The Large Scale Structure of Space-Time Physics by Stephen Hawking

Author:Stephen Hawking [Hawking, Stephen]
Language: eng
Format: epub
ISBN: 9781009253154
Barnesnoble:
Published: 2017-05-15T05:45:44+00:00

208

C A U S A L S T R U C T U R E

[6.6

a neighbourhood of y in C(p, q) consists of all the curves in C(p, q) whose points in .A lie in a neighbourhood if/ of the points of y in .A (figure 45). Leray’s definition is that an open set .At is globally hyperbolic if C(p, q) is compact for all p, q E .Af. These definitions are equivalent, as is shown by the following result.

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Yr

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I

I

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I

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FIGURE 45. A neighbourhood if” of the points of y in ..,/(. A neighbourhood of y in C(p, q) consists of all non -spacelike curves from p to q whose points lie in “fr.

Proposition 6.6.2 (Seifert ( 1 967), Gerock ( 1 970b)).

Let strong causality hold on an open set .At such that

.At J-(.At) n J+(.Af).

=

Then .At is globally hyperbolic if and only if C(p, q) is compact for all p, q E.Af.

Suppose first that C(p, q) is compact. Let r n be an infinite sequence of points in J+(p) n J-(q) and let An be a sequence ofnon-spacelike curves from p to q through the corresponding r n· 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 Oii be a neighbourhood of A in.A such that O/i is compact. Then Oii will contain all A’ n and hence all r’ n for n sufficiently large, and so there will be a point r E Oii which is a limit point of the r’ n·

Clearly r lies on A. Thus every infinite sequence in J+(p) n J-(q) has a limit point in J+(p) n J-(q) . Hence J+(p) n J-(q) is compact.

Conversely, suppose J+(p) n J-(q) is compact. Let An be an infinite sequence ofnon-spacelike curves from p to q. By lemma 6.2. 1 applied to the open set .A - q, there will be a future-directed non-spacelike curve A from p which is inextendible in .A - q, and is such that there is

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