Jl

Remove

Null geodesic

Remove J

Identify

Figure 38. A space-time satisfying the causality, future and past distinguishing conditions, but not satisfying the strong causality condition at p. Two strips have been removed from a cylinder; light cones are at ± 46°.

Proposition 6.4.6

If conditions (a) to (c) of proposition 6.4.5 hold and if in addition, (d) is null geodesically complete, then the strong causality condition holds on

Suppose the strong causality condition did not hold at peJ(. Let % be a convex normal neighbourhood oip and let Vn <= be an infinite sequence of neighbourhoods of p such that any neighbourhood of p contains all the Vn for n large enough. For each Vn there would be a future-directed non-spacelike curve An which left % and then returned to Vn. By lemma 6.2.1, there would be an inextendible non-spacelike curve A throughp which was a limit curve of the An. No two points of A could have timelike separation as otherwise one could join up some An to give a closed non-spacelike curve. Thus A must be a null geodesic. But by (a), (b) and (d) A would contain conjugate points and therefore points with timelike separation. □

Corollary

The past and future distinguishing conditions would also hold on since they are implied by strong causality.

Closely related to these three higher degree causality conditions is the phenomenon of imprisonment.

A non-spacelike curve y that is future-inextendible can do one of three things as one follows it to the future: it can

(i) enter and remain within a compact set ¿P,

(ii) not remain within any compact set but continually re-enter a compact set SP,

(iii) not remain within any compact set SP and not re-enter any such set more than a finite number of times.

In the third case y can be thought of as going off to the edge of space-time, that is either to infinity or a singularity. In the first and second cases we shall say that y is totally and partially future imprisoned in SP, respectively. One might think that imprisonment could occur only if the causality condition was violated, but the example due to Carter which is illustrated in figure 39 shows that this is not the case. Nevertheless one does have the following result:

Identify

Identify

Identify after shifting an irrational amount

Figure 39. A space with imprisoned non-spacelike lines but no closed non-spacelike curves. The manifold is R1xSlx S1 described by coordinates (t, y, z) where (t, y, z) and (t, y, z+1) are identified, and (t, y, z) and (y, y+ 1, z + a) are identified, where a is an irrational number. The Lorentz metric is given by dsl = (cosh t -1 )l (dtl - dyl) + dt Ay - dzK

(i) A section (z = constant} showing the orientation of the null cones.

(ii) The section t = 0 showing part of a null geodesic.

Proposition 6.4.7

If the strong causality condition holds on a compact set £7, there can be no future-inextendible non-spacelike curve totally or partially future imprisoned in £f.

£f can be covered by a finite number of convex normal coordinate neighbourhoods <25^ with compact closure, such that no non-spacelike curve intersects any etli more than once. (We shall call such neighbourhoods, local causality neighbourhoods.) Any future-inextendible non-spacelike curve which intersects one of these neighbourhoods must leave it again and not re-enter it. □

Proposition 6.4.8

If the future or past distinguishing condition holds on a compact set there can be no future-inextendible non-spacelike curve totally future imprisoned in (This result is included for its interest but is not needed for what follows.)

Let {"3^}, (a = 1,2,3,...), be a countable basis of open sets for Ji (i.e. any open set in can be represented as a union of the "Ta). As the future or past distinguishing condition holds on SP, any point peSP will have a convex normal coordinate neighbourhood % such that no future (respectively, past) directed non-spacelike curve fromp intersects & more than once. We define f(p) to be equal to the least value of a such that "Va contains p and is contained in some such neighbourhood °U.

Suppose there were a future-inextendible non-spacelike curve A which was totally future imprisoned in ¿P. Let qeA be such that A' = A fl J+(g) is contained in SP. Define s/0 to be the closed, nonempty set consisting of all points of SP which are limit points of A. Let p0esf0 be such that f(p0) is equal to the smallest value of f(p) on sf0. Through p0 there would be an inextendible non-spacelike curve y0 every point of which was a limit point of A'. No two points of y0 could have timelike separation since otherwise some segment of A' could be deformed to give a closed non-spacelike curve. Thus y0 would be an inextendible null geodesic which was totally imprisoned in SP in both the past and future directions. Let be the closed set consisting of all limit points of y0 fl J+(p0) (or, in the Case tfrat the past distinguishing condition holds on SP, y0 n J~(p0))- As every such point would also be a limit point of A', c sfQ. Since ,o) could contain no limit point of y0 n J+(Po) (respectively, y0 n J~(p0))> would be strictly smaller than j/0. We would thus obtain an infinite sequence of closed sets

Each would be non-empty, being the set of all limit points of the totally future (respectively, past) imprisoned null geodesic (\J+(Pp-i) (respectively, y^ n J~(Pp~i))-

Let Jf — H ¿¿b- As SP is compact, Jf" would be non-empty since the fi intersection of any finite number of the sfp would be non-empty (Hocking and Young (1961), p. 19). Suppose re Jf. Then f(r) =f(pfi) for some /?. But fl would be empty so r could not be in and so could not be in Jf. This shows that there can be no future-inextendible non-spacelike curve totally future imprisoned in SP. □

The causal relations on (Jt, g) may be used to put a topology OH dt called the Alexandrov topology This is the topology in which a set is defined to be open if and only if it is the union of one or more sets of thé form I+(p) fl I~(q), p,qeJ(. As I+[p) fl I~(q) is open in the manifold topology, any set which is open in the Alexandrov topology will be open in the manifold topology, though the converse is not necessarily true.

Suppose however that the strong causality condition holds on Jl.

Then about any point one can find a local causality neighbourhood The Alexandrov topology of {°U, regarded as a spacetime in its own right, is clearly the same as the manifold topology oitfl. Thus the Alexandrov topology of is the same as the manifold topology since can be covered by local causality neighbourhoods. This means that if the strong causality condition holds, one can determine the topological structure of space-time by observation of causal relationships.

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