It is sufficient to prove (i) since can also be regarded as the boundary of the past set {J(—SP). Let {a:n} be an infinite sequence of points in I+{q) n W which converge on q. If I+(q) <= I+{SP - it"), there will be a past-directed timelike curve An to SP — W from each xn. By lemma 6.2.1 there will be a past-directed limit curve A from q to — "W). As I~(q) is open and contained in —I~(q) n^is empty. Thus A must be a null geodesic and must lie in □

As an example of the above results, consider j+(JC) = 1+(JC), the boundary of the future of a closed set JC. By proposition 6.3.1 it is an achronal manifold and by the above lemma, every point of J (JC) — JC belongs to [j+(Jf)]w or [j+(Jf)]+. This means that is generated by null geodesic segments which may have future endpoints in J+(Jf) — Jf but which, if they do have past endpoints, can have them only on Jf itself. As figure 34 shows, there may be null geodesic generating segments which do not have past endpoints at all but which go out to infinity. This example is admittedly rather artificial but Penrose (1965a) has shown that similar behaviour occurs in something as simple as the plane wave solutions; the anti-de Sitter (§5.2) and Reissner-Nordstrom (§5.5) solutions provide other examples. We shall see in § 6.6 that this behaviour is connected with the absence of a Cauchy surface for these solutions.

We shall say that an open set °U is causally simple if for every compact set Jf" c: c%t j+(Jf) n ^ = ) n ^ and j-pf) n ^ = E-(JiT) n This is equivalent to saying that J+(Jf) and J~(Jf) are closed in

Postulate (a) of § 3.2 required only that causality should hold locally; the global question was left open. Thus we did not rule out the possibility that on a large scale there might be closed timelike curves (i.e. timelike Svs). However the existence of such curves would seem to lead to the possibility of logical paradoxes: for, one could imagine that with a suitable rocketship one could travel round such a curve and, arriving back before one's departure, one could prevent oneself from setting out in the first place. Of course there is a contradiction only if one assumes a simple notion of free will; but this is not something which can be dropped lightly since the whole of our philosophy of science is based on the assumption that one is free to perform any experiment. It might be possible to form a theory in which there were closed timelike curves and in which the concept of free will was modified (see, for example, Schmidt (1966)) but one would be much more ready to believe that space-time satisfies what we shall call the chronology condition: namely, that there are no closed timelike curves. One must however bear in mind the possibility that there might be points (maybe where the density or curvature was very high) of space-time at which this condition does not hold. The set of all such points will be called the chronology violating set of Ut and has the following character:

Proposition 6.4.1 (Carter)

The chronology violating set of Ul is the disjoint union of sets of the form I+(q) n I~(q), qeJt.

If q is in the chronology violating set of Ui, there must be a future-directed timelike curve A with past and future endpoints at q. If r e I~(q) n I+(q), there will be past- and future-directed timelike curves and ¡i2 from qtor. Then (fii)~l o Ao/i2 will be a future-directed timelike curve with past and future endpoints at r. Moreover if

To complete the proof, note that every point r at which chronology is then re[I-(q) n /+(?)] n [/-(y) n /+(?)] pel-(q)nl+(q) = l-(p)nl+(p).

Proposition 6.4.2

If Ut is compact, the chronology violating set of UK is non-empty.

Jl can be covered by open sets of the form I+(q), q e If the chronology condition holds at q, then q$I+(q). Thus if the chronology condition held at every point, ^ could not be covered by a finite number of sets of the form I+(q). □

From this result it would seem reasonable to assume that space-time is non-compact. Another argument against compactness is that any compact, four-dimensional manifold on which there is a Lorentz metric cannot be simply connected. (The existence of a Lorentz metric implies that the Euler number is zero (Steenrod (1951), p. 207). *

Now x = S (— l)n-Bn where Bn ^ 0 is the nth Betti number of Jf. By n-0

duality (Spanier (1966), p. 297) Bn = Bt_n. Since B0 = Bt = 1, this implies that Bt # 0 which in turn implies # 0 (Spanier (1966), p. 398).) Thus a compact space-time is really a non-compact manifold in which points have been identified. It would seem physically reasonable not to identify points but to regard the covering manifold as representing space-time.

We shall say that the causality condition holds if there are no closed non-spacelike curves. Similar to proposition 6.4.1, one has:

Proposition 6.4.3

The set of points at which the causality condition does not hold is the disjoint union of sets of the form J~(q) n J+(q), □

In particular, if the causality condition is violated at qe^tf but the chronology condition holds, there must be a closed null geodesic curve y through q. Let v be an affine parameter on y (regarded as a map of an open interval of JB1 to and let..., v_lt «;„, vlt v2>... be successive values of v at q. Then we may compare at q the tangent vector 3/0i>|„_„o and the tangent vector d/Sv]^^ obtained by parallelly transporting djdv] round y. Since they both point in the same direction, they must jbe proportional: 8/8v= a djdvThe factor a has the following significance: the affine distance covered in the nth circuit of y, (vn+1 — vn), is equal to «-»(^ - v0). Thus if a > 1, v never attains the value (Hj —«0)(1 —a-1)-1 and so y is geodesically incomplete in the future direction even though one can go round an infinite number of times. Similarly if a < 1, y is incomplete in ¿he past direction, while if a = 1, it is complete in both directions. In the two-dimensional model of Taub-NUT space described in § 5.7, there is a closed null geodesic which is an example with a > 1. Since the factor a is a conformal in variant, this incompleteness is independent of the conformal factor. This kind of behaviour, however, can happen only if there is a violation of causality in some sense; if the strong causality condition (see below) holds, a suitable conformal transformation of the metric will make all null geodesies complete (Clarke (1971)).

The factor a has a further significance from the following result.

If y is a closed null geodesic curve which is incomplete in the future direction then there is a variation of y which moves each point of y towards the future and which yields a closed timelike curve.

By §2.6, one can find on J( a timelike line-element field (V, — V) normalized so that g(V, V) = — 1. As we are assuming that is time-orientable, one can consistently choose one direction of (V, — V) and so obtain a future-directed timelike unit vector field V. One can then define a positive definite metric g' by

Let t be a (non-affine) parameter on y which is zero at some point gey and which is such that gr(V, 8/8t) = — 2~i. Then t measures proper distance along y in the metric g' and has the range — oo < t < oo. Consider a variation of y with variation vector 8j8u equal to xV, where x is a function x(t). By § 4.5, where fêjët = (Djët) (ê/êt). Now suppose v were an affine parameter on y. Then djdv would be proportional to 3/Si: djdv = h8/8t, where h~x dhjdt — —f. On going round one circuit of y, 8/8v increases by a factor a > 1. Thur -

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