Is Space-time a continuum?

Space-Time

Minkowski, Einstein’s old teacher of mathematics, inaugurated  the hybrid ‘Space-Time’ which is now on everyone’s lips. In an address delivered not long before his death in 1908 he said the now famous lines,

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

         But why should Minkowski, and whole generations of scientists, have ever thought that ‘space’ and ‘time’ could be completely separate in the first place? Certain consequences of a belief in ‘Space-Time’ in General Relativity do turn out to be  scarcely credible, but there is nothing weird or paradoxical per se about the idea of ‘time’ being a so-called fourth dimension. To specify an event accurately it is convenient to give three spatial coordinates which tell you how far the occurrence of this event is, or will be, along three different directions relative to an agreed  fixed point.  If I want to meet someone in a city laid out like a grid as New York is (more or less), I need to specify the street, say Fifth Avenue, the number of the building and the floor (how high above the ground it is). But this by itself will not be enough for a successful meet-up : I also need to give the time of the proposed rendez-vous, say, three o-clock in the afternoon. The wonder is, not that science has been obliged to bring time into the picture, but that it was possible for so long to avoid mentioning it (Note 1)

Succession

 Now, if you start off with ‘events’, which are by definition ‘punctual’ and impermanent, rather than things or ‘matter’ you cannot avoid bringing time into the picture from the start: indeed one  might be inclined to say that ‘time’ is a good deal more important than space. Events happen ‘before’ or ‘after’ each other; what happened yesterday preceded what happened this morning, and you read the previous sentence before you started on the current one. The very idea of ‘simultaneous’ events, events that have occurrence ‘at the same time’, is a tricky concept even without bringing Special Relativity into the picture. But  the idea of succession is both clearcut and basic and one could, as a first bash, even define ‘simultaneous’ events negatively as bona fide  occurrences that are not temporally ordered.

So, when I started trying to elaborate an ‘event-orientated’  world-view, I felt I absolutely had to have succession as a primary ingredient : if anything it came higher up the list than ‘space’. Originally I tried to kick off with a small number of basic assumptions (axioms or postulates) which seemed absolutely unavoidable. One such assumption was that most events are ‘ordered temporally’, that they have occurrence successively, ‘one after the other’ ─ with the small exception of so-called ‘simultaneous events’. Causality also seemed to be something I could not possibly do without and causality is very much tied up with succession since it is usually the prior event that is seen as ‘causing’ the other event in a causal pair. Again, one might tentatively  defined ‘simultaneous events’ as events which cannot have a direct causal bond, i.e. function as cause and effect (Note 2). And,  in an era innocent of Special Relativity and light cones, one might well define space as  the totality of all distinct events that are not temporally ordered.

From an ‘event-based’ viewpoint,  chopping up reality into ‘space’ and ‘time’ is not fundamental : all we require is a ‘place’ where events can and do have occurrence, an  Event Locality. Such a Locality starts off empty of events but has the capacity to receive them, indeed I have come to regard ultimate events as in some sense concretisations or condensations of an underlying substratum.

 Difference between Space and Time

 There is, however, a problem with having a single indivisible entity whether we call it ‘Space-Time’ or simply ‘the Locality’. The two parts or aspects of this creature are not at all equivalent. Although I believe, as some physicists have suggested, that, at a certain level, ‘space’ is ‘grainy’, it certainly appears to be continuous : we do not notice any dividing line, let alone a gap, between the different spatial ‘dimensions’ or between different spatial regions. We don’t have to ‘add’ the dimension height to pre-existing dimensions of length and width for example : experience always provides us with a three-dimensional physical block of reality (Note 3). And the fact that the choice of directions, up/down, left/right and so on, is more often than not completely arbitrary suggests that physical reality does not have inbuilt directions, is ‘all-of-a-piece’.

Another point worth mentioning is that we seem to have a strong sense of being ‘at rest’ spatially : not only are we ‘where we are’, and not where we are not, but we actually feel this to be the case. Indeed we tend to consider ourselves to be at rest even when we know we are moving : when in a train we consider that it is the other things, the countryside, that are in motion, not us. It is indeed this that gives Galileo’s seminal concept of inertia its force and plausibility; in practice all we notice is a flagrant disturbance of the ‘rest’ sensation, i.e. an ‘acceleration’.

What about time? Now it is true that time is often said to ‘flow’ and we do not notice any clearcut temporal demarcation lines any more than we notice spatial ones. Nonetheless, I would argue that it is much less natural and plausible to consider ‘time’ as a continuum because we have such a strong sense of sequence. We continually break up time into ‘moments’ which occur ‘one before the other’ even though the extent of the moment varies or is left vague. Sense of sequence is part of our world and since our impressions are themselves bona fide events even if only subjective ones, it would appear that sequence is a real feature of the physical world. There is in practice always an arrow of time, an arrow which points from the non-actual to the actual. Moreover, the process of ‘actualization’ is not reversible : an event that has occurrence cannot be ‘de-occurred’ as it were (Note 4).

And it is noteworthy that one very seldom feels oneself to be ‘at rest’ temporally, i.e. completely unaware of succession and variation. The sensation is so rare that it is often classed as  ‘mystical’, the feeling of being ‘out of time’ of which T.S. Eliot writes so eloquently in The Four Quartets. Heroin and certain other drugs,  by restricting one’s attention to the present moment and the recent past, likewise ‘abolish time’, hence their appeal. In the normal way,  even when deprived of all external physical stimuli, one still retains the sensation of there being a momentum and direction to one’s own thoughts and inner processes : one idea or internal sensation follows another and one never has any trouble assigning order (in the sense of sequence) to one’s inner feelings and thoughts. It is now thought that the brain uses parallel processing on a big scale but, if so, we are largely unaware of this incessant multi-tasking. Descartes in his thought experiment of being entirely cut off from the outside world and considering what he simply could not doubt, might well have concluded that sequence, rather than the (intemporal) thinking ego, was the one item that could not be dispensed with. For one can temporarily disbelieve in one’s existence as a particular person but not in the endless succession of thoughts and subjective sensations that stream through one’s mind/brain.

All this will be dismissed as belonging to psychology rather than physics. But our sense impressions and thoughts are rooted in our physiology and should not be waved aside for that very reason : in a sense they are the most important and inescapable ‘things’ we have for without them we would be zombies. Physical theories that deny sequence, that consider the laws of physics to be ‘perfectly reversible’, are both implausible and seemingly  unliveable, so great is our sense of ‘before and after’. Einstein towards the end of his life decided that it followed from General Relativity that everything happened in an ‘eternal present’. He took this idea seriously enough to mention it in a letter to the son of his college friend, Besso, on receiving news of the latter’s death, writing “For those of us who believe in physics, this separation between past, present and future is only an illusion, however tenacious”.

Breaks in Time

 If, then, we accept succession as an unavoidable feature of lived reality, are we to suppose that one moment shifts seamlessly into the next without any noticeable demarcation lines, let alone gaps? Practically all physicists, even those who toy with the idea that Space-time is in some sense ‘grainy’, seem to be stuck with the concept of a continuum. “There is time, but there is not really any notion of a moment in time. There are only processes that follow one another by causal necessity” as Lee Smolin puts it in Three Roads to Quantum Gravity..

But I cannot see how this can possibly be the case, and this is precisely why the ‘time dimension’ of the Event Locality is so different from the spatial one. If I shift my attention from two items in a landscape, from a rock and its immediate neighbourhood to a tree, there is no sense that the tree displaces the rock : the two items can peaceably co-exist and do not interfere with each other. But if one moment follows another, it displaces it, pushes it out of the way, as it were, since past and present moments, prior and subsequent events, cannot by definition co-exist ─ except perhaps in the inert way they might be seen to co-exist in an Einsteinian perpetual now. And all the attributes and particular features of a given moment must themselves disappear to make way for what follows. We do not usually see this happening, of course, because most of the time the very same objects are recreated and our senses do not register the transition. We only notice change when a completely different physical feature replaces another one, but the same principle must apply even if the same feature is recreated identically. Since a single moment is, in its physical manifestation, three-dimensional, all these three dimensions must go if a new moment comes into being.

Whether there is an appreciable gap between moments apart from there being a definite change is an open question. In the first sketch of Ultimate Event Theory I attribute a fixed extent to the minimal temporal interval, the ksana, and I allow for the possibility of flexible gaps between ksanas. The phenomenon of  time dilation is interpreted as the widening of the gap between ksanas rather than as an extension of the ‘length’ of a ksana itself. This feature, however, is not absolutely essential to the general theory.

What we actually perceive and consider to constitute  a ‘moment’ is, of course, a block containing millions of ksanas since the length of a ksana must be extremely small (around the Planck scale). However, it would seem that ksanas do form blocks and that there are transitions between blocks and that sometimes, if only subliminally, we are aware of these gaps. Instead of being a flowing river, ‘time’ is more like beads on a string though the best image would be a three-dimensional shape pricked out in coloured lights that is switched on and off incessantly.

. Mosaic Time

Temporal succession is either a real feature of the world or it is not, I cannot see that there is a possible third position. In Einstein’s universe “everything that can have occurrence already has occurrence” to put things in event terms. “In the ‘block universe’ conception of general relativity….the present moment has no meaning ─ all that exists is the whole history of the universe at once, timelessly. When laws of physics are represented mathematically, causal processes which are the activity of time are represented by timeless logical implications…. Mathematical objects, being timeless, don’t mhave present moments, futures or pasts”  (Lee Smolin, It’s Time to Rewrite time in New Scientist 20 April 2014).   

This means that there is no free will since what has occurrence cannot be changed, cannot be ‘de-occurred’. It also makes causality redundant as Lee Smolin states. One could indeed focus on certain pairs of events and baptise them ‘cause and effect’ but, since they both have occurrence, neither of them has brought the other about, nor has a third ‘previous’ event brought both of them about simultaneously. Causality becomes of no account since it is not needed.

Even a little acquaintance with Special Relativity leads one to conclude that it is impossible to establish a universally valid ‘now’. Instead we have the two light cones, one leading back to the past and one to the future (the observer’s future), and a large region classed as ‘elsewhere’. It is notorious that the order of events in ‘elsewhere’, viewed from inside a particular light cone, is not fixed for all observers : for one observer it can be said that event A precedes event B and for another that event B precedes A. This indeterminacy if of little or no practical consequence since there is (within SR) no possibility of interaction between the two regions. However, it does mean that it is on the face of it impossible to speak of a universally valid ‘now’ ─ although physicists do use expressions like the “present state of the universe”.

I personally cannot conceive of a ‘universe’ or a life or indeed anything at all without succession being built into it : the timeless world of mathematics is not reality but a ‘take’ on reality. The only way to conceptually save succession while accepting some of the more secure aspects of Relativity would seem to be to have some sort of ‘mosaic time’, physical reality split up into zones. How exactly these zones, which are themselves subjective in that they depend on a real or imagined ‘observer’, fit together is not a question I can answer though certain areas of research into general relativity can presumably be taken over into UET.  One could perhaps define the next best thing to a universal ‘now’ by taking a weighted average of all possible time zones : Eddington suggested something along these lines though he neglected to give any details. Note that if physical reality is a mosaic rather than a continuum, it would in principle be possible to shift the arrangement of particular tesserae in a small way, exchange one with another and so on.                     SH 23/01/15

 Note 1 Time was left out of the picture for so long, or at any rate neglected, because the first ‘science’ to be developed to a high degree of precision in the West was geometry. And the truths of (Euclidian) geometry, if they are truths at all, are ‘timeless’ which is why Plato prized geometry above all other branches of knowledge except philosophy. Inscribe a triangle in a circle with the diameter as base line and you will always find that it is right-angled. And if you don’t, this is to be attributed to careless drawing and measurement : in an ‘ideal’ Platonic world such an angle has to be a right angle. How do we know? Because the theorem has been proved.

This concentration on space rather than time meant that although the Greeks set out the basic principles  of statics, the West had to wait another 1,600 years or so before Galileo more or less invented the science of dynamics from scratch. And the prestige of Euclid and the associated static view of phenomena remained so great that Newton, perversely to our eyes, cast his Principia into a cumbrous geometrical mould using copious geometrical diagrams even though he had already invented a ‘mathematics of motion’, the Calculus.

 Note 2   Kant did in point of fact defend the idea of ‘simultaneous causation’ where each of two ‘simultaneous’ events affects the other ‘at the same time’. He gave the example of a ball resting on a cushion arguing that the ball presses down on the cushion for the same amount of time as the cushion is deformed by the presence of the ball. And if we take Newton’s Third Law as operating exactly at the same time on or between two different objects, we have to accept the possibility of simultaneous causation.

Within Ultimate Event Theory, what would normally be  called ‘causality’ is (sometimes) referred to as ‘Dominance’. I chose this term precisely because it signifies an unequal relation between two events, one event, referred to as the ‘cause’, as it were ‘dominating’ the other, the ‘effect’. In most, though perhaps not all, cases of causal relations I believe there really is priority and succession despite Newton’s Third Law. I would conceive of the ball pressing on the cushion as occurring at least a brief moment before its effect ─ though this is admittedly debatable. One could introduce the category of ‘Equal Dominance’ to cover cases of  Kant’s ‘simultaneous causality’ between two events.

Note 3  I have always found the idea of Flatland, which is routinely trotted out in popular books on Relativity, completely unconvincing. I can more readily conceive of there being more than three spatial dimensions as there being a world with less than three : a line, any line, always has some width and height.

 Note 4. If it is possible for an event in the future to have an effect ‘now’, this can only be because the ‘future’ event has already somehow already occurred, whereas intermediate events between ‘now’ and ‘then’ have not. I cannot conceive of a ‘non-event’ having any kind of causal repercussion — except, of course, in the trivial sense that current wishes or hopes about the future might affect our behaviour. Such wishes and desires belong to the present or recent past, not to the future.