# (p.295) Appendix 2 Branched Four-Dimensional Space-Time

# (p.295) Appendix 2 Branched Four-Dimensional Space-Time

In identifying the frame-invariant features of branched four-dimensional space-time, our point of departure will be that of Belnap (1992). Belnap defines ‘Our World’ as a huge set of *point-events,* i.e. space-time points each of which is characterized by attributes such as mass, charge, curvature, and various field properties. Point-events are not empty space-time locations, but are individual punctiform occurrences. We may, if we wish, speak of two point-events which share the same properties as instantianting the same *event-types,* the latter being universale while point-events are particulars.

As it is conceived in this appendix, the world is a *topological space* with a binary relation—the so-called causal relation of special relativity—defined on it. More precisely, the world is an ordered triple [**W**, *O*, ≤] where **W** is the set of all point-events, *O* is the set of open subsets of **W**, and ≤ is the causal relation which defines the light-cone structure of **W**. The relation ≤ is reflexive, transitive, and antisymmetric, so that ≤ is a partial ordering of **W**.

Two essential notions are those of *chain* and *slice.* A *chain* is a subset of **W**, any two members of which are comparable by ≤. A maximal chain is *lightlike* or *null* if for any two points *x* and *y* on it, every point *z* which is between *x* and *y,* i.e. which is such that *x 〉 z* and *z 〉 y,* is already a member of the chain. A *non-null timelike path* is a maximal chain with the property that for any two points on it, some point between them is not on the path. The surface of the double light cone associated with every point *x* of **W** is the set of all points connected by a null track to *x.*

In order to define the sense in which **W** is branched, the concept of a slice is required. Two points *x* and *y* are *incomparable* if *x* ≰ *y* and *y* ≰ *x.* A slice is then defined to be a maximal connected set of pairwise incomparable points. A slice is an infinitely thin hypersurface—only one instant thick. The requirement of connectedness is needed in order that a ‘slice’ should not consist of the union of two disjoint open sets, each belonging to a different branch, where ‘branch’ is defined as follows.

The relation ‘*x ≤ y’,* from which the strict relation *‘x 〉y’* (‘x precedes y’) can be derived, can also be read *‘y* is an upper bound of *x’.* An *upward directed set* is a set, any two of whose members have a common upper bound in the set. A *branch* of **W** can then be defined as a maximal upward directed set, i.e. one which is properly contained in no upward directed superset. **W** as a whole is a downward directed set, and to assert that the overall shape of **W** is treelike, i.e. that **W** (i) is branched
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and (ii) branches only towards the future, requires the notion of inaccessibility.

Two points are defined to be *mutually inaccessible* if (i) they are incomparable, and (ii) no slice contains them both. We can now say what it is for **W** to be a *branched* topological space. If **W** contains three points *x, y,* and *z* such that *x 〉 y, x 〉 z* and such that *y* and *z* are mutually inaccessible, then **W** is branched. To rule out branching towards the past, we stipulate that if two points are mutually inaccessible they have no common upper bound.

In this appendix I shall assume that all branching in **W** is lower cut branching (see Appendix 1). I believe that what I say holds also if upper cut branching is permitted, but the argumentation is more easily carried through if all branching is taken to be lower cut.

A point *x* of **W** is said to be a *choice point* if *x* lies on at least two distinct maximal chains *C* _{1} and *C* _{2}, and if the portion of G which is later than *x* and the portion of *C* _{2} which is later than *x* are mutually inaccessible. The *prism structure* of **W** rests on the fact that above any choice point *x,* and within the same prism, there stands a denumerable infinity of other choice points. The choice points on every chain above *x* form a discrete set, with an accumulation point which is the base node of the next prism. Choice points which stand at the base of prisms are *basic choice points,* and above them stand *subsidiary* choice points of level 1, level 2, … level *n*, …, before the next basic choice point (see Chapter 4, p. 89). The order type of the choice points is that of the ordinal numbers, so that every choice point has an immediate successor but not necessarily an immediate predecessor.

Up to now, nothing that has been said about **W** hangs upon the notion of a reference frame. However we now move from a frame-invariant description of **W** to a description which is associated with a frame of reference, where the particular frame can be arbitrarily chosen (within certain limits). The move is from a description of **W** as it existed at the beginning of time to a description of **W** at a later time, and the latter description will necessarily be associated with a frame of reference. So described, **W** assumes the form of one of the familiar frame-dependent branched models that have been featured throughout the book.

At the time of the big bang, the universe consisted of the full set W. But now, 15 billion years later, the size of **W** has been considerably reduced. The vast majority of the physically possible alternatives available at earlier times were not selected, and the corresponding unactualized point-events no longer form part of **W**. Let *S* be any spacelike hypersurface or slice passing through here-now which contains no subsidiary choice points, and which is such that it contains a basic choice point for every subsidiary choice point that is later than it (i.e. that is later than some member of S). Thus *S* contains the base node of every prism that it intersects. Each such *S* determines a subset **W**s of **W**, namely *S* plus all members of **W** which are either earlier or later than *S.* The set **W**s depends on the choice of *S,* which is arbitrary provided
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it includes the point here-now. With each *S* is associated a reference frame, determined by the simultaneity-classes of events which *S* defines.

The composition of the subset **W**s is in part arbitrary, depending on the choice of *S,* and in part non-arbitrary, depending on which of the original branches of **W** has turned out to contain the point here-now. What remains to be shown is that **W**s branches along parallel hypersurfaces.^{1}

We define first the relation of *proper precedence* for slices:

If

A 〉〉 B=_{Df}(x)(xɛA⊃ (∃y)(yɛB&x 〉 y)).

*A « B,*then

*A*and

*B*do not intersect and we shall speak of them as being parallel. We construct in succession the set {

*S*

_{12}} of all slices which are parallel to

*S*and which contain all subsidiary choice points one level above the basic choice points of

*S,*is followed by the set {

*S*

_{13}} of slices parallel to

*S*and containing subsidiary choice points two levels above 5, followed by …, followed by the set {

*S*

_{1}

_{n}} …, followed by the set {

*S*

_{21}} of slices containing the basic choice points of the next tier of prisms, … etc. This family {S

_{ij}} of parallel slices specifies the framework of the branching structure of

**W**s.

The family {S_{ij}} contains all the *branch points* of **W**s, i.e. the slices or hypersurfaces along which **W**s branches, but it does not contain all members of **W**s. To fill in the gaps we require a principle which states that if *T* and *U* are slices such that *T« U,* then there exists a set of pairwise parallel slices *{V* _{i} *}* such that every point between *T* and *U* belongs to one and only one member of *{V* _{i} *}.* That is, *{V* _{i} *}* partitions the space between *T* and *U.* Given this principle, the family *{S* _{ij} *}* of branch points above *S* is extended to a family of parallel hypersurfaces which partitions **W**s. The relation 〉〉 is a strict partial ordering of this family.

We have arrived at the conclusion we sought, namely that the set **W**s of points of **W** which is defined by *S* branches along hypersurfaces parallel to S, and is partitioned by a maximal family of non-intersecting hypersurfaces ordered by 〉〉. **W**s is therefore one of the branched models depicted earlier in the book. Furthermore, choice of a different hypersurface *Sʼ* leads to a different model **W**sʼ. Underlying and common to all such models is the branched topological space comprising the set of point-events **W**. What this appendix has provided is a timeless, frame-invariant description of **W**, together with various frame-dependent descriptions of **W**-at-a-time, i.e. the different subsets **W**s of **W**. These constitute the branched models introduced in Chapter 1.

## Notes:

(^{1})
[Added in proof.] It may be preferable to take the surfaces along which **W**s branches, linking spacelike separated basic choice points, to be not slices or hyperplanes but surfaces with alternating peaks and valleys, the slope of which is always that of a light ray. The advantage of such surfaces is that they are not imposed arbitrarily on **W**, but are objective features of **W** as structured by the causal relation ≤.