1

Identify

Figube 40. A space satisfying the strong causality condition, but in which the slightest variation of the metric would permit there to be closed timelike lines through p. Three strips have been removed from a cylinder; light cones are at ± 45°.

Identify

Figube 40. A space satisfying the strong causality condition, but in which the slightest variation of the metric would permit there to be closed timelike lines through p. Three strips have been removed from a cylinder; light cones are at ± 45°.

Even imposition of the strong casuality condition does not rule out all causal pathologies, as figure 40 shows one can still have a spacetime which is on the verge of violating the chronology condition in that the slightest variation of the metric can lead to closed timelike curves. Such a situation would not seem to be physically realistic since General Relativity is presumably the classical limit of some, as yet unknown, quantum theory of space-time and in such a theory the Uncertainty Principle would prevent the metric from having an exact value at every point. Thus in order to be physically significant, a property of space-time ought to have some form of stability, that is to say, it should also be a property of 'nearby' space-times. In order to give a precise meaning to' nearby' one has to define a topology on the set of all space-times, that is, all non-compact four-dimensional manifolds and all Lorentz metrics on them. We shall leave the problem of uniting in one connected topological space manifolds of different topologies (this can be done); and shall just consider putting a topology on the set of all Cr Lorentz metrics (r ^ 1) on a given manifold. There are various ways in which this can be done, depending on whether one requires a 'nearby' metric to be nearby in just its values (C° topology) or also in its derivatives up to the Ath order (Ck topology) and whether one requires it to be nearby everywhere (open topology) or only on compact sets (compact open topology).

For our purposes here, we shall be interested in the C° open topology. This may be defined as follows: the symmetric tensor spaces Ts%(jp) of type (0,2) at every point peUK form a manifold (with the natural structure) TS\(UK), the bundle of symmetric tensors of type (0,2) over UK. A Lorentz metric g on UK is an assignment of an element of Ts$ (UK) at each point peUK and so can be regarded as a map or cross-section ():Ut-+ T^(UK) such that ttoQ = 1 where 7ns the projection TB%(UK) UK which sends xeTs\(p) to p. Let °U be an open set in TS%(UK) and let 0(%) be the set of all C° Lorentz metrics g such that (j(UK) is contained in °U (figure 41). Then the open sets in the C° open topology of the Cr Lorentz metrics on UK are defined to be the union of one or more sets of the form 0(<%).

We say that the stable causality condition holds on UK if the spacetime metric g has an open neighbourhood in the C° open topology such that there are no closed timelike curves in any metric belonging to the neighbourhood. (It would not make any difference if one used the Ck topology here, but one could not use a compact open topology since in that topology each neighbourhood of any metric contains closed timelike curves.) In other words, what this condition means is that one can expand the light cones slightly at every point without introducing closed timelike curves.

Proposition 6.4.9

The stable causality condition holds everywhere on UK if and only if there is a function / on UK whose gradient is everywhere timelike.

Remark. The function/ can be thought of as a sort of cosmic time in the sense that it increases along every future-directed non-spacelike curve.

Proof. The existence of a function / with an everywhere timelike gradient implies the stable causality condition since there can be no closed timelike curves in any metric h which is sufficiently close to g that for every point p e the null cone oip in the metric h intersects the surface {/ = constant} through p only at p. To show that the converse is true we introduce a volume measure fi (unrelated to the volume measure defined by the metric g) on Jl such that the total volume of

Figure 41. An open set in the C" open topology on the space of symmetric tensors of type (0,2) on JK.

Ji is one. One way of doing this is as follows: choose a countable atlas <f>a) for JK such that is compact in Ri. Let fi0 be the natural

Euclidean measure on R* and let/a be a partition of unity for the atlas Then n may be defined as "Lf^l/i^)]-1^*/i0.

Now if the stable causality condition holds one can find a family of Cr Lorentz metrics h(a), ae [0,3], such that:

(2) there are no closed timelike curves in the metric h(o) for each ae[0,3];

(3) if av Oj e [0,3] with < a2, then every non-spacelike vector in the metric hfc^) is timelike in the metric h^).

For yeJ', let 6(j>, a) be the volume of I~{p, J(, h(a)) in the measure H where we use W, h) to denote the past of if relative to % in the metric h. For a given value of oe (0,3), 6(p, a) will be a bounded function which increases along every non-spacelike curve. It may not, however, be continuous: as figure 42 shows, it may be possible that a slight alteration of position may allow one to see past an obstruction and so greatly increase the volume of the past. One thus needs some way of smearing out 6(p, a) so as to obtain a continuous function which

Past of g

Past of g

Figure 42. A small displacement of a point from p to q results in a large ohange in the volume of the past of the point. Light cones are at ± 45° and a strip has been removed as shown.

increases along every curve which is future-directed and non-spacelike in the metric h(0). One can do this by averaging over a range of a: let

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