Chapter 2 Lorentzian Manifolds

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1 Chater 2 Lorentzian Manifolds Frank Pfäffle In this chater some basic notions from Lorentzian geometry will be reviewed. In articular causality relations will be exlained, Cauchy hyersurfaces and the concet of global hyerbolic manifolds will be introduced. Finally the structure of globally hyerbolic manifolds will be discussed. More comrehensive introductions can be found in [1] and [2]. 2.1 Preliminaries on Minkowski Sace Let V be an n-dimensional real vector sace. A Lorentzian scalar roduct on V isa nondegenerate symmetric bilinear form, of index 1. This means one can find a basis e 1,...,e n of V such that 1 ifi = j = 1, e i, e j = 1 ifi = j = 2,...,n, 0 otherwise. The simlest examle for a Lorentzian scalar roduct on R n is the Minkowski roduct, 0 given by x, y 0 = x 1 y 1 + x 2 y x n y n. In some sense this is the only examle because from the above it follows that any n-dimensional vector sace with Lorentzian scalar roduct (V,, ) is isometric to Minkowski sace (R n,, 0 ). We denote the quadratic form associated with, by γ : V R, γ(x) := X, X. A vector X V \{0} is called timelike if γ (X) > 0, lightlike if γ (X) = 0 and X 0, causal if timelike or lightlike, and sacelike if γ (X) > 0orX = 0. F. Pfäffle (B) Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, D Potsdam, Germany faeffle@math.uni-otsdam.de Pfäffle, F.: Lorentzian Manifolds. Lect. Notes Phys. 786, (2009) DOI / c Sringer-Verlag Berlin Heidelberg 2009

2 40 F. Pfäffle Fig. 2.1 Lightcone in Minkowski sace I + (0) C + (0) 0 C (0) I (0) For n 2thesetI (0) of timelike vectors consists of two connected comonents. We choose a time-orientation on V by icking one of these two connected comonents. Denote this comonent by I + (0) and call its elements future-directed. We ut J + (0) := I + (0), C + (0) := I + (0), I (0) := I + (0), J (0) := J + (0), and C (0) := C + (0). Causal vectors in J + (0) (or in J (0)) are called future-directed (or ast-directed resectively). (See Fig. 2.1.) Remark 1. Given a ositive number α>0 and a Lorentzian scalar roduct, on a vector sace V one gets another Lorentzian scalar roduct α,. One observes that X V is timelike with resect to, if and only if it is timelike with resect to α,. Analogously, the notion lightlike coincides for, and α,, and so do the notions causal and sacelike. Hence, for both Lorentzian scalar roducts one gets the same set I (0). If dim(v ) 2 and we choose identical time-orientations for, and α,, thesetsi ± (0), J ± (0), C ± (0) are determined indeendently whether formed with resect to, or α,. 2.2 Lorentzian Manifolds A Lorentzian manifold is a air (M, g) where M is an n-dimensional smooth manifold and g is a Lorentzian metric, i.e., g associates with each oint M a Lorentzian scalar roduct g on the tangent sace T M. One requires that g deends smoothly on : This means that for any choice of local coordinates x = (x 1,...,x n ):U V, where U M and V R n are oen subsets, and for any i, j = 1,...,n the functions g ij : V R defined by g ij = g( x i, x j ) are smooth. Here x i and x j denote the coordinate vector fields as usual

3 2 Lorentzian Manifolds 41 x 2 U x V R n x 1 M x() Fig. 2.2 Coordinate vectors x 1, x 2 (see Fig. 2.2). With resect to these coordinates one writes g = g ij dx i dx j or i, j shortly g = g ij dx i dx j. i, j Next we will give some rominent examles for Lorentzian manifolds. Examle 1. In cartesian coordinates (x 1,...,x n )onr n the Minkowski metric is defined by g Mink = (dx 1 ) 2 + (dx 2 ) 2 + +(dx n ) 2. This turns Minkowski sace into a Lorentzian manifold. Of course, the restriction of g Mink to any oen subset U R n yields a Lorentzian metric on U as well. Examle 2. Consider the unit circle S 1 R 2 with its standard metric (dθ) 2.The Lorentzian cylinder is given by M = S 1 R together with the Lorentzian metric g = (dθ) 2 + (dx) 2. Examle 3. Let (N, h) be a connected Riemannian manifold and I R an oen interval. For any t I, N one identifies T (t,) (I N) = T t I T N. Then for any smooth ositive function f : I (0, ) the Lorentzian metric g = dt 2 + f (t) 2 h on I M is defined as follows: For any ξ 1,ξ 2 T (t,) (I N) one writes ξ i = ( ) d α i dt ζi with α i R and ζ i T N, i = 1, 2, and one has g(ξ 1,ξ 2 ) = α 1 α 2 + f (t) 2 h(ζ 1,ζ 2 ). Such a Lorentzian metric g is called a wared roduct metric (Fig. 2.3). This examle covers Robertson Walker sacetimes where one requires additionally that (N, h) is comlete and has constant curvature. In articular Friedmann cosmological models are of this tye. In general relativity they are used to discuss big bang, exansion of the universe, and cosmological redshift; comare [2, Chas. 5 and 6] or [1, Cha. 12]. A secial case of this is desitter sacetime where I = R, N = S n 1, h is the canonical metric of S n 1 of constant sectional curvature 1, and f (t) = cosh(t). t M {t} N I N Fig. 2.3 Wared roduct

4 42 F. Pfäffle Examle 4. For a fixed ositive number m > 0 one considers the Schwarzschild function h :(0, ) R given by h(r) = 1 2m. This function has a ole at r = 0 r and one has h(2m) = 0. On both P I ={(r, t) R 2 r > 2m} and P II ={(r, t) R 2 0 < r < 2m} one defines Lorentzian metrics by g = h(r) dt dt + 1 dr dr, h(r) and one calls (P I, g) Schwarzschild half-lane and (P II, g) Schwarzschild stri.for a tangent vector α t + β r being timelike is equivalent to α2 > 1 β 2. Hence, one h(r) 2 can illustrate the set of timelike vector in the tangent saces T (r,t) P I, res., T (r,t) P II as in Fig The singularity of the Lorentzian metric g for r = 2m is not as crucial as it might seem at first glance, by a change of coordinates one can overcome this singularity (e.g., in the so-called Kruskal coordinates). One uses (P I, g) and (P II, g) to discuss the exterior and the interior of a static rotationally symmetric black hole with mass m, comare [1, Cha. 13]. For this one considers the two-dimensional shere S 2 with its natural Riemannian metric can S 2, and on both N = P I S 2 and B = P II S 2 one gets a Lorentzian metric by h(r) dt dt + 1 h(r) dr dr + r 2 can S 2. Equied with this metric, N is called Schwarzschild exterior sacetime and B Schwarzschild black hole, both of mass m. Examle 5. Let S n 1 = {(x 1,...,x n ) R n (x 1 ) (x n ) 2 = 1} be the n-dimensional shere equied with its natural Riemannian metric can S n 1. The restriction of this metric to S+ n 1 ={(x 1,...,x n ) S n 1 x n > 0} is denoted by can S n 1. Then, on R Sn one defines a Lorentzian metric by g AdS = 1 ( (x n ) 2 dt 2 + can S n 1 + ), t Fig. 2.4 Lightcones for Schwarzschild stri P II and Schwarzschild half-lane P I O m 2m r

5 2 Lorentzian Manifolds 43 and one calls (R S+ n 1, g AdS )then-dimensional anti-desitter sacetime. This definition is not exactly the one given in [1, Cha. 8,. 228f.], but one can show that both definitions coincide; comare [3, Cha. 3.5.,. 95ff.]. By Remark 1 we see that a tangent vector of R S+ n 1 is timelike (lightlike, sacelike) with resect to g AdS if and only if it is so with resect to the Lorentzian metric dt 2 + can S n 1. + In general relativity one is interested in four-dimensional anti-desitter sacetime because it rovides a vacuum solution of Einstein s field equation with cosmological constant Λ = 3; see [1, Cha. 14, Examle 41]. 2.3 Time-Orientation and Causality Relations Let (M, g) denote a Lorentzian manifold of dimension n 2. Then at each oint M the set of timelike vectors in the tangent sace T M consists of two connected comonents, which cannot be distinguished intrinsically. A time-orientation on M is a choice I + (0) T M of one of these connected comonents which deends continuously on. A time-orientation (Fig. 2.5) is given by a continuous timelike vector field τ on M which takes values in these chosen connected comonents: τ() I + (0) T M for each M. Definition 1. One calls a Lorentzian manifold (M, g) time-orientable if there exists a continuous timelike vector field τ on M. A Lorentzian manifold (M, g) together with such a vector field τ is called time-oriented. In what follows time-oriented connected Lorentzian manifolds will be referred to as sacetimes. It should be noted that in contrast to the notion of orientability which only deends on the toology of the underlying manifold the concet of time-orientability deends on the Lorentzian metric. T M M U Fig. 2.5 Time-orientation M

6 44 F. Pfäffle identify Fig. 2.6 Examle for orientable and time-orientable manifold If we go through the list of examles from Sect. 2.2, we see that all these Lorentzian manifolds are time-orientable (Fig. 2.7). Timelike vector fields can be given as follows: on Minkowski sace by x 1, on the Lorentzian cylinder by θ,on the wared roduct in Examle 3 by d dt, on Schwarzschild exterior sacetime by t, on Schwarzschild black hole by r, and finally on anti-desitter sacetime by t. From now on let (M, g) denote a sacetime of dimension n 2. Then for each oint M the tangent sace T M is a vector sace equied with the Lorentzian scalar roduct g and the time-orientation induced by the lightlike vector τ(), and in (T M, g ) the notions of timelike, lightlike, causal, sacelike, future-directed vectors are defined as exlained in Sect Definition 2. A continuous iecewise C 1 -curve in M is called timelike, lightlike, causal, sacelike, future-directed,or ast-directed if all its tangent vectors are timelike, lightlike, causal, sacelike, future-directed, or ast-directed, resectively. Fig. 2.7 Lorentzian manifold which is orientable, but not time-orientable Fig. 2.8 Lorentzian manifold which is not orientable, but time-orientable

7 2 Lorentzian Manifolds 45 The causality relations on M are defined as follows: Let, q M, then one has q : there is a future-directed timelike curve in M from toq, < q : there is a future-directed causal curve in M from toq, q : < q or = q. These causality relations are transitive as two causal (timelike) curves in M,saythe one from 1 to 2 and the other from 2 to 3, can be ut together to a iecewise causal (timelike) C 1 -curve from 1 to 3. Definition 3. The chronological future I+ M (x) of a oint x M is the set of oints that can be reached from x by future-directed timelike curves, i.e., I M + (x) = {y M x < y}. Similarly, the causal future J+ M (x)of a oint x M consists of those oints that can be reached from x by future-directed causal curves and of x itself: J M + (x) = {y M x y}. The chronological future of a subset A M is defined to be I+ M (A) := I+ M (x). x A Similarly, the causal future of A is J+ M (A) := J+ M (x). x A The chronological ast I M(x) res. I M(A) and the causal ast J M(x) res. J M(A) are defined by relacing future-directed curves by ast-directed curves. For A M one also uses the notation J M (A) = J M + (A) J M (A). Remark 2. Evidently, for any A M one gets the inclusion A I+ M (A) J + M (A). Examle 6. We consider Minkowski sace (R 2, g Mink ). Then for R 2 the chronological future I+ R2() R2 is an oen subset, and for a comact subset A of the x 2 -axis the causal ast J R2(A) R2 is a closed subset, as indicated in Fig Examle 7. By Examle 1 every oen subset of Minkowski sace forms a Lorentzian manifold. Let M be two-dimensional Minkowski sace with one oint removed. Then there are subsets A M whose causal ast is not closed as one can see in Fig

8 46 F. Pfäffle Fig. 2.9 Chronological future () and causal ast I R2 + J R2 (A) x 1 I 2 + () A x 2 J 2 (A) Fig Causal future and ast of subset A of M J M + (A) A J M (A) Examle 8. If one unwras Lorentzian cylinder (M, g) = (S 1 R, dθ 2 + dx 2 1 ) one can think of M as a stri in Minkowski sace R 2 for which the uer and lower boundaries are identified. In this icture it can easily be seen that I+ M () = J + M () = I M() = J M () = M for any M; see Fig Any connected oen subset Ω of a sacetime M is a sacetime in its own right if one restricts the Lorentzian metric of M to Ω. Therefore J+ Ω (x) and J Ω(x)arewell defined for x Ω. Definition 4. A domain Ω M in a sacetime is called causally comatible if for all oints x Ω one has J± Ω (x) = J ± M (x) Ω. Note that the inclusion always holds. The condition of being causally comatible means that whenever two oints in Ω can be joined by a causal curve in M this can also be done inside Ω (Fig. 2.12). Identify! Fig J M + () = M

9 2 Lorentzian Manifolds 47 Fig Causally comatible subset of Minkowski sace J M + () Ω = J Ω + () Ω Fig Domain which is not causally comatible in Minkowski sace J M + () Ω J Ω + () + + If Ω M is a causally comatible domain in a sacetime, then we immediately see that for each subset A Ω we have J± Ω (A) = J ± M (A) Ω. Note also that being causally comatible is transitive: If Ω Ω Ω,ifΩ is causally comatible in Ω, and if Ω is causally comatible in Ω, then so is Ω in Ω. Next, we recall the definition of the exonential ma: For M and ξ T M let c ξ denote the (unique) geodesic with initial conditions c ξ (0) = and ċ ξ (0) = ξ. One considers the set D = { ξ T M cξ can be defined at least on [0, 1] } T M and defines the exonential ma ex : D M by ex (ξ) = c ξ (1). One imortant feature of the exonential ma is that it is an isometry in radial direction which is the statement of the following lemma. Lemma 1 (Gauss Lemma). Let ξ D and ζ 1,ζ 2 T ξ (T M) = T M with ζ 1 radial, i.e., there exists t 0 R with ζ 1 = t 0 ξ, then g ex (ξ) ( ) d ex ξ (ζ 1 ), d ex ξ (ζ 2 ) = g (ζ 1,ζ 2 ). A roof of the Gauss lemma can be found, e.g., in [1, Cha. 5,. 126f.]. Lemma 2. Let M and b > 0 and let c :[0, b] T M be a iecewise smooth curve with c(0) = 0 and c(t) D for any t [0, b]. Suose that c := ex c : [0, b] M is a timelike future-directed curve, then c(t) I + (0) T M for any t [0, b].

10 48 F. Pfäffle Fig In radial direction the exonential ma reserves orthogonality T M ζ 2 ζ 1 ζ 0 1 ξ ex dex ξ (ζ 2 ) ex (ξ) dex ξ (ζ 1 ) M Proof. Suose in addition that c is smooth. On T M we consider the quadratic form induced by the Lorentzian scalar roduct γ : T M R, γ (ξ) = g (ξ,ξ), and we comute grad γ (ξ) = 2ξ. The Gauss lemma alied for ξ D and ζ 1 = ζ 2 = 2ξ yields g ex (ξ) ( ) d ex ξ (ζ 1 ), d ex ξ (ζ 2 ) = g (ζ 1,ζ 2 ) = 4γ (ξ). Denote P(ξ) = d ex ξ (2ξ). Then by the above formula P(ξ) is timelike whenever ξ is timelike. From c(0) = 0 and (d/dt) c(0) = d ex 0 ((d/dt) c(0)) = ċ(0) I + (0) we get for a sufficiently small ε>0 that c(t) I + (0) for all t (0,ε). Hence P( c(t)) is timelike and future-directed for t (0,ε). For ξ = c(t), ζ 1 = 2ξ = grad γ (ξ) and ζ 2 = (d/dt) c(t) the Gauss lemma gives d dt (γ c) (t) = g ( ) (ζ 1,ζ 2 ) = g ex (ξ) P( c(t)), ċ(t). If there were t 1 (0, b] with γ ( c(t 1 )) = 0, w.l.o.g. let t 1 be the smallest value in (0, b] with γ ( c(t 1 )) = 0, then one could find a t 0 (0, t 1 ) with 0 = d dt (γ c) (t 0) = g ex (ξ)( P( c(t0 )), ċ(t 0 ) ). On the other hand, having chosen t 1 minimally imlies that P( c(t 0 )) is timelike and future-directed. Together with ċ(t 0 ) I M + (c(t 0)) this yields g ex (ξ)( P( c(t0 )), ċ(t 0 ) ) < 0, a contradiction. Hence one has γ ( c(t)) > 0 for any t (0, b], and the continuous curve c (0,b] does not leave the connected comonent of I (0) in which it runs initially. This

11 2 Lorentzian Manifolds 49 finishes the roof if one suoses that c is smooth. For the roof in the general case see [1, Cha. 5, Lemma 33]. Definition 5. A domain Ω M is called geodesically starshaed with resect to a fixed oint Ω if there exists an oen subset Ω T M, starshaed with resect to 0, such that the Riemannian exonential ma ex x mas Ω diffeomorhically onto Ω. One calls a domain Ω M geodesically convex (or simly convex) if it is geodesically starshaed with resect to all of its oints. Remark 3. Every oint of a Lorentzian manifold (which need not necessarily be a sacetime) ossesses a convex neighborhood, see [1, Cha. 5, Pro. 7]. Furthermore, for each oen covering of a Lorentzian manifold one can find a refinement consisting of convex oen subsets, see [1, Cha. 5, Lemma 10]. Sometimes sets that are geodesically starshaed with resect to a oint are useful to get relations between objects defined in the tangent T M and objects defined on M. For the moment this will be illustrated by the following lemma. Lemma 3. Let M a sacetime and M. Let the domain Ω M be a geodesically starshaed with resect to (Fig. 2.15). Let Ω be an oen neighborhood of 0 in T M such that Ω is starshaed with resect to 0 and ex Ω : Ω Ω is a diffeomorhism. Then one has I± Ω () = ex ( I± (0) Ω ) and J± Ω () = ex ( J± (0) Ω ). Proof. We will only rove the equation I+ Ω () = ex ( I+ (0) Ω ). T M 0 Ω ex Ω Fig Ω is geodesically starshaed with resect to M

12 50 F. Pfäffle For q I+ Ω () one can find a future-directed timelike curve c :[0, b] Ω from to q. We define the curve c :[0, b] Ω T M by c = ex 1 c and get from Lemma 2 that c(t) I + (0) for 0 < t b, in articular ex 1 (q) = c(b) I + (0). This shows the inclusion I+ Ω () ex ( I+ (0) Ω ). For the other inclusion we consider ξ I + (0) Ω. Then the ma t t ξ takes its values in I + (0) Ω as t [0, 1]. Therefore ex (tξ) gives a timelike future-directed geodesic which stays in Ω as t [0, 1], and it follows that ex (ξ) I+ Ω (). For a roof of J± Ω () = ex ( J± (0) Ω ) we refer to [1, Cha. 14, Lemma 2]. For Ω and Ω as in Lemma 3 we ut C± Ω () = ex (C ±(0) Ω ). Proosition 1. On any sacetime M the relation is oen, this means that for every, q M with q there are oen neighborhoods U and V of and q, resectively, such that for any U and q V one has q (Fig. 2.16). Proof. For, q M with q there are geodesically convex neighborhoods Ũ, Ṽ, resectively. We can find a future-directed timelike curve c from to q. Then we choose Ũ and q Ṽ sitting on c such that q q. AsŨ is starshaed with resect to there is a starshaed oen neighborhood Ω of 0 in T M such that ex : Ω Ũ is a diffeomorhism. We set U = I Ũ ( ), and Lemma 3 shows that U = ex (I (0) Ω) is an oen neighborhood of in M. Analogously, one finds that V = I Ṽ+ ( q) is an oen neighborhood of q. Finally, for any U and q V one gets q q and hence q. Corollary 1. For an arbitrary A M the chronological future I+ M (A) and the chronological ast I M (A) are oen subsets in M. Proof. Proosition 1 imlies that for any M the subset I+ M () M is oen, and therefore I+ M (A) = A I + M () is an oen subset of M as well. On an arbitrary sacetime there is no similar statement for the relation. Examle 7 shows that even for closed sets A M the chronological future and ast are V q q q c Fig The relation is oen U

13 2 Lorentzian Manifolds 51 not always closed. In general one only has that I± M (A) is the interior of J ± M (A) and that J± M (A) is contained in the closure of I ± M (A). Definition 6. A domain Ω is called causal if its closure Ω is contained in a convex domain Ω and if for any, q Ω the intersection J+ Ω Ω () J (q) is comact and contained in Ω. Causal domains aear in the theory of wave equations: The local construction of fundamental solutions is always ossible on causal domains rovided their volume is small enough, see Proosition 3 on age 71. Remark 4. Any oint M in a sacetime ossesses a causal neighborhood, comare [4, Theorem 4.4.1], and given a neighborhood Ω of, one can always find a causal domain Ω with Ω Ω (Fig. 2.17). The last notion introduced in this section is needed if it comes to the discussion of uniqueness of solutions for wave equations: Definition 7. A subset A M is called ast-comact if A J M () is comact for all M. Similarly, one defines future-comact subsets (Fig. 2.18). Ω q Ω Ω q Ω convex, but not causal causal Fig Convexity versus causality A Fig The subset A is ast-comact J M ()

14 52 F. Pfäffle 2.4 Causality Condition and Global Hyerbolicity In general relativity worldlines of articles are modeled by causal curves. If now the sacetime is comact something strange haens. Proosition 2. If the sacetime M is comact, there exists a closed timelike curve in M. Proof. The family {I+ M ()} M is an oen covering of M. By comactness one has M = I+ M ( 1) I+ M ( k) for suitably chosen 1,..., k M. We can assume that 1 I+ M ( 2) I+ M ( k), otherwise 1 I+ M ( m) for an m 2 and hence I+ M ( 1) I+ M ( m) and we can omit I+ M ( 1) in the finite covering. Therefore we can assume 1 I+ M ( 1), and there is a timelike future-directed curve starting and ending in 1. In sacetimes with timelike loos one can roduce aradoxes as travels into the ast (like in science fiction). Therefore one excludes comact sacetimes, for hysically reasonable sacetimes one requires the causality condition or the strong causality condition (Fig. 2.19). Definition 8. A sacetime is said to satisfy the causality condition if it does not contain any closed causal curve. A sacetime M is said to satisfy the strong causality condition if there are no almost closed causal curves. More recisely, for each oint M and for each oen neighborhood U of there exists an oen neighborhood V Uofsuch that each causal curve in M starting and ending in V is entirely contained in U. Obviously, the strong causality condition imlies the causality condition. Examle 9. In Minkowski sace (R n, g Mink ) the strong causality condition holds. One can rove this as follows: Let U be an oen neighborhood of =( 1,..., n ) R n. For any δ>0denote the oen cube with center and edges of length 2δ by W δ = ( 1 δ, 1 +δ) ( n δ, n +δ). Then there is an ε>0with W 2ε U, and one can ut V = W ε. Observing that any causal curve c = (c 1,...,c n )inr n satisfies ( c 1 ) 2 ( c 2 ) 2 + +( c n ) 2 and ( c 1 ) 2 > 0, we can conclude that any causal curve starting and ending in V = W ε cannot leave W 2ε U. Remark 5. Let M satisfy the (strong) causality condition and consider any oen connected subset Ω M with induced Lorentzian metric as a sacetime. Then non-existence of (almost) closed causal curves in M directly imlies non-existence of such curves in Ω, and hence also Ω satisfies the (strong) causality condition. Examle 10. In the Lorentzian cylinder S 1 R the causality condition is violated. If one unwras S 1 R as in Examle 8 it can be easily seen that there are closed timelike curves (Fig. 2.20). Fig Strong causality condition forbidden! V U

15 2 Lorentzian Manifolds 53 Fig Closed timelike curve c in Lorentzian cylinder Identify! c Examle 11. Consider the sacetime M which is obtained from the Lorentzian cylinder by removing two sacelike half-lines G 1 and G 2 whose endoints can be joined by a short lightlike curve, as indicated in Fig Then the causality condition holds for M, but the strong causality condition is violated: For any on the short lightlike curve and any arbitrarily small neighborhood of there is a causal curve which starts and ends in this neighborhood but is not entirely contained. Definition 9. A sacetime M is called a globally hyerbolic manifold if it satisfies the strong causality condition and if for all, q M the intersection J+ M () J M(q) is comact. The notion of global hyerbolicity has been introduced by J. Leray in [5]. Globally hyerbolic manifolds are interesting because they form a large class of sacetimes on which wave equations ossess a very satisfying global solution theory; see Cha. 3. Examle 12. In Minkowski sace (R n, g Mink ) for any, q R n both J+ Rn() and J Rn Rn Rn (q) are closed. Furthermore J+ () J (q) is bounded (with resect to Euclidean norm), and hence comact. In Examle 9 we have already seen that for (R n, g Mink ) the strong causality condition holds. Hence, Minkowski sace is globally hyerbolic. Examle 13. As seen before, the Lorentzian cylinder M = S 1 R does not fulfill the strong causality condition and is therefore not globally hyerbolic. Furthermore the comactness condition in Definition 9 is violated because one has J+ M () J M(q) = M for any, q M. Examle 14. Consider the subset Ω = R (0, 1) of two-dimensional Minkowski sace (R 2, g Mink ). By Remark 5 the strong causality condition holds for Ω, but there Identify! G 1 Fig Causality condition holds but strong causality condition is violated G 2

16 54 F. Pfäffle Fig J+ Ω () J Ω (q) is not always comact in the stri Ω = R (0, 1) q Ω are oints, q Ω for which the intersection J+ Ω () J Ω (q) is not comact, see Fig Examle 15. The n-dimensional anti-desitter sacetime (R S+ n 1, g AdS )isnot globally hyerbolic (Fig. 2.23). As seen in Examle 5, a curve in M = R S+ n 1 is causal with resect to g AdS if and only if it is so with resect to the Lorentzian metric dt 2 + can S n 1. Hence for both g AdS and dt 2 + can + S n 1. one gets the same + causal futures and asts. A similar icture as in Examle 14 then shows that for, q M the intersection J+ M () J M (q) need not be comact. In general one does not know much about causal futures and asts in sacetime. For globally hyerbolic manifold one has the following lemma (see [1, Cha. 14, Lemma 22]). q M = S n 1 + Fig J+ M () J M (q) is not comact in anti-desitter sacetime

17 2 Lorentzian Manifolds 55 J M + (K) Fig For K is closed J+ M (K ) need not be oen K Lemma 4. In any globally hyerbolic manifold M the relation is closed, i.e., whenever one has convergent sequences i and q i q in M with i q i for all i, then one also has q. Therefore in globally hyerbolic manifolds for any M and any comact set K M one has that J± M () and J ± M (K ) are closed. If K is only assumed to be closed, then J± M (K ) need not be closed. In Fig a curve K is shown which is closed as a subset and asymtotic to to a lightlike line in two-dimensional Minkowski sace. Its causal future J+ M (K ) is the oen half-lane bounded by this lightlike line. 2.5 Cauchy Hyersurfaces We recall that a iecewise C 1 -curve in M is called inextendible, if no iecewise C 1 -rearametrization of the curve can be continuously extended beyond any of the end oints of the arameter interval. Definition 10. A subset S of a connected time-oriented Lorentzian manifold is called achronal (or acausal) if and only if each timelike (or causal, resectively) curve meets S at most once. A subset S of a connected time-oriented Lorentzian manifold is a Cauchy hyersurface if each inextendible timelike curve in M meets S at exactly one oint. Obviously every acausal subset is achronal, but the reverse is wrong. However, every achronal sacelike hyersurface is acausal (see [1, Cha. 14, Lemma 42]). Any Cauchy hyersurface is achronal. Moreover, it is a closed toological hyersurface and it is hit by each inextendible causal curve in at least one oint. Any two Cauchy hyersurfaces in M are homeomorhic. Furthermore, the causal future and ast of a Cauchy hyersurface is ast- and future-comact, resectively. This is a consequence of, e.g., [1, Cha. 14, Lemma 40]. Examle 16. In Minkowski sace (R n, g Mink ) consider a sacelike hyerlane A 1, hyerbolic saces A 2 ={x = (x 1,...,x n ) x, x = 1 and x 1 > 0} and A 3 = {x = (x 1,...,x n ) x, x = 0, x 1 0}. Then all A 1, A 2, and A 3 are achronal, but only A 1 is a Cauchy hyersurface; see Fig

18 56 F. Pfäffle Fig Achronal subsets A 1, A 2,andA 3 in Minkowski sace A 3 A 2 inextendible timelike curve which avoids both A 2 and A 3 A 1 Examle 17. Let (N, h) be a connected Riemannian manifold, let I R be an oen interval and f : I (0, ) a smooth function. Consider on M = I N the wared roduct metric g = dt 2 + f (t) h. Then {t 0 } N is a Cauchy hyersurface in (M, g) for any t 0 I if and only if the Riemannian manifold (N, h) is comlete (comare [3, Lemma A.5.14]). In articular, in any Robertson Walker sacetime one can find a Cauchy hyersurface. Examle 18. Let N be exterior Schwarzschild sacetime N and B Schwarzschild black hole, both of mass m, as defined in Examle 4. Then for any t 0 R a Cauchy hyersurface of N is given by (2m, ) {t 0 } S 2, and in B one gets a Cauchy hyersurface by {r 0 } R S 2 for any 0 < r 0 < 2m. Definition 11. The Cauchy develoment (Fig. 2.26) of a subset S of a sacetime M is the set D(S) of oints of M through which every inextendible causal curve in M meets S, i.e., D(S) = { M every inextendible causal curve assing through meets S }. Remark 6. It follows from the definition that D(D(S)) = D(S) for every subset S M. Hence if T D(S), then D(T ) D(D(S)) = D(S). Of course, if S is achronal, then every inextendible causal curve in M meets S at most once. The Cauchy develoment D(S)ofeveryacausal hyersurface S is oen, see [1, Cha. 14, Lemma 43]. If S M is a Cauchy hyersurface, then obviously D(S) = M. For a roof of the following roosition, see [1, Cha. 14, Thm. 38]. Proosition 3. For any achronal subset A M the interior int(d( A)) of the Cauchy develoment is globally hyerbolic (if nonemty). From this we conclude that a sacetime is globally hyerbolic if it ossesses a Cauchy hyersurface. In view of Examles 17 and 18, this shows that Robertson

19 2 Lorentzian Manifolds 57 Fig Cauchy develoment Cauchy develoment of S S Walker sacetimes, Schwarzschild exterior sacetime, and Schwarzschild black hole are all globally hyerbolic. The following theorem is very owerful and describes the structure of globally hyerbolic manifolds exlicitly: they are foliated by smooth sacelike Cauchy hyersurfaces. Theorem 1. Let M be a connected time-oriented Lorentzian manifold. Then the following are equivalent: (1) M is globally hyerbolic. (2) There exists a Cauchy hyersurface in M. (3) M is isometric to R S with metric βdt 2 + g t where β is a smooth ositive function, g t is a Riemannian metric on S deending smoothly on t R and each {t} S is a smooth sacelike Cauchy hyersurface in M. Proof. The crucial oint in this theorem is that (1) imlies (3). This has been shown by A. Bernal and M. Sánchez in [6, Theorem 1.1] using work of R. Geroch [7, Theorem 11]. See also [8, Preosition 6.6.8] and [2,. 209] for earlier mentionings of this fact. That (3) imlies (2) is trivial, and Proosition 3 rovides the imlication (2) (1). Corollary 2. On every globally hyerbolic Lorentzian manifold M there exists a smooth function h : M R whose gradient is ast-directed timelike at every oint and all of whose level sets are sacelike Cauchy hyersurfaces. Proof. Define h to be the comosition t Φ where Φ : M R S is the isometry given in Theorem 1 and t : R S R is the rojection onto the first factor. Such a function h on a globally hyerbolic Lorentzian manifold is called a Cauchy time function. Note that a Cauchy time function is strictly monotonically increasing along any future-directed causal curve. We conclude with an enhanced form of Theorem 1, due to A. Bernal and M. Sánchez (see [9, Theorem 1.2]).

20 58 F. Pfäffle Theorem 2. Let M be a globally hyerbolic manifold and S be a sacelike smooth Cauchy hyersurface in M. Then there exists a Cauchy time function h : M R such that S = h 1 ({0}). Any given smooth sacelike Cauchy hyersurface in a (necessarily globally hyerbolic) sacetime is therefore the leaf of a foliation by smooth sacelike Cauchy hyersurfaces. References 1. O Neill, B.: Semi-Riemannian Geometry. Academic Press, San Diego (1983) 2. Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984) 3. Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds and quantization. EMS Publishing House, Zürich (2007) 4. Friedlander, F.: The wave equation on a curved sace-time. Cambridge University Press, Cambridge (1975) 5. Leray, J.: Hyerbolic Differential Equations. Mimeograhed Lecture Notes, Princeton (1953) 6. Bernal, A.N., Sánchez, M.: Smoothness of time functions and the metric slitting of globally hyerbolic sacetimes. Commun. Math. Phys. 257, 43 (2005) 7. Geroch, R.: Domain of deendence. J. Math. Phys. 11, 437 (1970) 8. Ellis, G.F.R., Hawking, S.W.: The large scale structure of sace-time. Cambridge University Press, London-New York (1973) 9. Bernal, A.N., Sánchez, M.: Further results on the smoothability of Cauchy hyersurfaces and Cauchy time functions. Lett. Math. Phys. 77, 183 (2006)

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