Edge iy edge y

Let be a sequence of neighbourhoods of a point geedge (H+(£f))

such that any neighbourhood of q encloses all the akn for n sufficiently large. In each there will be points pn e I~(q, and rn e I+(q, which can be joined by a timelike curve A„ which does not intersect H+(£f). This means that A„ cannot intersect D+(£f). By proposition 6.5.1, qeD+(S?) and so I~(q) <= I~(D+(y)) <= J-(y) y D+(y). Thus pn must lie in I~{SP). Also every timelike curve from q which is inextend-ible in the past direction must intersects. Therefore for each n, there

Figure 43. The future Cauchy development D+(£f) and future Cauchy horizon H+(£f) of a closed set if which is partly null and partly spacelike. Note that H+(£f) is not necessarily connected. Null lines are'at ± 46° and a strip has been removed.

must be a point ofy on every timelike curve in <25'n between q andjp„ and so q must lie in P. As the curves An do not intersect q lies in edge (y). The proof the other way round is similar. □

Proposition 6.5.3

Let y be a closed achronal set. Then H+(y) is generated by null geodesic segments which either have no past endpoints or have past endpoints at edge (y).

The set & s D+(y) u I-(Sf) is a past set. Thus by proposition 6.3.1 & is an achronal C1- manifold. H+(y) is a closed subset of IF. Let q be a point of H+(y) - edge (y). If q is not in y then qeI+{y) since qeD+(y). A&y is achronal one can find a convex normal neighbour

Figure 43. The future Cauchy development D+(£f) and future Cauchy horizon H+(£f) of a closed set if which is partly null and partly spacelike. Note that H+(£f) is not necessarily connected. Null lines are'at ± 46° and a strip has been removed.

hood W oiq which does not intersect Alternatively if q is in Zf, let if be a convex normal neighbourhood of q such that no point of I+(q, W) can be joined to any point in I~(q, W) by a timelike curve in which does not intersect In either case, if p is any point in I+(q) there must be a past-directed timelike curve from p to some point of — — "W since otherwise p would be in D+(.Therefore by condition (i) of lemma 6.3.2, applied to the future set

Corollary

If edge vanishes, then H+(£?),if nonempty, is an achronal three-dimensional imbedded C1_ manifold which is generated by null geodesic segments which have no past endpoint.

We shall call an acausal set with no edge, a partial Cauchy surface. That is, a partial Cauchy surface is a spacelike hypersurface which no non-spacelike curve intersects more than once. Suppose there were a connected spacelike hypersurface y (with no edge) which some non-spacelike curve A intersected at points pt and p2. Then one could join pl and p2 by a curve /¿in^and/i U A would be a closed curve which crossed & once only. This Figure 44- & is a connected , , .. .. ii/. , spacelike hypersurface without curve couldnotbecontinuouslydeformed ¿ge ^ yjt is not a partial to zero since such a deformation could Cauchy surface; however each change the number of times it crossed image tt'HS^) of Sf in the uni-

by an even number only. Thus J( could v®rsal covering manifold^of . , . , ^k, is a partial Cauchy surface not be simply connected. This means we could' unwrap' Ji by going to the simply connected universal covering manifold jft in which each connected component of the image of if is a spacelike hypersurface (with no edge) and is therefore a partial Cauchy surface in jft (figure 44). However going to the universal covering manifold may unwrap J( more than is required to obtain a partial Cauchy surface and may result in

the partial Cauchy surface being non-compact even though was compact. For the purposes of the following chapters we would like a covering manifold which unwrapped Ui sufficiently so that each connected component of the image of if was a partial Cauchy surface but so that each such component remained homeomorphic to . Such a covering manifold may be obtained in at least two different ways.

Recall that the universal covering manifold may be defined as the set of all pairs of the form (p, [A]) where peUK and where [A] is an equivalence class of curves in from some fixed point qeUt to p, which are homotopic modulo q and p. The covering manifold U(H is defined as the set of all pairs (p, [A]) where now [A] is an equivalence class of curves from top homotopic modulo and p (i.e. the endpoints on if can be slid around). U(H may be characterized as the largest covering manifold such that each connected component of the image of if is homeomorphic to if. The covering manifold UiG (Geroch (19676)) is defined as the set of all pairs (p, [A]) where this time [A] is an equivalence class of curves from a fixed point q to p which cross if the same number of times, crossings in the future direction being counted positive and those in the past direction, negative. JiQ may be characterized as the smallest covering manifold in which each connected component of the image of if divides the manifold into two parts. In each case the topological and differential structure of the covering manifold is fixed by requiring that the projection which maps (p, [A]) to p is locally a diffeomorphism.

Define D(Sf) = D+(Sf) U D~(Sf). A partial Cauchy surface if is said to be a global Cauchy surface (or simply, a Cauchy surface) if D(Sf) equals UK. That is, a Cauchy surface is a spacelike hypersurface which every non-spacelike curve intersects exactly once. The surfaces {x* = constant} are examples of Cauchy surfaces in Minkowski space, but the hyperboloids

{(a;4)2 — (a;3)2 — (a;2)2 — (¡c1)2 = constant}

are only partial Cauchy surfaces since the past or future null cones of the origin are Cauchy horizons for these surfaces (see §5.1 and figure 13). Being a Cauchy surface is a property not only of the surface itself but also of the whole space-time in which it is imbedded. For example, if one cuts a single point out of Minkowski space, the resultant space-time admits no Cauchy surface at all.

If there were a Cauchy surface for Ui, one could predict the state of the universe at any time in the past or future if one knew the relevant data on the surface. However one could not know the data unless one was to the future of every point in the surface, which would be impossible in most cases. There does not seem to be any physically compelling reason for believing that the universe admits a Cauchy surface; in fact there are a number of known exact solutions of the Einstein field equations which do not, among them the anti-de Sitter space, plane waves, Taub-NUT space and Reissner-Nordstrom solution, all described in chapter 5. The Reissner-Nordstrom solution (figure 25) is a specially interesting case: the surface if shown is adequate for predicting events in the exterior regions I where r > r+ and in the neighbouring region II where r_ < r < r+, but then there is a Cauchy horizon at r = r_. Points in the neighbouring region III are not in D+(S?) since there are non-spacelike curves which are inextendible in the past direction and which do not cross r = r_ but approach the points i+ (which may be considered to be at infinity) or the singularity at r = 0 (which cannot be considered to be in the space-time; see §8.1). There could be extra information coming in from infinity or from the singularity which would upset any predictions made simply on the basis of data on if. Thus in General Relativity one's ability to predict the future is limited both by the difficulty of knowing data on the whole of a spacelike surface and by the possibility that even if one did it would still be insufficient. Nevertheless despite these limitations one can still predict the occurrence of singularities under certain conditions.

6.6 Global hyperbolicity

Closely related to Cauchy developments is the property of global hyperbolicity (Leray (1952)). A set is said to be globally hyperbolic if the strong causality assumption holds on Jf and if for any two points p,ge Jf, J+{p) f) J~{q) is compact and contained in Jf. In a sense this can be thought of as saying that J+(p) f) J~(q) does not contain any points on the edge of space-time, i.e. at infinity or at a singularity. The reason for the name 'global hyperbolicity' is that on Jf, the wave equation for a ¿-function source at peJf has a unique solution which vanishes outside Jf — J+[p,Jr) (see chapter 7).

Recall that Jf is said to be causally simple if for every compact set JT contained in Jf, J+pf") n Jf and J~(Jf) f) Jf are closed in Jf.

Proposition 6.6.1

An open globaliy hyperbolic set Jf is causally simple. Let p be any point oijV. Suppose there were a point qe(J+(p)-J+(p))(]^.

As jV is open, there would be a point re(I+(q) fl jV). But then q 6 J+(p) fl J~(r), which is impossible as J+(p) fl J~[r) would be compact and therefore closed. Thus J+(p) fl ^ and J~(p) fl Jf are closed in Jf.

Now suppose there exists a point qe(J+pT) — J+pf")) n Jf. Let qn be an infinite sequence of points in I+(q) fl Jf converging to q, with qn+iel~(qn). For each n, J~(qn)fl & would be a compact non-empty set. Therefore fl {^"(ffj fi -^O would be a non-empty set. Let p be a n point of this set. Then J+(p) would contain qn for all n. But J+(p) is closed. Therefore J+(p) contains q. □

Corollary

If and are compact sets in Jf, n J~(Jft) is compact.

One can find a finite number of points p^J/" such that

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