Causal structure

By postulate (a) of § 3.2, a signal can be sent between two points of ^ only if they can be joined by a non-spacelike curve. In this chapter we shall investigate further the properties of such causal relationships, establishing a number of results which will be used in chapter 8 to prove the existence of singularities.

By § 3.2, the study of causal relationships is equivalent to thatof the conformal geometry of i.e. of the set of all metrics g conformal to the physical metric g (g = fi2g, where fi is a non-zero, Cr function). Under such a conformal transformation of the metric a geodesic curve will not, in general, remain a geodesic curve unless it is null, and even in this case an affine parameter along the curve will not remain an affine parameter. Thus in most cases geodesic completeness (i.e. whether all geodesies can be extended to arbitrary values of their affine parameters) will depend on the particular conformal factor and so will not (except in certain special cases described in §6.4) be a property of the conformal geometry. In fact Clarke (1971) and Siefert (1968) have shown that, provided a physically reasonable causality condition holds, any Lorentz metric is conformal to one in which all null geodesies and all future-directed timelike geodesies are complete. Geodesic completeness will be discussed further in chapter 8 where it forms the basis of a definition of a singularity.

§ 6.1 deals with the question of the orientability of timelike and spacelike bases. In §6.2 basic causal relations are defined and the definition of a non-spacelike curve is extended from piecewise dif-ferentiable to continuous. The properties of the boundary of the future of a set are derived in § 6.3. In § 6.4 a number of conditions which rule out violations or near violations of causality are discussed. The closely related concepts of Cauchy developments and global hyperbolicity are introduced in § 6.5 and § 6.6, and are used in § 6.7 to prove the existence of non-spacelike geodesies of maximum length between certain pairs of points.

In §6.8 we describe the construction of Geroch, Kronheimer and

Penrose for attaching a causal boundary to space-time. A particular example of such a boundary is provided by a class of asymptotically flat space-times which are studied in § 6.9.

6.1 Orientability

In our neighbourhood of space-time there is a well-defined arrow of time given by the direction of increase of entropy in quasi-isolated thermodynamic systems. It is not quite clear what the relationship is between this arrow and the other arrows defined by the expansion of the universe and by the direction of electrodynamic radiation; the reader who is interested will find further discussion in Gold (1967), Hogarth (1962), Hoyle and Narlikar (1963) and Ellis and Sciama (1972). Physically it would seem reasonable to suppose that there is a local thermodynamic arrow of time defined continuously at every point of space-time, but we shall only require that it should be possible to define continuously a division of non-spacelike vectors into two classes, which we arbitrarily label future- and past-directed. If this is the case, we shall say that space-time is time-orientable. In some space-times it is not possible to define such a time-orientation. An example is the space-time obtained from de Sitter space (§5.2) in which points are identified by reflection through the origin of the five-dimensional imbedding space. In this space there are closed curves, non-homotopic to zero, on going round which the orientation of time is reversed. However this difficulty could clearly be resolved by simply unidentifying the points again, and in fact this is always the case: if a space-time (J(, g) is not time-orientable, then it has a double covering space (Ji, g) which is. Ji may be defined as the set of all pairs (p, a) where peUf and a is one of the two orientations of time at p. Then with the natural structure and the projection n : (p, a)->p, Jl is a double covering oiJ(. If Ji consists of two disconnected components then (Jl, g) is time-orientable. If connected, then (Jl, g) is not time-orientable but (Jl, g) is. In the following sections we shall assume that either (J(, g) is time-orientable or we are dealing with the time-orientable covering space. If one can prove the existence of singularities in this space-time then there must also be singularities in {Jl, g).

One may also ask whether space-time is space-orientable, that is whether it is possible to divide bases of three spacelike axes into right handed and left handed bases in a continuous manner. Geroch (1967 a)

has pointed out that there is an interesting connection between this and time-orientability which follows because some experiments on elementary particles are not invariant under charge or parity reversals, either singly or together. On the other hand there are theoretical reasons for believing that all interactions are invariant under the combination of charge, parity and time reversals (CPT theorem; see Streater and Wightman (1964)). If one believes that the non-invariance of weak interactions under charge and parity reversals is not merely a local effect but exists at all points of space-time, then it follows that going round any closed curve either the sign of a charge, the orientation of a basis of spacelike axes, and the orientation of time must all reverse, or none of them does. (The ordinary Maxwell theory, in which the electromagnetic field has a definite sign at every point, does not allow the sign of a charge to change on going around a closed curve non-homotopic to zero unless the orientation of time changes. However one could have a theory in which the field was double-valued and changed sign on going round such a curve. This theory would agree with all existing experimental evidence.) In particular if one assumes that space-time is time-orientable then it must also be space-orientable. (This in fact follows on using the experimental evidence alone without appealing to the CPT theorem.)

Geroch (1968c) has also shown that if it is possible to define two-component spinor fields at every point then space-time must be parallelizable, that is it must be possible to introduce a continuous system of bases of the tangent space at every point. (Further consequences of the existence of spinor structures are obtained in Geroch (1970a).)

6.2 Causal curves

Taking space-time to be time-orientable as explained in the previous section, one can divide the non-spacelike vectors at each point into future- and past-directed. For sets^ and the chronological future I+(Sf, ofSf relative to Ql can then be defined as the set of all points in % which can be reached from y by a future-directed timelike curve' in °ti. (By a curve we mean always one of non-zero extent, not just a single point. Thus I+(Sf,<%) may not contain y.) will be denoted by and is an open set, since if can be reached by a future-directed timelike curve from then there is a small neighbourhood of p which can be so reached.

This definition has a dual in which 'future' is replaced by 'past', and the + by a —; to avoid repetition, we shall regard dual definitions and results as self-evident.

The causal future of SP relative to % is denoted by J+(£P, it is defined as the union of SP n ^ with the set of all points in 'W which can be reached from SP by a future-directed non-spacelike curve in eii. We saw in § 4.5 that a non-spacelike curve between two points which was not a null geodesic curve could be deformed into a timelike curve between the two points. Thus if % is an open set andp, q, re 'W, then either qeJ+(p,®),reI+iq,®)} .

}• imply reI+(p,W). or qeI+(p,<%),reJ+(q,®)f ^

From this it follows that i+(p, <%) = J+(p, <%) and l+(p, <%) = J+(p, <%) where for any set JT, JT denotes the closure of JT and

Jf"=5fn M'-Jf) denotes the boundary of JT.

Chronological future I+(S>) ^ '^Null geodesic in J*(y)

which does not intersect J*(y) and has ho past endpoint in Jt

Null geodesies; through y generating past otJ*{y)

Figure 34. When a point has been removed from Minkowski space, the causal future J+(£f) of a closed set SP is not necessarily closed. Further parts of the boundary of the future of SP may be generated by null geodesic segments which have no past endpoints in

Causal—\ / "Point removed r i. / from Jt future —x - '

Null geodesies; through y generating past otJ*{y)

Figure 34. When a point has been removed from Minkowski space, the causal future J+(£f) of a closed set SP is not necessarily closed. Further parts of the boundary of the future of SP may be generated by null geodesic segments which have no past endpoints in

As before, J+(.SP, Jl) will be written simply as J+(SP). It is the region of space-time which can be causally affected by events in SP. It is not necessarily a closed set even when SP is & single point, as figure 34 shows. This example, incidentally, illustrates a useful technique for constructing space-times with given causal properties: one starts with some simple space-time (unless otherwise indicated this will be Minkowski space), cuts out any closed set and, if desired, pastes it together in an appropriate way (i.e. one makes identifications of points of J(). The result is still a manifold with a Lorentz metric and therefore still a space-time even though it may look rather incomplete where points have been cut out. As mentioned above, however, this incompleteness can be cured by an appropriate conformal transformation which sends the cut out points to infinity.

The future horismos of Sf relative to denoted by E+(.Sf, %), is defined as we write E+(Sf) for E+(^,JK). (In some papers the relations p el+(q), p e J+{q) and p e E+(q) are denoted by q ^ p, q < p and q-+p respectively.) If ^ is an open set, points of E+(Sf,<%) must lie on future-directed null geodesies from ¿f by proposition 4.5.10, and if ^ is a convex normal neighbourhood about p then it follows from proposition 4.5.1 that E+(p,^) consists of the future-directed null geodesies in ^ from p, and forms the boundary in °U of both I+(p, <%) and J+(p, °U). Thus in Minkowski space, the null cone ofp forms the boundary of the causal and chronological futures of p. However in more complicated space-times this is not necessarily the case (e.g. see figure 34).

For the purposes of what follows it will be convenient to extend the definition of timelike and non-spacelike curves from piecewise dif-ferentiable to continuous curves. Although such a curve may not have a tangent vector we can still say that it is non-spacelike if locally every two points of the curve can be joined by a piecewise differenti-able non-spacelike curve. More precisely, we shall say that a continuous curve y: F where F is a connected interval of R1, is future-directed and non-spacelike if for every teF there is a neighbourhood G of t in F and a convex normal neighbourhood °U of y{t) in J( such that for any txeO, y{ti)eJ~{y{t),<%)-y{t) if ^ < t, and y(ti)eJ+(y{t),'^) — y(t) if t < tv We shall say that y is future-directed and timdike if the same conditions hold with J replaced by I. Unless otherwise specified, we will in future mean by a timelike or non-spacelike curve such a continuous curve, and shall regard two curves as equivalent if one is a reparametrization of the other. With this generalization we can establish a result that will be used repeatedly in the rest of this chapter. We first give a few more definitions.

A point p will be said to be a. future endpoint of a future-directed non-spacelike curve y: if for every neighbourhood of p there ia&teF such that y{t^ e 'f for every t^eF with A non-spacelike curve is future-iveztendible (respectively, fvture-iveztendible in a set £?) if it has no future endpoint (respectively, no future endpoint in SP). A point p will be said to be a limit point of an infinite sequence of non-

spacelike curves An if every neighbourhood of p intersects an infinite number of the Xn. A non-spacelike curve A will be said to be a limit curve of the sequence A„ if there is a subsequence X'n of the Xn such that for every peX, X'n converges to p.

Lemma 6.2.1

Let be an open set and let An be an infinite sequence of non-spacelike curves inwhich are future-inextendible in^. If p is a limit point of An, then through p there is a non-spacelike curve A which is future-inextendible in S? and which is a limit curve of the A„.

It is sufficient to consider the case S? = JK since £f can be regarded as a manifold with a Lorentz metric. Let be a convex normal coordinate neighbourhood about p and let &8(q, a) be the open ball of coordinate radius a about q. Let b > 0 be such that SS(p, b) is defined and let A(l,0)n be a subsequence of Xn n ^ which converges to p. Since ¿%{p,b) is compact it will contain limit points of the A(l,0)n. Any such limit point y must lie either in J~(p, or J+{p, Ql^ since otherwise there would be neighbourhoods of y and "V^oip between which there would be no non-spacelike curve in Choose x^eJ+ip^/Jn^ip.b)

to be one of these limit points (figure 35), and choose A(l, l)n to be a subsequence of A(l, 0)n which converges to xlv The point xu will be a point of our limit curve A. Continue inductively, defining

as a limit point of the subsequence A(i— l,i— l)n for j = 0, A({,j — l)n for i^j^ 1, and defining X(i, j)n as a subsequence of the above subsequence which converges to xtj. In other words we are dividing the interval [0,6] into smaller and smaller sections and getting points on our limit curve on the corresponding spheres about p. As any two of the xi} will have non-spacelike separation, the closure of the union of all the xtj (j > i) will give a non-spacelike curve A from p = xi0 to xn = xu. It now remains to construct a subsequence A'„ of the An such that for each qeX, A'n converges to q. We do this by choosing A'm to be a member of the subsequence A(m, m)n which intersects each of the balls @{xmj, m-lb) for 0 < j < m. Thus A will be a limit curve of the Xn from p to xu. Now let be a convex normal neighbourhood about xxx and repeat the construction using this time the sequence A'n. Continuing in this fashion, one can extend A indefinitely. □

Figure 35. The non-spacelike limit curve A through p of a family of non-spacelike curves A„ for which p is a limit point.

6.3 Achronal boundaries

From proposition 4.6.1 it follows that in a convex normal neighbourhood W, the boundary of I+{p, Ql) or J+{j>, Ql) is formed by the future-directed null geodesies from p. To derive the properties of more general boundaries we introduce the concepts of achronal and future sets.

A set y is said to be achronal (sometimes referred to as 'semi-spacelike ' in the literature) if J+(y) n is empty, in other words if there are no two points oiif with timelike separation, if is said to be & future set if y ). Note that if y is a future set, J(—y is a past set. Examples of future sets include I+{jV) and J+{jV), where jV is any set. Examples of achronal sets are given by the following fundamental result.

Proposition 6.3.1

If y is a future set then the boundary of£f, is a closed, imbedded, achronal three-dimensional C1- submanifold.

liqeS?, any neighbourhood of q intersectsSf and Jl—lip el+{q), then there is a neighbourhood of q in I~(p). Thus I+(q) <= if. Similarly I~[q) <=■ If rel+{q), there is a neighbourhood V of r such that V c j+[q) <=. <?. Thus r cannot belong to One can introduce normal coordinates {x1, x2, xz, xi) in a neighbourhood about q with d/dx* timelike and such that the curves {a:1 = constant (i = 1,2,3)} intersect both I+(q, ^J and I~(q, Then each of these curves must contain precisely one point of The a^-coordinate of these points must be a Lipschitz function of the xi (i = 1,2,3) since no two points of & have timelike separation. Therefore the one-one map <f>a: SP n Wa-+R3 defined by <f>a(p) ~ x*(p) (i = 1,2,3) for is a homeomorphism. Thus n <j>a) is a Cl~ atlas for □

We shall call a set with the properties of^" listed in proposition 6.3.1, an achronal boundary. Such a set can be divided into four disjoint subsets ^ as follows: for a point qeS^ there may or may not exist points p,rewith peE~(q)—q, reE+(q)—q. The different possibilities define the subsets of ^ according to the scheme:

If qetfy, then reE+(p) since reJ+{p) and by proposition 6.3.1, r$I+(p). This means that there is a null geodesic segment in if through q.liqei?+ (respectively ¿f_) then q is the future (respectively, past) endpoint of a null geodesic in The subset ^ is spacelike (more strictly, acausal). These divisions are illustrated in figure 36.

A useful condition for a point to lie in ifN, or &L is given in the following lemma due to Penrose (Penrose (1968)):

Lemma 6.3.2

Let W be a neighbourhood of qei? where is a future set. Then

Bp fo

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