Archives for category: Einstein

In its present state, Ultimate Event Theory falls squarely between two stools : too vague and ‘intuitive’ to even get a hearing from professional scientists, let alone be  taken seriously, it is too technical and mathematical to appeal to the ‘ordinary reader’. Hopefully, this double negative can be eventually turned into a double positive, i.e. a rigorous mathematical theory capable of making testable predictions that nonetheless is comprehensible and has strong intuitive appeal. I will personally not be able to take the theory to the desired state because of my insufficient mathematical and above all computing expertise : this will be the work of others. What I can do is, on the one hand, to strengthen the mathematical, logical side as much as I can while putting the theory in a form the non-mathematical reader can at least comprehend. One friend in particular who got put off by the mathematics asked me whether I could not write something that gives the gist of the theory without any mathematics at all. Thus this post which recounts the story of how and why I came to develop Ultimate Event Theory in the first place some thirty-five years ago.

 Conflicting  beliefs

Although scientists and rationalists are loath to admit it, personal temperament and cultural factors play a considerable part in the development of theories of the universe. There are always individual and environmental factors at work although the accumulation of unwelcome but undeniable facts may eventually overpower them. Most people today are, intellectually speaking, opportunists with few if any deep personal convictions, and there are good reasons for this. As sociological and biological entities we are strongly impelled to accept what is ‘official doctrine’ (in whatever domain) simply because, as a French psycho-analyst whose name escapes me famously wrote, “It is always dangerous to think differently from the majority”.
At the same time, one is inclined, and in some cases compelled, to accept only those ideas about the world that make sense in terms of our own experience. The result is that most people spend their lives doing an intellectual balancing act between what they ‘believe’ because this is what they are told is the case, and what they ‘believe’ because this is what their experience tells them is (likely to be) the case. Such a predicament is perhaps inevitable if we decide to live in society and most of the time the compromise ‘works’; there are, however, moments in the history of nations and in the history of a single individual when the conflict becomes intolerable and something has to give.

The Belief Crisis : What is the basis of reality?

Human existence is a succession of crises interspersed with periods of relative stability (or boredom). First, there is the birth crisis (the most traumatic of all), the ‘toddler crisis’ when the infant starts to try to make sense of the world around him or her, the adolescent crisis, the ‘mid-life’ crisis which kicks in at about forty and the age/death crisis when one realizes the end is nigh. All these crises are sparked off by physical changes which are too obvious and powerful to be ignored with the possible exception of the mid-life crisis which is not so much biological as  social (‘Where am I going with my life?’ ‘Will I achieve what I wanted?’).
Apart from all these crises ─ as if that were not enough already ─  there is the ‘belief crisis’. By ‘crisis of belief’ I mean pondering the answer to the question ‘What is real?’ ‘What do I absolutely have to believe in?’. Such a crisis can, on the individual level, come at any moment, though it usually seems to hit one between the eyes midway between the adolescent ‘growing up’ crisis and the full-scale mid-life crisis. As a young person one couldn’t really care less what reality ‘really’ is, one simply wants to live as intensely as possible and ‘philosophic’ questions can just go hang. And in middle age, people usually find they want to find some ‘meaning’ in life before it’s all over. Now, although the ‘belief crisis’ may lead on to the ‘middle age meaning crisis’ it is essentially quite different. For the ‘belief crisis’ is not a search for fulfilment but simply a deep questioning about the very nature of reality, meaningful or not. It is not essentially an emotional crisis nor is it inevitable ─ many people and even entire societies by-pass it altogether without being any the worse off, rather the reverse (Note 1).
Various influential thinkers in history went through such a  ‘belief crisis’ and answered it in memorable ways : one thinks at once of the Buddha or Socrates. Of all peoples, the Greeks during the Vth and VIth centuries BC seem to have experienced a veritable epidemic of successive ‘belief crises’ which is what  makes them so important in the history of civilization  ─ and also what made the actual individuals and city-states so unstable and so quarrelsome. Several of the most celebrated answers to the ‘riddle of reality’ date back to this brilliant era. Democritus of Abdera answered the question, “What is really real?” with the staggering statement, “Nothing exists except atoms and void”. The Pythagoreans, for their part, concluded that the principle on which the universe was based was not so much physical as numerical, “All is Number”. Our entire contemporary scientific and technological ‘world-view’ (‘paradigm’) can  be traced back to the  two giant thinkers, Pythagoras and Democritus, even if we have ultimately ‘got beyond’  them since we have ‘split the atom’ and replaced numbers as such by mathematical formulae. In an equally turbulent era, Descartes, another major ‘intellectual crisis’ thinker, famously decided that he could disbelieve in just about everything but not that there was a ‘thinking being’ doing the disbelieving, cogito ergo sum (Note 2).
In due course, in my mid to late thirties, at about the time of life when Descartes decided to question the totality of received wisdom, I found myself with quite a lot of time on my hands and a certain amount of experience of the vicissitudes of life behind me to ponder upon. I too became afflicted by the ‘belief crisis’ and spent the greater part of my spare time (and working time as well) pondering what was ‘really real’ and discussing the issue interminably with the same person practically every evening (Note 3). 

Temperamental Inclinations or Prejudices

 My temperament (genes?) combined with my experience of life pushed me in certain well-defined philosophic directions. Although I only  started formulating Eventrics and Ultimate Event Theory (the ‘microscopic’ part of Eventrics) in the early nineteen-eighties and by then had long since retired from the ‘hippie scene’, the heady years of the late Sixties and early Seventies provided me with my  ‘field notes’ on the nature of reality (and unreality), especially the human part of it. The cultural climate of this era, at any rate in America and the West, may be summed up by saying that, during this time “a substantial number of people between the ages of fifteen and thirty decided that sensations were far more important than possessions and arranged their lives in consequence”. In practice this meant forsaking steady jobs, marriage, further education and so on and spending one’s time looking for physical thrills such as doing a ton up on the M1, hitch-hiking aimlessly around the world, blowing your mind with drugs, having casual but intense sexual encounters and so on. Not much philosophy here but when I and other shipwrecked survivors of the inevitable débâcle took stock of the situation, we retained a strong preference for a ‘philosophy’  that gave primary importance to sensation and personal experience.
The physical requirement ruled out traditional religion since most religions, at any rate Christianity in its later public  form, downgraded the body and the physical world altogether in favour of the ‘soul’ and a supposed future life beyond the grave. The only aspect of religion that deserved to be taken seriously, so I felt, was mysticism since mysticism is based not on hearsay or holy writ but on actual personal experience. The mystic’s claim that there was a domain ‘beyond the physical’ and that this deeper reality can to some degree actually be experienced within this life struck me as not only inspiring but even credible ─ “We are more than what we think we are and know more than what we think we know” as someone (myself) once put it.
At the same time, my somewhat precarious hand-to-mouth existence had given me a healthy respect for the ‘basic physical necessities’ and thus inclined to reject all theories which dismissed physical reality as ‘illusory’, tempting though this sometimes is (Note 4). So ‘Idealism’ as such was out. In effect I wanted a belief system that gave validity and significance to the impressions of the senses, sentio ergo sum to Descartes’ cogito ergo sum or, better, sentio ergo est :  ‘I feel therefore there is something’.

Why not physical science ?

 Why not indeed. The main reason that I didn’t decide, like most people around me,  that “science has all the answers” was that, at the time, I knew practically no science. Incredible though this seems today, I had managed to get through school and university without going to a single chemistry or physics class and my knowledge of biology was limited to one period a week for one year and with no exam at the end of it.
But ignorance was not the only reason for my disqualifying science as a viable ‘theory of everything’. Apart from being vaguely threatening ─ this was the era of the Cold War and CND ─ science simply seemed monumentally irrelevant to every aspect of one’s personal daily life. Did knowing about neutrons and neurons make you  more capable of making more effective decisions on a day to day basis? Seemingly not. Scientists and mathematicians often seemed to be less (not more) astute in running their lives than ordinary practical people.
Apart from this, science was going through a difficult period when even the physicists themselves were bewildered by their own discoveries. Newton’s billiard ball universe had collapsed into a tangled mess of probabilities and  uncertainty principles : when even Einstein, the most famous modern scientist, could not manage to swallow Quantum Theory, there seemed little hope for Joe Bloggs. The solid observable atom was out and unobservable quarks were in, but Murray Gell-Mann, the co-originator of the quark theory, stated on several occasions that he did not ‘really  believe in quarks’ but merely used them as ‘mathematical aids to sorting out the data’. Well, if even he didn’t believe in them, why the hell should anyone else? Newton’s clockwork universe was bleak and soulless but was at least credible and tactile : modern science seemed nothing more than a farrago of  abstruse nonsense that for some reason ‘worked’ often to the amazement of the scientists themselves.
There was another, deeper, reason why physical science appeared antipathetic to me at the time : science totally devalues personal experience. Only repeatable observations in laboratory conditions count as fact : everything else is dismissed as ‘anecdotal’. But the whole point of personal experience is that (1) it is essentially unrepeatable and (2) it must be spontaneous if it is to be worthwhile. The famous ‘scientific method’ might have a certain value if we are studying lifeless atoms but seemed unlikely to uncover anything of interest in the human domain — . the best ‘psychologists’ such as  conmen and dictators are sublimely ignorant of psychology. Science essentially treats everything as if it were dead, which is why it struggles to come up with any strong predictions in the social, economic and political spheres. Rather than treat living things as essentially dead, I was more inclined to treat ‘dead things’ (including the universe itself) as if they were in some sense alive. 

Descartes’ Thought Experiment 

Although I don’t think I had actually read Descartes’ Discours sur la méthode at the time, I had heard about it and the general idea was presumably lurking at the back of my mind. Supposedly, Descartes who, incredibly, was an Army officer at the time, spent a day in what is described in history books as a poêle (‘stove’) pondering the nature of reality. (The ‘stove’ must have been a small chamber close to a source of heat.) Descartes came to the conclusion that it was possible to disbelieve in just about everything except that there was a ‘thinking  being’, cogito ergo sum. To anyone who has done meditation, even in a casual way, Descartes’ conclusion appears by no means self-evident. The notion of individuality drops away quite rapidly when one is meditating and all one is left with is a flux of mental/physical impressions. It is not only possible but even ‘natural’ to temporarily disbelieve in the reality of the ‘I’ (Note 5)─ but one cannot and does not disbelieve in the reality of the various sensations/impressions that are succeeding each other as ‘one’ sits (or stands).

Descartes’ thought experiment nonetheless seemed  suggestive and required, I thought, more precise evaluation. Whether the ‘impressions/sensations’ are considered to be mental, physical or a mixture of the two, they are nonetheless always events and as such have the following features:

(1) they are, or appear to be, ‘entire’, ‘all of a piece’, there is no such thing as a ‘partial’ event/impression;

(2) they follow each other very rapidly;

(3) the events do not constitute a continuous stream, on the contrary there are palpable gaps between the events (Note 6);

(4) there is usually a connection between successive events, one thought ‘leads on’ to another and we can, if we are alert enough, work backwards from one ‘thought/impression’ to its predecessor and so on back to the start of the sequence;

(5) occasionally ‘thought-events’ crop up that seem to be  completely disconnected from all previous ‘thought-events’, arriving as it were ‘out of the blue.’.

Now, with these five qualities, I already have a number of features which I believe must be part of reality, at any rate individual ‘thought/sensation’  reality. Firstly, whether my thoughts/sensations are ‘wrong’, misguided, deluded or what have you, they happen, they take place, cannot be waved away. Secondly, there is always sequence : thought ‘moves from one thing to another’ by specific stages. Thirdly, there are noticeable gaps between the thought-events. Fourthly, there is  causality : one thought/sensation gives rise to another in a broadly predictable and comprehensible manner. Finally, there is an irreducible random element in the unfolding of thought-events — so not everything is deterministic apparently.
These are properties I repeatedly observe and feel I have to believe in. There are also a number of conclusions to be drawn from the above; like all deductions these ‘derived truths’ are somewhat less certain than the direct impressions, are ‘second-order’ truths as it were, but they are nonetheless compelling, at least to me. What conclusions? (1) Since there are events, there  must seemingly be a ‘place’ where these events can and do occur, an Event Locality. (2) Since there are, and continue to be, events, there  must be an ultimate source of events, an Origin, something distinct from the events themselves and also (perhaps) distinct from the Locality.
A further and more radical conclusion is that this broad schema can legitimately be generalized to ‘everything’, at any rate to everything in the entire known and knowable universe. Why make any hard and fast distinction between mental events and their features and ‘objective’ physical events and their features? Succession, discontinuity and causality are properties of the ‘outside’ world as well, not just that of the private world of an isolated thinking individual.
What about other things we normally assume exist such as trees and tables and ourselves? According to the event model, all these things must either be (1) illusory or irrelevant (same thing essentially) (2) composite and secondary and/or (3) ‘emergent’.
Objects are bundles of events that keep repeating more or less in the same form. And though I do indeed believe that ‘I’ am in some sense a distinct entity and thus ‘exist’, this entity is not fundamental, not basic, not entirely reducible to a collection of events. If the personality exists at all ─ some persons  have doubts on this score ─ it is a complex, emergent entity. This is an example of a ‘valid’ but not  fundamental item of reality.
Ideas, if they take place in the ‘mind’, are events whether true, false or meaningless. They are ‘true’ to the extent that they can ultimately be grounded in occurrences of actual events and their interactions, or interpretations thereof. I suppose this is my version of the ‘Verification Principle’ : whatever is not grounded in actual sensations is to be regarded with suspicion.  This does not necessarily invalidate abstract or metaphysical entities but it does draw a line in the sand. For example, contrary to most contemporary rationalists and scientists, I do not entirely reject the notion of a reality beyond the physical because the feeling that there is something ‘immeasurable’ and ‘transcendent’ from which we and the world emerge is a matter of experience to many people, is a part of the world of sensation though somewhat at the limits of it. This reality, if it exists, is ‘beyond name and form’ (as Buddhism puts it) is ‘non-computable’, ‘transfinite’. But I entirely reject the notion of the ‘infinitely large’ and the ‘infinitely small’ which has bedevilled science and mathematics since these (pseudo)entities are completely  outside  personal experience and always will be. With the exception of the Origin (which is a source of events but not itself an event), my standpoint is that  everything, absolutely everything, is made up of a finite number of ultimate events and an ultimate event is an event  that cannot be further decomposed. This principle is not, perhaps, quite so obvious as some of the other principles. Nonetheless, when considering ‘macro’ events ─ events which clearly can be decomposed into smaller events ─ we have two and only two choices : either the process comes to an end with an ‘ultimate’ event or it carries on interminably (while yet eventually coming to an end). I believe the first option is by far the more reasonable one.
With this, I feel I have the bare bones of not just a philosophy but a ‘view of the world’, a schema into which pretty well everything can be fitted ─ the contemporary buzzword is ‘paradigm’. Like Descartes emerging from his ‘stove’, I considered  I had a blueprint for reality or at least that part of it amenable to direct experience. To sum up, I could disbelieve, at least momentarily,  in just about everything but not that (1) there were events ; (2) that events occurred successively; (3) were subject to some sort of omnipresent causal force with  occasional lapses into lawlessness. Also, (4) these events happened somewhere (5) emerged from something or somewhere and (6) were decomposable into ‘ultimate’ events that could not be further decomposed.  This would do for a beginning, other essential features would be added to the mix as and when required.                                                                             SH

Note 1  Many extremely successful societies seem to have been perfectly happy in  avoiding the ‘intellectual crisis’ altogether : Rome did not produce a single original thinker and the official Chinese Confucian world-view changed little over a period of more than two thousand years. This was doubtless  one of the main reasons why these societies lasted so long while extremely volatile societies such as VIth century Athens or the city states of Renaissance Italy blazed with the light of a thousand suns for a few moments and then were seen and heard no more.

Note 2 Je pris garde que, pendant que je voulais ainsi penser que tout était faux, il fallait nécessairement que moi, qui le pensais, fusse quelquechose. Et remarquant que cette vérité : je pense, donc je suis, était si ferme et si assure, que toutes les autres extravagantes suppositions des sceptiques n’étaient capables de l’ébranler, je jugeai que je pouvais le reçevoir, sans scrupule, pour le premier principe de la philosophie que je cherchais.”
      René Descartes, Discours sur la Méthode Quatrième Partie
“I noted, however, that even while engaged in thinking that everything was false, it was nonetheless a fact that I, who was engaged in thought, was ‘something’. And observing that this truth, I think, therefore I am, was so strong and so incontrovertible, that the most extravagant proposals of sceptics could not shake it, I concluded that I could justifiably take it on  board, without misgiving, as the basic proposition of philosophy that I was looking for.”  [loose translation]

Note 3  The person in question was, for the record, a primary school teacher by the name of Marion Rowse, unfortunately now long deceased. She was the only person to whom I spoke about the ideas that eventually became Eventrics and Ultimate Event Theory and deserves to be remembered for this reason.

Note 4   As someone at the other end of the social spectrum, but who seemingly also went through a crisis of belief at around the same time, put it, “I have gained a healthy respect for the objective aspect of reality by having lived under Nazi and Communist regimes and by speculating in the financial markets” (Soros, The Crash of 2008 p. 40).
According to Boswell, Dr. Johnson refuted Bishop Berkeley, who argued that matter was essentially unreal, by kicking a wall. In a sense this was a good answer but perhaps not entirely in the way Dr. Johnson intended.  Why do I believe in the reality of the wall? Because if I kick it hard enough I feel pain and there is no doubt in my mind that pain is real — it is a sensation. The wall must be accorded some degree of reality because, seemingly, it was the cause of the pain. But the reality of the wall, is, as it were, a ‘derived’ or ‘secondary’  reality : the primary reality is the  sensation, in this case the pain in my foot. I could, I argued to myself, at a pinch, disbelieve in the existence of the wall, or at any rate accept that it is not perhaps so ‘real’ as we like to think it is, but I could not disbelieve in the reality of my sensation. And it was not even important whether my sensations were, or were not, corroborated by other people, were entirely ‘subjective’ if you like, since, subjective or not, they remained sensations and thus real.

Note 5 In the Chuang-tzu Book, Yen Ch’eng, a disciple of the philosopher Ch’i  is alarmed because his master, when meditating, appeared to be “like a log of wood, quite unlike the person who was sitting there before”. Ch’I replies, “You have put it very well; when you saw me just now my ‘I’ had lost its ‘me’” (Chaung-tzu Book II. 1) 

Note 6 The practitioner of meditation is encouraged to ‘widen’ these gaps as much as possible (without falling asleep) since it is by way of the gaps that we can eventually become familiar with the ‘Emptiness’ that is the origin and end of everything.

 

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 Although, in modern physics,  many elementary particles are extremely short-lived, others such as protons are virtually immortal. But either way, a particle, while it does exist, is assumed to be continuously existing. And solid objects such as we see all around us like rocks and hills, are also assumed to be ‘continuously existing’ even though they may undergo gradual changes in internal composition. Since solid objects and even elementary particles don’t appear, disappear and re-appear, they don’t have a ‘re-appearance rate ’ ─ they’re always there when they are there, so to speak.
However, in UET the ‘natural’ tendency is for everything to flash in and out of existence and virtually all  ultimate events disappear for ever after a single appearance leaving a trace that would, at best, show up as a sort of faint background ‘noise’ or ‘flicker of existence’. All apparently solid objects are, according to the UET paradigm, conglomerates of repeating ultimate events that are bonded together ‘laterally’, i.e. within  the same ksana, and also ‘vertically’, i.e. from one ksana to the next (since otherwise they would not show up again ever). A few ultimate events, those that have acquired persistence ─ we shall not for the moment ask how and why they acquire this property ─ are able to bring about, i.e. cause, their own re-appearance : in such a case we have an event-chain which is, by definition,  a causally bonded sequence of ultimate events.
But how often do the constituent events of an event-chain re-appear?  Taking the simplest case of an event-chain composed of a single repeating ultimate event, are we to suppose that this event repeats at every single ksana (‘moment’ if you like)? There is on the face of it no particular reason why this should be so and many reasons why this would seem to be very unlikely.    

The Principle of Spatio-Temporal Continuity 

Newtonian physics, likewise 18th and 19th century rationalism generally, assumes what I have referred to elsewhere as the Postulate of Spatio-temporal Continuity. This postulate or principle, though rarely explicitly  stated in philosophic or scientific works,  is actually one of the most important of the ideas associated with the Enlightenment and thus with the entire subsequent intellectual development of Western society. In its simplest form, the principle says that an event occurring here, at a particular spot in Space-Time (to use the current term), cannot have an effect there, at a spot some distance away without having effects at all (or at least most?/ some?) intermediate spots. The original event sets up a chain reaction and a frequent image used is that of a whole row of upright dominoes falling over one by one once the first has been pushed over. This is essentially how Newtonian physics views the action of a force on a body or system of bodies, whether the force in question is a contact force (push/pull) or a force acting at a distance like gravity.
As we envisage things today, a blow affects a solid object by making the intermolecular distances of the surface atoms contract a little and they pass on this effect to neighbouring molecules which in turn affect nearby objects they are in contact with or exert an increased pressure on the atmosphere,  and so on. Moreover, although this aspect of the question is glossed over in Newtonian (and even modern) physics, each transmission of the original impulse  ‘takes time’ : the re-action is never instantaneous (except possibly in the case of gravity) but comes ‘a moment later’, more precisely at least one ksana later. This whole issue will be discussed in more detail later, but, within the context of the present discussion, the point to bear in mind is that,  according to Newtonian physics and rationalistic thought generally, there can be no leap-frogging with space and time. Indeed, it was because of the Principle of Spatio-temporal Continuity that most European scientists rejected out of hand Newton’s theory of universal attraction since, as Newton admitted, there seemed to be no way that a solid body such as  the Earth could affect another solid body such as the Moon thousands  of kilometres with nothing in between except ‘empty space’.   Even as late as the mid 19th century, Maxwell valiantly attempted to give a mechanical explanation of his own theory of electro-magnetism, and he did this essentially because of the widespread rock-hard belief in the principle of spatio-temporal continuity.
The principle, innocuous  though it may sound, has also had  extremely important social and political implications since, amongst other things, it led to the repeal of laws against witchcraft in the ‘advanced’ countries ─ the new Legislative Assembly in France shortly after the revolution specifically abolished all penalties for ‘imaginary’ crimes and that included witchcraft. Why was witchcraft considered to be an ‘imaginary crime’? Essentially because it  offended against the Principle of Spatio-Temporal Continuity. The French revolutionaries who drew the statue of Reason through the streets of Paris and made Her their goddess, considered it impossible to cause someone’s death miles away simply by thinking ill of them or saying Abracadabra. Whether the accused ‘confessed’ to having brought about someone’s death in this way, or even sincerely believed it, was irrelevant : no one had the power to disobey the Principle of Spatio-Temporal Continuity.
The Principle got somewhat muddied  when science had to deal with electro-magnetism ─ Does an impulse travel through all possible intermediary positions in an electro-magnetic field? ─ but it was still very much in force in 1905 when Einstein formulated the Theory of Special Relativity. For Einstein deduced from his basic assumptions that one could not ‘send a message’ faster than the speed of light and that, in consequence,  this limited the speed of propagation of causality. If I am too far away from someone else I simply cannot cause this person’s death at that particular time and that is that. The Principle ran into trouble, of course,  with the advent of Quantum Mechanics but it remains deeply entrenched in our way of thinking about the world which is why alibis are so important in law, to take but one example. And it is precisely because Quantum Mechanics appears to violate the principle that QM is so worrisome and the chief reason why some of the scientists who helped to develop the theory such as Einstein himself, and even Schrodinger, were never happy with  it. As Einstein put it, Quantum Mechanics involved “spooky action at a distance” ─ exactly the same objection that the Cartesians had made to Newton.
So, do I propose to take the principle over into UET? The short answer is, no. If I did take over the principle, it would mean that, in every bona fide event-chain, an ultimate event would make an appearance at every single ‘moment’ (ksana), and I could see in advance that there were serious problems ahead if I assumed this : certain regions of the Locality would soon get hopelessly clogged up with colliding event-chains. Also, if all the possible positions in all ‘normal’ event-sequences were occupied, there would be little point in having a theory of events at all, since, to all intents and purposes, all event-chains would behave as if they were solid objects and one might as well just stick to normal physics. One of the main  reasons for elaborating a theory of events in the first place was my deep-rooted conviction ─ intuition if you like ─ that physical reality is discontinuous and that there are gaps between ksanas ─ or at least that there could be gaps given certain conditions. In the theory I eventually roughed out, or am in the process of roughing out, both spatio-temporal continuity and infinity are absent and will remain prohibited.
But how does all this square with my deduction (from UET hypotheses) that the maximum propagation rate of causality is a single grid-position per ksana, s0/t0, where s0 is the spatial dimension of an event capsule ‘at rest’ and t0 the ‘rest’ temporal dimension? In UET, what replaces the ‘object-based’ image of a tiny nucleus inside an atom, is the vision of a tiny kernel of fixed extent where every ultimate event occurs embedded in a relatively enormous four-dimensional event capsule. Any causal influence emanates from the kernel and, if it is to ‘recreate’ the original ultimate event a ksana later, it must traverse at least half the ‘length’ (spatial dimesion) of one capsule plus half of the next one, i.e. ½ s0 + ½ s0 = 1 s0 where s0 is the spatial dimension of an event-capsule ‘at rest’ (its normal state). For if the causal influence did not ‘get that far’, it would not be able to bring anything about at all, would be like a messenger who could not reach a destination receding faster than he could run flat out. The runner’s ‘message’, in this case the recreation of a clone of the original ultimate event, would never get delivered and nothing would ever come about at all.
This problem does not occur in normal physics since objects are not conceived as requiring a causal force to stop them disappearing, and, on top of that, ‘space/time’ is assumed to be continuous and infinitely divisible. In UET there are minimal spatial and temporal units (that of the the grid-space and the ksana) and ‘time’ in the UET sense of an endless succession of ksanas, stops for no man or god, not even physicists who are born, live and die successively like everything else. I believe that succession, like causality, is built into the very fabric of physical reality and though there is no such thing as continuous motion, there is and always will be change since, even if nothing else is happening, one ksana is being replaced by another, different, one ─ “the moving finger writes, and, having writ, moves on” (Rubaiyat of Omar Khayyam). Heraclitus said that “No man ever steps into the same river twice”, but a more extreme follower of his disagreed, saying that it was impossible to step into the same river once, which is the Hinayana  Buddhist view. For ‘time’ is not a river that flows at a steady rate (as Newton envisaged it) but a succession of ‘moments’ threaded like beads on an invisible  chain and with minute gaps between the beads.

Limit to unitary re-appearance rate

So, returning to my repeating ultimate event, could the ‘re-creation rate’ of an ultimate event be  greater than the minimal rate of 1 s0/t0 ? Could it, for example, be  2, 3 or 5 spacesper ksana? No. For if and when the ultimate event re-appeared, say  5 ksanas later, the original causal impulse would have covered a distance of 5 s0   ( s0 being the spatial dimension of each capsule) and would have taken 5 ksanas to do  this. Consequently the space/time displacement rate would be the same (but not in this case the individual distances). I note this rate as c* in ‘absolute units’, the UET equivalent of c, since it denotes an upper limit to the propagation of the causal influence (Note 1). For the very continuing existence of anything depends on causality : each ‘object’ that does persist in isolation does so because it is perpetually re-creating itself (Note 2).

But note that it is only the unitary rate, the distance/time ratio taken over a single ksana,  that cannot be less (or more) than one grid-space per ksana or 1 s0/t0 : any fractional (but not irrational) re-appearance rate is perfectly conceivable provided it is spread out over several ksanas. A re-appearance rate of m/n s0/t0  simply means that the ultimate event in question re-appears in an equivalent spatial position on the Locality m times every n ksanas where m/n ≤ 1. And there are all sorts of different ways in which this rate be achieved. For example, a re-appearance rate of 3/5 s0/t0 could be a repeating pattern such as

Reappearance rates 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

and one pattern could change over into the other either randomly or, alternatively, according to a particular rule.
As one increases the difference between the numerator and the denominator, there are obviously going to be many more possible variations : all this could easily be worked out mathematically using combinatorial analysis. But note that it is the distribution of ™the black and white at matters since, once a re-appearance rhythm has begun, there is no real difference between a ‘vertical’ rate of 0™˜™˜●0● and ˜™˜™™˜™˜™˜™˜●0™˜™˜●0 ™˜™™˜™˜ ˜™˜™ ─ it all depends on where you start counting. Patterns with the same repetition rate only count as different if this difference is recognizable no matter where you start examining the sequence.
Why does all this matter? Because, each time there is a blank line, this means that the ultimate event in question does not make an appearance at all during this ksana, and, if we are dealing with large denominators, this could mean very large gaps indeed in an event chain. Suppose, for example, an event-chain had a re-appearance rate of 4/786. There would only be four appearances (black dots) in a period of 786 ksanas, and there would inevitably be very large blank sections of the Locality when the ultimate event made no appearance.

Lower Limit of re-creation rate 

Since, by definition, everything in UET is finite, there must be a maximum number of possible consecutive gaps  or non-reappearances. For example, if we set the limit at, say, 20 blank lines, or 200, this would mean that, each time this blank period was observed, we could conclude that the event-chain had terminated. This is the UET equivalent  of the Principle of Spatio-Temporal Continuity and effectively excludes phenomena such as an ultimate event in an event-chain making its re-appearance a century later than its first appearance. This limit would have to be estimated on the  basis of experiments since I do not see how a specific value can be derived from theoretical considerations alone. It is tempting to estimate that this value would involve c* or a multiple of c* but this is only a wild guess ─ Nature does not always favour elegance and simplicity.
Such a rule would limit how ‘stretched out’ an event-chain can be temporally and, in reality , there may not after all be a hard and fast general rule  : the maximal extent of the gap could decline exponentially or in accordance with some other function. That is, an abnormally long gap followed by the re-appearance of an event, would decrease the possible upper limit slightly in much the same way as chance associations increase the likelihood of an event-chain forming in the first place. If, say, there was an original limit of a  gap of 20 ksanas, whenever the re-appearance rate had a gap of 19, the limit would be reduced to 19 and so on.
It is important to be clear that we are not talking about the phenomenon of ‘time dilation’ which concerns only the interval between one ksana and the next according to a particular viewpoint. Here, we simply have an event-chain where an ultimate event is repeating at the same spot on the spatial part of the Locality : it is ‘at rest’ and not displacing itself laterally at all. The consequences for other viewpoints would have to be investigated.

Re-appearance Rate as an intrinsic property of an event-chain  

Since Galileo, and subsequently Einstein, it has become customary in physics to distinguish, not between rest and motion, but rather between unaccelerated motion and  accelerated motion. And the category of ‘unaccelerated motion’ includes all possible constant straight-line speeds including zero (rest). It seems, then,  that there is no true distinction to be made between ‘rest’ and motion just so long as the latter is motion in a straight line at a constant displacement rate. This ‘relativisation’ of  motion in effect means that an ‘inertial system’ or a particle at rest within an inertial system does not really have a specific velocity at all, since any estimated velocity is as ‘true’ as any other. So, seemingly, ‘velocity’ is not a property of a single body but only of a system of at least two bodies. This is, in a sense, rather odd) since there can be no doubt that a ‘change of velocity’, an acceleration, really is a feature of a single body (or is it?).
Consider a spaceship which is either completely alone in the universe or sufficiently remote from all massive bodies that it can be considered in isolation. What is its speed? It has none since there is no reference system or body to which its speed can be referred. It is, then, at rest ─ or this is what we must assume if there are no internal signs of acceleration such as plates falling around or rattling doors and so on. If the spaceship is propelling itself forward (or in some direction we call ‘forward’) intermittently by jet propulsion the acceleration will be note by the voyagers inside the ship supposing there are some. Suppose there is no further discharge of chemicals for a while. Is the spaceship now moving at a different and greater velocity than before? Not really. One could I suppose refer the vessel’s new state of motion to the centre of mass of the ejected chemicals but this seems rather artificial especially as they are going to be dispersed. No matter how many times this happens, the ship will not be gaining speed, or so it would appear. On the other hand, the changes in velocity, or accelerations are undoubtedly real since their effects can be observed within the reference frame.
So what to conclude? One could say that ‘acceleration’ has ‘higher reality status’ than simple velocity since it does not depend on a reference point outside the system. ‘Velocity’ is a ‘reality of second order’ whereas acceleration is a ‘reality of first order’. But once again there is a difference between normal physics and UET physics in this respect. Although the distinction between unaccelerated and accelerated motion is taken over into UET (re-baptised ‘regular’ and ‘irregular’ motion), there is in Ultimate Event Theory, but not in contemporary physics, a kind of ‘velocity’ that has nothing to do with any other body whatsoever, namely the event-chain’s re-appearance rate.
When one has spent some time studying Relativity one ends up wondering whether after all “everything is relative” and quite a lot of physicists and philosophers seems to actually believe something not far from this : the universe is evaporating away as we look it and leaving nothing but a trail of unintelligible mathematical formulae. In Quantum Mechanics (as Heisenberg envisaged it anyway) the properties of a particular ‘body’ involve the properties of all the other bodies in the universe, so that there remain very few, if any, intrinsic properties that a body or system can possess. However, in UET, there is a reality safety net. For there are at least two  things that are not relative, since they pertain to the event-chain or event-conglomerate itself whether it is alone in the universe or embedded in a dense network of intersecting event-chains we view as matter. These two things are (1) occurrence and (2) rate of occurrence and both of them are straight numbers, or ratios of integers.
An ultimate event either has occurrence or it does not : there is no such thing as the ‘demi-occurrence’ of an event (though there might be such a thing as a potential event). Every macro event is (by the preliminary postulates of UET) made up of a finite number of ultimate events and every trajectory of every event-conglomerate has an event number associated with it. But this is not all. Every event-chain ─ or at any rate normal or ‘well-behaved’ event-chain ─ has a ‘re-appearance rate’. This ‘re-appearance rate’ may well change considerably during the life span of a particular event-chain, either randomly or following a particular rule, and, more significantly, the ‘re-appearance rates’ of event-conglomerates (particles, solid bodies and so on) can, and almost certainly do, differ considerably from each other. One ‘particle’ might have a re-appearance rate of 4, (i.e. re-appear every fourth ksana) another with the same displacement rate  with respect to the first a rate of 167 and so on. And this would have great implications for collisions between event-chains and event-conglomerates.

Re-appearance rates and collisions 

What happens during a collision? One or more solid bodies are disputing the occupation of territory that lies on their  trajectories. If the two objects miss each other, even narrowly, there is no problem : the objects occupy ‘free’ territory. In UET event conglomerates have two kinds of ‘velocity’, firstly their intrinsic re-appearance rates which may differ considerably, and, secondly, their displacement rate relative to each other. Every event-chain may be considered to be ‘at rest’ with respect to itself, indeed it is hard to see how it could be anything at all if this were not the case. But the relative speed of even unaccelerated event-chains will not usually be zero and is perfectly real since it has observable and often dramatic consequences.
Now, in normal physics, space, time and existence itself is regarded as continuous, so two objects will collide if their trajectories intersect and they will miss each other if their trajectories do not intersect. All this is absolutely clearcut, at least in principle. However, in UET there are two quite different ways in which ‘particles’ (small event conglomerates) can miss each other.
First of all, there is the case when both objects (repeating event-conglomerates) have a 1/1 re-appearance rate, i.e. there is an ultimate event at every ksana in both cases. If object B is both dense and occupies a relatively large region of the Locality at each re-appearance, and the relative speed is low, the chances are that the two objects will collide. For, suppose a relative displacement rate of 2 spaces to the right (or left)  at each ksana and take B to be stationary and A, marked in red, displacing itself two spaces at every ksana.

Reappearance rates 2

Clearly, there is going to be trouble at the  very next ksana.
However, since space/time and existence and everything else (except possibly the Event Locality) is not continuous in UET, if the relative speed of the two objects were a good deal greater, say 7 spaces per 7 ksanas (a rate of 7/7)  the red event-chain might manage to just miss the black object.

This could not happen in a system that assumes the Principle of Spatio-Temporal Continuity : in UET there is  leap-frogging with space and time if you like. For the red event-chain has missed out certain positions on the Locality which, in principle could have been occupied.

But this is not all. A collision could also have been avoided if the red chain had possessed a different re-appearance rate even though it remained a ‘slow’ chain compared to the  black one. For consider a 7/7 re-appearance rate i.e. one appearance every seven ksanas and a displacement rate of two spaces per ksana relative to the black conglomerate taken as being stationary. This would work out to an effective rate of 14 spaces to the right at each appearance ─ more than enough to miss the black event-conglomerate.

Moreover, if we have a repeating event-conglomerate that is very compact, i.e. occupies very few neighbouring grid-spaces at each appearance (at the limit just one), and is also extremely rapid compared to the much larger conglomerates it is likely to come across, this ‘event-particle’ will miss almost everything all the time. In UET it is much more of a problem how a small and ‘rapid’ event-particle can ever collide with anything at all (and thus be perceived) than for a particle to apparently disappear into thin air. When I first came to this rather improbable conclusion I was somewhat startled. But I did not know at the time that neutrinos, which are thought to have a very small mass and to travel nearly at the speed of light, are by far the commonest particles in the universe and, even though millions are passing through my fingers as I write this sentence, they are incredibly difficult to detect because they interact with ordinary ‘matter’ so rarely (Note 3). This, of course, is exactly what I would expect ─ though, on the other hand, it is a mystery why it is so easy to intercept photons and other particles. It is possible that the question of re-appearance rates has something to do with this : clearly neutrinos are not only extremely compact, have very high speed compared to most material objects, but also have an abnormally high re-appearance rate, near to the maximum.
RELATIVITY   Reappeaance Rates Diagram         In the adjacent diagram we have the same angle sin θ = v/c but progressively more extended reappearance rates 1/1; 2/2; 3/3; and so on. The total area taken over n ksanas will be the same but the behaviour of the event-chains will be very different.
I suspect that the question of different re-appearance rates has vast importance in all branches of physics. For it could well be that it is a similarity of re-appearance rates ─ a sort of ‘event resonance’ ─ that draws disparate event chains together and indeed is instrumental in the formation of the very earliest event-chains to emerge from the initial randomness that preceded the Big Bang or similar macro events.
Also, one suspects that collisions of event conglomerates  disturb not only the spread and compactness of the constituent events-chains, likewise their ‘momentums’, but also and more significantly their re-appearance rates. All this is, of course, highly speculative but so was atomic theory prior to the 20th century event though atomism as a physical theory and cultural paradigm goes back to the 4th century BC at least.        SH  29/11/13

 

 

Note 1  Compared to the usual 3 × 108 metres/second the unitary  value of s/t0 seems absurdly small. But one must understand that s/t0 is a ratio and that we are dealing with very small units of distance and time. We only perceive large multiples of these units and it is important to bear in mind that s0is a maximum while t0 is a minimum. The actual kernel, where each ultimate event has occurrence, turns out to be s0/c* =  su so in ‘ultimate units’ the upper limit is c* su/t0.  It is nonetheless a surprising and somewhat inexplicable physiological fact that we, as human beings, have a pretty good sense of distance but an incredibly crude sense of time. It is only necessary to pass images at a rate of about eight per second for the brain to interpret the successive in images as a continuum and the film industry is based on this circumstance. Physicists, however, gaily talk of all sorts of important changes happening millionths or billionths of a second and in an ordinary digital watch the quartz crystal is vibrating thousands of times a second (293,000 I believe).

 

Note 2  Only Descartes amongst Western thinkers realized there was a problem here and ascribed the power of apparent self-perpetuation to the repeated intervention of God; today, in a secular world, we perforce ascribe it to ‘ natural forces’.

In effect, in UET, everything is pushed one stage back. For Newton and Galileo the  ‘natural’ state of objects was to continue existing in constant straight line motion whereas in UET the ‘natural’ state of ultimate events is to disappear for ever. If anything does persist, this shows there is a force at work. The Buddhists call this all-powerful causal force ‘karma but unfortunately they were only interested in the moral,  as opposed to physical, implications of karmic force otherwise we would probably have had a modern theory of physics centuries earlier than we actually did.

Note 3  “Neutrinos are the commonest particles of all. There are even more of them flying around the cosmos than there are photons (…) About 400 billion neutrinos from the Sun pass through each one of us every second.”  Frank Close, Particle Physics A Very Short Introduction (OUP) p. 41-2 

In Ultimate Event Theory (UET) the basic building-block of physical reality is not the atom or elementary particle (or the string whatever that is) but the ultimate event enclosed by a four-dimensional  ‘Space/Time Event-capsule’. This capsule has fixed extent s3t = s03t0 where s0 and t0 are universal constants, s0 being the maximum ‘length’ of s, the ‘spatial’ dimension,  and t0 being the minimal ‘length’ of t, the basic temporal interval or ksana. Although s3t = s03 t0  = Ω (a constant), s and t can and do vary though they have maximum and minimum values (as does everything in UET).
All ultimate events are, by hypothesis, of the same dimensions, or better they occupy a particular spot on the Event Locality, K0 , whose dimensions do not change (Note 1). The spatial region occupied by an ultimate event is very small compared to the dimensions of the ‘Event capsule’ that contains it and, as is demonstrated in the previous post (Causality and the Upper Limit), the ratio of ‘ultimate event volume’ to ‘capsule volume’ or  su3/s03 is
1: (c*)3 and of single dimension to single dimension 1 : c* (where c* is the space/time displacement rate of a causal impulse (Note 2)). Thus, s3 varies from a minimum value su3, the exact region occupied by an ultimate event, to a maximum value of  s03  where s0 = c* su. In practice, when the direction of a force or velocity is known, we only need bother about the ‘Space/Time Event Rectangle’  st = constant but we should not forget that this is only a matter of convenience : the ‘event capsule’ cannot be decomposed and  always exists in four dimensions (possibly more).

Movement and ‘speed’ in UET     If by ‘movement’ we mean change, then obviously there is movement on the physical level unless all our senses are in error. If, however, by ‘movement’ we are to understand ‘continuous motion of an otherwise unchanging entity’, then, in UET, there is no movement. Instead there is succession : one event-capsule is succeeded by another with the same dimensions. The idea of ‘continuous motion’ is thus thrown into the trash-can along with the notion of ‘infinity’ with which it has been fatally associated because of the conceptual underpinning of the Calculus. It is admittedly difficult to avoid reverting to traditional science-speak from time to time but I repeat that, strictly speaking, in UET there is no ‘velocity’ in the usual sense : instead there is a ‘space/time ratio’ which may remain constant, as in a ‘regular event-chain, or may change, as in the case of an ‘irregular (accelerated) event-chain. For the moment we will restrict ourselves to considering only regular event-chains and, amongst regular event-chains, only those with a 1/1 reappearance rate, i.e. when one or more constituent ultimate event recurs at each ksana.
An event-chain is a bonded sequence of events which in its simplest form is simply a single repeating ultimate event. We associate with every event-chain an ‘occupied region’ of the Locality constituted by the successive ‘event-capsules’. This region is always increasing since, at any ksana,  any ‘previous spots’ occupied by members of the chain remain occupied (cannot be emptied). This is an important feature of UET and depends on the Axiom of Irreversibility which says that once an event has occurrence on the Locality there is no way it can be removed from the Locality or altered in any way. This property of ‘irreversible occurrence’ is, if you like, the equivalent of entropy in UET since it is a quantity that can only increase ‘with time’.
So, if we have two regular event-chains, a and d , each with the same 1/1 re-appearance rhythm, and which emanate from the same spot (or from two adjacent spots), they define a ‘Causal Four-Dimensional Parallelipod’ which increases in volume at each ksana. The two event-chains can be represented as two columns, one  strictly vertical, the other slanting, since we only need to concern ourselves with the growing Space-Time Rectangle.

So, if we have two regular event-chains, a and d , each with the same 1/1 re-appearance rhythm, and which emanate from the same spot (or from two adjacent spots), they define a ‘Causal Four-Dimensional Parallelipod’ which increases in volume at each ksana. The two event-chains can be represented as two columns, one  strictly vertical, the other slanting, since we only need to concern ourselves with the growing Space-Time Rectangle.

•   

•        

•    •    •    

•    •    •    •    

The two bold dotted lines (black and  red) thus define the limits of the ‘occupied region’ of the Locality, although these ‘guard-lines’ of ultimate events standing there like sentinels are not capable of preventing other events from occurring within the region whose extreme limits they define. Possible emplacements for ultimate events not belonging to these two chains are marked by grey points. The red dotted line may be viewed as displacing itself by so many spaces to the right at each ksana (relative to the vertical column). If we consider the vertical distance from bold black dot to dot to represent t0 , the ‘length’ of a single ksana (the smallest possible temporal interval), and the distance between neighbouring dots in a single row to be 0  then, if there are v spaces in a row (numbered 0, 1,2…..v) we have a Space/Time Event Rectangle of v s­0  × 1 t­0  , the ‘Space/time ratio’ being v grid-spaces per ksana.

It is important to realize what v signifies. ‘Speed’ (undirected velocity) is not a fundamental unit in the Système Internationale but a ratio of the fundamental SI units of spatial distance and time. For all that, v is normally conceived today as an independent  ‘continuous’ variable which can take any possible value, rational or irrational, between specified limits (which are usually 0 and c). In UET v is, in the first instance, simply a positive integer  which indicates “the number of simultaneously existing neighbouring spots on the Event Locality where ultimate events can have occurrence between two specified spots”. Often, the first spot where an ultimate event does or can occur is taken as the ‘Origin’ and the vth spot in one direction (usually to the right) is where another event has occurrence (or could have). The spots are thus numbered from 0 to v where v is a positive integer. Thus

0      1      2       3      4       5                v
•       •       •       •       •       • ………….•     

There are thus v intervals, the same number as the label applied to the final event ─ though, if we include the very first spot, there are (v + 1) spots in all where ultimate events could have (had) occurrence. This number, since it applies to ultimate events and not to distances or forces or anything else, is absolute.
      A secondary meaning of v is : the ratio of ‘values of lateral spatial displacement’ compared to ‘vertical’ temporal displacement’. In the simplest case this ratio will be v : 1 where the ‘rest’ values 0  and 0 are understood. This is the nearest equivalent to ‘speed’  as most of you have come across it in physics books (or in everyday conversation). But, at the risk of seeming pedantic, I repeat that there are (at least) two significant  differences between the meaning of v in UET and that of v  in traditional physics. In UET, v is (1) strictly a static space/time ratio (when it is not just a number) and (2) it cannot ever take irrational values (except in one’s imagination). If we are dealing with event-chains with a 1/1 reapperance rate, v is a positive integer but the meaning can be intelligibly extended to include m/n where m, n are integers. Thus  v = m/n spaces/ksana  would mean that successive events in an event-chain are displaced laterally by m spaces every nth ksana. But, in contrast to ‘normal’ usage, there is no such thing as a displacement of m/n spaces per (single) ksana. For both the ‘elementary’ spatial interval, the ‘grid-space’, and the ksana are irreducible.
One might suppose that the ‘distance’ from the 0th  to the vth spot does not change ─ ‘v is v’ as it were. However, in UET, ‘distance’ is not an absolute but a variable quantity that depends on ‘speed’ ─ somewhat the reverse of how we see things in our everyday lives since we usually think of distances between spots as fixed but the time it takes to get from one spot to the other as variable.

The basic ‘Space-Time Rectangle’ st can be represented thus

Relativity cos diagram

Rectangle   s0 × t0   =   s0 cos φ  × t0 /cos φ
where  PR cos φ = t0     

sv = s0 cos φ        tv = t0 /cos φ       sv = s0 cos φ       tv = t0 /cos φ    sv /s0  =  cos φ     tv /t0  =  1/cos φ s0 /t0  = tan β = constant       tv2  =  t02 + v2 s02     v2 s02 = t02 ( (1/cos2φ) – 1) s02/ t02  tan2 β  =  (1/v2) ((1/cos2φ) – 1) =  (1/v2) tan2 φ  

tan β  = s0 /t0   =    (tan θ)/(v cos θ)      since  sin φ =  tan θ = (v/c)                          

    v =    ( tan θ)/ (tan β cos φ)                  

 

So we have s = s0 cos φ  where φ ranges from 0 to the highest possible value that gives a non-zero length, in effect that value of  φ for which cos φ = s0/c* = su . What is the relation of s to v ? If sv is the spacing associated with the ratio v , and dependent on it, we have sv = s0 cos φ  and so sv /s0  = cos φ. So cos φ is the ‘shrink factor’ which, when applied to  any distance reckoned in terms of s0, converts it by changing the spacing. The ‘distance’ between two spots on the Locality is composed of two parts. Firstly, there is the number of intermediate spots where ultimate events can/could have/had occurrence and this ‘Event-number’ does not change ever. Secondly, there is the spacing between these spots which has a minimum value su which is just the diameter of the exact spot where an ultimate event can occur, and s0 which is the diameter of the Event capsule and thus the maximum distance between one spot where an ultimate event can have occurrence and the nearest neighbouring spot. The spacing  varies according to v  and it is now incumbent on us to determine the ‘shrink factor’ in terms of v.
The spacing s is dependent on so s = g(v) . It is inversely proportional to v since as v increases, the spacing is reduced while it is at a maximum when v = 0. One might make a first guess that the function will be of the form s = 1 – f(v)/h(c)   where f(v) ranges from 0 to h(c) . The simplest such function is just  s = (1 – v/c).
As stated, v in UET can only take rational values since it denotes a specific integral number of spots on the Locality. There  is a maximum number of such spots between  any two ultimate events or their emplacements, namely c –1  such spots if we label spot A as 0 and spot B as c. (If we make the second spot c + 1 we have c intermediate positions.) Thus v  = c/m where m is a rational number.  If we are concerned only with event-chains that have a 1/1 reappearance ratio, i.e. one event per ksana, then m  is an integer. So v  = c/n

We thus have tan θ = n/c  where n  varies from 0 to c* =  (c – 1) (since in UET a distinction is made between the highest attainable space/time displacement ratio and the lowest unattainable ratio c) .
So 0 ≤ θ < π/4  ─ since tan π/4 = 1. These are the only permissible values for tan θ .

Relativity tangent diagram

  

   

 

 

 

 

 

 

 

 

 

If now we superimpose the ‘v/c’ triangle above on the st Rectangle to the previous diagram we obtain

Relativity Circle Diagram tan sin

 

Thus tan θ = sin φ which gives
                cos φ = (1 – sin2 θ)1/2  = (1 – (v/c)2 )1/2  

This is more complicated than our first guess, cos φ = (1 – (v/c), but it has the same desired features that it goes to cos φ = 1 as v goes to zero and has a maximum value when v approaches c.
(This maximum value is 1/c √2c – 1  =  √2/√c )

     

 

 

         cos φ = (1 – (v/c)2 )1/2  is thus a ‘shrinkage factor’ which is to be applied to all lengths within event-chains that are in lateral motion with respect to a ‘stationary’ event chain. Students of physics will, of course, recognize this factor as the famous ‘Fitzgerald contraction’ of all lengths and distances along the direction of motion within an ‘inertial system’ that is moving at constant speed relative to a stationary one (Note 3)

Parable of the Two Friends and Railway Stations

It is important to understand what exactly is happening. As all books on Relativity emphasize, the situation is exactly symmetrical. An observer in system A would judge all distances within system B to be ‘contracted’, but an observer situated within system B would think exactly the same about the distances in system A. This symmetricality is a consequence of Einstein’s original assumption that  ‘the laws of physics take the same form in all inertial frames’. In effect, this means  that one inertial frame is as good as any other because if we could distinguish between two frames, for example by carrying out identical  mechanical or optical experiments, the two frames would not be equivalent with respect to  their physical behaviour. (In UET, ‘relativity’ is a consequence of the constancy of the area on the Locality occupied by the Event-capsule, whereas Minkowski deduced an equivalent principle from Einstein’s assumption of relativity.)
As an illustration of what is at stake, consider two friends undertaking train journeys from a station which spans the frontier between two countries. The train will stop at exactly the same number of stations, say 10, and both friends are assured that the stations are ‘equally spaced’ along each line. The friends start at the same time in Grand Central Station but go to platforms which take passengers to places in different countries.
In each journey there are to be exactly ten stops (including the final destination) of the same duration and the friends are assured that the two trains will be running at the ‘same’ constant speed. The two  friends agree to stop at the respective tenth station along the respective lines and then relative to each other. The  tracks are straight and close to the border so it is easy to compare the location of one station to another.
Friend B will thus be surprised if he finds that friend A has travelled a lot further when they  both get off at the tenth station.  He might conclude that the tracks were not straight, that the trains were  dissimilar or that they didn’t keep to the ‘same’ speed. Even might conclude  that, even though the distances between stations as marked on a map were the same for both countries, say 20 kilometres, the map makers had got it wrong. However, the problem would be cleared up if the two friends eventually learned that, although the two countries assessed distances in metres, the standard metre in the two countries was not the same. (This could not happen today but in the not sp distant past measurements of distance, often employing the same terms, did differ not only from one country to another but, at any rate within the Holy Roman Empire, from one ‘free town’ to another. A Leipzig ‘metre’ (or other basic unit of length) was thus not necessarily the same as a Basle one. It was only since the advent of the French Revolution and the Napoleonic system that weights and measures were standardized throughout most of Europe.’)

    This analogy is far from exact but makes the following point. On each journey, there are exactly the same number of stops, in this case 10, and both friends would agree on this. There is no question of a train in one country stopping at a station which did not exist for the friend in the other country. The trouble comes because of the spacing between stations which is not the same in the two countries, though at first sight it would appear to be because the same term is used.
    The stops correspond to ultimate events : their number and the precise region they occupy on the Locality is not relative but absolute. The ‘distance’ between one event and the next is, however, not absolute but varies according to the way you ‘slice’ the Event capsules and the region they occupy, though there is a minimum distance which is that of a ‘rest chain’.  As Rosser puts it, “It is often asked whether the length contraction is ‘real’?  What
the principle of relativity says is that the laws of physics are the same in all inertial frames, but the actual measures of particular quantities may be
different in different systems” (Note 4)

Is the contraction real?  And, if so,  why is the situation symmetrical? 

   What is not covered in the train journey analogy is the symmetricality of the situation. But if the situation is symmetrical, how can there be any observed discrepancy?
This is a question frequently asked by students and quite rightly so. The normal way of introducing Special Relativity does not, to my mind, satisfactorily answer the question. First of all, rest assured that the discrepancy really does exist : it is not a mathematical fiction invented by Einstein and passed off on the public by the powers that be.
μ mesons from cosmic rays hitting the atmosphere get much farther than they ought to — some even get close to sea level before decaying. Distance contraction explains this and, as far as I know, no other theory does. From the point of view of UET, the μ meson is an event-chain and, from its inception to its ‘decay’, there is a finite number of constituent ultimate events. This number is absolute and has nothing to do with inertial frames or relative velocities or anything you like to mention. We, however, do not see these occurrences and cannot count the number of ultimate events — if we could there would be no need for Special Relativity or UET. What we do register, albeit somewhat unprecisely, is the first and last members of this (finite) chain : we consider that the μ meson ‘comes into existence’ at one spot and goes out of existence at another spot on the Locality (‘Space/Time’ if you like). These events are recognizable to us even though we are not moving in sync with the μ meson (or at rest compared to it). But, as for the distance between the first and last event, that is another matter. For the μ meson (and us if we were in sync with it) there would be a ‘rest distance’ counted in multiples of s (or su).  But since we are not in sync with the meson, these distances are (from our point of view) contracted — but not from the meson’s ‘point of view’. We have thus to convert ‘his’ distances back into ours. Now, for the falling μ meson, the Earth is moving upwards at a speed close to that of light and so the Earth distances are contracted. If then the μ meson covers n units of distance in its own terms, this corresponds to rather more in our terms. The situation is somewhat like holding n strong dollars as against n debased dollars. Although the number of dollars remains the same, or could conceivably remain the same, what you can get with them is not the same : the strong dollars buy more essential goods and services. Thus, when converting back to our values we must increase the number. We find, then, that the meson has fallen much farther than expected though the number of ultimate events in its ‘life’ is exactly the same. We reckon, and must reckon, in our own distances which are debased compared to that of a rest event-chain. So the meson falls much farther than it would travel (longitudinally) in a laboratory. (If the meson were projected downwards in a laboratory there would be a comparable contraction.) This prediction of Special relativity has been confirmed innumerable times and constitutes the main argument in favour of its validity.
From the point of view of UET, what has been lost (or gained) in distance is gained (or lost) in ‘time’, since the area occupied by the event capsule or event capsules remains constant (by hypothesis).  The next post will deal with the time aspect.        SH  1 September 2013

 

Note 1  An ultimate event is, by definition, an event that cannot be further decomposed. To me, if something has occurrence, it must have occurrence somewhere, hence the necessity of an Event Locality, K0, whose function is, in the first instance, simply to provide a ‘place’ where ultimate events can have occurrence and, moreover, to stop them from merging. However, as time went on I found it more natural and plausible to consider an ultimate event, not as an entity in its own right, but rather as a sort of localized ‘distortion’ or ‘condensation’ of the Event Locality. Thus attention shifts from the ultimate event as primary entity to that of the Locality. There has been a similar shift in Relativity from concentration on isolated events and inertial systems (Special Relativity) to concentration on Space-Time itself. Einstein, though he pioneered the ‘particle/finitist’ approach ended up by believing that ‘matter’ was an illusion, simply being “that part of the [Space/Time] field where it is particularly intense”. Despite the failure of Einstein’s ‘Unified Field Theory’, this has, on the whole,  been the dominant trend in cosmological thinking up to the present time.
But today, Lee Smolin and others, reject the whole idea of ‘Space/Time’ as a bona fide entity and regard both Space and Time as no more than “relations between causal nodes”. This is a perfectly reasonable point of view which in its essentials goes back to Leibnitz, but I don’t personally find it either plausible or attractive. Newton differed from Leibnitz in that he emphatically believed in ‘absolute’ Space and Time and ‘absolute’ motion ─ although he accepted that we could not determine what was absolute and what was relative with current means (and perhaps never would be able to). Although I don’t use this terminology I am also convinced that there is a ‘backdrop’ or ‘event arena’ which ‘really exists’ and which does in effect provide ‘ultimate’ space/time units. 

Note 2. Does m have to be an integer? Since all ‘speeds’ are integral grid-space/ksana ratios, it would seem at first sight that m must be integral since c  (or c*) is an exact number of grid-spaces per ksana and v = (c*/m). However, this is to neglect the matter of reappearance ratios. In a regular event-chain with a 1/1 reappearance ratio, m would have to be integral ─ and this is the simplified event-chain we are considering here. However, if a certain event-chain has a space/time ratio of 4/7 , i.e. there is a lateral displacement of 4 grid-spaces every 7  ksanas, this can be converted to an ‘ideal’ unitary rate of 4/7 sp/ks.
In contemporary physics space and time are assumed to be continuous, so any sort of ‘speed’ is possible. However, in UET there is no such thing as a fractional unitary rate, e.g. 4/7th of a grid-space per ksana since grid-spaces cannot be chopped up into anything smaller. An ‘idealfractional rate per ksana is intelligible but it does not correspond to anything that actually takes place. Also, although a rate of m/n is intelligible, all rates, whether simple or ideal, must be rational numbers ─ irrational numbers are man-made conventions that do not represent anything that can actually occur in the  real world.

Note 3  Rosser continues :
     “For example, in the example of the game of tennis on board a ship going out to sea, it was reasonable to find that the velocity of the tennis ball was different relative to the ship than relative to the beach. Is this change of velocity ‘real’? According to the theory of special relativity, not only the measures of the velocity of the ball relative to the ship and relative to the seashore will be different, but the measures of the dimensions of the tennis court parallel to the direction of relative motion and the measures of the times of events will also be different. Both the reference frames at rest relative to the beach and to the ship can be used to describe the course of the game and the laws of physics will be the same in both systems, but the measures of certain quantities will differ.”                          W.G.V. Rosser, Introductory Relativity

 

 

1. Anomalous nature of causality

Causality has a peculiar status in science. Without it, there could hardly be science at all and Claude Bernard, the 19th century French biologist, went so far as to define science as determinism. Quantum Mechanics has, of course, somewhat dented the privileged position of causality in scientific thinking ─ but much less than is commonly thought. We still have ‘statistical determinism’ and, for most practical purposes the difference between absolute and statistical determinism is academic. Moreover, contrary to what many people think, chaos theory does not dispense with determinism : in principle chaotic behaviour, though unpredictable, is held to be nonetheless deterministic (Note 1).
I consider that we just have to take Causality on board.  Apart from the idea that there is some sort of objective reality ‘out there’, causality tops the list of concepts I would be the most reluctant to do without. The trouble is that belief in causality is essentially an act of faith since there seems  no way to demonstrate whether it is operative or not. If there were actually some sort of test whereby we could show that causality was at work, in the sort of way we can test whether an electric current is flowing through a circuit, questions of causality could be resolved rapidly and decisively. But no such test exists and, seemingly, can’t exist. Scientists and engineers generally believe in causality because they can’t do without it, but several philosophers have questioned whether there really is such a thing, notably Hume and Wittgenstein. The fact  that event A has up to now always been succeeded by event B does not constitute proof as Hume correctly observed. Scientists, engineers and practical people need to believe in causality and so they simply ignored Hume’s attack, though his arguments have never been refuted. Hume himself said that he abandoned his scepticism concerning the reality of causality when playing billiards ─ as well he might.

Relativity and the Upper Limit to Causal Processes

What we can do, since the advent of Relativity, is to decide when causality is not operative by appealing to the well-known test of whether two events lie within the ‘light cone’. But all this talk of light and observers and sending messages is misleading : it is putting the cart before the horse. What we should concentrate on is causality.
Eddington once said that one could deduce from a priori reasons (i.e. without carrying out any experiments) that there must be an upper limit to the speed of light in any universe, though one could not deduce a priori what the value of this limit had to be. Replace ‘speed of light’ by ‘speed of propagation of a causal influence’ and I agree with him. Certainly I can’t conceive of any ‘world’ where the operation of causality was absolutely simultaneous.
I thus propose introducing as an axiom  of Ultimate Event Theory  that

        There exists an upper limit to the ‘speed’ (event-space/ksana ratio) for the propagation of all causal influences

  In traditional physics, since Relativity, this upper limit is noted as c ≈ 3 × 108 metres per second. Without wishing to be pedantic, I think it is worthwhile at the outset distinguishing between an attainable upper limit and the lowest unattainable upper limit. The latter will be noted c* while the attainable limit will be noted as c in accordance with tradition. In Ultimate Event Theory the units are not metres and seconds : the standard unit of length is s0 , the distance between two adjacent spots on the Event Locality K0, and t0 , the ‘length’ of a ksana, the distance between two successive ultimate events in an event-chain. Thus  c* is an integer = c* s0/t0  and c, the greatest actually attainable ‘speed’ ─ better, displacement ratio ─ is (c* – 1) s0/t0   
        In modern physics, since Einstein’s 1905 paper, c, the maximum rate of propagation of causality is equated with the actual speed of light. I argued in an article some years ago that there was no need to exactly identify c, the upper limit of the propagation of causality with the actual speed of light, but only to conclude that the speed of light was very close to this limit (Note 2)

Revised Rule for Addition of Velocities

 Einstein realized that his assumption  (introduced as an Axiom) that “the speed of light in vacuo is the same for all observers irrespective of the movement of the source” invalidated the usual rule for adding velocities. Normally, when considering ‘motion within motion’ as when, for example, I walk down a moving train, we just add my own speed relative to the train to the current speed of the train relative to the Earth. But, if  V = v + w  we can easily push V over the limit simply by making v and w very close to c. For example, if v = (3/4)c and w = (2/3)c  the combined speed will be greater than c.

Since c is a universal constant, the variables v and w may  be defined in terms of c. So, let  v = c/m   w = c/n  where m, n > 1   (though not necessarily integral).

Thus, using the normal (Galileian) Rule for adding velocities

V = c/m + c/n  = c( 1   +  1  ) = c (m + n) 
                   m       n           mn

The factor (m + n)   <if m > 1, n > 1
         mn

For, let m = (1 + d), n = (1 + e)  with d, e > 0    then

      mn =  (1 + d)(1 + e)  = (1 + (d+e) + de) = (2 + (d+e)) – (1 – de)

                                                        = (m + n) – (1 – de)

                        So  (m + n) – mn = (1– de)  <  1

So, in order to take V beyond c, all we have to do is make  de < 1 and this will be true whenever 0 < d < 1  and 0 < e < 1. For example, if we have m = 3/2 = 1 + ½    and n = 9/8 = 1 + 1/8   we obtain a difference of (1 – 1/16) = 15/16. And in fact

If v = c/(3/2)   w = c/(9/8)  we have

V = (2/3)c + (8/9) c = (14/9) c >  c

The usual formula for ‘adding’ velocities is thus no good since it allows the Upper Limit to be exceeded and this is impossible (by assumption). So we must look for another formula, involving m and n only, which will stop V from exceeding c. We need a factor which, for all possible non-zero quantities m and n (excluding m = 1, n = 1) will make the product < 1.

Determining the New Rule for Adding Velocities

 The first step would seem to be to cancel the mn in the expression (m + n)/mn . So for the multiplying factor  we want mn divided by some expression involving m and n only but which (1) is larger than mn and (2) has the effect of making

           (m + n)  × mn    <  1  for all possible m, n > 1
          mn       f(mn)

The simplest function is just (1 + mn) since this is > mn and also has the desired result that

                (m + n)  ×     mn        <  1  for all possible m, n > 1
                  mn          (1 + mn)

This is so because (m + n) > (1 + mn)  for all m, n > 1

        Again, we set m = (1 + d), n = (1 + e)  so

(m + n)  = 2 + (c + d)  and 1 + mn = 1 + (1 + (d + e) + de)

                                                        =  (2 + (d + e) + de)

                                                        =  (m + n) + de

So  (m + n)/(1 + mn) < 1  (since denominator is larger than the nominator).

Moreover, this function (1 + mn) is the smallest such function that fits the bill for all legitimate values of m  and n. For, if we set f(m, n) = (e + mn)  we must have e > (1 – de) for all values d, e > 0 . The smallest such e is just 1 itself.

I start by assuming that there is an unattainable Upper Limit to the propagation of causal influences, call it c*. This being the case, the ‘most extended’ regular event-chain can have at most a spatial distance/temporal distance ratio of c . Anything beyond this is not possible (Note 3).
This value c is a universal constant and any ‘ordinary’ speeds (space/time ratios) within event-chains ought, if possible to be defined with reference to c, i.e. as c/m, c/n and so on. What are the units of c? The ‘true’ units are s0 and t0, the inter-ultimate event spatial and temporal distances, i.e. the ‘distance’ from the emplacement of one ultimate event to its nearest simultaneously existing neighbour (in one spatial dimension) and the distance from one ultimate event to its immediate successor in a chain. These distances are those of an event-chain at rest with regard to the Locality K0. These values are, by hypothesis, constants.
A successor event in an event-chain can only displace itself by integral units since every event must occupy a spot on the Locality. The smallest displacement would just be that of a single grid-space, 1 s­0 . Using the c/m notation this is a displacement ratio of c/c  = 1 0/t0    And the smallest ‘combined speed’ V would be V = c/c + c/c  = 2  using the ‘traditional’ method of combining velocities. But, using the new formula we have

V = c (c + c)        =   2c2     s­0/t
       (1 + c2)         (1 + c2)

        This is very slightly less than c(2c)/c2 = 2 . We may consider the second fraction       c2  .    =       1   
                                                                                                                                                      (1 + c2)       1 + 1/c2
  as a ‘shrinkage factor’. Since c is so large 1/c2 is minute and the shrinkage factor is correspondingly small.
More generally, for  u = c/m   and  w = c/n  we have a ‘shrinkage factor’ of  1/(1 + 1/mn)
      This should be interpreted as follows. By the Space/Time capsule Axiom, the region s3t is constant and = s03 t0  where s0 and  t0  are constants. We neglect two of the spatial dimensions and concentrate only on the ‘rectangle’ st which is s0 by  t0  for a ‘stationary’ event chain. Since the sides of the ‘rest’ rectangle are fixed, so is the mixed space/time ratio s0 /t0   This in principle gives the ratio width to height of the region occupied by a single ultimate event in a rest chain ─ but, of course, we do not at present know the values of s0 and t0 .
Associated with a single event-chain is the region of the Locality it occupies. If an ultimate event conglomerate repeats at every ksana (has a reappearance rate of 1/1), the event-chain effectively monopolises the available space, i.e. stops any other ultimate events from having occurrence within this region. If there are N events in the chain the total area of the occupied region is N (s0)3 t0 . Note that if the event-chain contains N events, there are  (N – 1) intervals whereas if we number the ultimate events 0, 1, 2, 3….  there are N temporal intervals, i.e. N ksanas in all. Also, it is important to note that, in this model, each ultimate event itself only occupies a small part of this ‘Space/Time capsule’ of size (s0)3 t0 ─ but its occupancy is enough to exclude all other possible ultimate events.
As stated before, when dealing with simple event-chains with a fixed ‘direction’, we can neglect two of the three ‘spatial’ dimensions (the y and z dimensions), so we only need to bother about the ‘Space-Time rectangle’ of fixed size s0 t0 . Thus, when dealing with a simple regular event-chain we only need to bother about the region occupied by N such rectangles. Although the area  of the rectangle s0 t0 is constant (= R), the ratio of the sides need not be. However, for all s, t  st = s0 t0   the lengths s0  and t0  of this mixed ‘Space-Time’ rectangle are the ‘rest’ lengths, the dimensions of each capsule when considered in isolation ─ ‘rest’ lengths because, by the Rest Axiom (or definition) every event-chain is at rest relative to the Event Locality K0 (Note 4) .  Although there is no such thing as absolute movement relative to the Locality, there certainly is relative movement (displacement ksana by ksana) of one event-chain with respect to another which may be considered to be stationary. And this relative movement changes the distances distances of the event capsules and so of the entire chain. The changed distances are noted sv and tv  and, since the product sv tv is constant and equal to the ‘rest area’ s0 t0, ­the sides of the rectangle, sv and tv change inversely i.e. s0 /sv   = tv/tv  so if the ‘spatial dimension’ of the rectangle decreases, the ‘time dimesnion’, the length of a ksana in absolute terms increases. I had in a previous post introduced tentatively as an axiom that the rest length of a ksana, t0 , was a minimum. But, in fact as I hoped, this is a consequence of the behaviour of the s dimension. The Upper Limit Assumption and the consequent discussion of the rule for adding velocities, shows that s0 is a maximum which in turn makes t0 a maximum as required. And practically speaking, in ‘normal conditions’, s and t will also have a maximum and minimum, namely the values they take when the displacement ratio s/t  = c the upper limit. Thus s0 > sv ≥ sc and t0 < tv ≤ tc  

Revised Rule for Addition of Velocities

 Einstein realized that his assumption  (introduced as an Axiom) that “the speed of light in vacuo is the same for all observers irrespective of the movement of the source” invalidated the usual rule for adding velocities. Normally, when considering ‘motion within motion’ as when, for example, I walk down a moving train, we just add my own speed relative to the train to the current speed of the train relative to the Earth. But, if  V = v + w  we can easily push V over the limit simply by making v and w very close to c. For example, if v = (3/4)c and w = (2/3)c  the combined speed will be greater than c.
Since c is a universal constant, the variables v and w may  (and should) be defined in terms of c. So, let  v = c/m   w = c/n  where m, n > 1   (though not necessarily integral).

Thus, using the normal (Galileian) Rule for adding velocities

          V    = c/m + c/n  = c( 1   +  1  ) = c (m + n) 
                                           m      n           mn

The factor (m + n)   <if m > 1, n > 1
                     mn

For, let m = (1 + d), n = (1 + e)  with d, e > 0    then

         mn =  (1 + d)(1 + e)  = (1 + (d+e) + de) = (2 + (d+e)) – (1 – de)

                                                        = (m + n) – (1 – de)

               Thus          (m + n) – mn = (1– de)  <  1

So, in order to take V beyond c, all we have to do is make  de < 1 and this will be true whenever 0 < d < 1  and 0 < e < 1. For example, if we have m = 3/2 = 1 + ½    and n = 9/8 = 1 + 1/8   we obtain a difference of (1 – 1/16) = 15/16. And in fact

If v = c/(3/2)   w = c/(9/8)  we have

V = (2/3)c + (8/9) c = (14/9) c >  c

The usual formula for ‘adding’ velocities is thus no good since it allows the Upper Limit to be exceeded and this is impossible (by assumption). So we must look for another formula, involving m and n only, which will stop V from exceeding c. We need a factor which, for all possible non-zero quantities m and n (excluding m = 1, n = 1) will make the product < 1.
The first step would seem to be to get rid  of the mn in the expression (m + n)/mn . So for the multiplying factor  we want mn divided by some expression involving m and n only but which (1) is larger than mn and (2) has the effect of making

(m + n)  × mn    <  1  for all possible m, n > 1
 mn       f(mn)

The simplest function having the desired properties is just mn/(1 + mn) since this is > mn for m, n > 1.and also has the desired result that

(m + n)  ×     mn        <  1  for all possible m, n > 1
 mn        (1 + mn)

For, let m = (1 + d), n = (1 + e)  so that

(m + n)  = 2 + (c + d)  and 1 + mn = 1 + (1 + (d + e) + de)
                                                        =  (2 + (d + e) + de)
                                                        =  (m + n) + de

So  (m + n)/(1 + mn) < 1  (since denominator is larger than the nominator).

Moreover, this function (mn)/(1 + mn) is the smallest such function that fits the bill for all legitimate values of m  and n. For, if we set f(m, n) = (e + mn)  we must have e > (1 – de) for all values d, e > 0 . The smallest such e is just 1 itself.

The factor     mn       =        1   .       should be regarded as
                   (1 + mn)        1 + (1/mn)

a ‘shrinkage factor’ which gets applied automatically when velocities are combined. It is not a mathematical fiction but something that  is really operative in the physical world and which excludes  ‘runaway’ speeds which otherwise would wreck the system ─ much as a thermostat stops a radiator from overheating. It is not today helpful to view such procedures as ‘physical laws’ ─ though this is how Newton and possibly even Einstein viewed them. Rather, they are automatic procedures that ‘kick in’ when appropriate.
Mathematics is a tool for getting a handle on reality, no more, no less, and it is essential to distinguish between mathematical procedures which are simply aids to calculation or invention and those which correspond to actual physical mechanisms. I believe that the factor (mn)/(1 + mn)  ─ and likewise γ = (1/(1 – v2/c2))1/2  that we shall come to later─ fall into the latter category. How and why such mechanisms got developed in the first place, we do not know and perhaps will never know, though it is quite conceivable that they developed like so much else by ‘trial and error’ over a vast expanse of ‘time’ in much the same way as biological mechanisms developed without the users of these mechanisms knowing what they were doing or where they were heading.

The Space/Time Capsule and the units of c. 

This value c is a universal constant and any ‘ordinary’ speeds (space/time ratios) within event-chains ought, if possible, to be defined with reference to c, i.e. as c/m, c/n and so on. But what are the units of c in Ultimate Event Theory?
          As stated in the previous post, I take as axiomatic that “the region of the Space/Time  capsule s3t is constant and equal to the ‘rest’ value of so3 t0.  But, although the product is fixed, s and t can and do vary. When dealing with a (resolved) force or motion which thus has a unique spatial direction, we only need bother about the rectangle of area s × t which can be plotted as a hyperbola of the form st =  constant.
         However, unlike most rectangular hyperbolae, the graph does not extend to infinity along both axes ─ nothing in UET extends to infinity. So s and t must have minimal and maximal values. I have assumed so far that s0 is a maximum and this is in accord with Special Relativity. So this makes t0 a minimum since st = s0 t0  = Ω . Actually, while writing this post, it has occurred to me that one could do things the other way round and have s0 as a minimum and t0 a maximum since there does not seem any reason a priori why this should not be possible. But I shall not pursue this line of thought at the moment.
So, if we wish to convert to ‘ultimate’ units of distance and time, we can use t0  , the minimal length of a ksana which it attains in a ‘rest chain’ as the appropriate temporal unit. But what about spatial distance? Since s0 is the maximum value for the spatial dimension of the mixed Space-Time capsule’ of fixed volume  s3 t  = s03 t0, we must ask whether s has a minimum? The answer is yes. In UET there is no infinity and everything has a minimum and a maximum with the single exception of the Event Locality itself, K0, which has neither because it is intrinsically ‘non-metrical’. s03 t0 represents the volume of the ‘Space/time capsule’ enclosing an ultimate event and which in UET is the ultimate basic building-block of physical reality. But an ultimate event itself occupies a non-zero, albeit minuscule, region. Since there is nothing smaller than an ultimate event, this value, we can take the dimensions of the region occupied as the ultimate volume and each of the spatial lengths as the ultimate unit of distance. So su  will serve as the ‘ultimate’ unit of distance where the subscript u means ‘ultimate’. And, since st = s0 t0  = Ω for all permissible values of s  and t, we have su tu  =  s0 t   thus su /s= t0/tu . So the ratios of the extreme values of spatial and temporal units are inversely related. Thus s0 = M su  where m is an integer since every permissible length must be a whole number multiple of the base unit. Thus s0/t0  which we have noted as M, is, in ultimate units, (M su)/t0  so M = c
This was, to me, an unexpected and rather satisfying,  result. Instead of c appearing, as it were, from nowhere, the UET treatment gives a clear physical meaning to the idea that light travels at (or near to) the limiting value for all causal processes. We can argue the matter in this way.
I view an ultimate event as something akin to a hard seed in a pulpy fruit or the nucleus in an atom, where the fruit as a whole or the atom is the space/time  capsule. Suppose a causal influence emanating from the kernel of a Space/Time Relativity  Upper Limitcapsule where there is an ultimate event. Then, if it going to have any effect on the outside world, it must at least travel a distance of ½ so to get outside its capsule and another  ½ so to get to the centre of the next capsule where it repeats (or produces a different ultimate event). And in the case of a regular event-chain with a 1/1 reappearance ratio (i.e. one ultimate event at each and every ksana) the causal force must traverse this distance within the space of a single ksana. If the chain is considered in isolation and thus at rest, the length of every ksana will be the minimal temporal distance t0. The causal influence must thus have a space/time ratio of  (½ s+ ½ so )/t0  =  so /t0  = c.
Thus, c is not only the limiting speed for causal processes, but turns out to be the only possible speed in a rest chain since a causal influence must get outside a Space/Time capsule if it is going to have any effects.  And, since every event-chain is itself held together by causal forces, it makes sense that the electro-magnetic event-chain commonly known as ‘light’ cannot exceed this limit ─ an event-chain which exceeded the limit, supposing this to be conceivable, would immediately terminate since any subsequent events would be completely dissociated from prior events. What this means is that if, say, two light rays were sent out at right angles to each other, each event in the ‘moving chain’ would be displaced a distance of s0 at each ksana relative to the ‘stationary‘ chain, while the causal influence in the stationary chain would have traversed exactly the same distance in absolute units. In General Relativity, the constant c is often replaced by 1 to make calculations easier : this interpretation justifies the practice. For in the ‘capsule’ units  s9 and t0 the ratio is csu/t0 = 1 s0 /t0   

Asymmetry of Space/Time

It would seem that there is a serious asymmetry between the ‘spatial’ dimension(s) and the temporal. Since s/s0 = t0/t , spatial  and temporal distances ─ ‘space’ and ‘time’ ─ are inversely proportional (Note 3). We are a species with a very acute sense of spatial distance and a very crude sense of time ─ the film industry is based on the fact that the eye, or rather the eye + brain, ‘runs together’ images that are flashed across the screen at a rate of more than a few frames per second (8, I believe). And we do not have too much difficulty imagining huge spatial distances such as the diameter of the galaxy, while we find it difficult to conceive of anything at all of any interest happening within a hundredth, let alone a hundred billionth, of a second. Yet cosmologists happily talk about all sorts of complicated processes going on within minute fractions of a second after the Big Bang, so we know that there can be quite a lot of activity in what is, to us, a very small temporal interval.
For whatever reason, one feels that  the smallest temporal interval (supposing there is one) and which in UET is t0, must be extremely small compared to the maximum unitary ‘spatial’ distance s0 . This may be an illusion based on our physiology but I suspect that there is something in it : ‘time’ would seem to be much more finely tuned than space. This goes some way to explaining why we are unaware of their being any ‘gaps’ between ksanas as I believe there must be, while there are perhaps no gaps between contemporaneous spatial capsules. I believe there must be gaps between ksanas because the whole of the previous spatial set-up has to vanish to give way to the succeeding set-up, whereas ‘length’ does not have to vanish to pass on to width or depth. These gaps, if they exist, are probably extremely small and so do not show up even with our contemporary precision instruments. However, at extremely high (relative) speeds, gaps between ksanas should be observable and one day, perhaps quite soon, will be observed and their minimal and maximum extent calculated at least approximately.

 Strange Consequence of the Addition Rule

Curiously, the expression

        (m + n)  ×     mn        <  1  for all possible values of m, n
          mn        (1 + mn)

except when m = 1, n = 1 (or both). I have already shown that

(m + n)  ×     mn        <  1  when m, n > 1
 mn        (1 + mn)

But the inequality holds even when we are dealing with (possibly imaginary) speeds greater  than the limit. For consider c/m + c/n where m < 1  n < 1   i.e. when c/m and c/n  are each > 1

Let m = (1 – d)   n = (1 – e)   d, e > 0

        Then (m + n)     = 2 – (d + e) 

                  1 + mn   = 1 + {1 – d)(1 – e)}
                                = 1 + {1 –(d + e) + de)}
                                = 2 – (d + e) + de
                                = (m + n) + de  > (m + n) since d, e > 0

This means that even if both velocities exceed c, their combination (according to the new addition rule) is less than c !

For take c/(1/2) + c/(1/5) = 2c + 5 c = 7c by the ‘normal’ addition rule. But, according to the new rule, we have

c ((1/2) + (1/5))  =   c (0.5 + 0.2) = c (0.7)  <  c
(1 + (1/2)(1/5)           (1.1)              1.1

         I am not sure how to interpret this in the context of Ultimate Event Theory and causality. It would seem to imply that ‘event sequences’ ─ one cannot call them ‘chains’ because there is no bonding between the constituent events ─ which have displacement rates above  the causal limit, when combined, are somehow dragged back below the upper limit and become a bona fide event-chain. So, independently, such loose sequences can exist by themselves ‘above the limit’, but if they get entangled with each other, they get pulled back into line. In effect, this makes a sort of sense. Either causality is not operative at all, or, if it is operative, it functions at or below the limit.
This curious result has, of course, been noted many times in discussions about Special Relativity and given rise to all sorts of fantasies such as particles propagating backwards in time, effects preceding causes and so on. Although Ultimate Event Theory may itself appear far-fetched to a lot of people, it does not accommodate such notions : sequence and causality, though little else, are normally conserved and the ‘arrow of time’ only points one way. There must, however, be some good physical reason for this ‘over the speed limit’ anomaly and it will one day be of use technologically.

Random Events   

A random event by definition does not have a causal predecessor, although it can have a causal successor event. Random events are thus causal anomalies,  spanners in the works : they should not exist but apparently they do ─ indeed I suspect that they heavily outnumber well-behaved events which belong to recognizable event-chains (but individual random ultimate events are so short-lived they are practically speaking unobservable at present).
One explanation for the occurrence of random events ─ and they certainly seem to exist ─ is that they are events that have got dissociated from the event-chains to which they ‘by right’ belonged because “they arrived too fast”. If this is so, these stray events could pop up more or less anywhere on the Locality where there was an unoccupied space and they would appear completely out of place (i.e. ‘random’) because they would have no connection at all  with neighbouring events. This is indeed how many so-called ‘random’ events do appear to us : they look for all the world as if they have been wrongly assembled by an absent-minded worker. One might  draw a parallel with ‘jumping genes’ where sections of DNA get fitted into sections where they have no business to be (as far as we can tell).                                                                        S. H.     7/8/13

Note 1 Whether considering that a chaotic system is both unpredictable and yet deterministic is ‘having your cake and eating it’ I leave to others to decide. There is a generic difference between “being unable to make exact predictions because we can never know exactly the original situation” and “being unable to make predictions because the situation evolves in a radically unpredictable, i.e. random, manner”. No one disputes that in cases of non-linear dynamics the situation is inclined to ‘blow up’ because small variations in the original conditions can have vast consequences. Nor does anyone dispute that we will most likely never be able to know the initial conditions with the degree of certainty we would like to have. Therefore, ‘chaotic systems’ are unpredictable in practice ─ though they follow certain contours because of the existence of ‘strange attractors’.
But are the events that make up chaotic systems unpredictable in principle? Positivists sweep the whole discussion under the carpet with the retort, “If we’ll never be able to establish complete predictability, there’s no point in discussing the issue”. But for people of a ‘realistic’ bent, amongst whom I include myself, there is all the reason in the world to discuss the issue and come to the most ‘reasonable’ conclusion. I believe  there is a certain degree of  randomness ‘hard-wired’ into the workings of the universe anyway, even in ‘well-behaved’ linear systems. Nessim Taleb is, in my view, completely right to insist that there is a very real and important difference between the two cases. He believes there really is an inherent randomness in the workings of the universe and so nothing will ever be absolutely predictable. In consequence, he argues that, instead of bothering about how close we can get to complete predictability, it makes more sense to ‘prepare for the worst’ and allow in advance for the unforeseen.

Note 2. If you don’t identify the upper limit with the observed speed of an actual process, this allows you, even in ‘well-behaved’ linear systems. Nessim Taleb is, in my view, completely right to insist that there is a very real and important difference between the two cases. He believes there really is an inherent randomness in the workings of the universe and so nothing will ever be absolutely predictable. In consequence, he argues that, instead of bothering about how close we can get to complete predictability, it makes more sense to ‘prepare for the worst’ and allow in advance for the unforeseen.

Note 2. If you don’t identify the two exactly, this allows you to attribute a small mass to the ‘object under consideration and, as a matter of fact, at the time it was thought that the neutrino, which travels at around the same speed as light, was massless whereas we now have good reason to believe that the neutrino does have a small mass. But this issue is not germane to the present discussion and, for the purposes of this article, it is not necessary to make too much distinction between the two. When there is possible confusion, I shall use c* to signify the strictly unattainable limit and c to signify the upper limit of what can be attained. Thus v ≤ c  but v < c.

Note 3 Although hardly anyone seems to have been bothered by the issue, it questionable that it is legitimate to have mixed space/time values since this presupposes that there is a shared basic unit.

Note 4   If ‘random’ events greatly outnumber well-behaved causal events, why do we not record the fact with our senses and conclude that ‘reality’ is completely unpredictable? The ancients, of course, did believe this to a large extent and, seemingly, this was one reason why the Chinese did not forge ahead further than they actually did : they lacked the Western notion of ‘physical law’ (according to Needham). There may have been some subliminal perception of underlying disorder that surfaced in such ancient beliefs. But the main reason why the horde of ‘random’ ultimate events passes unnoticed is that these events flash in and out of existence without leaving much of a trace : only very few form recognizable event-chains and our senses are only responsive to relatively large conglomerates of events.

 

Almost everyone schoolboy these days has heard of the Lorentz transformations which replace the Galileian transformations in Special Relativity. They are basically a means of dealing with the relative motion of two bodies with respect to two orthogonal co-ordinate systems. Lorentz first developed them in an ad hoc manner somewhat out of desperation in order to ‘explain’ the null result of the Michelson-Morley experiment and other puzzling experimental results. Einstein, in his 1905 paper, developed them from first principles and always maintained that he did not at the time even know of Lorentz’s work. What were Einstein’s assumptions?

  1. 1.  The laws of physics take the same form in all inertial frames.
  2. 2.  The speed of light in free space has the same value for all observers in inertial frames irrespective of the relative motion of the source and the observer.

As has since been pointed out, Einstein did, in fact, assume rather more than this. For one thing, he assumed that ‘free space’ is homogeneous and isotropic (the same in all directions) (Note 1). A further assumption that Einstein seems to have made is that ‘space’ and ‘time’ are continuous ─ certainly all physicists at the time  assumed this without question and the wave theory of ele tro-magnetism required it as Maxwell was aware. However, the continuity postulate does not seem to have played much of a part in the derivation of the equations of Special Relativity  though it did stop Einstein’s successors from thinking in rather different ways about ‘Space/Time’. Despite everything that has happened and the success of Quantum Mechanics and the photo-electric effect and all the rest of it, practically all students of physics think of ‘space’, ‘time’ and electro-magnetism as being ‘continuous’, rather than made up of discrete bits especially since Calculus is primarily concerned with ‘continuous functions’. Since nothing in the physical world is continuous, Calculus is in the main a false model of reality.

Inertial frames, which play such a big role in Special Relativity, as it is currently taught, do not exist in Nature : they are entirely man-made. It was essentially this realisation that motivated Einstein’s decision to try to formulate physics in a way that did not depend on any particular co-ordinate system whatsoever. Einstein assumed relativity and the constancy of the speed of light and independently deduced the Lorentz  transformations. This post would be far too long if I went into the details of Special Relativity (I have done this elsewhere) but, for the sake of the general reader, a brief summary can and should be given. Those who are familiar with Special Relativity can skip this section.

The Lorentz/Einstein Transformations     Ordinary people sometimes find it useful, and physicists find it indispensable, to locate an object inside a real or imaginary  three dimensional box. Then, if one corner of the imaginary box (e.g. room of house, railway carriage &c.) is taken as the Origin, the spot to which everything else is related, we can pinpoint an object by giving its distance from the corner/Origin, either directly or by giving the distance in terms of three directions. That is, we say the object is so many spaces to the right on the ground, so many spaces on the ground at right angles to this, and so many spaces upwards. These are the three co-ordinate axes x, y and z. (They do not actually need to be at right angles but usually they are and we will assume this.)

Also, if we are locating an event rather than an object, we will need a fourth specification, a ‘time’ co-ordinate telling us when such and such an event happened. For example, if a balloon floating around the room at a particular time, to pinpoint the event, it would not be sufficient to give its three spatial co-ordinates, we would need to give the precise time as well. Despite all the hoo-ha, there is nothing in the least strange or paradoxical about us living in a ‘four-dimensional universe’. Of course, we do done  : the only slight problem is that the so-called fourth dimension, time, is rather different from the other three. For one thing, it seems to only have one possible direction instead of two; also the three ‘spatial’ directions are much more intimately connected to each other than they are to the ‘time’ dimension. A single unit serves for the first three, say the metre, but for the fourth we need a completely different unit, the second, and we cannot ‘see’ or ‘touch’ a second whereas we can see and touch a metre rod or ruler.
Now, suppose we have a second ‘box’ moving within the original box and moving in a single direction at a constant speed. We select the x axis for the direction of motion. Now, an event inside the smaller box, say a pistol shot, also takes place within the larger box : one could imagine a man firing from inside the carriage of a train while it has not yet left the station. If we take the corner of the railway carriage to be the origin, clearly the distance from where the shot was fired to the railway carriage origin will be different from the distance from where the buffers train are. In other words, relative to the railway carriage origin, the distance is less than the distance to the buffers. How much less? Well, that depends on the speed of the train as it pulls out. The difference will be the distance the train has covered since it pulled out. If the train pulls out at constant speed 20 metres/second  metres/second and there has been a lapse of, say, 4 seconds, the distance will be  80 metres. More generally, the difference will be vt where t starts at 0 and is counted in seconds. So, supposing relative to the buffers, the distance is x, relative to the railway carriage the distance is v – xt a rather lesser distance.
Everything else, however, remains the same. The time is the same in the railway carriage as what is marked on the station clock. And, if there is only displacement in one dimension, the other co-ordinates don’t change : the shot is fired from a metre above ground level for example in both systems and so many spaces in from the near side in both systems. This all seems commonsensical and, putting this in formal mathematical language, we have the Galilean Transformations so-called

x = x – vt    y  = y    z  – z     t= t 

All well and good and nobody before the dawn of the 20th century gave much more thought to the matter. Newton was somewhat puzzled as to whether there was such a thing as ‘absolute distance’ and ‘absolute time’, hence ‘absolute motion’, and though he believed in all three, he accepted that, in practice, we always had to deal with relative quantities, including speed.
If we consider sound in a fluid medium such as air or water, the ‘speed’ at which the disturbance propagates differs markedly depending on whether you are yourself stationary with respect to the medium or in motion, in a motor-boat for example. Even if you are blind, or close your eyes, you can tell whether a police car is moving towards or away from you by the pitch of the siren, the well-known Doppler effect. The speed of sound is not absolute but depends on the relative motion of the source and the observer. There is something a little unsettling in the idea that an object does not have a single ‘speed’ full stop, but rather a variety of speeds depending on where you are and how you are yourself moving. However, this is not too troublesome.
What about light? In the latter 19th century it was viewed as a disturbance rather like sound that was propagated in an invisible medium, and so it also should have a variable speed depending on one’s own state of motion with respect to this background, the ether. However, no differences could be detected. Various methods were suggested, essentially to make the figures come right, but Einstein cut the Gordian knot once and for all and introduced as an axiom (basic assumption) that the speed of light in a vacuum (‘free empty space’) was fixed and completely independent of the observer’s state of motion. In other words, c, the speed of light, was the same in all co-ordinate systems (provided they were moving at a relative constant speed to each other). This sounded crazy and brought about a completely different set of ‘transformations’, known as the Lorentz Transformations  although Einstein derived them independently from his own first principles. This derivation is given by Einstein himself in the Appendix to his ‘popular’ book “Relativity : The Special and General Theory”, a book which I heartily recommend. Whereas physicists today look down on books which are intelligible to the general reader, Einstein himself who was not a brilliant student at university (he got the lowest physics pass of his year) and was, unlike Newton, not a particularly gifted pure mathematician, took the writing of accessible ‘popular’ books extremely seriously. Einstein is the author of the staggering put-down, “If you cannot state an issue clearly and simply, you probably don’t understand it”.
If we use the Galileian Tranformations and set v = c , the speed of light (or any form of electro-magnetism) in a vacuum, we have x = ct  or with x in metres and t in seconds, x = 3 × 108 metres (approximately) when t = 1 second. Transferring to the other co-ordinate system which is moving at v metres/sec relative to the first, we have  x’  x – vt  and, since t is the same as t, when dividing we obtain for x’ /t ,  (x – vt)/t = ((x/t) – v)  = (c – v), a somewhat smaller speed than c. This is exactly what we would expect if dealing with a phenomenon such as sound in a fluid medium. However, Einstein’s postulate is that, when dealing with light, the ratio distance/time is constant in all inertial frames, i.e. in all real or imaginary ‘boxes’ moving in a single direction with a constant difference in their speeds.

One might doubt whether it is possible to produce ‘transformations’ that do keep c the same for different frames. But it is. We need not bother about the y and z co-ordinates because they are most likely going to stay the same ─ we can arrange to set both at zero if we are just considering an object moving along in one direction. However, the x and t equations are radically changed. In particular, it is not possible to set t = t, meaning that ‘time’ itself (whatever that means) will be modified when we switch to the other frame.           The equations are

         x = γ (x – vt)     t = γ(t – vx/c2)  where γ = (1/(1 – v2/c2)1/2 )

The reader unused to mathematics will find them forbidding and they are indeed rather tiresome to handle though one gets used to them. If you take the ratio If  x /t you will find ─ unless you make a slip ─ that, using the Lorentz Transformations you eventually obtain c as desired.

We have x = ct  or t = x/c  and the Lorentz Transformations

                    x = γ (x – vt)     t = γ(t – vx/c2)  where γ = (1/(1 – v2/c2)1/2 )

Then  x/t  = γ (x – vt)        =   (x – vt)       =    c2(x – vt)
γ
(t – vx/c2)         (t – vx/c2)         (c2t – vx)   

               = c2(x – vt)      =  c2x – cv(ct)
                 
(c2t – vx)            (c(ct) – vx)

                                        =  c2x – cvx)       = (cx)(c – v)
                                            (cx – vx)            x(c – v)                  

                                          =   c

The amazing thing that this is true for any value of v ─ provided it is less than c ─ so it applies to any sort of system moving relative to the original ‘box’, as long as the relative motion is constant and in a straight line. It is true for v = 0 , i.e. the two boxes are not moving relatively to each other : in such a case the complicated Lorentz Transformations reduce to x = x      t = t   and so on.
The Lorentz/Einstein Transformations have several interesting and revealing properties. Though complicated, they do not contain terms in x2 or t2 or higher powers : they are, in mathematical parlance, ‘linear’. This is what we want for systems moving at a steady pace relatively to each other : squares and higher powers rapidly produce erratic increases and a curved trajectory on a space/time graph. Secondly, if v is very small compared to c, the ratio v/c which appears throughout the formulae is negligible since c is so enormous. For normal speeds we do not need to bother about these terms and the Galileian formulae give  satisfactory results.
Finally, and this is possibly the most important feature : the Lorentz/Einstein Transformations are ‘symmetric’. That is, if you work backwards, starting with the ‘primed’ frame and x and t, and convert to the original frame, you end up with a mirror image of the formulae with a single difference, a change of sign in the xto formula denoting motion in the opposite direction (since this time it is the original frame that is moving away). Poincaré was the first to notice this and could have beaten Einstein to the finishing line by enunciating the Principle of Relativity several years earlier ─ but for some reason he didn’t, or couldn’t, make the conceptual leap that Einstein made. The point is that each way of looking at the motion is equally valid, or so Einstein believed, whether we envisage the countryside as moving towards us when we are in the train, or the train moving relative to the static countryside.

Relativity from Ultimate Event Theory?

    Einstein assumed relativity and the constancy of the speed and deduced the Lorentz Transformations : I shall proceed somewhat in the opposite direction and attempt to derive certain well-known features of Special Relativity from basic assumptions of Ultimate Event Theory (UET). What assumptions?

To start with, the Event Number Postulate  which says that
  Between any two  events in an event-chain there are a fixed number of ultimate events. 
And (to recap basic definitions) an ultimate event is an event that cannot be further decomposed — this is why it is called ultimate.
Thus, if the ultimate events in a chain, or subsection of a chain, are numbered 0, 1, 2, 3…….n  there are n intervals. And if the event-chain is ‘regular’, sort of equivalent of an intertial system, the ‘distance’  between any two successive events stays the same. By convention, we can treat the ‘time’ dimension as vertical — though, of course, this is no more than a useful convention.   The ‘vertical’ distance between the first and last ultimate events of a  regular event-chain thus has the value n × ‘vertical’ spacing, or n × t.  Note that whereas the number indicating the quantity of ultimate events and intervals, is fixed in a particular case,  t turns out to be a parameter which, however, has a minimum ‘rest’ value noted t0. This minimal ‘rest’ value is (by hypothesis) the same for all regular event-chains.

….        Likewise, between any two ‘contemporary’ i.e. non-successive, ultimate events there are a fixed number of spots where ultimate events could have (had) occurrence. If there are two or more neighbouring contemporary ultimate events bonded together we speak of an event-conglomerate and, if this conglomerate repeats or gives rise to another conglomerate of the same size, we have a ‘conglomerate event-chain’. (But normally we will just speak of an event-chain).
A conglomerate is termed ‘tight’, and the region it occupies within a single ksana (the minimal temporal interval) is ‘full’ if one could not fit in any more ultimate events (because there are no available spots). And, if all the contemporary ultimate events are aligned, i.e. have a single ‘direction’, and are labelled   0, 1, 2, 3…….n  , then, there are likewise n ‘lateral’ intervals along a single line.

♦        ♦       ♦       ♦       ♦    ………

If the event-conglomerate is ‘regular’, the distance between any two neighbouring events will be the same and, for n events has the value n × ‘lateral’ inter-event spacing, or n × s. Although s, the spacing between contemporary ultimate events must obviously always be greater than the spot occupied by an ultimate event, for all normal circumstances it does not have a minimum. It has, however, a maximum value s0 .

The ‘Space-Time’ Capsule

Each ultimate event is thus enclosed in a four-dimensional ‘space-time capsule’ much, much larger than itself — but not so large that it can accommodate another ultimate event. This ‘space-time capsule’ has the mixed dimension s3t.
In practice, when dealing with ‘straight-line’ motion, it is only necessary to consider a single spatial dimension which can be set as the x axis. The other two dimensions remain unaffected by the motion and retain the ‘rest’ value, s­0.  Thus we only need to be concerned with the ‘space-time’ rectangle st.
We now introduce the Constant Size Postulate

      The extent, or size, of the ‘space-time capsule’ within which an ultimate event can have occurrence (and within which only one can have occurrence) is absolute. This size is completely unaffected by the nature of the ultimate events and their interactions with each other.

           We are talking about the dimensions of the ‘container’ of an ultimate event. The actual region occupied by an ultimate event, while being non-zero, is extremely small compared to the dimensions of the container and may for most purposes be considered negligible, much as we generally do not count the mass of an electron when calculating an atom’s mass. Just as an atom is mainly empty space, a space time capsule is mainly empty ‘space-time’, if the expression is allowed.
Note that the postulate does not state that the ‘shape’ of the container remains constant, or that the two ‘spatial’ and ‘temporal’ dimensions should individually remain constant. It is the extent of the space-time parallelipod’ s3t which remains the same or, in the case of the rectangle it is the product st ,that is fixed, not s and t individually.  All quantities have minimum and maximum values, so let the minimum temporal interval be named  t0 and, Space time Area diagramconversely, let s0 be the maximum value of s. Thus the quantity s0 t0 ,  the ‘area’ of the space-time rectangle, is fixed once and for all even though the temporal and spatial lengths can, and do, vary enormously. We have, in effect a hyperbola where xy = constant but with the difference that the hyperbola is traced out by a series of dots (is not continuous) and does not extend indefinitely in any direction (Note 3).
         This quantity s0 t0  is an extremely important constant, perhaps the most important of all. I would guess that different values of s0 t0   would lead to very different universes. The quantity is mixed so it is tacitly assumed that there is a common unit. What this common unit is, is not clear : it can only be  based on the dimensions of an ultimate event itself, or its precise emplacement (not its container capsule), since K0 , the backdrop or Event Locality does not have a metric, is elastic, indeterminate in extent.
         Although one can, in imagination, associate or combine all sorts of events with each other, only events that are bonded sequentially constitute an event-chain, and only bonded contemporary events remain contemporary in successive ksanas. This ‘bonding’ is not a mathematical fiction but a very real force, indeed the most basic and most important force in the physical universe without which the latter would collapse at any and every moment — or rather at every ksana.
         Now, within a single ksana one and only one ultimate event can have occurrence. However, the ‘length’ of a ksana varies from one event-chain to another since, although the size of the emplacements where the ultimate events occur is (by hypothesis) fixed, the spacing is not fixed, is indeterminate though the same in similar conditions (Note 5). The length of a ksana has a minimum and this minimal length is attained only when an event-chain is at rest, i.e. when it is considered without reference to any other event-chain. This is the equivalent of a ‘proper interval’ in Relativity. So t is a parameter with minimal value t0. It is not clear what the maximum value is though there must be one.
         The inter-space distance s does not have a minimum, or not one that is, in normal conditions ever attained — this minimum would be the exact ‘width’ of the emplacement of an ultimate event, an extremely small distance. It transpires that the inter-space distance s is at a maximum in a rest-chain taking the value s0. I am not absolutely sure whether this needs to be stated as an assumption or whether it can be derived later from the assumptions already made.)

         Thus, the ‘space-time’ paralleliped s3t has the value (s0)3t0 , an absolute value.

The Rest Postulate

This says that

          Every event-chain is at rest with respect to the Event Locality K0 and may be considered to be ‘stationary’.

          Why this postulate and what does it mean? We all have experience of objects immersed in a fluid medium and there can also be events, such as sounds, located in this medium. Now, from experience, it is possible to distinguish between an object ‘at rest’ in a fluid medium such as the ocean and ‘in motion’ relative to this medium. And similarly there will be a clear difference between a series of siren calls or other sounds emitted from a ship in a calm sea, and the same sequence of sounds when the ship is in motion. Essentially, I envisage ultimate events as, in some sense, immersed in an invisible omnipresent ‘medium’, 0, — indeed I envisage ultimate events as being localized disturbances of K0. (But if you don’t like this analogy, you can simply conceive of ultimate events having occurrence on an ‘Event Locality’ whose purpose is simply to allow ultimate events to have occurrence and to keep them separate from one another.) The Rest Postulate simply means that, on the analogy with objects in a fluid medium, there is no friction or drag associated with chains of ultimate events and the medium in or on which they have occurrence. This is basically what Einstein meant when he said that “the ether does not have a physical existence but it does have a geometric existence”.

What’s the point of this constant if no one knows what it is? Firstly, it by no means follows that this constant s0 t0 is unknowable since we can work backwards from experiments using more usual units such as metres and seconds, giving at least an approximate value. I am convinced that the value of s0 t0  will be determined experimentally within the next twenty years, though probably not in my lifetime unfortunately. But even if it cannot be accurately determined, it can still function as a reference point. Galileo was not able to determine the speed of light even approximately with the apparatus at his disposal (though he tried) but this did not stop him stating that this speed was finite and using the limit in his theories without knowing what it was.

Diverging Regular Event-chains

Imagine a whole series of event-chains with the same reappearance rate which diverge from neighbouring spots — ideally which fork off from a single spot. Now, if all of them are regular with the same reappearance rate, the nth member of Event-chain E0 will be ‘contemporaneous’ with the nth members of all the other chains, i.e. they will have occurrence within the same ksana. Imagine them spaced out so that each nth ultimate event of each chain is as close as possible to the neighbouring chains. Thus, we imagine E0 as a vertical column of dots (not a continuous vertical line) and E1 a slanting line next to it, then E2 and so on. The first event of each of these chains (not counting the original event common to all) will thus be displaced by a single ‘grid-space’ and there will be no room for any events to have occurrence in between. The ‘speed’ or displacement distance of each event-chain relative to the first (vertical one) is thus lateral distance in successive ksanas/vertical distance in successive ksanas.  For a ‘regular’ event-chain the ‘slant’ or speed remains the same and is tan θ   =  1 s/t0 , 2 s/t0  and so on where, if θ is the slant angle,

tan θr  = vr  = 1, 2, 3, 4……   ­­

“What,” asked Zeno of Elea “is the speed of a particular chariot in a chariot race?”  Clearly, this depends on what your reference body is. We usually take the stadium as the reference body but the charioteer himself perceives the spectators as moving towards or away from him and he is much more concerned about his speed relative to that of his nearest competitor than to his speed relative to arena. We have grown used to the idea that ‘speed’ is relative, counter-intuitive though it appears at first.
But ‘distance’ is a man-made convenience as well : it is not an ‘absolute’ feature of reality. People were extremely put out by the idea that lengths and time intervals could be ‘relative’ when the concept was first proposed but scientists have ‘relatively’ got used to the idea. But everything seems to be slipping away — is there anything at all that is absolute, anything at all that is real? Ultimate Event Theory evolved from my attempts to ponder this question.
The answer is, as far as I am concerned, yes. To start with, there are such things as events and there is a Locality where events occur. Most people would go along with that. But it is also one of the postulates of UET that every macroscopic ‘event’ is composed of a specific number of ultimate events which cannot be further decomposed. Also, it is postulated that certain ultimate events are strongly bonded together into event-chains temporally and event-conglomerates laterally. There is a bonding force, causality.
Also, associated with every event chain is its Event Number, the number of ultimate events between the first event A and the last Z. This number is not relative but absolute. Unlike speed, it does not change as the event-chain is perceived in relation to different bodies or frames of reference. Every ultimate event is precisely localised and there are only a certain number of ultimate events that can be interposed between two events both ‘laterally’ (spatially) and ‘vertically’ (temporally). Finally, the size of the ‘space-time capsule’ is fixed once and for all. And there is also a maximum ‘space/time displacement ratio’ for all event-chains.
This is quite a lot of absolutes. But the distance between ultimate events is a variable since, although the dimensions of each ultimate event are fixed, the spacing is not fixed though it will remain the same within a so-called ‘regular’ event-chain.
It is important to realize that the ‘time’ dimension, the temporal interval measured in ksanas, is not connected up to any of the three spatial dimensions whereas each of the three spatial dimensions is connected directly to the other two. It is customary to take the time dimension as vertical and there is a temptation to think of t, time, being ‘in the same direction’ as the z axis in a normal co-ordinate system. But this is not so : the time dimension is not in any spatial direction but is conceived as being orthogonal (at right angles) to the whole lot. To be closer to reality, instead of having a printed diagram on the page, i.e. in two dimensions, we should have a three dimensional optical set-up which flashes on and off at rhythmic intervals and the trajectory of a ‘particle’ (repeating event-chain) would be presented as a repeating pinpoint of light in a different colour.
Supposing we have a repeating regular event-chain consisting for simplicity of just one ultimate event. We [resent it as a column of dots, i.e. conceive of it as vertical though it is not. The dots are numbered 0, 1, 2….    and the vertical spacing does not change (since this is a regular event-chain) and is set at  t0 since this is a ‘rest chain’.  Similar regular event-chains can then be presented as slanting lines to the right (or left) regularly spaced along the x axis. The slant of the line represents the ‘speed’. Unlike the treatment in Calculus and conventional physics, increasing v does not ‘pass through a continuous set of values’, it can only move up by a single ‘lateral’ space each time. The speeds of the different event-chains are thus 0s/t0  (= 0) ;  1s/t0 ;
2s/t0 ; 
 3s/t0 ;  4s/t0 ;……  n s/t0 and so on right up to  c s/t0 .  But to what do we relate the spacing s ?  To the ‘vertical’ event-chain or to slanting one? We must relate s to the event-chain under consideration so that its value depends on v so v =  v sv    The ratio  s/t0 is thus a mixed ratio sv/t0 .   tv  gives the intervals between successive events in the ‘moving’ event-chains and the number of these intervals does not increase because there are only a fixed number of events in any event-chain evaluated in any way. These temporal intervals thus undoubtedly increase because the hypotenuse gets larger. What about the spacing along the horizontal ? Does it also increase? Stay the same?  If we now introduce the Constant Size Postulate which says that the product  sv  tv  = s0 t0    we find that   sv  decreases with increasing v since tv  certainly increases. There is thus an inverse ratio and one consequence of this is that the mixed ratio sv/t0 = s0/tv    and we get symmetry. This leads to relativity whereas any other relation does not and we would have to specify which regular event-chain ‘really’ is the vertical one. One can legitimately ask which is the ‘real’ spatial distance between neighbouring events? The answer is that every distance is real and not just real for a particular observer. Most phenomena are not observed at all but they still occur and the distances between these events are real : we as it were take our pick, or more usually one distance is imposed on us.

Now the real pay off is that each of these regular event-chains with different speeds v is an equally valid description of the same event-chain. Each of these varying descriptions is true even though the time intervals and distances vary. This is possible because the important thing, what really matters, does not change : in any event-chain the number and order of the individual events is fixed once and for all although the distances and times are not. Rosser, in his excellent book Introductory Relativity, when discussing such issues gives the useful analogy of a gamer of tennis being played on a cruise liner in calm weather. The game would proceed much as on land, and if in a covered court, exactly as on land. And yet the ‘speed’ of the ball varies depending on whether you are a traveller on the boat or someone watching with a telescope from another boat or from land. The ‘real’ speed doesn’t actually matter, or, as I prefer to put it, is indeterminate though fixed within a particular inertial frame (event system). Taking this one step further, not just the relative speed but the spacing between the events of a regular  event-chain  ‘doesn’t matter’ because the constituent events are the same and appear in the same order. It is interesting that. on this interpretation, a certain indeterminacy with regard to distance is already making its appearance before Quantum Theory has even been mentioned. 

Which distance or time interval to choose?

Since, apparently, the situation between regular event-chains is symmetric (or between inertial systems if you like) one might legitimately wonder how there ever could be any observed discrepancy since any set of measurements a hypothetical observer can make within his own frame (repeating event system) will be entirely consistent and unambiguous. In Ultimate Event Theory, the situation is, in a sense, worse since I am saying that, whether or not there is or can be an observer present, the time-distance set-up is ‘indeterminate’ — though the number and order of events in the chain is not. Any old ‘speed’ will do provided it is less than the limiting value c. So this would seem to make the issue quite academic and there would be no need to learn about Relativity. The answer is that this would indeed be the case if we as observers and measurers or simply inhabitants of an event-environment could move from one ‘frame’ to another effortlessly and make our observations how and where we choose. But we can’t : we are stuck in our repeating event-environment constituted by the Earth and are at rest within it, at least when making our observations. We are stuck with the distance and time units of the laboratory/Earth event-chain and cannot make observations using the units of the electron event-chain (except in imagination). Our set of observations is fully a part of our system and the units are imposed on us. And this does make a difference, a discernible, observable difference when dealing with certain fast-moving objects.
Take the µ-meson. µ-mesons are produced by cosmic rays in the upper reaches of the atmosphere and are normally extremely short-lived, about  2.2 × 10–6 sec.  This is the (average) ‘proper’ time, i.e.  when the µ-meson is at rest — in my terms it would be N × t0 ksanas. Now, these mesons would, using this t value, hardly go more than 660 metres even if they were falling with the speed of light (Note 4). But a substantial portion actually reach sea level which seems impossible. Now, we have two systems, the meson event-chain which flashes on and off N times whatever N is before terminating, i.e. not reappearing. Its own ‘units’ are t0 and s0 since it is certainly at rest with itself. For the meson, the Earth and the lower atmosphere is rushing up with something approaching the limiting speed towards it. We are inside the Earth system and use Earth units : we cannot make observations within the meson. The time intervals of the meson’s existence are, from our rest perspective, distended : there are exactly the same number of ksanas for us as for the meson but, from our point of view, the meson is in motion and each ‘motion’ ksana is longer, in this case much much  longer. It thus ‘lives’ longer, if by living longer we mean having a longer time span in a vague absolute way,  rather than having more ‘moments of existence’. The meson’s ksana is worth, say, eight of our ksanas. But the first and last ultimate event of the meson’s existence are events in both ‘frames’, in ours as well as its. And if we suppose that each time it flashed into existence there was a (slightly delayed) flash in our event-chain, the flashes would be much more spaced out and so would be the effects. So we would ‘observe’, say, a duration of, say, eight of ‘our’ ksanas between consecutive flashings instead of one. And the spatial distance between flashes would also be evaluated in our system of metres and kilometres : this is imposed on us since we cannot measure what is going on within the meson event-chain. The meson actually would travel a good deal further in our system — not ‘would appear to travel farther’. Calculations show that it is well within the meson’s capacity to reach sea level (see full discussion in Rosser, Introductory Relativity pp. 71-3).
What about if we envisaged things from the perspective of the meson? Supposing, just supposing, we could transfer to the meson event-chain or its immediate environment and could remember what things were like in the world outside, the familiar Earth event-frame. We would notice nothing strange about ‘time’, the intervals between ultimate events, or the brain’s merging of them, would not surprise us at all. We would consider ourselves to be at rest. What about if we looked out of the window at the Earth’s atmosphere speeding by? Not only would we recognize that there was relative motion but, supposing there were clear enough landmarks (skymarks rather), the distances between these marks would appear to be far closer than expected — in effect there would be a double or triple sense of motion since our perception of motion is itself based on estimates of distance. As the books say, the Earth and its atmosphere would be ‘Lorentz contracted’. There would be exactly the same number of ultimate events in the meson’s trajectory, temporarily our trajectory also. The first and last event of the meson’s lifetime would be separated by the same number of temporal intervals and if these first and last events left marks on the outside system, these marks would also be separated by exactly the same number of spatial intervals. Only these spatial intervals — distances — would be smaller. This would very definitely be observed : it is as if we were looking out at the countryside on a familiar journey in a train fantastically speeded up. We would still consider ourselves at rest but what we saw out of the window would be ludicrously foreshortened and for this reason we would conclude that we were travelling a good deal faster than on our habitual journey. I do not think there would be any obvious way to recognize the time dilation of the outside system.

One is often tempted to think that the time dilation and the spatial contraction cancel each other out so all this talk of relativity is purely academic since any discrepancies should cancel out. This would indeed be the case if we were able to make our observations inside the event-chain we are observing, but we make the measurements (or perceptions) in a single frame. Although it is the meson event-chain that is dictating what is happening, both the time and spatial distance observations are made in our system. It is indeed only because of this that there is so much talk about ‘observers’ in Special Relativity. The point is not that some intelligent being observes something because usually he or she doesn’t : the point is that the fact of observation, i.e. the interaction with another system seriously confuses the issue. The ‘rest-motion’ situation is symmetrical but the ‘observing’ situation is not symmetrical, nor can it be in such circumstances.

This raises an important point.  In Ultimate Event Theory, as in Relativity, the situation is ‘kinematically’ symmetrical. But is it causally symmetrical? Although Einstein stressed that c was a limit to the “transfer of causality”  he was more concerned with light and electro-magnetism than causality. UET is concerned above all with causality — I have not mentioned the speed of light yet and don’t need to. In situations of this type, it is essential to clearly identify the primary causal chain. This is obviously the meson : we observe it, or rather we pick up indications of its flashings into and out of existence. The observations we make, or simply perceptions,  are dependent on the meson, they do not by themselves constitute a causal chain. So it looks at first sight as if we have a fundamental asymmetry : the meson event-chain is the controlling one and the Earth/observer event chain  is essentially passive. This is how things first appeared to me. But on reflection I am not so sure. In many (all?) cases of ‘observation’ there is interaction with the system being observed and it is inevitably going to be affected by this even if it has no senses or observing apparatus of its own. One could thus argue that there is causal symmetry after all, at least in some cases. There is thus a kind of ‘uncertainty principle’ due to the  interaction of two systems latent in Relativity before even Quantum Mechanics had been formulated. This issue and the related one of the limiting speed of transmission of causality will be dealt with in the subsequent post.

Sebastian Hayes  26 July
Note 1. And in point of fact, if General Relativity is to be believed, ‘free space’ is not strictly homogeneous even when empty of matter and neither is the speed of light strictly constant since light rays deviate from a straight path in the neighbourhood of massive bodies.

Note 2  For those people like me who wish to believe in the reality of 0 — rather than seeing it as a mere mathematical convenience like a co-ordinate system —  the lack of any ‘friction’ between the medium or backdrop and the events or foreground would, I think. be quite unobjectionable, even ‘obvious’, likewise the entire lack of any ‘normal’ metrical properties such as distance. The ‘backdrop’, that which lies ‘behind’ material reality though in some sense permeating it, is not physical and hence is not obliged to possess familiar properties such as a shape, a metric, a fixed distance between two points and so on. Nevertheless, this backdrop is not completely devoid of properties : it does have the capacity to receive (or produce) ultimate events and to keep them separate which involves a rudimentary type of ‘geometry’ (or topology). Later, as we shall see, it would seem that it is affected by the material points on it, so that this ‘geometry’, or ‘topology’, is changed, and so, in turn,  affects the subsequent patterning of events. And so it goes on in a vicious or creative circler, or rather spiral.
            The relation between K0, the underlying substratum or omnipresent medium, and the network of ultimate events we call the physical universe, K1  is somewhat analogous to the distinction between nirvana and samsara in Hinayana Buddhism. Nirvana  is completely still and is totally non-metrical, indeed non-everything (except non-real), whereas samsara is turbulence and is the world of measure and distancing. It is alma de-peruda, the ‘domain of separation’, as the Zohar puts it.  The physical world is ruled by causality, or karma, whereas nirvana is precisely the extinction of karma, the end of causality and the end of measurement.

Note 3   The ‘Space-time hyperbola’ , as stated, does not extend indefinitely either along the ‘space’ axis s (equivalent of x) or indefinitely upwards Space time hyperbolaalong the ‘time’ axis (equivalent of y).  — at any rate for the purposes of then present discussion. The variable t has a minimum t0   and the variable s a maximum s0  which one suspects is very much greater than  tc  .  Since there is an upper limit to the speed of propagation of a causal influence, c , there will in practice be no values of t greater than tc  and no s values smaller than sc  .   It thus seems appropriate to start marking off the s axis at the smallest value sc  =   s0/ c  which can function as the basic unit of distance.  Then s0 is equal to c of these ‘units’. We thus have a hyperbola something like this — except that the curve should consist of a string of separate dots which, for convenience I have run together.

Note 4  See Rosser, Introductory Relativity pp. 70-73. Incidentally, I cannot recommend too highly this book.

Note 5   I have not completely decided whether it is the ‘containers’ of ultimate events that are elastic, indeterminate, or the ‘space’ between the containers (which have the ultimate events inside them)’. I am inclined to think that there really are temporal gaps not just between ultimate events themselves but even between their containers, whereas this is probably not so in the case of spatial proximity. This may be one of the reasons, perhaps even the principal reason, why ‘time’ is felt to be a very different ‘dimension’. Intuitively, or rather experientially, we ‘feel’ time to be different from space and all the talk about the ‘Space/Time continuum’ — a very misleading phrase — is not enough to dispel this feeling.

To be continued  SH  18 July 2013

 

The Theory of Special Relativity is based on two simple postulates, that “1. the laws of physics take the same form in all inertial frames” and “2. the observed speed of light in a vacuum is constant for all (inertial) observers irrespective of their relative motion”. I shan’t say much about the first postulate now or define an ‘inertial frame’ — basically a ‘frame’ where you can’t say whether you’re moving or not except by looking out of the window — but we need to look at the second.
It is important to realize that (2) is an extremely surprising claim. The speed of a train, for example, is by no means the same for all observers : for the person inside the train the speed is essentially zero since he/she considers himself quite rightly to be at rest unless there is a sudden jolt, but for someone standing alongside the track the speed of the train is, say, 120 miles an hour.  And for an observer in a spacecraft navigating the Earth it is different again (Note 1). Normally, we add speeds together and, if I rolled a marble along the corridor of an unaccelerating train in the direction of travel, the marble’s speed, judged by someone outside would be its speed in the train plus the speed of the train. How is it possible for light to have a constant recorded speed whether the emitter is in a spaceship receding from you or in your own train or spacecraft?
According to Ultimate Event Theory, light, like everything that “has occurrence” is composed of a finite number of ultimate events (Axiom of Finitude). Suppose simply for the sake of argument that the ‘reappearance rate’ of a photon (a specific type of repeating ultimate event) is 1 space/ksana (Note 2). We can represent this by
The blue block represents a repeating event that (rightly or wrongly) we consider to be ‘stationary’from ksana to ksana. My position from ksana to ksana is given by the green blocks and I consider myself to be drifting eastwards away from the blue blocks by one grid-position at each ksana, or, more likely would consider the blue blocks to be drifting steadily away from me westwards. The red blocks represent the positions of some other repeating event that I judge to be moving steadily away from me at a rate of 1 grid-position per ksana. Note that all three coloured blocks joined up give straight lines (they are, in traditional parlance, inertial systems). From the standpoint of the blue blocks, which arbitrarily we take as our ‘landmark sequence’, both the green and red event-chains are moving steadily to the right and the red ‘event-chain’ is moving ‘faster’ since it has a shallower gradient. The ‘speed’ (reappearance rate) of the red line can be calculated by noting the speed of the green blocks relative to the blue and adding on the speed of the green relative to the green. Whereas the green blocks are gaining an extra space each ksana, the red are gaining rather more but the increase (acceleration) is regular. All this is what one would expect.
However, according to Einstein’s Theory of Special Relativity, if light is emitted from the green blocks and the red simultaneously (i.e. within the same ksana), when we eventually pick up the signals at the blue block, compare distances and so on, we do not judge the speed of the light ray from red to be any different from the speed of the light ray from green.  This is extremely unexpected  but will have to be accepted, not because modern physics textbooks say this is so, but because countless actual experiments have (allegedly) failed to detect any difference in the observed speed of light irrespective of the relative movement of the source. Instruments have measured the speed of a light beam projected from an aircraft moving towards the observer and the speed of a light beam projected backwards from the tail of an aircraft moving away ─ and there is no appreciable difference (within experimental error). To see how astonishing this is, imagine a fighter aircraft gunning you down : if it is travelling towards you, the bullets will hit you rather sooner than if you were both travelling at around the same speed. And if the fighter aircraft is moving away from you faster than the ‘muzzle velocity’ of the machine-gun, the bullets from the tail-gun will never reach you at all! Light clearly behaves unlike material objects.
Assuming that Einstein’s prediction about the observed speed of light is substantially correct (which I believe), how can this anomaly be explained in terms of Ultimate Event Theory?  Certainly, there is nothing in my preliminary postulates or my original ‘universe model’ that would lead me to expect anything of the kind, quite the reverse. Since everything that has occurrence is composed of a finite number of ultimate events (the Axiom of Finitude) any and every apparently continuous burst of light is made up of so many individual ‘photonic events’. And the number of these events between two recognizable end-points is fixed once and for all. Also, I absolutely refuse to countenance the notion that the occurrence or not of an ultimate event depends on my personal state of motion or anything else pertaining to me since I consider this the worst kind of subjectivism. If we accept this, we have the absurd consequence that all sorts of things can be conjured into existence just by jumping into a train or a spaceship while they simply never happen at all for someone left behind on the ground !
It is true that I could account for the observed constancy of the photonic event-chain we call light by making the ultimate events themselves larger or smaller according to the relative motion of the observer and observed. But once again I am very reluctant to do this since the advantage of having truly elementary entities is that they have a minumum of attributes and these attributes (such as size) are fixed, are ‘absolute’. It would be equivalent to making the size or charge of a proton changeable in differing situations in ordwer to make certain observations come out right, something one would only wish to do if there was no alternative. The merit of the basic assumptions of Ultimate Event Theory is that they provide a comprehensible, simple framework (or so I would claim) and certainly the simplest and most reasonable assumption is to suppose that all ultimate events are of fixed size (supposing it makes any sense to talk of their having a size) and likewise that the positions available on the Locality are also of fixed size. And finally, for reasons of simplicity and also perhaps aesthetics, I insist on the ‘ksana’, the ‘temporal’ dimension of every event block  as being of fixed size.
If I were stuck with a strictly continuous model of reality, I would now be in an impossible situation. But my Event Locality — which the reader may envisage as, very roughly, the equivalent of ‘Space/Time’ in normal physics — is radically discontinuous, that is, there are gaps. The Locality is not a continuum but a connected dis-continuum, at any rate that section of it that is available to ultimate events. To make Ultimate Event Theory square with Special relativity (which I certainly consider desirable) the only possibility is to consider the ‘gaps’ between events, i.e. the ‘interval’ between co-existing grid-positions and also between successive grid-positions (i.e. between ksanas) as being ‘elastic’, ‘flexible’. These gaps are ‘non-metrical’, have no objective fixed extent and may thus function differently in different event-chains, or rather the same event-chain envisaged from a different perspective (Note 3).

Now, it is possible to maintain the same gradient in the diagram by adjusting the lateral and vertical spacings. Suppose I increase the drift to the right of the red square to represent an increase in speed of the spaceship as perceived by me.  Instead of the original speed of ‘one space to right per ksana’ we have, say, ‘two spaces/ksana’   i.e. we go from

 

 

 

 

However, if I compensate by spacing out the rows, representing the situation at successive ksanas, we have something more likespeed visuals final

The increased gap between rows, i.e. between successive ksanas,  corresponds to the famous ‘time dilation’ of Special Relativity.
There is, however, still an ‘extra space’ between the red squares in any row, a space which,  by hypothesis cannot be filled ─ since, if so, we would have something travelling faster than light which (according to Einstein) cannot occur. If we want to keep the ‘one space per ksana’ as the maximum ‘speed’ (reappearance rhythm) we can adjust matters by ‘spacing out’ the grid-positions within each ksana, in effect by suppressing the extra black square. This gives something like

c visualwhere the diagonal line red squares has roughly the same slant as in my original diagram ─ the difference is due to the deficiencies of my computer graphics. Spacing out the black squares (which correspond to possible locations of ultimate events) is equivalent to a ‘space contraction’, also a standard alleged effect of Special Relativity.
It must be stressed that there is a significant difference between this model and that of Special Relativity, at least as commonly understood. While the ‘length’ and ‘duration’ of objects (event conglomerates) or trajectories (event chains) are, as in SR, dependent on relative states of motion (reappearance rates), the number of ultimate events in any event chain is not relative but is ‘absolute’. Every trajectory between two marker events will have associated with it an ‘Ultimate Event Number’ which is completely independent of states of motion or material cosntituents or anything else you like to mention. We will not normally know this number — though we will perhaps one day be able to make an informed guess much as we can make an informed guess as to the number of molecules in a given piece of chalk — but it suffices to know that (according to the postulates of UET) this number exists and is unchangeable. I have enshrined this in one of the fundamental assumptions of the theory, the Axiom of Occurrence, “Once an ultimate event has occurrence, there is no way in which it can be altered or prevented from having occurrence : its occurrence is absolute.”
It is not yet entirely clear to me what consequences this principle would have in actual physical situations. It would mean, for example, that the ‘event number’ for the voyage of the twin who goes off on a trip at nearly the speed of light would be the same for both brothers : simply travelling around is not going to conjure into existence events which do not exist for the stay at home brother. If the twin is indeed ‘younger’ when he returns (as Special relativity predicts) this can only be because the gaps between the two twins’ biological events such as heart beats are relatively shorter or longer. Of course, no such experiment could ever be carried out and the occurrence is not in fact covered by the theory of Special Relativity since accelerations are involved when the space traveller takes off, turns round and lands. However, there may be a way to test the independence of the event number in cases of the decay of particles entering the Earth’s orbit, the usual example given of differing time scales because of SR.        SH  26/11/12

___________________________________________

Note 1  Zeno of Elea pointed out the relativity of motion in his Paradox of the Chariot. What, he asked, was the ‘true’ speed of a chariot in a chariot race? This differed according to whether you adopt the standpoint of the spectator in the stand or that of the different charioteers in the race. By this thought experiment, Zeno seems to have been attempting to show that there was no such thing as ‘absolute motion’ since the perceived motion depended on the observer’s own state of motion. Newton was deeply bothered by the problem and came to the strange sounding conclusion that ‘absolute motion’ could only mean the motion an object had “relative to the fixed stars”. But today we know that the position of the stars is not at all fixed because of expansion, galactic rotation and so on.

Note 2 This ‘speed’ is, I must emphasize, purely illustrative. The actual speed or rather ‘reappearance rate’ of a photonic event chain would be far, far greater than this : a photonic event would have to shift billions of grid positions to the right or left from ksana to ksana relative to a ‘stationary’ event-chain. It would be interesting to know if there is an event chain whose reappearance rate is exactly 1 space/ksana. This is, incidentally, not the slowest possible rate since, as will be discussed subsequently, I envisage reappearance rates where, during many ksanas, the event does not repeat at all. For example, there could be a reappearance rate of  1 space/7 ksanas or 1 space/100 ksanas and so on. This could be expressed as a reappearance rate of “1/7 spaces per ksana” but this would give the unwary the wrong impression : neither grid positions not ksanas can be subdivided and that is that.

Note 3  This solution would seem to be closest to the spirit of the Special Theory of Relativity. Einstein and his followers continually emphasize that an observer within a given inertial frame would notice nothing untoward : he or she would consider himself to be at rest and the other inertial frame to be ‘moving’. There is only ever a problem when, at a later date, the two observers, one within a given frame and one outside it and in a second inertial frame, confront each other with their meticulous observations. In my terms, each observation is ‘correct’ for the individual concerned because the gaps  between events “have no intrinsic length” and thus may legitimately ‘vary’ according to the standpoint adopted. Are these discrepancies ‘real’ or sim,ply how things appear? There is general agreement that the viewpoint of any and every ‘inertial observer’ is equally legitimate :“there is no truth of the matter” as Martin Gardner put it. I am not sure that this answer is sufficient but I cannot improve on it : I ‘resolve’ the problem by simply positing that the Locality is non-metrical and so all sorts of different metrics can be legitimately ascribed to it provided we keep to the chosen metric.
But what is there between ultimate events? Just the emptiness between adjacent grid-positions. This may remind some readers of the so-called ‘ether’ in which all 19th century physicists believed. It is commonly stated that Einstein ‘did away with the ether’ but this is not strictly true. In a quote that unfortunately I cannot at present trace, he said that “the ether has no physical properties but does have geometrical properties”. By this one should understand that the background ether does not, for example, offer any noticeable resistance to the passage of bodies through it but can (and does) affect space-time the direction of trajectories. After banning mention of the ether for over sixty or so years, the ‘ether’ is well and truly back in physics again, re-baptised the vacuum and far from being empty it is vibrant with quantum energy.  “The modern conception of the vacuum is one of a seething ferment of quantum field activity, with waves surging randomly this way and that. In quantum mechanics waves also have characteristics of particles — photons for the electro-magnetic field, gravitons for the gravitational field and so on — popping out of nowhere and disappearing again. Wave or particle, what one gets is a picture of the vacuum that is reminiscent , in some respects of the ether. It does not provide a special frame of rest against which bodies may be said to move, but it does fill all of space and have measurable physical properties such as energy density and pressure.”    Paul Davies, article NS  19 Nov 2011


Newton’s Third Law states rather cryptically that

“To every action there is an opposite and equal reaction.”

This law is the most misunderstood (though probably most employed) Law of the three since it suggests at first sight that everything is in a permanent deadlock !   Writers of Mechanics textbooks hasten to point out that the action and reaction apply to  two different bodies, call them body A and body B.  The Third Law claims that the force exerted by body A on body B is met by an equivalent force, equal in magnitude but opposite in direction, which body B exerts on body A.
Does this get round the problem? Not entirely. The schoolboy or schoolgirl who somehow feels uneasy with the Third Law is on to something. What is either completely left out of the description, or not sufficiently  emphasized by physics and mechanics textbooks, is the timing of the occurrences. It is my push against the wall that is the prior occurrence, and the push back from the wall is a re-action. Without my decision to strike the wall, this ‘reaction’ would never have come about. What in fact is happening at the molecular level is that the molecules of the wall have been squeezed together by my blow and it is their attempt to recover their original conformation that causes the compression in my hand, or in certain other circumstances, pushes me away from the wall altogether. (The ‘pain’ I feel is a warning message sent to the brain to warn it/me that something is amiss.) The reaction of the wall is a restoring force and its effectiveness depends on the elasticity or plasticity of the material substance from which the wall is made — if the ‘wall’ is made of putty I feel practically nothing at all but my hand remains embedded in the wall. As a reliable author puts it, “The force acting on a particle should always be thought of as the cause and the associated  change of momentum as the effect” (Heading, Mathematical methods in Science and Engineering).
In cases where the two bodies remain in contact, a lengthy toing and froing goes on until both sides subside into equilibrium (Note 1). For the reaction of the wall becomes the action in the subsequent cause/event pair, with the subsequent painful compression of the tissues in my hand being the result. It is essential to realize that we are in the presence not of ‘simultaneous’ events, but of a clearly differentiated event-chain involving two ‘objects’  namely the wall and my hand. It is this failure to distinguish between cause and effect, action and reaction, that gives rise to the conceptual muddle concerning  centrifugal ‘force’. It is a matter of common experience that if objects are whirled around but restrained from flying away altogether, they seem to keep to the circumference of an imaginary circle — in the case of s spin dryer, the clothes press themselves against the inside wall of the cylinder while a conker attached to my finger by a piece of string follows a roughly circular path with my finger as centre (only roughly because gravity and air pressure deform the trajectory from that of a perfect circle) . At first sight, it would seem, then, that there is a ‘force’ at work pushing the clothes or the conker outwards  since the position of the clothes on the inside surface of the dryer or of the conker some distance away from my finger is not their ‘normal’ position. However, the centrifugal ‘force’ (from Latin fugo ‘I flee’) is not something applied to the clothes or the conker but is entirely a response to the initiating centripetal force (from Latin peto ‘I seek’) without which it would never have come into existence. The centrifugal ‘force’ is thus entirely secondary in this action/reaction couple and, for this reason, is often referred to as a ‘fictitious’ force — though this is somewhat misleading since the effects are there for all to see, or rather to  feel.
Newton does in certain passages make it clear that there is a definite sequence of events but in other passages he is ambivalent because, as he fully realized, according to his assumptions, gravitational influences seemed to propagate themselves over immense distances instantaneously (and in both directions) — which seemed extremely far-fetched and was one reason why continental scientists rejected the theory of gravitational attraction. Leaving gravity aside since it is ‘action at a distance’, what we can say is that in cases of direct contact, there really is an explicit, and often visible, sequencing of events. In the well-known Ball with Two Strings experiment (Note 2) we have a heavy lead ball suspended from the ceiling by a cotton thread with a second thread hanging underneath the ball. Where will the thread break? According to Newton’s Laws it should break just underneath the ceiling since the upper thread has to support the weight of the ball as well as responding to my tug. However, if you pull smartly enough the lower thread will break first and the ball will stay suspended. Why is this? Simply because there is not ‘time enough’ for my pull to be transmitted right up through the ‘ball plus thread’ system to the ceiling and call forth a reaction there. And, if it is objected that this is a somewhat untypical case because there is a substantial speed of transmission involved, an even more dramatic demonstration is given by high speed photographs of a golf club striking a ball. We can actually see the ball still in contact with the club massively deformed in shape and it is the ball’s recovery of its original configuration (the reaction) that propels it into the air. As someone said, all (mechanical) propulsion is ‘reaction propulsion’, not just that of jet planes.
In Ultimate Event Theory the strict sequencing of events, which is only implicit in Newtonian mechanics, becomes explicit.  If we leave aside for the moment the question of ‘how far’ a ksana extends (Note 3), it  is possible to give a clearcut definition of simultaneous (ultimate) events : Any two events are simultaneous if they have occurrence within the same ksana. A ‘ksana’ (roughly ‘instant’) is a three-dimensional ‘slice’ of the Locality and, within this slice, everything is still because there is, if you like, noit enough ‘time’ for anything to change. Consequently, an ultimate event which has occurrence within or at a particular ksana cannot possibly influence another event having occurrence within this same ksana : any effect it may have on other event-chains will have to wait until at least the next ksana. The entire chain of cause and effect is thus strictly consecutive (cases of apparent ‘causal reversal’ will be considered later.) In effect when bodies are in contact there is a ceaseless toing and froing, sort of ‘passing the buck’ from one side to the other, until friction and other forces eventually dampen down the activity to almost nothing (while not entirely destroying it).
S.H. 21/08/12

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Note 1 Complete static equilibrium does not and cannot exist since what we call ‘matter’ is always in a state of vibration and bodies in contact affect each other even when apparently completely motionless. What can and does exist, however. is a ‘steady state’ when the variations in pressure of two bodies in contact more or less cancel each other out over time (over a number of ksanas). We are, in chemistry, familiar with the notion that two fluids in solution are never equally mixed and that, for example, oxidation and reduction reactions take place continually; when we say a fluid is ‘in equilibrium’ we do not mean that no chemical reactions are taking place but that the changes more or less equal out over a certain period of time. The same applies to solid bodies in contact though the departures from the mean are not so obvious. Although it is practical to divide mechanics into statics and dynamics, there is in reality no hard and fast division.

Note 2  I am indebted to Den Hartog for pointing this out in his excellent book Mechanics (Dover 1948).

Note 3  It is not yet the moment — or maybe I should say ksana — to see how Ultimate Event Theory squares with Relativity : it is hard enough seeing how it squares with Newtonian Mechanics. However, this issue will absolutely be tackled head on at a later date. Einstein, in his 1905 paper, threw a sapnner in the works by querying the then current understanding of ‘simultaneity’ and physics has hardly recovered since. In his latter days, Einstein adhered to the belief that everything takes place in an eternal present so that what is ‘going to’ happen has, in a sense, already been — in my terms already has occurrence on the Locality. I am extremely reluctant to accept such a theory which flies in the face of all our perceptions and would sap our will to live (mine at any rate). On the other hand, it would, I think, be fantastic to consider a single ksana (instant) stretching out across the known universe so that, in principle, all events are either ‘within’ this same ksana or within a previous one. At the moment I am inclined to think there is a sort of mosaic of ‘space/time’ regions and it is only within a particular circumscribed region that we can talk meaningfully of (ultimate) events having occurrence within or at the same ksana. Nonetheless, if you give up sequencing, you give up causality and this is to give up far too much. As Keith Devlin wrote, “It seems to me that there is nothing for it but to take as fundamental the relationh of one event causing another” (Devlin, Logic and Information p. 184)

 

In daily life we do not use co-ordinate systems unless we are engineers or scientists and even they do not use them outside the laboratory or factory. If we wish to be passed a certain book or utensil, we do not say it has x, y and z co-ordinates of (3, 5, 7)  metres relative to the left hand bottom corner of the room ― anyone who behaved in such a way would be considered half-mad. We specify the position of an object by saying it is “on the table”, “below the sink”, “near the Church”, “to the right of the Post Office” and so on. As Bohm pointed out in an interview, these are, mathematically speaking, topological concepts since they do not involve distance or angles. In practice, in our daily life, we define an object’s position by referring it to some prominent object or objects whose position(s) we do know. Aborigines and other roving peoples start off by referring their position to a well-known landmark visible for miles around and refer subsequent focal points to it, in effect using a movable origin or set of origins. In this way one advances  step by step from the known to the unknown instead of plunging immediately into the unknown as we do when we refer everything to a ‘point’ like the centre of the Earth, something of which we have no experience and never will have. We do much the same when directing someone to an object in a room : we relate a hidden or not easily visible object by referring to large objects whose localization is well-known, is imprinted permanently on our mental map, such as a particular table, chair, sink and so on. Even when we do not know the exact localization of the object, a general indication will at least tell us where to look ― “It is on the floor”. Such a simple and informative (but inexact) statement would be nearly impossible to put into mathematical/scientific language precisely because the latter is exact, too exact for everyday use.
I have gone into this at some length because it is important to bear in mind how unnatural scientific and mathematical co-ordinate systems are. Such systems, like so much else in an ‘advanced’ culture, are patterns that we impose on natural phenomena for our convenience and which have no  independent existence whatsoever (though scientists are rather loath to admit this). So why bother with them ? Well, for a long time humanity did not bother with such things, getting along perfectly well with more rough and ready but also more user-friendly systems like the local reference point directional system, or the ‘person who looks like so-and-so’ reference system. It is only when society became urban and started manufacturing its own goods rather than taking them directly from nature that such things as  geometrical systems and co-ordinate systems became necessary. The great advantage of the GPS or rectangular  three-dimensional co-ordinate system is that such systems are universal, not local, though this is also their drawback. Such artifices give us a way of fixing the position of  any object anywhere,  by using three, and only three, numbers. Using topological concepts such as ‘on’, ‘under’, ‘behind’ and so on, we commonly need more than three directional terms and the specifications tend to differ markedly depending on the object we are looking for, or the person we are talking to. But the ‘scientific’ co-ordinate system works everywhere ― though it is useless for practical purposes if we do not know, cannot see or remember the point to which everything is related. When out walking, the scientific system is only necessary when you are lost, i.e. when the normal local reference point system has broken down. Anyone who went hiking and looked at their computer every ten minutes to check on their position would be a fool and, if ever deprived of electronic devices, would never be able to find his or her way in the wilderness because he would not be able to pick up the natural cues and clues.
Why rectangular axes and co-ordinates? As a matter of fact, we  sometimes do use curved lines instead of straight ones since this is what the lines of latitude and longitude are, but human beings, when they do think quantitatively, almost always tend to think in terms of straight lines, squares, cubes and rectangles, shapes that do not exist in Nature (Note 1). The ‘Method of Exhaustion’, ancestor of the Integral Calculus, was essentially a means of reducing the areas and volumes of irregular figures to so many squares (Note 2). I have indeed sometimes wondered whether there might be an intelligent species for whom circles were much more natural shapes than straight lines and who would evaluate the area of a square laboriously in terms of epicycles whereas we evaluate the area of a circle by turning it into so many half rectangles, i.e. triangles. Be that as it may, it seems that human beings cannot take too much curved reality and I doubt if even a student of General Relativity ever thinks in curvilinear Gaussian co-ordinates.
Now, if we wish to accurately pinpoint the position of an object, we can do so, as stated, using only three distances plus the specification of the origin. (In the case of an object on the surface of the Earth we use latitude and longitude with the assumed origin being the centre of the Earth, the height above sea level being the third ‘co-ordinate’.) However, this is manifestly inadequate if we wish to specify the position, not of an object, but of an event. It would be senseless to specify an occurrence such as a tap on the window or a knife thrust to the heart by giving the distance of the occurrence from the right hand corner of the room in which it took place. It shows what a space-orientated culture we live in that it is only relatively recently that it has been found necessary to tack on a ‘fourth’ dimension to the other three and a lot of people still find this somewhat bizarre. For certain cultures, Indian especially, time seems to have been more significant than space (inasmuch as the two can be separated) and, had modern science developed there rather than in the West, it would doubtless have been very different. For a long time the leading science and branch of mathematics in the West was Mechanics, which studies the motions of rigid bodies that change little over brief periods of time. But from the point of view of Eventrics, what we familiarly call an ‘object’ is simply a relatively persistent event-cluster and the only reason we do not need to specify a time co-ordinate is that this object is assumed to be unchanging at least over ‘small’ intervals of time. Even the most stable objects are always changing, or rather they flash into existence, disappear and (sometimes) reoccur in a more or less identical shape and position with respect to nearby ‘objects’.
Instead of somehow tacking on a mysterious ‘fourth dimension’ to the familiar three spatial dimensions, Ultimate Event Theory posits discrete ‘globules’ or three-dimensional grids spreading out in all possible directions, each of which can receive one, and only one, ultimate event. The totality of possible positions for ultimate events constitutes the enduring  base-entity which I shall refer to as K0, or rather the only part of K0 with which we need to concern ourselves at the moment. It is misleading, if not meaningless, to refer to  this backdrop or substratum as ‘Space-Time’. Although I believe that ‘succession’ and ‘co-existence’ really do exist ― since events can and do occur ‘in succession’ and can also exist ‘at the same moment’  ― ‘Space’ and ‘Time’ have  no objective existence though one understands (sometimes) what people have in mind when they use the terms. Forf me ‘Space’ and ‘Time’ are basically mental constructs but I believe that the ultimate events themselves really do exist and likewise I believe that there really is an ‘entity’ on whose ‘surface’ ultimate events have occurrence. Newton fervently believed in the ‘absolute’ nature of Space and Time but his contemporary Leibnitz viewed  ‘Space’ as nothing but the sum-total of instantaneous relations between objects and some  contemporary physicists such as Lee Smolin (Note 3) take a similar line. For me, however, if there are events there must be a ‘somewhere’ on or in which these events can and do occur. Indeed, I take the view that the backdrop is more fundamental than the ultimate events since they emerge from it and are  essentially just momentary surface disturbances on it, froth on the ocean of K0.
For the present purposes it is, however, not so very important how one views this underlying entity, and what one calls it, it is sufficient to assume that it exists and that ultimate events are localized on or within it. K0 is assumed to be featureless and homogeneous, stretching indefinitely in all possible directions. For most of the time its existence can be neglected since all that we can observe and experiment with are the events themselves and their inter-relations. In particular, Kdoes not exert any ‘pressure’ on event-clusters or offer any  noticeable resistance to their apparent movements although it does seem to restrict them  to specific trajectories. As Einstein put it, referring to the ether, “It [the ether] has no physical effects, only geometrical ones”. (Note 4) In the terms of Ultimate Event Theory, what this means is that there are, or at least might be, ‘preferred pathways’ on the surface of K0 and, other things being equal, persisting event-clusters will pursue these pathways rather than others. Such  pathways and their inter-connections are inherent to K0  but are not fixed for all time because the landscape and its topology is itself affected by the event-clusters that have occurrence on and within it.
Even though I have argued that co-ordinate systems are entirely man-made and have no independent reality, in practiced I have found it impossible to proceed without an image at the back of my mind of a sort of fluid rectangular co-ordinate system consisting of an indefinite number of positions where ultimate events can and sometimes do occur. Ideally, instead of using two dimensional diagrams for a four-dimensional reality, we ought to have a three-dimensional framework, traced out by lights for example, and which appears and reappears at intervals ― possibly something like this is already in use. The trajectory of an object (i.e. repeating event-chain or event-cluster) would then be traced out, frame  by frame,  on this repeating three-dimensional co-ordinate backdrop. This would be a far more truthful image than the more convenient two dimensional representation.
One point should be made at once and cannot be too strongly stressed. Whereas the three spatial dimensions co-exist and, as it were, run into each other ― in the sense that a position (x, y, z) co-exists alongside a position (x1, y1, z1) ― ‘moments of time’ do not co-exist. This may seem obvious enough since if ‘moments of time’ really did co-exist they would be simultaneous, in effect the ‘same’ moment. And if all moments co-existed there would be nothing but an eternal present and no ‘time’ at all (Note 4).  But there is an unexpected and drastic consequence : it means that for the next ‘moment in time’ to come about, the previous one must disappear and along with it everything that existed at that moment. If we had an accurate three–dimensional optical model, when the lights defining the axes were turned off, everything framed by the optical co-ordinate system, pinpoints of coloured light for example, would by rights also disappear.
Rather few Western thinkers and scientists have ever realized that there is a problem here, let alone resolved it. (And there is no problem if we assume that existence and ‘Space-Time’ and everything else is ‘continuous’ but I do not see how this can possibly be the case and Ultimate Event Theory is based on the hypothesis that it is not the case.) Most scientists and philosophers in the West have assumed that it is somehow inherent in the make-up of objects and, above all, human beings to carry on existing, at least for a certain ‘time’. Descartes was the great exception : he concluded that it required an effort that could only come from God Himself to stop the whole universe disintegrating at every single instant. To Indian Buddhists, of course, the ephemeral nature of reality was taken for granted, and they ascribed the re-appearance and apparent continuity of ‘objects’, not to a supernatural Being,  but to the operation of a causal Force, that of ‘Dependent Origination’ (Note 4). Similarly, in Ultimate Event Theory, it is not the appearance or disappearance of ultimate events that requires explanation ― it is their ‘nature’, if you like,  to flash into and out of existence ― but rather it is the apparent solidity and continuous existence of ‘things’ that requires explanation. (Note 5) This is taking the Newtonian schema one step back : instead of ascribing the altered motion of a particle to an external force, it is the continuing existence of a ‘particle’ that requires a ‘force’, in this case a self-generated one.
Although Relativity and other modern theories have done away with all sorts of things that classical physicists thought  practically self-evident, the idea of a physical/temporal continuum is not one of them. Einstein, no less than Newton, believed that Space and Time were continuous. “The surface of a marble table is spread out in front of me. I can get from any point on this table to any othe point by passing continuously from one point to a ‘neighbouring’ one and, repeating this process a (large) number of times, or, in other words, by going from point to point without executing ‘jumps’. (…) We express this property of the surface by describing the latter as a continuum” (Einstein, Relativity p. 83).  To me, however, it is not possible to go from one point to another without a ‘jump’ as Einstein he puts it — quite the reverse, physical reality is made up of ‘jumps’. Also, the idea of a neighbourhood is quite different in Ultimate Event Theory : there are not an ‘infinite’ number of positions between a point on where an ultimate event has occurrence and another point where a different ultimate event has occurrence (or will have, has had, occurrence) but only a finite number. This number is not relative but absolute (though the perceived or inferred ‘distances’ may differ according to one’s standpoint and state of motion). And, of course, the three dimensional co-ordinate system we find appropriate need not necessarily be rectangular but might be curvilinear as in General Relativity.   S.H.   8 July 2012

Note 1 :  Extremely few natural objects have the appearance of our standard geometrical shapes, and the only ones that do are microscopic like rock crystals and radiolaria.

Note 2 : Geometry means literally ‘land measurement’ and was first developed for practical reasons —“According to most accounts, geometry was first discovered in Egypt, having had its origin in the measurement of areas. For this was a necessity for the Egyptians owing to the rising of the Nile which effaced the proper boundaries of everyone’s lands” (Proclus, Summary). Herodotus says something similar, claiming that the Pharaoh Ramses II distributed land in equal rectangular plots and levied an annual tax on them but that, subsequently, owners applied for tax reductions when their land got swept away by the overflowing Nile. To settle such disputes surveyors toured the country and had to work out accurately how much land had been lost. See Heath, A History of Greek Mathematics Vol. 1 pp. 119-22 from which these quotations were taken.

Note 3: “Space is  nothing apart from the things that exist; it is only an aspect of the relationships that hold between things” (Lee Smolin, Three Roads to Quantum Gravity, p. 18)

Note 4 : In the terms of Ultimate Event Theory, what this means is that there are, or at least might be, ‘preferred pathways’ on the surface of K0 and, other things being equal, persisting event-clusters will pursue these pathways rather than others. Such  pathways and their inter-connections are inherent to K0  but are not fixed for all time because the landscape and its topology is itself affected by the event-clusters that have occurrence on and within it.

  Note 5 : This is the same force that operates within a single existence, or causal chain of individual existences, in which case it is named Karma (literally ‘activity’). The entire aim of meditation and related practices is to eliminate, or rather to still, this force which drives the cycle of death and rebirth. The arhat (Saint?) succeeds in doing this and is thus able to enter the state of complete quiescence that is nirvana ― a state to which, eventually, the entire universe will return. The image of something completely still, like the surface of a mountain lake, being disturbed and these disturbances perpetuating themselves could prove to be a useful schema for a future physics. It is a very different paradigm from that of indestructible atoms moving about in the void which we inherit from the Greeks. In the  new paradigm, it is the underlying and invisible ‘substance’ that endures while everything we think of as material is a passing eruption on the surface of this something. The enorm ous event-cluster we currently call the ‘universe’ will thus not expand for ever, nor contract back again into a singularity : it will simply evaporate, return to the nothingness (that is also everything) from which it once emerged. In my unfinished SF novel The Web of Aoullnnia, the future mystical sect the Yther make this idea the cornerstone of their cosmology and activities ― Yther  is a Lenwhil Katylin term which signifies ‘ebbing away’. Interested readers are referred to my personal site www.sebastianhayes.com