U Jtii

Then J+( JQ fl will be contained in

Leray (1952) did not, in fact, give the above definition of global hyperbolicity but an equivalent one which we shall present: for points p,qeUK such that strong causality holds on J+(p) fl J~(q), we define C(p, q) to be the space of all (continuous) non-space-like curves from p to q, regarding two curves y(t) and A(u) as representing the same point oiC(p, q) if one is a reparametrization of the other, i.e. if there is a continuous monotonic function f(u) such that y(f(u)) = A(u). (C(p, q) can be defined even when the strong causality condition does not hold on J+{p) fl J~(q), but we shall only be interested in the case in which its does hold.) The topology of C(p, q) is defined by saying that a neighbourhood of y in C(p, q) consists of all the curves in C(p, q) whose points in he in a neighbourhood W of the points of y in Ji (figure 45). Leray's definition is that an open set Jf is globally hyperbolic if C{p,q) is compact for allp,qejV. These definitions are equivalent, as is shown by the following result.

Figure 46. A neighbourhood W of the points of y in uf. A neighbourhood oty in C(p, q) consists of all non-spacelike curves from p to 5 whose points lie in W.

Proposition 6.6.2 (Sei/ert (1967), Geroch (1970fc)). Let strong causality hold on an open set Jf such that

Then Jf is globally hyperbolic if and only if C(p, q) is compact for all p, qeJ^.

Suppose first that C(p, q) is compact. Let rn be an infinite sequence of points in J+{p) fl J~(q) and let An be a sequence of non-spacelike curves from p to q through the corresponding rn. As C(p, q) is compact, there will be a curve A to which some subsequence A'n converges in the topology on C(p, q). Let ^ be a neighbourhood of A iiLstf such that is compact. Then will contain all A'n and hence all r'n for n sufficiently large, and so there will be a point retft which is a limit point of the r'n. Clearly r lies on A. Thus every infinite sequence in J+(p) f) J~(q) has a limit point in J+(p) fl J~(q)- Hence J+(p) fl J~{q) is compact.

Conversely, suppose J+(p) fl J~(q) is compact. Let An be an infinite sequence of non-spacelike curves fromjp to q. By lemma 6.2.1 applied to the open set JK — q, there will be a future-directed non-spacelike curve A from p which is inextendible in Ji — q, and is such that there is a subsequence A'n which converges to r for every re A. The curve A must have a future endpoint at q since by proposition 6.4.7 it cannot be totally future imprisoned in the compact set J+(p) n J~(q), and it cannot leave the set except at q.

Let be any neighbourhood of A in Ji and let rt (1 < i < k) be a finite set of points on A such that = p, rk = q and each ri has a neighbourhood ^ with J+ (iQ f| contained in Then for sufficiently large n, A'n will be contained in Thus A'n converge to A in the topology on C(p, q) and so C(p, q) is compact. □

The relation between global hyperbolicity and Cauchy developments is given by the following results.

Proposition 6.6.3

Ify is a closed achronal set, then int (D(Sf)) = if non empty, is globally hyperbolic.

We first establish a number of lemmas. Lemma 6.6.4

If peD+(Sf) — H+(Sf), then every past-inextendible non-spacelike curve through p intersects I~(£P).

Let y be in D+{Sf) — H+(Sf) and let y be a past-inextendible non-spacelike curve through p. Then one can find a point q e D+(£f) n I+(p) and a past-inextendible non-spacelike curve A through q such that for each point are A there is a point yey with y e I~( x). As A will intersect if at some point xx there will be a y1 ey n □

Corollary lip eint (D(y)) then every inextendible non-spacelike curve through p intersects and int(Z)(^)) = D(y)-{H+{y)vH-{y)}. If peI+[P) or 1-{S?) the result follows immediately. If ye - then peif <=■ D~{Sf)

and the result again follows. □

Lemma 6.6.5

The strong causality condition holds on int7)(5^).

Suppose there were a closed non-spacelike curve A through ye int (7)(5^)). By the previous result there would be points g<e An and re An 7+(^). As reJ~(q), it would also be in which would contradict the fact that y is achronal. Thus the causality condition holds on int (D(.¥)). Now suppose that the strong causality condition did not hold at p. Then as in lemma 6.4.6 there would be an infinite sequence of future-directed non-spacelike curves An which converged to an inextendible null geodesic y through p. There would be points qey(\ I~{£f) and reyfl I+(Sf) and so there would be some An which intersected and then I~(y), which would contradict the fact that y was achronal. □

Proof of proposition 6.6.3. We wish to show that C(p,q) is compact for p,qe int (D(y)). Consider first the case that p, q eI~(Sf) and suppose peJ~(q). Let An be an infinite sequence of non-spacelike curves from q to p. By lemma 6.2.1 there will be a future-directed non-spacelike limit curve from p which is inextendible in — q. This must have a future endpoint at q since otherwise it would intersect y which would be impossible as qel~(y). Consider now the case that peJ~(y), qeJ+(y) fl J+(p)- If the limit curve A has an endpoint at q, it is the desired limit point in C(p, q). If it does not have an endpoint at q, it would contain a point yel+(y) since it is inextendible in Ji — q. Let A'n be a subsequence which converges to r for every point r on A between p and y. Let A be a past-directed limit curve from q of the A'n. If X has a past endpoint at p, it would be the desired limit point in C(p, q). If X passed through y, it could be joined up with A to provide a non-spacelike curve from p to q which would be the desired limit point in C(p, q). Suppose X does not have endpoint at p and does not pass through y. Then it would contain some point zel~(y). Let A"n be a subsequence of the A'n which converges to r for every point r on X between q and z. Let "V be an open neighbourhood of X which does not contain y. Then for sufficiently large n, all A"n D would be con tained in "V. This would be impossible as y is a limit point of the A"n. Thus there will be a non-spacelike curve from ptoq which is a limit point of the An in C(p, q).

The cases p,qeI~(S'f) and peJ~(y), qeJ+(y) together with their duals cover all possible combinations. Thus in all cases we get a non-spacelike curve from p to q which is a limit point of the An in the topology on C(p, q). □

By a similar procedure one can prove: Proposition 6.6.6

If int (D(y)), then fl J~(q) is compact or empty. □

To show that the whole of D(S?) and not merely its interior is globally hyperbolic, one has to impose some extra conditions.

Proposition 6.6.7

If y is a closed achronal set such that J+(Sf) fi J~(£f) is both strongly causal and either

(1) acausal (this is the case if and only if y is acausal), or

(2) compact, then -D(y) is globally hyperbolic.

Suppose that strong causality did not hold at some point q e D(£f). Then by an argument similar to lemma 6.6.5, there would be an inextendible null geodesic through q at each point of which strong causality did not hold. This is impossible, since it would intersect y. Therefore strong causality holds on D{SP).

lip, qeI~{Sf), the argument of proposition 6.6.3holds. lipe J~(£f), qeJ+{£f) one can as in proposition 6.6.3 construct a future-directed limit curve A from p and a past-directed limit curve 'X from q, and choose a subsequence A"n which converges to r for every point r on A or In case (1), A would intersect y in a single point x. Any neighbourhood of a; would contain points of A"n for n sufficiently large, and so would contain x"n, defined as A"n fi y, since y is achronal. Therefore x"n would converge to x. Similarly x"n would converge to & s A n y. Thus £ = x and so one could join A and % to give a non-spacelike limit curve in C(p, q).

In case (2), suppose that A did not have a future endpoint at q. Then A would leave since it would intersect y and by proposi tion 6.4.7 it would have to leave the compact set J+(y) fi J~(y). Thus one could find a point a; on A which was not in J~{SP). For each n, choose a point x"ne£ff) A"n. Since y is compact, there will be some point y ey and a subsequence A"'n such that the corresponding points x"'n converge to y. Suppose that y does not lie on A. Then for sufficiently large n each ¡»¡"'„-would lie to the future of any neighbourhood of a:. This would imply xeJ~{£f). This is impossible as x is in J+(y) but is not in the compact set </+(y) fi J~(Sf). Therefore A would pass through y. Similarly X would pass through y. One could then join them to obtain a limit curve. □

Proposition 6.6.3 shows that the existence of a Cauchy surface for an open set jV implies global hyperbolicity of jV. The following result shows that the converse is also true:

Proposition 6.6.8 (Oeroch (1970fc))

If an open BetjV is globally hyperbolic, then Jf, regarded as a manifold, is homeomorphic to RlxSf where is a three-dimensional manifold, and for each aeR1, {a} xS? is a Cauchy surface for Jf.

As in proposition 6.4.9, put a measure ft on Jf such that the total volume of Jf in this measure is one. For peJf define f+(p) to be the volume of J+(p,Jf) in the measure ¡i. Clearly f+(p) is a bounded function on Jf which decreases along every future-directed non-spacelike curve. We shall show that global hyperbolicity implies that f+(p) is continuous on Jf so that we do not need to 'average' the volume of the future as in proposition 6.4.9. To do this it will be sufficient to show that f+(p) is continuous on any non-spacelike curve A.

Let re A and let xn be an infinite sequence of points on A strictly to the past of r. Let be fl J+(xn> Suppose that f+(p) was not upper n semi-continuous on A at r. There would be a point qe^ — J+(r,Jr). Then r$J~(q,Jf); but each xneJ~(q,Jf) and so reJ~{q,Jf), which is impossible as J~(q, Jf) is closed in Jf by proposition 6.6.1. The proof that it is lower semi-continuous is similar

As p is moved to the future along an inextendible non-spacelike curve A in Jf the value off+(p) must tend to zero. For suppose there were some point q which lay to the future of every point of A. Then the future-directed curve A would enter and remain within the compact set J+(r) n J~(q) for any reX which would be impossible by proposition 6.4.7 as the strong causality condition holds on Jf.

Now consider the function f(p) defined on Jf by f(p) = f~(p)lf+(p). Any surface of constant / will be an acausal set and, by proposition 6.3.1, will be a three-dimensional C1" manifold imbedded in Jf. It will also be a Cauchy surface for Jf since along any non-spacelike curve, /- will tend to zero in the past and/+ will tend to zero in the future. One can put a timelike vector field V on Jf and define a continuous map fi which takes points of Jf along the integral curves of V to where they intersect the surface Sf (/ = 1). Then (logf(p), fl(p)) is a homeo-morphism of Jf onto R x if. If one smoothed/as in proposition 6.4.9, one could improve this to a diffeomorphism. Cf

Thus if the whole of space-time were globally hyperbolic, i.e. if there were a global Cauchy surface, its topology would be very dull.

6.7 The existence of geodesies

The importance of global hyperbolicity for chapter 8 lies in the following result:

Proposition 6.7.1

Let p and q lie in a globally hyperbolic set jV with qeJ+(p). Then there is a non-spacelike geodesic from p to q whose length is greater than or equal to that of any other non-spacelike curve from p to q.

Figure 46. tfl is an open neighbourhood of the timelike curve A from p to q. There exist in Ql timelike curves from p to q which approximate broken null curves and are of arbitrarily small length.

We shall present two proofs of this result : the first, due to Avez (1963) and Seifert (1967), is an argument from the compactness of C(p, q), and the second (applicable only when Jf is open) is a procedure whereby the actual geodesic is constructed.

The space C{p, q) contains a dense subset C'(p, q) consisting of all the timelike C1 curves from p to q. The length of one of these curves A is defined (cf. § 4.5) as where t is a C1 parameter on A. The function L is not continuous on G'(p, q) since any neighbourhood of A contains a zig-zag piecewise almost null curve of arbitrarily small length (figure 46). This lack of continuity arises because we have used the C° topology which says that two curves are close if their points in JK, but not necessarily their

Figure 46. tfl is an open neighbourhood of the timelike curve A from p to q. There exist in Ql timelike curves from p to q which approximate broken null curves and are of arbitrarily small length.

Almost broken almost null curve from p to q in ft tangent vectors, are close. We could put a C1 topology on C'(p, q) and so make L continuous but we do not do this because C'(p, q) is not compact; one gets a compact space only when one includes all the continuous non-spacelike curves. Instead, we use the C° topology and extend the definition of L to C{p, q).

Because of the signature of the metric, putting wiggles in a timelike curve reduces its length. Thus L is not lower semi-continuous. However one has:

Lemma 6.7.2

L is upper semi-continuous in the C° topology on C'(p, q).

Consider a C1 timelike curve A(i) from p to q, where the parameter t is chosen to be the arc-length from p. In a sufficiently small neighbourhood of A, one can find a function / which is equal to t on A and is such that the surfaces {/ = constant} are spacelike and orthogonal to djdt (i.e. gra6/;6|A = (0/at)a)- One way to define such an / would be to construct the spacelike geodesies orthógonal to A. For a sufficiently small neighbourhood W of A, they will give a unique mapping of to A, and the value of / at a point in can be defined as the value of t at the point on A into which it is mapped. Any curve fi in can be parametrized by /. The tangent vector (0/0/)A to fi can be expressed as where k is a spacelike vector lying in the surface {/ = constant}, i.e. = 0. Then

However on A, <7a6/;a/;¡) = — 1. Thus given any e > 0, one can choose <T c q¿ sufficiently small that on <r, gabf-af.b > -1 +e. Therefore for any curve ¡x in <T, ^ ^ (1 +e)iL[Á] Q

We now define the length of a continuous non-spacelike curve A from p to q as follows: let be a neighbourhood of A in ^and let be the least upper bound of the lengths of timelike curves in W from p to q. Then we define L[A] as the greatest lower bound of l(^) for all neighbourhoods of A in ^. This definition of length will work for all curves A from p to q which have a C1 timelike curve in every neighbour hood, i.e. it will work for all points in C(p, q) which lie in the closure of C'(p, q). By §4.5, a non-spacelike curve from p to q which is not an unbroken null geodesic curve can be varied to give a piecewise C1 timelike curve from p to q, and the corners of this curve can be rounded off to give a C1 timelike curve from p to q. Thus points in C(P> ?)— C'(p, q) are unbroken null geodesies (containing no conjugate points), and we define their length to be zero.

This definition of L makes it an upper semi-continuous function on the compact space C'(p, q). (Actually, as a continuous non-spacelike curve satisfies a local Lipschitz condition, it is differentiable almost everywhere. Thus the length could still be defined as and this would agree with the definition above.) If C'(p, q) is empty but C(p,q) is non-empty, p and q are joined by an unbroken null geodesic and there are no non-spacelike curves from p to q which are not unbroken null geodesies. If C'(p, q) is non-empty, it will contain some point at which L attains its maximum value, i.e. there will be a non-spacelike curve y from p to q whose length is greater than or equal to that of any other such curve. By proposition 4.5.3, y must be a geodesic curve as otherwise one could find points x,yey which lay in a convex normal coordinate neighbourhood and which could be joined by a geodesic segment of greater length than the portion of y between x and y. □

For the other, constructive, proof, we first define d(p,q) for p,qeJ? to be zero if q$J+(p) and otherwise to be the least upper bound of the lengths of future-directed piecewise non-spacelike curves from p to q. (Note that d(p,q) may be infinite.) For sets and we define d{£P, °U) to be the least upper bound of d(p, q), p e^, jet.

Suppose qel+(p) and that d(p,q) is finite. Then for any S > 0 one can find a timelike curve A of length d(p, q) — faS from p to q and a neighbourhood Qi of q such that A can be deformed to give a timelike curve of length d(p, q) — S fromp to any point ref. Thus d(p, q), where finite, is lower semi-continuous. In general d(p, q) is not upper semi-continuous but:

Lemma 6.7.3

d(p, q) is finite and continuous in p and q when p and q are contained in a globally hyperbolic set ¿V.

We shall first prove d{p, q) is finite. Since strong causality holds on the compact set J+(p) fl J~{q), one can cover it with a finite number of local causality sets such that each set contains no non-spacelike curve longer than some bound e. Since any non-spacelike curve from p to q can enter each neighbourhood at most once, it must have finite length.

Now suppose that for p,qeJT, there is a 8 > 0 such that every neighbourhood of q contains a point reJf such that d(p,r) > d(p,q) + 8.

Let xn be an infinite sequence of points in Jf converging to q such that d(p, xn) > d(p, q) + 8. Then from each xn one can find a non-spacelike curve An to p of length > d(p, q) 4- 8. By lemma 6.2.1 there will be a past-directed non-spacelike curve A through q which is a limit curve of the An. Let be a local causality neighbourhood of q. Then A cannot intersect I~[q) n Qi since if it did one of the An could be deformed to give a non-spacelike curve fromp to q of length > d(p,q). Thus An must be a null geodesic from q and at each point x of A n Ql, d(p, x) will have a discontinuity greater than 8. This argument can be repeated to show that A is a null geodesic and at each point xeX, d(p,x) has a discontinuity greater than 8. This shows that A cannot have an endpoint at p, since by proposition 4.5.3, d(p, x) is continuous on a local causality neighbourhood ofp. On the other hand, A would be inextend-ible in JK—p and so if it did not have an endpoint at p, it would have to leave the compact set J+(p) n J~(q) by proposition 6.4.7. This shows that d(p, q) is upper semi-continuous on N. □

In the case that Jf is open, one can easily construct the geodesic of maximum length from p to q by using the distance function. Let % <=■ ^ be a local causality neighbourhood olp which does not contain q and let xeJ+(p)C\ J~(q) be such that d(p,r) + d(r,q), re<2?, is maximized for r — x. Construct the future-directed geodesic y from p through x. The relation d(p, r) + d(r, q) = d(p, q) will hold for all points r on y between p and x. Suppose there were a point y e J~(q) — q which was the last point on y at which this relation held. Let "T <= N be a local causality neighbourhood of y which does not contain q and let ze J+(y) n J~{q) n be such that d(y, r) + d(r, q),reT?', attains its maximum value d(y, q) for r = z. If z did not lie on y, then d(p, z) > d(p, y) + d(y, z) and d(p, z) + d(z, q) > d(p, q) which is impossible. This shows that the relation d(p,r) + d{r,q) = d(p,q)

must hold for all rey n J~(q). As J+(p) n J~(q) is compact, y must leave J~{q) at some point y. Suppose y + q\ then y would lie on a past-directed null geodesic A from q. Joining y to A would give a non-spacelike curve from p to q which could be varied to give a curve longer than d(p, q), which is impossible. Thus y is a geodesic curve from p to q of length d(p, q). □

Corollary

If Sf is a C2 partial Cauchy surface, then to each point qeD+(£f) there is a future-directed timelike geodesic curve orthogonal to Sf of length d{Sf,q), which does not contain any point conjugate to ¿f between if and q.

By proposition 6.5.2, H+(£f) and H~(£f) do not intersect if and so are not in D(£f). Thus D^) = intD(y) is globally hyperbolic by proposition 6.6.3. By proposition 6.6.6, ^n J~(q) is compact and so d(p,q), peS?, will attain its maximum value of d(Sf,q) at some point There will be a geodesic curve y from r to q of length d(£f, q) which by lemma 4.5.5 and proposition 4.5.9 must be orthogonal to if and not contain a point conjugate to if between if and q. □

6.8 The causal boundary of space-time

In this section we shall give a brief outline of the method of Geroch, Kronheimer and Penrose (1972) for attaching a boundary to spacetime. The construction depends only on the causal structure of (UK, g). This means that it does not distinguish between boundary points at a finite distance (singular points) and boundary points at infinity. In § 8.3 weshall describe a different construction which attaches a boundary which represents only singular points. Unfortunately there does not seem to be any obvious relation between the two constructions.

We shall assume that (UK, g) satisfies the strong causality condition. Then any pointy in (UK, g) is uniquely determined by its chronological past I~(p) or its future I+(p), i.e.

The chronological past if = I~(p) of any point peUK has the properties:

(3) cannot be expressed as the union of two proper subsets which have properties (1) and (2).

We shall call a set with properties (1), (2) and (3) an indecomposable past set, abbreviated as IP. (The definition given by Geroch, Kron-heimer and Penrose does not include property (1). However it is equivalent to the definition given here, since by 'a past set' they mean a set which equals its chronological past, rather than merely containing it.) One can define an IF, or indecomposable future set, similarly.

One can divide IPs into two classes: proper IPs (PIPs) which are the pasts of points in and terminal IPs (TIPs) which are not the past of any point in The idea is to regard these TIPs and the similarly defined TIFs as representing points of the causal boundary (c-bounda~y) of g). For instance, in Minkowski space one would regard the shaded region in figure 47 (i) as representing the point p on Note that in this example, the whole of Jt is itself a TIP and also a TIF. These can be thought of as representing the points i+ and i~ respectively. In fact all the points of the conformal boundary of Minkowski space, except i°, can be represented as TIPs or TIFs. In some cases, such as anti-de Sitter space, where the conformal boundary is timelike, points of the boundary will be represented by both a TIP and a TIF (see figure 47 (ii)).

One can also characterize TIPs as the pasts of future-inextendible timelike curves. This means that one can regard the past I~(y) of a future-inextendible curve y as representing the future endpoint of y on the c-boundary. Another curve y' has the same endpoint if and only if/"(y) =/-(/).

Proposition 6.8.1 (Geroch, Kronheimer and Penrose) A set W is a TIP if and only if there is a future-inextendible timelike curve y such that I~(y) = "W.

Suppose first that there is a curve y such that I~(y) = W. Let = W U where W and are open past sets. One wants to show that either W is contained in y, or contained in Suppose that, on the contrary, is not contained in y and "V not contained in Then one could find a point q in W — "V and a point r in "V — °U. Now q,rel~(y), so there would be points q'.r'ey such that qel~(q') and rsl~(r'). But whichever of % or V contained the futuremost of q', r' would also contain both q and r, which contradicts the original definitions of q and r.

TIP representing i>omt p

TIF representing point p

TIP representing point p

Figure 47. Penrose diagrams of Minkowski space and anti-de Sitter space (cf. figures 16 and 20), showing (i) the TIP representing a point p onS+ in Minkowski space, and (ii) the TIP and the TIF representing a point p on J in anti-de Sitter space.

TIF representing point p

TIP representing point p

Figure 47. Penrose diagrams of Minkowski space and anti-de Sitter space (cf. figures 16 and 20), showing (i) the TIP representing a point p onS+ in Minkowski space, and (ii) the TIP and the TIF representing a point p on J in anti-de Sitter space.

Conversely, suppose ^ is a TIP. Then one must construct a timelike curve y such that if — I~(y)- Now if p is any point of if, then if = I~(if n I+(p)) U I~(if-I+(p)). However if is indecomposable, so either if = I~{if n I+(p)) or if = I~(if - I+(p)). The point p is not contained in I~(if — I+(p)), so the second possibility is eliminated. The conclusion may be restated in the following form: given any pair of points of^T, then if contains a point to the future of both of them. Now choose a countable dense family pn of points of if. Choose a point qa in "W to the future of p0. Since qQ andf^ are in if, one can choose a point in to the future of both of them. Since qt andf>2 are in "W, one can choose q% in if to the future of both of them, and so on. Since each point qn obtained in this way lies in the past of its successor, one can find a timelike curve yin^ through all the points of the sequence. Now for each point p e if, the set if n I+(p) is open and non-empty, and so it must contain at least one of the pn, since these are dense. But for each k, pk lies in the past of qk, whence p itself lies in the past of y. This shows that every point of if lies to the past of y, and so since y is contained in the open past set if, one must have if = /-(y). □

We shall denote by Jl the set of all IPs of the space g). Then represents the points of Ji plus a future c-boundary; similarly, Jl, the set of all IFs of g), represents plus a past c-boundary. One can extend the causal relations I, J and E to JC and Jt in the following way. For each W, "f <= J(t we shall say if e /-(■f, Ji) if c I-(q) for some point qe"f,

With these relations, the IP-space is a causal space (Kronheimer and Penrose (1967)). There is a natural infective map I—. J? which sends the point psji into This map is an iso morphism of the causality relation J~ as peJ~(q) if and only if I~(p) e J~{I~(q), J?). The causality relation is preserved by I~ but not by its inverse, i.e. pel~(q) => I-(p)€l~(I~(q),Jf). One can define causal relations on Jf similarly.

The idea now is to write and in some way to form a space JK* which has the form J'U A where A will be called the c-boundary of (JK, g). To do so, ane needs a method of identifying appropriate IPs and IFs. One starts by forming the space which is the union of and Ji, with each PIF identified with the corresponding PIP. In other words, corresponds to the points of JK together with the TIPs and TIFs. However as the example of anti-de Sitter space shows, one also wants to identify some TIPs with some TIFs. One way of doing this is to define a topology on And then to identify some points to make this topology Hausdorff.

As was mentioned in § 6.4, a basis for the topology of the topological space is provided by sets of the form I+(p) fl I~(q). Unfortunately one cannot use a similar method to define a basis for the topology of UK# as there may be some points of UK# which are not in the chronological past of any points of UK#. However one can also obtain a topology of UK from a sub-basis consisting of sets of the form I+(p), I~(p), UK — I+(p) and UK — I~(p). Following this analogy, Geroch, Kromheimer and Penrose have shown how one can define a topology on UKn. For an IF si ^ UK, one defines the sets

¿/tat _ yr-. fei and ^n^* 0}, and j/ext = {tT : iT sJt and "T = I~{1T) => 7+( )T) $ si).

For an IP 08eUK, the sets and are defined similarly. The open sets of UK# are then defined to be the unions and finite intersections of sets of the form si™, si°xt, and The sets sf^ and are the analogues in UK# of the sets I+(p) and I~(g). If in particular si = I+(p) and -T = l~(q) then fe si™ if and only if qe I+(p). However the definitions enable one also to incorporate TIPS into sim. The sets siext and ^ext are the analogues of UK-I+(p) and UK-Tig).

Finally one obtains UK* by identifying the smallest number of points in the space UK# necessary to make it a Hausdorff space. More precisely UK* is the quotient space UK#\Rh where Rh is the intersection of all equivalence relations R <=■ UK# x UK# for which UK#jR is Hausdorff. The space UK* has a topology induced from UK# which agrees with the topology of UK on the subset UK of UK*. In general one cannot extend the differentiable structure of UK to A, though one can on part of A in a special case which will be described in the next section.

6.9 Asymptotically simple spaces

In order to study bounded physical systems such as stars, one wants to investigate spaces which are asymptotically flat, i.e. whose metrics approach that of Minkowski space at large distances from the system. The Schwarzschild, Reissner-Nordstrom and Kerr solutions are examples of spaces which have asymptotically flat regions. As we saw in chapter 5, the conformal structure of null infinity in these spaces is similar to that of Minkowski space. This led Penrose (1964, 19656, 1968) to adopt this as a definition of a kind of asymptotic flatness. We shall only consider strongly causal spaces. Penrose does not make the requirement of strong causality. However it simplifies matters and implies no loss of generality in the kind of situation we wish to consider.

A ti me- and space-orientable space (Jt, g) is said to be asymptotically simple if there exists a strongly causal space g) and an imbedding 8 : -> ^ which imbeds as a manifold with smooth boundary d^tf lH »/^j such that:

(1) there is a smooth (say C3 at least) function Q on ^ such that on CI is positive and C22g = 0#(g) (i.e. g is conformal to g on 6{J())\

(3) every null geodesic in has two endpoints on "bJt.

In fact this definition is rather more general than one wants since it includes cosmological models, such as de Sitter space. In order to restrict it to spaces which are asymptotically flat spaces, we will say that a space {^k, g) is asymptotically empty and simple if it satisfies conditions (1), (2), and (3), and

(4) Rai, = 0 on an open neighbourhood of in jM . (This condition can be modified to allow the existence of electromagnetic radiation near <L#).

The boundary can be thought of as being at infinity, in the sense that any affine parameter in the metric g on a null geodesic in Jl attains unboundedly large values near This is because an affine parameter v in the metric g is related to an affine parameter v in the metric g by dv/dv = Q-2. Since Q = 0 at d^f, fdv diverges.

From conditions (2) and (4) it follows that the boundary d^tf is a null hypersurface. This is because the Ricci tensor Jt^ of the metric gab is related to the Ricci tensor J?^ of g^ by

Rab = Cl-2Rab - 2Q-1(Q)loc^+{ - Cl-micd + 3 Q-2Qle C\d}g°%b where | denotes covariant differentiation with respect to g^. Thus

Since the metric g^ is C3, R is C1 at d^f where £1 = 0. This implies that Q|c Clidgcd = 0. However by condition (2), fl|e * 0. Thus is a null vector, and the surface (Cl = 0) is a null hypersurface.

In the case of Minkowski space, consists of the two null surfaces and each of which has the topology R1 x S2. (Note that it does not include the points i°, i+ and since the conformal boundary is not a smooth manifold at these points.) We shall show that in fact has this structure for any asymptotically simple and empty space.

Since is a null surface, lies locally to the past or future of it. This shows that must consist of two disconnected components:

on which null geodesies in ^ have their future endpoints, and on which they have their past endpoints. There cannot be more than two components of since there would then be some point p for which some future-directed null geodesies would go to one component and others to another component. The set of null directions at p going to each component would be open, which is impossible, since the set of future null directions at p is connected. We next establish an important property.

Lemma 6.9.1

An asymptotically simple and empty space (JC, g) is causally simple.

Let iV be a compact set of One wants to show that every null geodesic generator of ) has past endpoint at W. Suppose there were a generator that did not have endpoint there. Then it could not have any endpoint in so it would intersect which is impossible. □

Proposition 6.9.2

An asymptotically simple and empty space g) is globally hyperbolic.

The proof is similar to that of proposition 6.6.7. One puts a volume element on such that the total volume of ^ in this measure is unity. Since g) is causally simple, the functions f+(p), f~(p) which are the volumes of I+{p), I~(p) are continuous on Since strong causality holds on f+(p) will decrease along every future-directed non-spacelike curve. Let A be a future-inextendible timelike curve. Suppose that= D I+(P) was non-empty. Then & would be a future set peA

and the null generators of the boundary of ¡F in Jl would have no past endpoint in Ji. Thus they would intersect which again leads to a contradiction. This shows that f+(p) goes to zero as p tends to the future on A. Prom this it follows that every inextendible non-spacelike curve intersects the surface={p: f+(p) = f~(p)}, which is therefore a Cauchy surface for □

Lemma 6.9.3

Let "W be a compact set of an asymptotically empty and simple space (yti, g). Then every null geodesic generator of intersects ¿ft)

once, where ' indicates the boundary in jM .

LetpeA, where A is a null geodesic generator of •/+. Then the past set (in J?) J~(p, n must be closed in since every null geodesic generator of its boundary must have future endpoint on at p. Since strong causality holds on , — J~(p, will be non-empty. Now suppose that A were contained in J+(if~, M). Then the past set PI (J~(P> H would be non-empty. This would be impossible, peA

since the null generators of the boundary of the set would intersect

Suppose on the other hand that A did not intersect J(). Then

U (J~(p,Jt)C\ JK) would be non-empty. This would again lead peA

to a contradiction, as the generators of the boundary of the past set U (J~(P> n would intersect J+. □

jjeA

Corollary

We shall now show that J* (and and ^ are the same topologically as they are for Minkowski space.

Proposition 6.9.4 (Oeroch (1971))

In an asymptotically simple and empty space , g),./+ and are topologically R1 x S2, and Jl is R*.

Consider the set N of all null geodesies in Since these all intersect the Cauchy surface MP, one can define local coordinates on N by the local coordinates and directions of their intersections with Jff. This makes N into a fibre bundle of directions over 34? with fibre S2. However every null geodesic also intersects •/+. Thus N is also a fibre bundle over In this case, the fibre is S2 minus one point which corresponds to the null geodesic generator of which does not enter In other words, the fibre is R2. Therefore N is topologically J* x R2. However S+ is R1 x (j+C>r, n dJ(). This is consistent with N only if Jf » Ra and J* » R1 x S2. □

Penrose (1965fc) has shown that this result implies that the Weyl tensor of the metric g vanishes on J+ and This can be interpreted as saying that the various components of the Weyl tensor of the metric g 'peel off', that is, they go as different powers of the affine parameter on a null geodesic near or Further Penrose (1903), Newman and Penrose (1968) have given conservation laws for the energy-momentum as measured from in terms of integrals on ./+.

The null surfaces ./+ and J~ form nearly all the c-boundary A of {■Ji, g) defined in the previous section. To see this, note first that any point ye/+ defines a TIP I~(p, J() n J(. Suppose A is a future-

inextendible curve in Jt. If A has a future endpoint atpe J+, then the TIP I~(A) is the same as the TIP defined by p. If A does not have a future endpoint on J+, then —1~(A) must be empty, since if it were not, the null geodesic generators of /"(A) would intersect J+ which is impossible as A does not intersect •/+. The TIPs therefore consist of one for each point of and one extra TIP, denoted by i+, which is itself. Similarly, the TIFs consist of one for each point of and one, denoted by which again is itself.

One now wants to verify that one does not have to identify any TIPs or TIFs, i.e. that JC.t is Hausdorff. It is clear that no two TIPs or TIFs corresponding to or J~ are non-Hausdorff separated. If psJ+ then one can find qeJf such that p$I+(q,J(). Then (I+(q, J?))ext is a neighbourhood in of the TIP I~{p, Jfi) n and (/+(<?, is a disjoint neighbourhood of the TIP i+. Thus i+ is

Hausdorff separated from every point of Similarly it is Hausdorff separated from every point of Thus the c-boundary of any asymptotically simple and empty space , g) is the same as that of Minkowski space-time, consisting of •/+, J~ and the two points i+, i~.

Asymptotically simple and empty spaces include Minkowski space and the asymptotically flat spaces containing bounded objects such as stars which do not undergo gravitational collapse. However they do not include the Schwarzschild, Reissner-Nordstrom or Kerr solutions, because in these spaces there are null geodesies which do not have endpoints on ./+ or J~. Nevertheless these spaces do have asymptotically flat regions which are similar to those of asymptotically empty and simple spaces. This suggests that one should define a space (JK, g) to be weakly asymptotically simple and empty if there is an asymptotically simple and empty space (JC, g') and a neighbourhood W of dJt' in Jl' such that W n is isometric to an open set W oiJ(. This definition covers all the spaces mentioned above. In the Reissner-Nordstrom and Kerr solutions there is an infinite sequence of asymptotically flat regions fy which are isometric to neighbourhoods W of asymptotically simple spaces. There is thus an infinite sequence of null infinities J* and However we shall consider only one asymptotically flat region in these spaces. One can then regard {Jt, g) as being conformally imbedded in a space {¿ft, §) such that a neighbourhood Qi of BJK in is isometric to . The boundary dJt consists of a single pair of null surfaces and

We shall discuss weakly asymptotically simple and empty spaces in §9.2 and §9.3.

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