, g ) lacks nulllines (cf. footnote for Theorem 1 of [6]), but since those details never appeared we include them forcompleteness. The future horismos of p is defined as J + \ I + ( p ). MARTIN LESOURD related outer points p i , q i / ∈ K i , q i ∈ J − ( p i ) − I − ( p i ) that are connected by a causalcurve, necessarily achronal, that intersects K . In considering larger compact sets, wemake these causal curves longer in the sense of an auxiliary Riemannian metric. Allof these curves intersect K . Taking the limit, by compactness of K , there is a limitpoint for the sequence of points lying in K for each causal curve linking p i , q i , whichmoreover is in K . By a standard limit curve arguments, cf. Proposition 3.1 of [1],there passes an inextendible limit curve through this limit point. The limit curve isalso straightforwardly seen to be achronal. Thus we have a null line. (cid:3)
With the same proof as in Corollary 1 of [6], Lemma 3.7 implies the following. Lemma 3.8.
Let ( M, g ) be a spacetime with compact Cauchy surface that does notadmit any null lines. Then ( M, g ) admits a point p ∈ M such that S ⊂ I − ( p ) for someCauchy surface S .Proof. We include this for completeness, but this argument is exactly as in [6] exceptthat Lemma 3.7 plays the role of Theorem 1 of [6]. Since (
M, g ) is globally hyperbolic,there exists a continuous global time function t : M → R , such that each surface ofconstant t is a Cauchy surface. Let K = Σ and K ′ be as in Lemma 3.7. Let t and t denote, respectively, the minmum and maximum values of t on K ′ . Let Σ beany Cauchy surface with t < t and let Σ denote the Cauchy surface t = t . Let q ∈ I + (Σ ), p ∈ Σ and suppose that p ∈ ∂I − ( q ). Since ( M, g ) is globally hyperbolic, J − ( q ) is closed and so p ∈ J − ( q ) \ I − ( q ) and thus there is a causal curve connecting p and q . It follows from Lemma 3.7 that this causal curve does not intersect Σ. However,this contradicts the fact that Σ is a Cauchy surface. Consequently, there cannot exista p ∈ Σ and such that p ∈ ∂I − ( q ), i.e., ∂I − ( q ) ∩ Σ = ∅ . But I − ( q ) is open and since ∂I − ( q ) ∩ Σ = ∅ , the complement of I − ( q ) in Σ is also open. Since we have I − ( q ) ∩ Σ and Σ is connected, this implies Σ ⊂ I − ( q ). (cid:3) We now finish the proof of Proposition 3.6. By Lemma 3.8 there is a point p ∈ M with S ⊂ I − ( p ) where S is a Cauchy surface of M . φ ( S ) is compact . By weak observational indistinguishability, there is an isometricembedding φ : I − ( p ) ֒ → M ′ . Because φ is a diffeomorphism onto its image, the map φ | S , φ restricted to S , is a diffeomorphism of S onto its image and thus φ ( S ) is compactin M ′ . φ ( S ) is achronal. Suppose otherwise that γ ′ is a past directed timelike curvefrom x ′ to y ′ with x, y ∈ φ ( S ). Extend γ ′ to σ ′ so that σ ′ is a past directed timelikecurve from q ′ to x ′ to y ′ . Note that the isometric embedding forces φ ( S ) to be locallyachronal; that is, there is an open neighborhood O around around S with two con-nected boundary components ∂ + O ( ⊂ I + ( S )), ∂ − O (allocated using the orientation in M ), such that no two distinct points in O can be joined by a timelike curve in O . Since O isometrically embeds into M ′ , a locally achronal neighborhood O ′ exists around φ ( S )in M ′ . As such, the curve σ ′ must leave ∂ − O ′ and re-enter O ′ via either ∂ − O ′ or ∂ + O ′ . See [11] for significantly stronger limit curve statements. Lemma 3.8 strengthens the main result of [6].
BSERVATIONS AND PREDICTIONS FROM PAST LIGHTCONES 7
In the former case, we can build a closed piecewise smooth timelike curve from q ′ and back, which violates causality of M ′ .In the latter case, we will obtain a contradiction with the inextendibility of M . Since I − ( S ) ⊂ D − ( S ) ⊂ I − ( p ), we must have that σ ′ leaves φ ( D − ( S )), say at some point r ′ ∈ ∂φ ( D − ( S )), and re-enter φ ( I − ( p )) ∩ I + ( φ ( S )). Since the global hyperbolicity of M implies that every future directed timelike curve in φ ( I − ( p )) must eventually leave φ ( I − ( p )) when sufficiently extended in the future direction, we can take the re-entrypoint to lie on the ‘future’ boundary of φ ( I − ( p )); that is, which is the endpoint of atimelike curve whose φ − pre-image has endpoint on ∂I − ( p ). Now consider an openneighborhood Z ′ of r ′ in M ′ ∩ ∂φ ( D − ( S )). Consider the open subset φ ( I − ( p )) ∪ Z ′ of M ′ . Now define a new spacetime M ′′ by φ ( I − ( p )) ∪ Z ′ ∪ J + ( ∂I − ( p )). We know this canbe done because the global hyperbolicity of M and the locally Cauchy property of M and M ′ imply that the regions I − ( p ) and φ ( I − ( p )) \ ∂φ ( D − ( S )) are isometric. We nowhave a spacetime M ′′ into which M can be isometrically embedded as a proper sub-set (in virtue of the extra Z ′ beyond φ ( D − ( S ))), contradicting the inextendibility of M . φ ( S ) is edgeless. Compactness of S and achronality of φ ( S ) straightforwardlyimplies that φ ( S ) is edgeless. Remark . At this point if ( M ′ , g ′ ) is assumed globally hyperbolic, φ ( S ) being anedgeless compact connected achronal set means that φ ( S ) can be taken to be a Cauchysurface of M ′ . In that case, both ( M, g ) and ( M ′ , g ′ ) are representatives of the unique,up to isometry, maximal globally hyperbolic development of S , and are thus isometric.We will instead show that ( M ′ , g ′ ) is globally hyperbolic.In ( M, g ) we can consider an ǫ -development D ǫ ( S ) ⊂ M . Within D ǫ ( S ), we can thenfind an acausal hypersurface S ǫ which is still Cauchy for M . We know that D ǫ ( S ) ⊂ M isometrically embeds in M ′ as a small neighborhood around φ ( S ). The image φ ( S ǫ )of S ǫ , now denoted S ′ ǫ , is acausal in M ′ (by a causal version of the argument for theachronality of φ ( S )) and edgeless. We now have partial Cauchy surfaces S ǫ and S ′ ǫ in M and M ′ respectively. We have the inclusion D − ( S ′ ǫ ) ⊇ M ′ \ I + ( S ′ ǫ ). Since ( M, g ) is globally hyperbolicand inextendible, we have D − ( S ǫ ) ⊇ J − ( S ǫ ). Since J − ( S ǫ ) isometrically embeds into( M ′ , g ′ ), if D − ( S ′ ǫ ) fails to cover M ′ \ I + ( S ′ ǫ ), then S ′ ǫ must have a past Cauchy horizon H − ( S ′ ǫ ) = ∅ in M ′ . In that case, by the locally Cauchy property we can isometricallyembed S ′ ǫ and ˜ D − ( S ′ ǫ ) back into M using φ − , and by the global hyperbolicity andinextendibility of M , we have that D − ( φ − ( S ′ ǫ )) ⊃ φ − ( H − ( S ′ ǫ )), which contradictsthe past hole-freeness of ( M ′ , g ′ ). We have the inclusion D + ( S ′ ǫ ) ⊇ M ′ \ I − ( S ′ ǫ ) . This follows by the same argument.Since we now have D ( S ǫ ) = M and D ( S ′ ǫ ) = M ′ , the conclusion follows from lo-cally Cauchy. (cid:3) In view of the role of null lines, we note the rigidity theorem of Galloway-Solis [5].
Theorem 3.10 (Galloway-Solis [5]) . Assume that the -dimensional spacetime ( M , g ) MARTIN LESOURD (i) satisfies (3.1) with T = 0 and Λ > ,(ii) is asymptotically de-Sitter ,(iii) is globally hyperbolic,(iv) there is a null line with endpoints on J + and J − .Then ( M , g ) isometrically embeds as an open subset of the de-Sitter spacetime con-taining a Cauchy surface. Together, Proposition 3.6 and Theorem 3.10 imply the following.
Corollary 3.11.
Given two -dimensional spacetimes ( M, g ) and ( M ′ , g ′ ) assume that(i) ( M, g ) and ( M ′ , g ′ ) satisfy (3.1) with T = 0 and Λ > ,(ii) ( M, g ) is inextendible and has a compact Cauchy surface,(iii) ( M, g ) is asymptotically de-Sitter but not isometric to de-Sitter,(iv) ( M ′ , g ′ ) is inextendible, causal and hole-free.Then ( M, g ) and ( M ′ , g ′ ) are isometric iff they are weakly observationally indistinguish-able. Note that the assumptions of Corollary 3.11 just falls short of astrophysical relevanceon account of the assumption that (
M, g ) be past asymptotically de-Sitter, which isnot supported by current data. A more desirable statement would be welcome. (B) . We say that a spacetime ( M, g ) is
Cauchy friendly if it is weakly locally Cauchy and there are no points p ∈ M such that J − ( p ) ⊇ M . We now show the following,which guarantees that genuine predictions extend a little beyond what is suggested inProposition 2.6. Proposition 3.12.
Given two Cauchy friendly spacetimes ( M, g ) and ( M ′ , g ′ ) , assumethat(i) there is a partial Cauchy surface S ⊂ J − ( q ) ⊆ D ( S ) for some point q ∈ M ,(ii) there is an isometry φ : J − ( q ) → J − ( q ′ ) .Then there is an isometric embedding ψ : A ֒ → M ′ for some A ) J − ( q ) such that • ψ | J − ( q ) = φ , • A and ψ ( A ) contain points in the domain of verifiable prediction of q and q ′ .Proof. First we make some basic observations, and throughout we denote S ′ ≡ φ ( S ). We have J − ( q ) ( D ( S ). Since J − ( q ) lies in a globally hyperbolic set D ( S ), J − ( q ) is closed. Moreover, since S is acausal and edgeless, D ( S ) must be open. Fromthis it follows that the inclusion of J − ( q ) ⊆ D ( S ) is strict. Suppose otherwise that J − ( q ) = D ( S ). In that case D ( S ) is both open and closed in M , and since M is con-nected, that implies D ( S ) = M . But then M = J − ( q ), in contradiction with Cauchy Generalizations to Einstein-Maxwell are possible. cf. [5] for definitions of asymptotically de-Sitter and the associated hypersurfaces J +( − ) in thatcontext. [13] contains results precluding the existence of null lines based on astrophysically interestingassumptions. Weakly locally Cauchy replaces ˜ D ( S ) with D ( S ). BSERVATIONS AND PREDICTIONS FROM PAST LIGHTCONES 9 friendly. Thus J − ( q ) is a closed proper subset of the open set D ( S ). S ′ is a partial Cauchy hypersurface. Since φ is a diffeomorphism onto itsimage, we know that S ′ is compact. Unlike the arguments given in Proposition 3.6, φ ( S ) is acausal and edgeless by the fact that φ is an isometry (as opposed to merely anembedding). Since S ′ belongs to J − ( q ′ ), if S ′ is acausal in M ′ , then S is also acausalin M , which contradicts (ii). Thus S ′ is a partial Cauchy surface in M ′ . We have J − ( x ′ ) ∩ J + ( S ′ ) ⊆ D + ( S ′ ) for any x ′ ∈ φ ( J − ( q )). We seek to show thatany past inextendible causal curve in J − ( x ′ ) with with future endpoint x ′ ∈ φ ( J − ( q ))intersects S ′ . By assumption, we have an isometric embedding φ : J − ( q ) ∩ D + ( S ) ֒ → M ′ where S is a closed achronal set. Consider any point x ∈ J − ( q ) ∩ J + ( S ). By theisometry φ , we have φ ( J − ( x )) = J − ( x ′ ). Note first that by definition, there is a neigh-borhood of 0 ∈ T x M such that the exponential map of the past non-spacelike vectorsin that neighborhood is contained in D + ( S ). By the isometry φ , the same is true for x ′ with respect to φ ( D + ( S )), in particular there is a neighborhood U x ′ of x ′ such that U x ′ ∩ J − ( x ′ ) ⊂ φ ( D + ( S )). Seeking a contradiction, suppose there is a past inextendiblecausal curve γ ′ with future endpoint x ′ that does not intersect S ′ . By the propertyaforementioned, there is at least a segment of γ ′ contained in φ ( D + ( S )). The curvedefined by γ ≡ φ − ( γ ′ ) is causal and ends at x , and is thus entirely contained in D + ( S ).But then γ does not intersect S , which is a contradiction. Similarly, we have J − ( S ) ⊆ D − ( S ′ ) . This follows from (ii), the isometry φ ,and the argument just above. We have J − ( q ′ ) ( D ( S ′ ) . This proceeds as above, which now leads to a con-tradiction with the Cauchy friendliness of M ′ .We have two closed sets J − ( q ′ ) and J − ( q ) each strictly contained in the open sets D ( S ′ ) and D ( S ). We now seek to show the existence of ψ . Although there may be noisometric embedding of D ( S ) into M ′ , we need only show that it is possible to non-trivially extend the pre-image of φ beyond J − ( q ), which will thus enter D ( S ) \ J − ( q ).Consider now the unique (up to isometry) maximal globally hyperbolic development X ( S ) of S , where X ( S ) denotes one representative among all isometric developments.By the isometry φ , we know that S and and S ′ are isometric as initial data sets, andthus that X ( S ) is the unique (up to isometry) maximal globally hyperbolic develop-ment of both S and S ′ . By locally Cauchy, it follows that both D ( S ) and D ( S ′ ) canbe isometrically embedded into X ( S ). Denote these isometries by ρ : D ( S ) ֒ → X ( S )and ρ ′ : D ( S ′ ) ֒ → X ( S ).It is obvious that ρ ( D ( S )) ∩ ρ ′ ( D ( S ′ )) = ∅ and since ρ and ρ ′ are local diffeomor-phisms, both ρ ( D ( S )) and ρ ( D ( S ′ )) are open in X ( S ), and moreover both ρ ( J − ( q ))and ρ ′ ( J − ( q ′ ) are closed in X ( S ). We now define the following set in X ( S )[ ρ ( D ( S )) − ρ ( J − ( q ))] ∩ [ ρ ′ ( D ( S ′ )) − ρ ( J − ( q ′ ))] ≡ I Note here that we could use a weaker version of locally Cauchy here that involves D ( S ) ratherthan ˜ D ( S ). The openness of ρ ( D ( S )) , ρ ′ ( D ( S ′ )), and the fact that we may choose ρ, ρ ′ such that ρ ( J − ( q )) = ρ ′ ( J − ( q ′ )) means that the strict inclusions D ( S ) \ J − ( q ) , D ( S ′ ) \ J − ( q ′ ) and = ∅ extend to ρ and ρ ′ , i.e., ρ ( D ( S )) \ ρ ( J − ( q )) = ∅ and ρ ( D ( S ′ )) \ ρ ( J − ( q ′ )) = ∅ . Itfollows that I = ∅ .We can now identify the set A ≡ ρ − [ ρ ( J − ( q )) ∪ I ] as having the desired properties,i.e. there exists an isometric embedding of ψ : A ֒ → M ′ such that ψ | J − ( q ) = φ . (cid:3) We can also consider what happens after lifting the compactness assumption on S . Proposition 3.13.
Let S be a partial Cauchy hypersurface in M and D ǫ ( S ) an ǫ -development of S . Let φ : D ǫ ( S ) ֒ → M ′ be an isometric embedding into a hole-freespacetime M ′ . Then either • φ ( S ) is causal in M ′ , • or φ ( S ) is a partial Cauchy hypersurface in M ′ .In the latter case, if M, M ′ are locally Cauchy and M ′ is inextendible, then there is anisometric embedding ψ : D ( S ) ֒ → M ′ with ψ | D ǫ ( S ) = φ . Thus, after basic assumptions like hole-freeness, locally Cauchy and inextendibility,the only obstruction concerns the acausality of φ ( S ). It may be that φ ( S ) is alwaysacausal if M ′ satisfies some causality assumption, eg. causally simple, causally contin-uous , etc.Note also that φ ( S ) need not be a partial Cauchy hypersurface if we replace D ǫ ( S )by ‘an open globally hyperbolic subset of D ( S )’ (delete a half-space at t = 0 from( R ,n , η )). Proof.
First we show the second statement. If φ ( S ) is a partial Cauchy hypersurface, weknow that D ( S ) and D ( φ ( S )) are both open subsets of M and M ′ respectively, and since M, M ′ are locally Cauchy, we know that D ( S ) and D ( φ ( S )) share a common (isometric)subset extending beyond D ǫ ( S ). Let D ( S ) denote the maximal open globally hyperbolicsubset of D ( S ) for which there is an isometric embedding ψ ′ : D ( S ) ֒ → M ′ .Now suppose that D ( S ) does not isometrically embed into M ′ , i.e. D ( S ) ( D ( S ).If H ( ψ ′ ( S )) = ∅ then by locally Cauchy we can use ψ ′− to embed ˜ D ( ψ ′ ( S )) into M and contradict the hole-freeness of M ′ . If H ( ψ ′ ( S )) = ∅ , then M ′ = D ( ψ ′ ( S )). Butin that case, by locally Cauchy, M ′ isometrically embeds into M as a proper subset of M , contradicting the inextendibility of M ′ .Now we show the first part: if φ ( S ) is acausal, then it is a partial Cauchy hypersurfacein M ′ . Supposing that edge( φ ( S )) = ∅ , we will show that edge( φ ( S )) ∩ φ ( S ) = ∅ , andwe then show that this implies ( M ′ , g ′ ) is holed.Let q ′ be a point in edge( φ ( S )) ∩ φ ( S ). In that case denote q = φ − ( q ′ ) ∈ S and takea future directed timelike curve σ from q to q ǫ ∈ I + ( S ) ∩ D ǫ ( S ), and set σ to be pastinextendible in M . Then there is a timelike curve σ ′ = φ ( σ ) ⊂ M ′ passing through q ′ .Take U i ( q ′ ) to be a system of increasingly small neighborhoods U i ( q ′ ) ) U i +1 ( q ′ ), eachcontaining points in I + ( q ′ ) and I − ( q ′ ), such that { U i ( q ′ ) } has accumulation point q ′ .Define a collection of curves { γ ′ i } by taking σ ′ , removing from σ ′ the portion σ ′ ∩ U i ( q ′ ),and replacing that portion with timelike segments with endpoints in I + ( q ′ ) and I − ( q ′ )which miss φ ( S ). Although this might produce only piecewise smooth timelike curves, cf. pg. 59 of [1] BSERVATIONS AND PREDICTIONS FROM PAST LIGHTCONES 11 the curves { γ ′ i } can be approximated by C causal curves (still missing φ ( S )), which werelabel as { γ ′ i } . Now consider { φ − ( γ ′ i ) } . This defines a collection of C causal curvesin D ǫ ( S ) that approach σ but which do not intersect S .We now recall a well known fact. In any spacetime L , there exists a sufficiently smallneighborhood N ( x ) ⊂ L around some point x such that J + ( y ) ∩ J − ( z ) is compact forall y , z ∈ N ( x ). Since a sufficiently small N ( x ) is causal, every point in a spacetimelives in a small globally hyperbolic neighborhood.Consider such a globally hyperbolic neighborhood N ( q ) ⊂ M centered at q . Withoutloss of generality, we can take N ( q ) to have Cauchy surface S ∩ N ( q ). For somesufficiently large n ∈ N , the causal curves { φ − ( γ ′ i ≥ n ) } lie in N ( q ) and are inextendibletherein. But since these do not intersect S , we contradict the global hyperbolicity of N ( q ).It now follows that edge( φ ( S )), if not empty, lies outside of φ ( S ). By standard resultsin causal theory, H ( φ ( S )) is ruled by null geodesics intersecting edge( φ ( S )). By thedefinition of D ǫ ( S ), we can use φ to pull back H ( φ ( S )) ∩ φ ( D ǫ ( S )) into D ǫ ( S ) ∩ M . Bythe definition of D ǫ ( S ), it is then clear that φ − [ H ( φ ( S )) ∩ φ ( D ǫ ( S )] ∩ D ( S ) = ∅ , andthus ( M ′ , g ′ ) is holed. (cid:3) References [1] Beem, J., Ehrlich, P., Easley, K.,
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