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Therefore if we take x(t) to be where b — j di, this will give a variation of y to the future and gives a closed timelike curve.

Proposition 6.4.5

(6) the generic condition holds, i.e. every null geodesic contains a point at which K[aRbhd[eKf}KcKd is non-zero, where K is the tangent vector;

(c) the chronology condition holds on Jl, then the causality condition holds on J(.

If there were closed null geodesic curves which were incomplete, then by the previous result they could be varied to give closed timelike curves. If they were complete, then by proposition 4.4.5 they would contain conjugate points and so by proposition 4.5.12 they could again be varied to give closed timelike curves. □

This shows that in physically realistic solutions, the causality and chronology conditions are equivalent.

As well as ruling out closed non-spacelike curves, it would seem reasonable to exclude situations in which there were non-spacelike curves which returned arbitrarily close to their point of origin or which passed arbitrarily close to other non-spacelike curves which then passed arbitrarily close to the origin of the first curve-and so on. In fact Carter (1971 a) has pointed out that there is a more than countably infinite hierarchy of such higher degree causality conditions depending on the number and order of the limiting processes involved. We shall describe the first three of these conditions and shall then give the ultimate in causality conditions.

The future (respectively, past) distinguishing condition (Kronheimer and Penrose (1967)) is said to hold at e J( if every neighbourhood oip contains a neighbourhood of p which no future (respectively, past) directed non-spacelike curve from p intersects more than once. An equivalent statement is that I+(q) = I+(p) (respectively, I~(q) =I~(p)) implies that q = p. Figure 37 shows an example in which the causality and past distinguishing conditions hold everywhere but the future distinguishing condition does not hold at p.

The strong causality condition is said to hold at^> if every neighbourhood of p contains a neighbourhood of p which no non-spacelike curve intersects more than once. Figure 38 shows an example of violation of this condition.

Null geodesic Remove strip

Figure 37. A space in which the causality and past distinguishing conditions hold everywhere, but the future distinguishing condition does not hold at p or q (in fact, I+(p) = /+(s)). The light cones on the cylinder tip over until one null direction is horizontal, and then tip back up; a strip has been removed, thus breaking the closed null geodesic that would otherwise occur.

Null geodesic Remove strip

Figure 37. A space in which the causality and past distinguishing conditions hold everywhere, but the future distinguishing condition does not hold at p or q (in fact, I+(p) = /+(s)). The light cones on the cylinder tip over until one null direction is horizontal, and then tip back up; a strip has been removed, thus breaking the closed null geodesic that would otherwise occur.

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