We shall show that d(p) is continuous on

First to show that it is upper semi-continuous: given e > 0, let SS be a ball about p such that the volume of ¿¡§ in the measure fi is less than By property (3), for av a2 e [0,3] with a^ < a2 one can find a neighbourhood ^(c^, a2) ofp in such that

Let n be a positive integer greater than 2e-1. Then we define the set to be ^ = fl ^{i + tirr1, 1 + \{i +l)«"1), i = 0, 1,..., 2n. ^ will be d(p) = j a neighbourhood of p and will be contained in ^(a.a + vr1) for any oe fl, 2], Therefore l~(q,Ul, h(a)) — 3S will be contained in

Thus 6{q, a) < 6{p, a + £e) + \e and so D(q) < fl(p) + e, showing that 5 is upper semi-continuous. The proof that it is lower semi-continuous is similar. To obtain a differenti-able function one can average 5 over a neighbourhood of each point with a suitable smoothing function. By taking the neighbourhood small enough one can obtain a function/which has everywhere a timelike gradient in the metric g. Details of this smoothing procedure are given in Seifert (1968). □

The spacelike surfaces {/ = constant} may be thought of as surfaces of simultaneity in space-time, though of course they are not unique. If they are all compact they are all diffeomorphic to each other, but this is not necessarily true if some of them are non-compact.

In Newtonian theory there is instantaneous action-at-a-distance and so in order to predict events at future points in space-time one has to know the state of the entire universe at the present time and also to assume some boundary conditions at infinity, such as that the potential goes to zero. In relativity theory, on the other hand, it follows from postulate (a) of § 3.2 that events at different points of space-time can be causally related only if they can be joined by a non-spacelike curve. Thus a knowledge of the appropriate data on a closed set SP (if one knew data on an open set, that on its closure would follow by continuity) would determine events in a region D+(£P) to the future oi£P called the future Cauchy development or domain of dependence of SP, and defined as the set of all points peUK such that every past-inextendible non-spacelike curve through p intersects $P (N.B. D+{£P) => £P).

Penrose (1966, 1968) defines the Cauchy development of £f slightly differently, as the set of all points psJi such that every past-inextendible timelike curve through p intersects SP. We shall denote this set by D+(£P). One has the following result:

Proposition 6.5.1

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