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8.6 Necessary and Contingent Ordering

If there were no constraints on the ordering of actual events, then there would be a free-for-all for all events at all times, as described above.   Although physical theories have been proposed with this feature (see e.g. Cramer [1986] for a somewhat similar scheme), this does not appear to be the case in nature. We expect, for example, some natural constraints on the occurrence of events which are definitely past. We would like, for instance, to have a `Principle of the Definite Past', to the effect that all past events have immutably and irretrievably happened, and that no new events can spring up among them. We therefore expect that the order in which actual past-events are `laid down' in spacetime will depend on the relations of precedes and succeeds between the places for these events. This should be the case whether these relations are defined for Newtonian, relativistic, or general-relativistic metrics.

We have also the general intuition that objects in nature are extended more in three dimensions than in four dimensions. The options for interactions of a given body at a given stage in its development seem to be arrayed in only three dimensions, whereas the general philosophy of nature here, so far at least, seems to allow for events to occur anywhere in spacetime, in any order, subject only to some general rules that have yet to be elucidated. This general intuition cannot be exactly correct, because the actually existing things in the world, we saw in chapter 6, had to be past-events with finite time intervals between them. There can not be three dimensional objects which exist continuously and actually at all times. Therefore, at least on the scale of actual events, there has to be finite `extension' of objects into the fourth (time) dimension of spacetime. If this extension is small, however, the general intuition of three dimensionality should be approximately true, and therefore the rules which order actual events in spacetime should give rise, at least approximately, to a world of three dimensional objects which change their states through time. We therefore proceed by finding a minimal set of event-ordering rules which produce the appearance of objects in three dimensions along with a realistic account of the relativity and quantum theories.

    Thus I propose a Principle of Definite Past:

An actualising at place p occurs after all places which precede p are either definitely filled or definitely not filled. C. This principle states the necessary relations which hold between events which either precede or succeed each other in spacetime. If p precedes q in spacetime, then an event at p necessarily is in the past of an event at place q Similarly, if p succeeds q in spacetime, then an event at p necessarily is to the future of the event at place q If, however, p alternates with q in spacetime, then the event at p has only a contingent order with the event at place q The two events in the last case do have an actual order (once they actually exist, of course), but this order is not dictated by any physical law.

The Principle of Definite Past implies that, if there is an actualising at p, then there can be no actual events at places succeeding p This is because if a succeeding event were actual, then the event preceding it at p would have to have been already `frozen out', i.e. definitely occurred or definitely not occurred. The Principle also stops some causal chains of events `running ahead' into the future light cones of other neigbouring chains, because if these other events are not yet definitely decided or complete, then the chain ahead will be held up for as long as necessary. Similarly, it stops some chains `getting behind' of their neigbours, because it stops their neighbours getting ahead more than is allowable.

This Principle of Definite Past will clearly have to be supplemented by further description of how potentialities change in spacetime, and how the ordering of these changes is related to the order of any actual past-events produced. These matters will be addressed in chapter 10, once we know how to represent potentialities in spacetime by wave functions.

Appearance of a `Current Space'

On the basis of the Principle of Definite Past, it turns out that under several plausible assumptions, we can show that there is a sequence of spacelike hypersurfaces which define some global ordering of actual events. These hypersurfaces are only approximations to the actual order, because there is a certain amount of arbitrariness in their construction. Nevertheless, they do enable a simple description of the process of actualisation in the universe.

Let us compare the sequences of actualising here and in, say, the Andromeda galaxy. That galaxy is five million light years away from our own ( $ 4.7 \times 10^{19}$ km.). That is small enough on a cosmological scale that we can neglect any effects due to the general expansion of the universe. A distance of five million light years, however, means according to ordinary relativity theory, that there is an uncertainty of ten million years in any simultaneity relation we may set up between the two galaxies. The conventionality of distant simultaneity means that we can set of many different definitions of `legal simultaneity', and that these definitions give times for `now in Andromeda' that can vary from five million years ago to five million years in the future, with respect to the notion of `legal time' with $ \epsilon = \frac{1}{2}$ we usually adopt.

With special relativity that enormous range is the best we can give. Problems occur however if we have nonlocal correlations over that distance.   We can image an EPR experiment with correlated photons emitted from a point half way to the Andromeda galaxy, and with one detector in each galaxy. Quantum mechanics tells us that there are still (in theory!) correlations between the spin directions of the paired photons, even when they are five million light years apart. It appears that as soon as one photon is detected, the results of this actual event are immediately `transmitted' to the other photon of the correlated pair. According to quantum physics, however, it could just as easily been the `other' photon which was detected first and whose result was `transmitted' to the first photon to influence its state before it could be observed. We cannot ever tell which order it might have been, so we are tempted to think that there was never any real order of the two events.

If these detection events are actual events (as will be investigated in chapter 12), then they must have a definite order despite being so far apart. This means that it is in fact true that one detection event has a definite result before the other, and that this definite result changes the potentialities available for the occurrence of the second event. Which event of the two occurred before the other is left to some contingent property of the potentialities and/or events concerned. The actual ordering is contingent in the sense that no physical law could predict it, as physical laws the same for all reference frames.   The actual ordering is thus related to the residual indeterminism that indeterministic physical laws leave for the events to resolve. The whole scheme of course can only work if the physical laws are indeterministic, but this is what we are lead to believe from quantum physics.

If there is an actual ordering of events that relates processes here and in other galaxies, even though they may be many millions of light years away, then we can construct, we will see below, a global process time $\tau $ that is valid throughout the universe. For each allowed value of process time $\tau $, there will be a corresponding spacelike hypersurface stretching across spacetime which separates all actual events to date from where events are yet to become definite.


Appearance of a `Global Process Time'

The construction of a global process time proceeds as follows. We could reasonably assume that the mean time interval between successive actualising events would be similar in the two galaxies, that is that actualising proceeds at approximately the same rate. By integrating this rate, we deduce that there is an average time shift between the two regions, in the sense that for a given process time $\tau $, the most recent actual events here have a time coordinate t which differs by an approximately constant amount from the time t' of the most recent actual events in the Andromeda galaxy. If we then apply this comparison to all regions of the universe, we can for a given process time $\tau $construct a function $t = \sigma ( \vec{x} ; \tau )$ which gives the time t of the most recent actual events at position $\vec{x}$ within the universe. If we then apply the Principle of Definite Past given above, we see that the maximum rate of variation of t with coordinates $\vec{x}_i$ is 1/c, and that t strictly increases for every increase in $\tau $. That is, we have constructed a sequence of spacelike hypersurfaces labelled by different values of $\tau $, our global process time.

This construction, however, is only an approximate one, even for Newtonian space and time. We might have expected in that case that the function $t = \sigma ( \vec{x} ; \tau )$ reduce to t = $\tau $, as c has effectively become infinite in this case, and hence 1/c zero. The difficulty is that $\tau $ is not essentially a continuous time. Rather, $\tau $ is a variable that is used to count actual events, and, by the Finiteness Postulate of chapter 6, these have finite time intervals between them. Actual events can therefore be counted by an integer variable, so process time is essentially discrete, not continuous. It is therefore not really an additional `hypertime' superimposed on the spacetime continuum in addition to ordinary time, but is rather a simple counting of actual events as they actually happen. Newton could still have regained the identity t = $\tau $, however, as he did not accept our Finiteness Postulate. According to him, as we have seen earlier, actual physical processes were really continuous in the strict sense, for variations both in space and in time.

Because actual events only occur intermittently, process time $\tau $ is a discrete variable which we take (for convenience) to have only integer values. For a given process time $\tau $, the construction of the spacelike hypersurface $t = \sigma ( \vec{x} ; \tau )$ is only approximate because the exact value of $ \sigma ( \vec{x} ; \tau )$ does not matter as long as it gives some time t between the places of the most recent actual event and the place next to be realised. There is a finite interval between those places, as we saw at the end of chapter 6, and hence the precise values of the function $ \sigma ( \vec{x} ; \tau )$is arbitrary to this extent. This arbitrariness describes the basic inaccuracy of our description of the natural world in terms of three dimensional objects changing through time, and stems from the irreducible quantum nature of the world.

The construction of the spacelike hypersurfaces is derivative from the basic order of actual events, which depends in the individual process themselves. But, to a good approximation on the large scale where quantum effects are not noticable, we then have a sequence of `current' spatial regions which separate past actualities from `future potentialities' (strictly: from potentialities which are not yet realised). These spatial regions need not be according to anyone's legal time, but they will behave as if they were everywhere the local space for some local velocity.


What Difference the Order Makes

The fact that the actual ordering of events is contingent, and not determined uniquely by any law, means that not a lot can be done with it.   You can't use the actual order, for example, for communicating between here and the neighbouring galaxies, because you can never be certain in advance that your transmission event will be before8.8 the reception event there. Because the order is contingent, it is just as likely to turn out that the alleged `reception event' was definite and fixed before your `transmission event' occurred. It is impossible to give laws or rules for determining in advance the actual order of any events which are only contingently ordered.

Similarly, because the actual order is independent of any physical law, the details of the ordering can only have an influence on future states of affairs within the range permitted by those laws. If the actual ordering of events here and in our surrounding galaxies did have a significant effect on the future course of events, then, because the actual ordering is contingent, those future processes would be contingent in the same way. If the actual ordering was objectively random, for example, then this randomness would make itself effect in the subsequent developments. The physical laws, therefore, such as the probability laws of quantum mechanics, would have to allow for a indeterministic component of this order. Using this arguments in reverse, the limited component of indeterminism and randomness in quantum mechanics means that the actual ordering of space-like events has severely limited effects.  

Conclusion: Actualisation and Special Relativity

Relativity limits only `communications' (i.e. law-like signals) to the speed of light -- it leaves open the possibility that the effects of purely contingent orderings could be felt simultaneously over large spatial regions.    

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Prof Ian Thompson


Author: I.J. Thompson (except as stated)