Archives for category: Special Relativity

What is time? Time is succession. Succession of what? Of events, occurrences, states. As someone put it, time is Nature’s way of stopping everything happening at once.

In a famous thought experiment, Descartes asked himself what it was not possible to disbelieve in. He imagined himself alone in a quiet room cut off from the bustle of the world and decided he could, momentarily at least, disbelieve in the existence of France, the Earth, even other people. But one thing he absolutely could not disbelieve in was that there was a thinking person, cogito ergo sum (‘I think, therefore I am’).
Those of us who have practiced meditation, and many who have not, know that it is quite possible to momentarily disbelieve in the existence of a thinking/feeling person. But what one absolutely cannot disbelieve in is that thoughts and bodily sensations of some sort are occurring and, not only that, that these sensations (most of them anyway) occur one after the other. One outbreath follows an inbreath, one thought leads on to another and so on and so on until death or nirvana intervenes. Thus the grand conclusion: There are sensations, and there is succession.  Can anyone seriously doubt this?

 Succession and the Block Universe

That we, as humans, have a very vivid, and more often than not  acutely painful, sense of the ‘passage of time’ is obvious. A considerable body of the world’s literature  is devoted to  bewailing the transience of life, while one of the world’s four or five major religions, Buddhism, has been well described as an extended meditation on the subject. Cathedrals, temples, marble statues and so on are attempts to defy the passage of time, aars long vita brevis.
However, contemporary scientific doctrine, as manifested in the so-called ‘Block Universe’ theory of General Relativity, tells us that everything that occurs happens in an ‘eternal present’, the universe ‘just is’. In his latter years, Einstein took the idea seriously enough to mention it in a letter of consolation to the son of his lifelong friend, Besso, on the occasion of the latter’s death. “In quitting this strange world he [Michel Besso] has once again preceded me by a little. That doesn’t mean anything. For those of us who believe in physics, this separation between past, present and future is an illusion, however tenacious.”
Never mind the mathematics, such a theory does not make sense. For, even supposing that everything that can happen during what is left of my life has in some sense already happened, this is not how I perceive things. I live my life day to day, moment to moment, not ‘all at once’. Just possibly, I am quite mistaken about the real state of affairs but it would seem nonetheless that there is something not covered by the ‘eternal present’ theory, namely my successive perception of, and participation in, these supposedly already existent moments (Note 1). Perhaps, in a universe completely devoid of consciousness,  ‘eternalism’ might be true but not otherwise.

Barbour, the author of The End of Time, argues that we do not ever actually experience ‘time passing’. Maybe not, but this is only because the intervals between different moments, and the duration of the moments themselves, are so brief that we run everything together like movie stills. According to Barbour, there exists just a huge stack of moments, some of which are interconnected, some not, but this stack has no inherent temporal order. But even if it were true that all that can happen is already ‘out there’ in Barbour’s Platonia (his term), picking a pathway through this dense undergrowth of discrete ‘nows’ would still be a successive procedure.

I do not think time can be disposed of so easily. Our impressions of the world, and conclusions drawn by the brain, can be factually incorrect ― we see the sun moving around the Earth for example ― but to deny either that there are sense impressions and that they appear successively, not simultaneously, strikes me as going one step too far. As I see it, succession is an absolutely essential component  of lived reality and either there is succession or there is just an eternal now, I see no third possibility.

What Einstein’s Special Relativity does, however, demonstrate is that there is seemingly no absolute ‘present moment’ applicable right across the universe (because of the speed of light barrier). But in Special Relativity at least succession and causality still very much exist within any particular local section, i.e. inside a particular event’s light cone. One can only surmise that the universe as a whole must have a complicated mosaic successiveness made up of interlocking pieces (tesserae).

Irreversibility
In various areas of physics, especially thermo-dynamics, there is much discussion of whether certain sequences of events are reversible or not, i.e. could take place other than in the usual observed order. This is an important issue but is a quite different question from whether time (in the sense of succession) exists. Were it possible for pieces of broken glass to spontaneously reform themselves into a wine glass, this process would still occur successively and that is the point at issue.

Time as duration

‘Duration’ is a measure of how long something lasts. If time “is what the clock says” as Einstein is reported to have once said, duration is measured by what the clock says at two successive moments (‘times’). The trick is to have, or rather construct, a set of successive events that we take as our standard set and relate all other sets to this one. The events of the standard set need to be punctual and brief, the briefer the better, and the interval between successive events must be both noticeable and regular. The tick-tock of a pendulum clock provided such a standard set for centuries though today we have the much more regular expansion and contraction of quartz crystals or the changing magnetic moments of electrons around a caesium nucleus.

Continuous or discontinuous?

 A pendulum clock records and measures time in a discontinuous fashion: you can actually see, or hear, the minute or second hand flicking from one position to another. And if we have an oscillating mechanism such as a quartz crystal, we take the extreme positions of the cycle which comes to the same thing.
However, this schema is not so evident if we consider ‘natural’ clocks such as sundials which are based on the apparent continuous movement of the sun. Hence the familiar image of time as a river which never stops flowing. Newton viewed time in this way which is why he analysed motion in terms of ‘fluxions’, or ‘flowings’. Because of Calculus, which Newton invented, it is the continuous approach which has overwhelmingly prevailed in the West. But a perpetually moving object, or one perceived as such, is useless for timekeeping: we always have to home in on specific recurring configurations such as the longest or shortest shadow cast. We have to freeze time, as it were, if we wish to measure temporal intervals.

Event time

The view of time as something flowing and indivisible is at odds with our intuition that our lives consist of a succession  of moments with a unique orientation, past to future, actual to hypothetical. Science disapproves of the latter common sense schema but is powerless to erase it from our thoughts and feelings: clearly the past/present/future schema is hard-wired and will not go away.
If we dispense with continuity, we can also get rid of  ‘infinite divisibility’ and so we arrive at the notion, found in certain early Buddhist thinkers, that there is a minimum temporal interval, the ksana. It is only recently that physicists have even considered the possibility that time  is ‘grainy’, that there might be ‘atoms of time’, sometimes called chronons. Now, within a minimal temporal interval, there would be no possible change of state and, on this view, physical reality decomposes into a succession of ‘ultimate events’ occupying  minimal locations in space/time with gaps between these locations. In effect, the world becomes a large (but not infinite) collection of interconnected cinema shows proceeding at different rates.

Joining forces with time 

The so-called ‘arrow of time’ is simply the replacement of one localized moment by another and the procedure is one-way because, once a given event has occurred, there is no way that it can be ‘de-occurred’. Awareness of this gives rise to anxiety ― “the moving finger writes, and having writ/ Moves on, nor all thy piety or wit/Can lure it back to cancel half a line….”  Most religious, philosophic and even scientific systems attempt to allay this anxiety by proposing a domain that is not subject to succession, is ‘beyond time’. Thus Plato and Christianity, the West’s favoured religion. And even if we leave aside General Relativity, practically all contemporary scientists have a fervent belief in the “laws of physics” which are changeless and in effect wholly transcendent.
Eastern systems of thought tend to take a different approach. Instead of trying desperately to hold on to things such as this moment, this person, this self, Buddhism invites us to  ‘let go’ and cease to cling to anything. Taoism goes even further, encouraging us to find fulfilment and happiness by identifying completely with the flux of time-bound existence and its inherent aimlessness. The problem with this approach is, however, that it is not clear how to avoid simply becoming a helpless victim of circumstance. The essentially passive approach to life seemingly needs to be combined with close attention and discrimination ― in Taoist terms, Not-Doing must be combined with Doing.

Note 1 And if we start playing with the idea that  not only the events but my perception of them as successive is already ‘out there’, we soon get involved in infinite regress.

 

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Galileo’s Ship

 It was Galileo who opened up the whole subject of ‘inertial frames’ and ‘relativity’, which has turned out to be of the utmost importance in physics. Nonetheless, he does not actually use the term ‘inertial frame’ or formulate a ‘Principle of Relativity’ as such.

Galileo wrote his Dialogue Concerning the Two World Systems, Ptolemaic and Copernican in 1616 to defend the revolutionary Copernican view that the Earth and the planets moved round the Sun. The Dialogue, modelled on Plato’s writings, takes the form of a three day long discussion where Salviati undertakes to explain and justify the heliocentric system to two friends, one of whom, Simplicius, advances various arguments against the heliocentric view. One of his strongest objections is, “If the Earth is moving, why do we not feel this movement?” Salviati’s reply is essentially this, “There are many other circumstances when we do not feel we are moving just so long as our motion is steady and in a straight line”.

Salviati asks his friends to conduct a ‘thought experiment’, ancestor of innumerable modern Gedanken Experimenten. They are to imagine themselves in “the main cabin below decks on some large ship” and this, given the construction at the time, meant there would have been no portholes so one would not be able to see out. The cabin serves as a floating laboratory and Galileo’s homespun apparatus includes “a large bowl of water with some fish in it”, “a bottle that empties drop by drop into a narrow-mouthed vessel beneath it”, a stick of incense, some flies and butterflies, a pair of scales and so on. The ship, presumably a galley, is moving steadily on a calm sea in a dead straight line. Galileo (via Salviati) claims that the occupants of the cabin would not be able to tell, without going up on deck to look, whether the ship was at rest or not. Objects will weigh just the same, drops of water from a tap will take the same time to fall to the ground, the flies and butterflies will fly around in much the same way, and so on — “You will discover not the least difference in all the effects named, nor could you tell from any of them whether the ship was moving or standing still” (Note 1).

Now, it should be said at once that this is not at all what one would expect, and not what Aristotle’s physics gave one to expect. One might well, for example, expect the flies and butterflies flying about to be impelled towards the back end of the cabin and even for human beings to feel a pull in this direction along with many other noticeable effects if the ship were in motion, effects that one would not perceive if the ship were safely in the dock.

What about if one conducted experiments on the open deck?  It is here that Galileo most nearly anticipates Newton’s treatment of motion and indeed Einstein himself. Salviati specifies that it is essential to decide whether a ‘body’ such as a fly or butterfly falls, or does not fall, within the confines of the system ‘ship + immediate environment’ ─ what we would call the ship’s ‘inertial frame’. Salviati concedes that flies and butterflies “separated from it [the ship] by a perceptible distance” would indeed be prevented from participating in the ship’s motion but this would simply be because of air resistance. “Keeping themselves near it, they would follow it without effort or hindrance, for the ship, being an unbroken structure, carries with it a part of the nearby air”. This mention of an ‘unbroken structure’ is the closest Galileo comes to the modern concept of an ‘inertial frame’ within which all bodies behave in the same way. As Salviati puts it, “The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things contained within it, and to the air also” (Dialogue p. 218 ).

Now, the claim that all bodies on and in the ship are and remain ‘in the same state of motion’ is, on the face of it, puzzling and counter-intuitive. For one might ask how an object ‘knows’ what ‘frame’ it belongs to and thus how to behave, especially since the limits of the frame are not necessarily, or even usually, physical barriers. Galileo does not seem to have conducted any actual experiments relating to moving ships himself, but other people at the time did conduct experiments on moving ships, dropping cannon balls, for example, from the top of a mast and noting where it hit the deck. According to Galileo’s line of argument, a heavy object should strike the deck very nearly at the foot of the mast if the ship continued moving forward at exactly the same speed in a straight line whereas the Aristotelians, on their side, expected the cannon ball to be shifted backwards from the foot of the mast by an appreciable distance. The issue  depended on which ‘structure’, to use Galileo’s term, a given object belonged to. For example, a cannonball dropped by a helicopter that happened to be flying over the ship at a particular moment, belongs to the helicopter ‘system’ and not to the system ‘ship’. In consequence, its trajectory would not be the same as that of a cannonball dropped from the top of a mast ─ unless the helicopter and ship were, by some fluke, travelling at an identical speed and in exactly the same direction.

By his observations and reflexions Galileo thus laid the foundations for the modern treatment of bodies in motion though this was not really his intention, or at any rate not at this stage in the argument. Newton was to capitalize on his predecessor’s observations by making a clearcut distinction between the velocity of a body which, other things being equal, a body retains indefinitely and a body’s acceleration which is always due to an outside force.

Families of Inertial Frames 

In the literature, ‘inertial frame’ has come to mean a ‘force-free frame’, that is, a set-up where a body inside some sort of a, usually box-like, container remains at rest unless interfered with or, if considered to be already in straight line constant motion, retains this motion indefinitely. But neither Galileo nor Newton used the term ‘inertial reference frame’ (German: Inertialsystem) which seems to have been coined by Ludwig Lange in 1885.

The peculiarity of inertial frames is, then, that they are, physically speaking, interchangeable and cannot be distinguished from one another ‘from the inside’. Mathematically speaking, ‘being an inertial frame’ is a ‘transitive’ relation : if A is an inertial frame and B is at rest or moves at constant speed in a straight line relative to A, then B is also an inertial frame. We have, then, a vast family of ‘frames’ within which objects allegedly behave in exactly the same way and which, when one  is inside such a frame, ‘feel’ no different from one another.

It is important to be clear that the concept of ‘inertial frame’ implies (1) that it is not possible to tell, from the inside, whether the ‘frame’ (such as Galileo’s cabin or Einstein’s railway coach) is at rest or in straight line constant motion, and (2) that it is not possible to distinguish between two or more frames, neither of which are considered to be stationary, provided their motion remains constant and in a straight line. These two cases are distinct: we might, for example, be able to tell whether we were moving or not but be unable to decide with precision what sort of motion we were in ─ to distinguish, for example, between two different straight-line motions at constant speed. As it happens, Galileo was really only concerned with the distinction between being ‘at rest’ and in constant straight-line motion, or rather with the alleged inability to make such a distinction from inside such a ‘frame’, since it was this inability which was relevant to his argument. But the lumping together of a whole host of different straight-line motions is actually a more important step conceptually though Galileo himself did not perhaps realize this.

So. Were Galileo in the cabin of a ship moving at a steady pace of, say, 10 knots, he would, so he claims, not be able to differentiate between what goes on inside such a cabin from what goes on in a similar cabin of a similar ship not moving at all or one moving at a speed of 2 or 20 or 200 or even 2,000 knots supposing this to be possible. Now, this is an extremely surprising fact (if it is indeed a fact) since Ship A and Ship B are not ‘in the same state of motion’ : one is travelling at a certain speed relative to dry land and the second at a quite different speed relative to the same land. One would, on the face of it, expect it to be possible to tell whether a ship were ‘in motion’ as opposed to being at rest, and, secondly, to be able to distinguish between two states of straight line constant motion with different speeds relative to the same fixed mass of land. Newton himself felt that it ought to be possible to distinguish between ‘absolute rest’ and ‘absolute motion’ but conceded that this seemed not to be possible in practice. He was obviously somewhat troubled by this point as well he might be.

 Galileo’s Ship is not a true Inertial Frame

 As a matter of fact, it would not only be possible but fairly easy today to tell whether we are at rest or in motion when, say, locked up without radio or TV communication in a windowless cabin of an ocean liner. All I would need to carry out the test successfully would be a heavy pendulum, a means to support it so that it can revolve freely, a good compass, and a certain amount of time. Foucault demonstrated that a heavy pendulum, suspended with the minimum possible friction from the bearings so that it can move freely in any direction, will appear to swing in a circle : the Science Museum in London and countless other places have working Foucault pendulums. The time taken to make a complete circuit depends on one’s latitude — or, more correctly, the time it takes the Earth to revolve around the pendulum depends on what we choose to call latitude. A Foucault pendulum suspended at the North Pole would, so we are assured, take 24 hours to make a full circuit and a similar one at the Equator would not change its direction of swing at all, within the margins of experimental error. By timing the swings carefully one could thus work out whether the ship was changing its latitude, i.e. moving ‘downwards’ in the direction of the South Pole, or ‘upwards’ in the direction of the North (geographical) pole. On the other hand, a ship at rest, whatever its latitude, would show no variation in the time of swing ─ again within the limits of scientific error.

However, suppose I noted no change in the period of the Foucault pendulum. I would now have to decide whether my ship, galley or ocean liner, was stationary relative to dry land or was moving at constant speed along a great circle of latitude. This is rather more difficult to determine but could be managed nonetheless even with home-made instruments. One could examine  the ‘dip’ of a compass needle which points downwards in regions above the Equator and upwards in regions south of the Equator ─ because the compass needle aligns itself according to the lines of force of the Earth’s magnetic field. Again, any change in the angle of dip would be noticeable and there would be changes as the ship moved nearer the magnetic south or north poles. Nor is this all. The magnetic ‘north pole’ differs appreciably from the geographical north pole and this discrepancy changes as we pursue a great circle path along a latitude : so-called isoclinics, lines drawn through places having the same angle of dip, are different from lines of latitude. There are also variations in g, the acceleration due to gravity at the Earth’s surface, because of the Earth’s slightly irregular shape, its ‘oblateness’ which makes the circumference of the Earth measured along the Equator markedly different from that measured along a great circle of longitude passing through the poles. And so, despite Galileo’s claim to the contrary, there would be slight differences in the weight of objects in the cabin at different moments if the ship were wandering about. Only if the Earth were a perfect sphere with the magnetic poles precisely aligned with the geographical poles, would such tests be inconclusive. But a perfect sphere does not exist in Nature and never will exist unless it is manufactured by humans or some other intelligent species.
Galileo’s claim is thus not strictly true : it is a typical case of an ‘ideal situation’ to which actual situations approximate but which they do not, and cannot, attain.

Einstein’s Generalizations

But, one might go on to argue, the discrepancies mentioned above only  arose because Galileo’s ship was constrained to move on a curved surface, that of the ocean : what about a spaceship in ‘empty space’?

The full Principle of Relativity, Galileian or early Einsteinian,  asserts that there is no way to distinguish from the inside between conditions inside a rocket stationary with respect to the Earth, and conditions inside one travelling at any permissible constant ‘speed’ in a straight line relative to the Earth. It is routinely asserted in textbooks on the Special Theory of Relativity that there would indeed be no way to distinguish the two cases provided one left gravity out of the picture.

Newton made Galileo’s idealized ship’s cabin into the arena where his laws of motion held sway. An object left to its own devices inside a recognizable container-like set-up (an inertial system) would either remain stationary or, if already moving relative to the real or imagined frame, would keep moving in a straight line at constant speed indefinitely. This is Newton’s First Law. Any deviation from this scenario would show that there was an outside force at work ─ and Newton, knowing nothing of interior chemical or nuclear forces, always assumed that any supposed force would necessarily come from the outside. Thus, Newton’s Second Law.

So, supposing I let go of a piece of wood I hold in my hand in this room, which I take as my inertial frame, what happens to it? Instead of remaining where it was when I had it in my hand, the piece of wood falls to the ground and its speed does not stay the same over the time of its trajectory but increases as it falls, i.e. is not constant. And if I throw a ball straight up into the air, not only does it not continue in a vertical line at constant speed but slows down and reverses direction while a shot fired in the air roughly northwards will be deflected markedly to the right because of the Earth’s rotation (if I am in the northern hemisphere). Neither this room nor the entire Earth are true inertial frames : if they were Newton’s laws would apply without any tinkering about. To make sense of the bizarre trajectories just mentioned it is necessary to introduce mysterious forces such as the gravitational pull of the Earth or the Coriolis ‘force’ produced by its rotation on its own axis.

As we know, Einstein’s theory of Special Relativity entirely neglects gravity, and introducing the latter eventually led on to the General Theory which is essentially a theory of Gravitation. Einstein’s aim, even in 1905, was quite different from Galileo’s. Whereas Galileo was principally concerned to establish the heliocentric theory and only introduced his ship thought-experiments to deal with objections, Einstein was concerned with identifying the places (‘frames’) where the ‘laws of physics’ would hold in their entirety, and by ‘laws’ he had in mind not only Newton’s laws of motion but also and above all Maxwell’s laws of electro-magnetism. Einstein’s thinking led him on to a search for a ‘true’ inertial frame as opposed to a merely stationary frame such as this room since the latter is certainly not a ‘force-free’ frame. Einstein, reputedly after speculating about what would happen to a construction worker falling from the scaffolding around a building, decided that a real or imaginary box falling freely under the influence of gravitation was a ‘true’ inertial   frame. Inside such a frame, not only would the ‘normal’ Newtonian laws governing mechanics hold good but the effects of gravity would be nullified and so could be legitimately left out of consideration. Such a ‘freely-falling frame’ would thus be the nearest thing to a spaceship marooned in the depths of space far away from the influence of any celestial body.

A freely falling frame is not a true inertial frame

So, would it in fact be impossible to distinguish from the inside between a box falling freely under the gravitational influence of the Earth and a spaceship marooned in empty space? The answer is, perhaps surprisingly, no. In a ‘freely falling’ lift dropping towards the Earth, or the centre of any other massive body, there would be so-called ‘tidal effects’ because the Earth’s gravitational field is not homogeneous (the same in all localities) and isotropic (the same in all directions). If one released a handful of ball-bearings or a basketful of apples in a freely falling lift, the ball-bearings or apples at the ‘horizontal’ extremities would curve slightly towards each other as they fell since their trajectories would be directed towards the centre of the Earth rather than straight downwards. Likewise, the top and bottom apples would not remain the same distance apart since the forces on them, dependent as they are on the distances of the two apples from the Earth’s centre of mass, would be different and this difference would increase as the falling lift accelerated.

It turns out, then, that, at the end of the day, Einstein’s freely falling lift is not a great deal better than Galileo’s ship ─ although both are good enough approximations to inertial frames, or rather are very good imitations of inertial frames. One can, of course, argue in Calculus manner that the strength of the Earth’s gravitational field will be the same over an ‘infinitesimally small region’ ─ though without going into further details about the actual size of such a region. Newton’s Laws in their purity and integrity are thus only strictly applicable to such ‘infinitesimal’ regions in which case there will inevitably be abrupt transitions, i.e. ‘accelerations’, as we move from one infinitesimal region to another. The trajectory of any free falling object will thus not be fluent and continuous but jerky at a small enough scale.

For that matter, it is by no means obvious that a spaceship marooned in the  middle of ‘empty’ space is a true ‘inertial frame’. According to Einstein’s General Theory of Relativity, Space-time is ‘warped’ or distorted by the presence of massive objects and this space-time curvature apparently extends over the whole of the universe ─ albeit with very different local effects. If the universe is to be considered a single entity, then strictly speaking there is nowhere inside it which is completely free of ‘curvature’, and so there is nowhere to situate a ‘true’ inertial frame.

What to Conclude?

 So where does all this leave us? Or, more specifically, what bearing does all this discussion have on the theory I am attempting to develop ?

In Ultimate Event Theory, the basic entities are not bodies but point-like ultimate events which, if they are strongly bonded together and keep repeating more or less identically, constitute what we view as objects. In its most simplistic form, the equivalent of an ‘object’ is a single ultimate event that repeats indefinitely, i.e. an event-chain, while several ‘laterally connected’ event-chains make up an event cluster. There is no such thing as continuous motion in UET and, if this is what we understand by motion, there is no motion. There is, however, succession and also causal linkage between successive ultimate events which belong to the ‘same’ event-chain.

Although I did not realize this until quite recently, one could say that the equivalent of an ‘inertial frame’ in UET is the basic ‘event-capsule’, a flexible though always finite region of the event Locality within which every ultimate event has occurrence. There is no question of the basic ‘building block’ in Eventrics ‘moving’ anywhere : it has occurrence at a particular spot, then disappears and, in some cases, re-appears in a similar (but not identical) spot a ksana (moment) later. One can then pass on to imagining a ‘rest event-chain’ made up of successive ultimate events sufficiently far removed from the influence of massive event-clusters for the latter to have no influence on what occurs. This is the equivalent, if you like, of the imaginary spaceship marooned in the midst of empty space.

So, where does one go from here? One thing to have come out of the endless discussions about inertial frames and their alleged indistinguishability (at least from the inside), is that the concept of ‘motion’ has little if any meaning if we are speaking of a single object whether this object or body is a boat, a particle, ocean liner or spaceship. We thus need at least two ‘objects’, one of which is traditionally seen as ‘embedded’ in the other more or less like an object in a box. In effect, Galileo’s galley is related to the enclosing dry land of the Mediterranean or, at the limit, to the Earth itself including its atmosphere. The important point is being able to relate an object which ‘moves’ to a larger, distinctive object that remains still, or is perceived to remain so.

In effect, then, we need a system composed of at least two very different ‘objects’, and the simplest such system in UET is a ‘dual event-system’ made up of just two event-chains, each of which is composed of a single ultimate event that repeats at every ksana. Now, although any talk of such a system ‘moving’ is only façon de parler , we can quite properly talk of such a system expanding, contracting or doing neither. If our viewpoint is event-chain A , we conceive event-chain B to be, for example, the one that is ‘moving further away’ at each ksana, while if we take the viewpoint of event-chain B, it is the other way round. The important point, however, is that the dual system is expanding if this distance increases, and by distance increasing we mean that there is a specified, finite number of ultimate events that could be ‘fitted into’ the space between the two chains at each ksana.

This is the broad schema that will be investigated in subsequent posts. How much of Galileo’s, Newton’s and early Einstein’s assumptions and observations do I propose to carry over as physical/philosophic baggage into UET?

To start with, what we can say in advance is that the actual distance (in terms of possible positions for ultimate events) between two event-chains does not seem to matter very much. Although Galileo, or Salviati, does not see fit to mention the point ─ he doubtless thinks it too ’obvious’ ─ it is notable that, whether the ship is in motion or not, the objects inside Galileo’s cabin do not change wherever the ship is, neglecting the effects of sun and wind, i.e. that position as such does not bring about changes in physical behaviour. This is not a trivial matter. It amounts to a ‘law’ or ‘principle’ that carries over into UET, namely that the Event Locality does not by itself seem to affect what goes on there, i.e. we have the equivalent of the principle of the ‘homogeneity’ and ‘isotropy’ of Space-time. As a contemporary author puts it : “The homogeneity of space means that all points in space are physically equivalent, i.e. a transportation of any object in space does not affect in any way the processes taking place in this object. The homogeneity of time must be understood as the physical indistinguishability of all instants of time for free objects. (By a free object we mean an object which is far from all surrounding objects so that their interaction can be neglected.)”  Saxena, Principles of Modern Physics  2.2)   

What about the equivalent of velocity? Everything we know about so-called ‘inertial systems’ in the Galileian sense suggests that, barring rather recondite magnetic and gravitational effects, the velocity of a system does not seem to matter very much, provided it is constant and in a straight line. Now, what this means in UET terms is that if successive members of two event-chains get increasingly separated along one spatial direction, this does not affect what goes on in each chain or cluster so long as this increase remains the same. What does affect what goes on in each chain is when the rate of increase or decrease changes : this not only means the system as a whole has changed, but that this change is reflected in each of the two members of the dual system. When travelling in a car or train we often have little idea of our speed but our bodies register immediately any abrupt substantial change of speed or direction, i.e. an acceleration.  This is, then, a feature to be carried over into UET since it is absolutely central to traditional physics.

Finally, that there is the question of there being a limit to the possible increase of distance between two event-chains. This principle is built into the basic assumptions of UET since everything in UET, except the extent of the Locality itself, has an upper and lower limit. Although there is apparently nothing to stop two event-chains which were once adjacent from becoming arbitrarily far apart at some subsequent ksana provided they do this by stages, there is a limit to how much a dual system can expand within the ‘space’ of a single ksana. This is the (now) well-known concept of there being an upper limit to the speed of all particles. Newton may have thought there had to be such a limit but if so he does not seem to have said so specifically : in Newtonian mechanics a body’s speed can, in principle, be increased without limit. In UET, although there is no continuous movement, there is a (discontinuous) ‘lateral space/time displacement rate’ and this, like everything else is limited. In contrast to orthodox Relativity theory, I originally attempted to make a distinction between such an unattainable upper limit, calling it c, and the highest attainable rate which would be one space less per ksana. This means one does not have the paradox of light actually attaining the limit and thus being massless (which it is in contemporary physics). However, this finicky separation between c s0/t0 and c* = (c – 1) s0/t0 (where s0 and t0 are ‘absolute’ spatial and temporal units) may well prove to be too much of a nuisance to be worth maintaining.  SH 21/11/14

 NOTES

 Note 1  This extract and following ones are taken from Drake’s translation of Dialogue concerning two world systems by Galileo Galilei (The Modern Library)

Note : Recent posts have focused on ‘macroscopic’ events and event-clusters, especially those relevant to personal ‘success’ and ‘failure’. I shall be returning to such themes eventually, but the point has now come to review the basic ‘concepts’ of ‘micro’ (‘ultimate’) events. The theory ─ or rather paradigm ─ seems to  know where it wants to go, and, after much trepidation, I have decided to give it its head, indeed I don’t seem to have any choice in the matter.  An informal ─ but nonetheless tolerably stringent ─ treatment now seems more appropriate than my original attempted semi-axiomatic presentation. SH   26/6/14

 Beginnings  

It is always necessary to start somewhere and assume certain things, otherwise you can never get going. Contemporary  physics may be traced back to Democritus’ atomism, that is to the idea that ‘everything’ is composed of small ‘bodies’ that cannot be further divided and which are indestructible ─ “Nothing exists except atoms and void” as Democritus put it succinctly. What Newton did was essentially to add in the concept of a ‘force’ acting between atoms and which affects the motions of the atoms and the bodies they form. ‘Classical’, i.e. post-Renaissance  but pre twentieth-century physics, is based on the conceptual complex atom/body/force/motion.

Events instead of things  

Ultimate Event Theory (UET), starts with the concept of the ‘event’. An event is precisely located : it happens at a particular spot and at a particular time, and there is nothing ‘fuzzy’ about this place and time. In contrast to a solid object an ‘event’ does not last long, its ‘nature’ is to appear, disappear and never come back again. Above all, an event does not ‘evolve’ : it is either not at all or ‘in one piece’. Last but not least, an ultimate event is always absolutely still : it cannot ‘move’ or change, only appear and disappear. However, in certain rare cases it can give rise to other ultimate events, either similar or dissimilar.

Rejection of Infinity 

The spurious notion of ‘infinity’ is completely excluded from UET: this clears the air considerably and allows one to deduce at once certain basic properties about events. To start with, macroscopic events, the only ones we are directly aware of, are not (in UET) made up of an ‘infinite’ number of ‘infinitely small’ micro-events: they are composed of a particular, i.e. finite, number of ‘ultimate events’ ─ ultimate because such micro-events cannot be further broken down (Note 1).

 Size and shape of Ultimate Events

Ultimate events may well  vary in size and shape and other characteristics but as a preliminary simplifying assumption, I assume that they are of the same shape and size, (supposing these terms are even meaningful at such a basic level). All ultimate events thus have exactly the same ‘spatio/temporal extent’ and this extent is an exact match for the ‘grid-spots’ or  ‘event-pits’ that ultimate events occupy on the Event Locality. The occupied region may be envisaged as a cuboid of dimensions su × su × su , or maybe a sphere of radius su ,  or indeed any shape of fixed volume which includes three dimensions at right angles to each other.
Every ultimate event occupies such a ‘space’ or ‘place’ for the duration of a single ksana of identical ‘length’ t0. Since everything that happens is reducible to a certain number of  ultimate events occupying fixed positions on the Locality, ‘nothing can happen’ within a spatial region smaller than su3 or within a ‘lapse of time’ smaller than t0. Though there may conceivably be smaller spatial and temporal intervals, they are irrelevant since Ultimate Event Theory is a theory about ‘events’ and their interactions, not about the Locality itself.

Event Kernels and Event Capsules 

The region  su3 t0  corresponds to the precise region occupied by an individual ultimate event. As soon as I started playing around with this simple model of precisely located ultimate events, I saw that it would be necessary to introduce the concept of the ‘Event Capsule’. The latter normally has a much greater spatial extent than that occupied by the ultimate event itself : it is only the small central region known as the ‘kernel’ that is of spatial extent su3, the relation between the kernel and the capsule as a whole being somewhat analogous to that between the nucleus and the enclosing atom. Although each ‘emplacement’ on the Locality can only receive a single ultimate event, the vast spatial region surrounding the ‘event-pit’ itself is, as it were, ‘flexible’. The essential point is that the Event Capsule, which completely fills the available ‘space’, is able to expand and contract when subject to external (or possibly also internal) forces.
There are, however, fixed limits to the size of an Event Capsule ─ everything except the Event Locality itself has limits in UET (because of the Anti-Infinity Axiom). The Event Capsule varies in spatial extent from the ‘default’, maximal size of s03 to the  absolute minimum size of u3which it attains when the Event Capsule has shrunk to the dimensions of the ‘kernel’ housing a single ultimate event.

Length of a ksana 

The ‘length’ of a ksana, the duration or ‘temporal dimension’ of an ultimate event, likewise of an Event Capsule, does not expand or contract but, by hypothesis, always stays the same. Why so? One could in principle make the temporal interval flexible as well but this seems both unnecessary and, to me, unnatural. The size of the enveloping capsule should not, by rights, have anything to do what actually occurs inside it, i.e. with the ultimate event itself, and, in particular, should not affect how long an ultimate event lasts. A gunshot is the same gunshot whether it is located within an area of a few square feet, within a square kilometre or a whole county, and it lasts the same length of time whether we record it as simply having taken place in such and such a year, or between one and one thirty p.m. of a particular day within this year.

Formation of Event-Chains and Event-clusters 

In contrast to objects, a fortiori organisms, it is in the nature of an ultimate event to appear and then disappear for ever : transience and ephemerality are of the very essence of Ultimate Event Theory. However, for reasons that we need not enquire into at present, certain ultimate events acquire the ability to repeat more or less identically during (or ‘at’) subsequent ksanas, thus forming event-chains. If this were not so, there would be no universe, no life, nothing stable or persistent, just a “big, buzzing confusion” of ephemeral ultimate events firing off at random and immediately subsiding into darkness once again.
Large repeating clusters of events that give the illusion of permanence are commonly known as ‘objects’ or ‘bodies’ but before examining these, it is better to start with less complex entities. The most rudimentary  type of event-chain is that composed of a single ultimate event that repeats identically at every ksana.

‘Rest Chains’

Classical physics kicks off with Galileo’s seminal concept of inertia which Newton later developed and incorporated into his Principia (Note 2). In effect, according to Galileo and Newton,  the ‘natural’ or ‘default’ state of a body is to be “at rest or in constant straight-line motion”. Any perceived deviation from this state is to be attributed to the action of an external force, whether this force be a contact force like friction or a force which acts from a distance like gravity.
As we know, Newton also laid it down as a basic assumption that all bodies in the universe attract all others. This means that, strictly speaking, there cannot be such a thing as a body that is exactly at rest (or moving exactly at a constant speed in a straight line) because the influence of other massive bodies would inevitably make such a body deviate from a state of perfect rest or constant straight-line motion. And for Newton there was only one universe and it was not empty.
However, if we  consider a body all alone in the depths of space, it is reasonable to dismiss the influence of all other bodies as entirely negligible ─ though the combined effect of all such influences is never exactly zero in Newtonian Mechanics. Our ideal isolated body will then remain at rest for ever, or if conceived as being in motion, this ‘motion’ will be constant and in a straight line. Thus Newtonian Mechanics. Einstein replaced the classical idea of an ‘inertial frame’ with the concept of a ‘free fall frame’, a region of Space/Time where no external forces could trouble an object’s state of rest ─ but also small enough for there to be no variation in the local gravitational field.
EVENT CAPSULE IMAGEIn a similar spirit, I imagine an isolated event-chain completely removed from any possible interference from other event-chains. In the simplest possible case, we thus have a single ultimate event which will carry on repeating indefinitely (though not for ‘ever and ever’) and each time it re-appears, this event will occupy an exactly similar spatial region on the Locality of size s03 and exist for one ksana, that is for a ‘time-length’ of to.  Moreover, the interval between successive appearances, supposing there is one, will remain the same. The trajectory of such a repeating event, the ‘event-line’ of the chain, may, very crudely, be modelled as a series of dots within surrounding boxes all of the same size and each ‘underneath’ the other.

True rest?

Such an event-chain may be considered to be ‘truly’ at rest ─ inasmuch as a succession of events can be so considered. In such a context, ‘rest’ means a minimum of interference from other event-chains and the Locality itself.
Newton thought that there was such a thing as ‘absolute rest’ though he conceded that it was apparently not possible to distinguish a body in this state from a similar body in an apparently identical state that was ‘in steady straight-line motion’ (Note 3). He reluctantly conceded that there were no ‘preferential’ states of motion and/or rest.
But Newton dealt in bodies, that is with collections of  atoms which were eternal and did not change ever. In Ultimate Event Theory, ‘everything’ is at rest for the space of a single ksana but ‘everything’ is also ceaselessly being replaced by other ‘things’ (or by nothing at all) over the ‘space’ of two or more ksanas. In the next post I will investigate what meaning, if any, is to be given to ‘velocity’ ‘acceleration’ and ‘inertia’ in Ultimate Event Theory.       SH  26/6/14

 Note 1  One could envisage the rejection of infinity as a postulate, one of the two or three most important postulates of Ultimate Event Theory, but I simply regard the concept of infinity as completely meaningless, as ‘not even wrong’.         I do, however,  admit the possibility of the ‘para-finite’ which is a completely different and far more reasonable concept. The ‘para-finite’ is a domain/state where all notions of measurement and quantity are meaningless and irrelevant : it is essentially a mystical concept (though none the worse for that) rather than a mathematical or physical one and so should be excluded from natural science.
The Greeks kept the idea of actual infinity firmly at arm’s length. This was both a blessing and, most people would claim, also a curse. A blessing because their cosmological and mathematical models of reality made sense, a curse because it stopped them developing the ‘sciences of motion’, kinematics and dynamics. But it is possible to have a science of dynamics without bringing in infinity and indeed this is one of the chief aims of Ultimate Event Theory.

Note 2  Galileo only introduced the concept of an ‘inertial frame’ to meet the obvious objection to the heliocentric theory, namely that we never feel the motion of the Earth around the Sun. Galileo’s reply was that neither do we necessarily detect the regular motion of a ship on a calm sea ─ the ship is presumably being rowed by well-trained galley-slaves. In his Dialogue Concerning the Two World Systems, (pp. 217-8 translator Drake) Galileo’s spokesman, Salviati, invites his friends to imagine themselves in a makeshift laboratory, a cabin below deck (and without windows) furnished with various homespun pieces of equipment such as a bottle hung upside down with water dripping out, a bowl of water with goldfish in it, some flies and butterflies, weighing apparatus and so on. Salviati claims that it would be impossible to know, simply by observing the behaviour of the drips from the bottle, the flight of insects, the weight of objects and so on, whether one was safely moored at a harbour or moving in a straight line at a steady pace on a calm sea.
        Galileo does not seem to have realized the colossal importance of this thought-experiment. Newton, for his part, does realize its significance but is troubled by it since he believes ─ or at least would like  to believe ─ that there is such a thing as ‘absolute motion’ and thus also ’absolute rest’. The question of whether Galileo’s principle did, or did not, cover optical (as opposed to mechanical) experiments eventually gave rise to the theory of Special Relativity. The famous Michelsen-Morley experiment was, to everyone’s surprise at the time, unable to detect any movement of the Earth relative to the surrounding ‘ether’. The Earth itself had in effect become Galileo’s ship moving in an approximately straight line at a steady pace through the surrounding fluid.
Einstein made it a postulate (assumption)  of his Special Theory that “the laws of physics are the same in all inertial frames”. This implied that the observed behaviour of objects, and even living things, would be essentially the same in any ‘frame’ considered to be ‘inertial’. The simple ‘mind-picture’ of a box-like container with objects inside it that are free to move, has had tremendous importance in Western science. The strange thing is that in Galileo’s time vehicles  ─ even his ship ─ were very far from being ‘inertial’, but his idea has, along with other physical ideas, made it possible to construct very tolerable ‘inertial frames’ such as high-speed trains, ocean liners, aeroplanes and space-craft.

Note 3  Newton is obviously ill at ease when discussing the possibility of ‘absolute motion’ and ‘absolute rest’. It would seem that he believed in both for philosophical (and perhaps also religious) reasons but he conceded that it would, practically speaking, be impossible to find out whether a particular state was to be classed as ‘rest’ or ‘straight-line motion’. In effect, his convictions clashed with his scientific conscience.

“Absolute motion, is the translation of a body from one absolute place into another. Thus, in a ship under sail, the relative place of a body is that part of the ship which the body possesses, or that part of its cavity which the body fills, and which therefore moves together with the ship, or its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space in which the ship itself, its cavity and all that it contains, is moved. (…) It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. (…) Instead of absolute places and motions we use relative ones; and that without any inconvenience in common affairs: but in philosophical disquisitions, we ought to abstract from our senses and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.”
Newton, Principia, I, 6 ff.

 

As related in the previous post, Einstein, in his epoch-making 1905 paper, based his theory of Special Relativity on just two postulates,

  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.

I asked myself if I could derive the main results of the Special Theory, the Rule for the Addition of Velocities, Space Contraction, Time Dilation and the ‘Equivalence’ of Mass and Energy from UET postulates.
Instead of Einstein’s Postulate 2, the ‘absolute value of the speed of light’, I employ a more general but very similar principle, namely that there is a ‘limiting speed’ for the propagation of causal influences from one spot on the Locality to another. In the simplest case, that of an  event-chain consisting of a single ultimate event that repeats at every ksana, this amounts to asking ourselves ‘how far’ the causal influence can travel ‘laterally’ from one ksana to the next. I see the Locality as a sort of grid extending indefinitely in all directions where  each ‘grid-position’ or ‘lattice-point’ can receive one, and only one, ultimate event (this is one of the original Axioms, the Axiom of Exclusion). At each ksana the entire previous spatial set-up is deftly replaced by a new, more or less identical one. So, supposing we can locate the ‘same’ spot, i.e. the ‘spot’ which replaces the one where the ultimate event had occurrence at the last ksana, is there a limit to how far to the left (or right) of this spot the ultimate event can re-occur? Yes, there is. Why? Well, I simply cannot conceive of there being no limit to how far spatially an ‘effect’ ─ in this case the ‘effect’ is a repetition of the original event ─ can be from its cause. This would be a holographic nightmare where anything that happens here affects, or at least could affect, what happens somewhere billions of light years away. One or two physicists, notably Heisenberg, have suggested something of the sort but, for my part, I cannot seriously contemplate such a state of affairs.  Moreover, experience seems to confirm that there is indeed a ‘speed limit’ for all causal processes, the limit we refer to by the name of c.
However, this ‘upper speed limit’ has a somewhat different and sharper meaning in Ultimate Event Theory than it does in matter-based physics because c (actually c*) is an integer and corresponds to a specific number of adjacent ‘grid-positions’ on the Locality existing at or during a single ksana. It is a distance rather than a speed and even this is not quite right : it is a ‘distance’ estimated not in terms of ‘lengths’ but only in terms of the number of the quantity of intermediary ultimate events that could conceivably be crammed into this interval.
In UET a distinction is made between an attainable limiting number of grid-positions to right (or left) denoted c* and the lowest unattainable limit, c, though this finicky distinction in many cases can be neglected. But the basic schema is this. A  ‘causal influence’, to be effective, must not only be able to at least traverse the distance between one ksana and the next ‘vertically’ (otherwise nothing would happen) but must also stretch out ‘laterally’ i.e. ‘traverse’ or rather ‘leap over’ a particular number of  grid-positions. There is an upper limit to the number of positions that can be ‘traversed’, namely c*, an integer. This number, which is very great but not infinite ─ actual infinity is completely banished from UET ─ defines the universe we (think we) live in since it puts a limit to the operation of causality (as  Einstein clearly recognized), and without causality there can, as far as I am concerned, be nothing worth calling a universe. Quite conceivably, the value of this constant c i(or c*) is very different in other universes, supposing they exist, but we are concerned only  with this ‘universe’ (massive causally connected more or less identically repeating event-cluster).
So far, so good. This sounds a rather odd way of putting things, but we are still pretty close to Special Relativity as it is commonly taught. What of Einstein’s other principle? Well, firstly, I don’t much care for the mention of “laws of physics”, a concept which Einstein along with practically every other modern scientist inherited from Newton and which harks back to a theistic world-view whereby God, the supreme law-giver, formulated a collection of ‘laws’ that everything must from the moment of Creation obey ─ everything material at any rate. My concern is with what actually happens whether or not what happens is ‘lawful’ or not. Nonetheless, there do seem to be certain very general principles that apply across the board and which may, somewhat misleadingly, be classed as laws. So I shall leave this question aside for the moment.
The UET Principle that replaces Einstein’s First Principle (“that the laws of physics are the same in all inertial frames”) is rather tricky to formulate but, if the reader is patient and broad-minded enough, he or she should get a good idea of what I have in mind. As a first formulation, it goes something like this:

The occupied region between two or more successive causally related positions on the Locality is invariant. 

         This requires a little elucidation. To start with, what do I understand by ‘occupied region’? At least to a first approximation, I view the Locality (the ‘place’ where ultimate events can and do have occurrence) as a sort of three-dimensional lattice extending in all directions which  flashes on and off rhythmically. It would seem that extremely few ‘grid-spots’ ever get occupied at all, and even less spots ever become the seats of repeating events, i.e. the location of the  first event of an event-chain. The ‘Event Locality’ of UET, like the Space/Time  of matter-based physics, is a very sparsely populated place.
Now, suppose that an elementary event-chain has formed but is marooned in an empty region of the Locality. In such a case, it makes no sense to speak of ‘lateral displacement’ : each event follows its predecessor and re-appears at the ‘same’ ─ i.e.  ‘equivalent’ ─ spot. Since there are no landmark events and every grid-space looks like every other, we can call such an event-chain ‘stationary’. This is the default case, the ‘inertial’ case to use the usual term.
We concentrate for the moment on just two events, one the clone of the other re-appearing at the ‘same spot’ a ksana later. These two events in effect define an ‘Event Capsule’ extending from the centre (called ‘kernel’ in UET) of the previous grid-space to the centre of the current one and span a temporal interval of one ksana. Strictly speaking, this ‘Event Capsule’ has two parts, one half belonging to the previous ksana and the other to the second ksana, but, at this stage, there is no more than a thin demarcation line separating the two extremities of the successive ksanas. Nonetheless, it would be quite wrong (from the point of view of UET) to think of this ‘Event Capsule’ and the whole underlying ‘spatial/temporal’ set-up as being ‘continuous’. There is no such thing as a ‘Space/Time Continuum’ as Minkowski understood the term.  ‘Time’ is not a dimension like ‘depth’ which can seamlessly be added on to ‘length’ or ‘width’ : there is a fundamental opposition between the spatial and temporal aspect of things that no physical theory or mathematical artifice can completely abolish. In the UET  model, the demarcations between the ‘spatial’ parts of adjacent Event Capsules do not widen, they  remain simple boundaries, but the demarcations between successive ksanas widen enormously, i.e. there are gaps in the ‘fabric’ of time. To be sure there must be ‘something’ underneath which persists and stops everything collapsing, but this underlying ‘substratum’ has no physical properties whatsoever, no ‘identity’, which is why it is often referred to, not inaccurately, both in Buddhism and sometimes even in modern physics, as ‘nothing’.
To return to the ‘Constant Region Postulate’. The elementary ‘occupied region’ may be conceived as a ‘Capsule’ having the dimensions  s0 × s0  × s= s03  for the spatial extent  and t0 ­for time, i.e. a region of extent s03 × t0 ­. These dimensions are fixed once and for all and, in the simplest UET model, s0 is a maximum and t0 ­is a minimum. Restricting ourselves for simplicity to a single spatial dimension and a single temporal dimension, we  thus have an ‘Event Rectangle’ of  s0  by t0­ .  
        For anything of interest to happen, we need more than one event-chain and, in particular, we need at least three ultimate events, one of which is to serve as a sort of landmark for the remaining pair. It is only by referring to this hypothetical or actual third event, occurring as it does at a particular spot independently of the event-pair, that we can meaningfully talk of the ‘movement’ to left or right of the second ultimate event in the pair with relation to the first. Alternatively, one could imagine an ultimate event giving rise to two events, one occurring ‘at the same spot’ and the other so many grid-spaces to the right (or left). In either case, we have an enormously expanded ‘Event Capsule’ spatially speaking compared to the original one. The Principle of the Constancy of the Area of the Occupied Region asserts that this ‘expanded’ Event Capsule which we can imagine as a ‘Space/Time rectangle’ (rather than Space/Time parallelipod), always has the ‘same’ area.
How can this be possible? Quite simply by making the spatial and temporal ‘dimensions’ inversely proportional to each other. As I have detailed in previous posts, we have in effect a ‘Space/Time Rectangle’ of sides sv and tv (subscript v for variable) such that sv × tv  = s0 × t0  = Ω = constant. Just conceivably, one could make s0  a minimum and t0 a maximum but this would result in a very strange universe indeed. In this model of UET, I take s0 as a maximum and t0 as a minimum. These dimensions are those of the archetypal ‘stationary’ or ‘inertial’ Event Capsule, one far removed from the possible influence of any other event-chains. I do not see how the ‘mixed ratio’ s0 : t0 can be determined on the basis of any fundamental physical or logical considerations, so this ratio just ‘happens to be’ what it is in the universe we (think we) live in. This ratio, along with the determination of c which RELATIVITY  HYPERBOLA DIAGRAMis a number (positive integer), are the most important constants in UET and different values would give rise to very different universes. In UET s0/t0 is often envisaged  in geometrical terms : tan β = s0/t0 = constant.    s0  and   t0   also have minimum and maximum values respectively, noted as  su    and tu  respectively, the subscript u standing for ‘ultimate’. We thus have a hyperbola but one constrained within limits so that there is no risk of ‘infinite’ values.

 

 

What is ‘speed’?   Speed is not one of the basic SI units. The three SI mechanical units are the metre, the standard of length, the kilogram, the standard of mass, and the second, the standard of time. (The remaining four units are the ampere, kelvin, candela and mole). Speed is a secondary entity, being the ratio of space to time, metre to second. For a long time, since Galileo in fact, physicists have recognized the ‘relational’ nature of speed, or rather velocity (which is a ‘vector’ quantity, speed + direction). To talk meaningfully about a body’s speed you need to refer it to some other body, preferably a body that is, or appears to be, fixed (Note 1). This makes speed a rather insubstantial sort of entity, a will-o’-the-wisp, at any rate compared to  ‘weight’, ‘impact’, ‘position’, ‘pain’ and so forth. The difficulty is compounded by the fact that we almost always consider ourselves to be ‘at rest’ : it is the countryside we see and experience whizzing by us when seated in a train. It requires a tremendous effort of imagination to see things from ‘the other object’s point of view’. Even a sudden jolt, an acceleration, is registered as a temporary annoyance that is soon replaced by the same self-centred ‘state of rest’. Highly complex and contrived set-ups like roller-coasters and other fairground machines are required to give us the sensation of ‘acceleration’ or ‘irregular movement’, a sensation we find thrilling precisely because it is so inhabitual. Basically, we think of ourselves as more or less permanently at rest, even when we know we are moving around. In UET everything actually is at rest for the space of a single ksana, it does not just appear to be and everything that happens occurs ‘at’ or ‘within’ a ksana (the elementary temporal interval).
I propose to take things further ─ not in terms of personal experience but physical theory. As stated, there is in UET no such thing as ‘continuous motion’, only succession ─ a succession of stills. An event takes place here, then a ksana or more later, another event, its replica perhaps, takes place there. What matters is what occurs and the number and order of the events that occur, everything else is secondary. This means not only that ultimate events do not move around ─ they simply have occurrence where they do have occurrence ─  but also that the distances between the events are in a sense ‘neither here nor there’, to use the remarkably  apt everyday expression. In UET v signifies a certain number of grid-spaces to right or left of a fixed point, a shift that gets repeated every ksana (or in more complex cases with respect to more than one ksana). In the case of a truncated event-chain consisting of just two successive events, v is the same as d, the ‘lateral displacement’ of event 2 with respect to the position of event 1 on the Locality (more correctly, the ‘equivalent’ of such a position a ksana later). Now, although the actual number of ‘grid-positions’ to right or left of an identifiable spot on the Locality is fixed, and continues to be the same if we are dealing with a ‘regular’ event-chain, the distance between the centres (‘kernels’) of adjacent spots is not fixed but can take any number (sic) of permissible values ranging from 0 to c* according to the circumstances. The ‘distance’ from one spot to another can thus be reckoned in a variety of legitimate ways ─ though the choice is not ‘infinite’. The force of the Constancy of the Occupied Region Principle is that, no matter how these intra-event distances are measured or experienced, the overall ‘area’ remains the same and is equal to that of the ‘default’ case, that of a ‘stationary’ Event Capsule (or in the more extended case a succession of such capsules).
This is a very different conception from that which usually prevails within Special Relativity as it is understood and taught today. Discussing the question of the ‘true’ speed of a particular object whose speed  is different according to what co-ordinate system you use, the popular writer on mathematics, Martin Gardner, famously wrote, “There no truth of the matter”. Although I understand what he meant, this is not how I would put it. Rather, all permissible ‘speeds’, i.e. all integral values of v, are “the truth of the matter”. And this does not lead us into a hopeless morass of uncertainty where “everything is relative” because, in contrast to ‘normal’ Special Relativity, there is in UET always a fixed framework of ultimate events whose number within a certain region of the Locality and whose individual ‘size’ never changes. How we evaluate the distances between them, or more precisely between the spots where they can and do occur, is an entirely secondary matter (though often one of great interest to us humans).

Space contraction and Time dilation 

In most books on Relativity, one has hardly begun before being launched into what is pretty straightforward stuff for someone at undergraduate level but what is, for the layman, a completely indigestible mass of algebra. This is a pity because the actual physical principle at work, though it took the genius of Einstein to detect its presence, is actually extreme simple and can much more conveniently be presented geometrically rather than, as usual today, algebraically. As far as I am concerned, space contraction and time dilation are facts of existence that have been shown to be true in any number of experiments : we do not notice them because the effects are very small at our perceptual level. Although it is probably impossible to completely avoid talking about ‘points of view’ and ‘relative states of motion’ and so forth, I shall try to reduce such talk to a minimum. It makes a lot more sense to forget about hypothetical ‘observers’ (who most of the time do not and could not possibly exist) and instead envisage length contraction and time dilation as actual mechanisms which ‘kick in’ automatically much as the centrifugal governor on Watt’s steam-engine kicks in to regulate the supply of heat and the consequent rate of expansion of the piston. See things like this and keep at the back of your mind a skeletal framework of ultimate events and you won’t have too much trouble with the concepts of space contraction and time dilation. After all why should the distances between events have to stay the same? It is like only being allowed to take photographs from a standing position. These distances don’t need to stay the same provided the overall area or extent of the ‘occupied region’ remains constant since it is this, and the causally connected events within it, that really matters.
Take v to represent a certain number of grid-spaces in one direction which repeats; for our simple truncated event-chain of just two events it is d , the ‘distance’ between two spots. d is itself conceived as a multiple of the ‘intra-event distance’, that  between the ‘kernels’ of any two adjacent ‘grid-positions’ in a particular direction. For any specific case, i.e. a given value of d or v, this ‘inter-possible-event’ distance does not change, and the specific extent of the kernel, where every ultimate event has occurrence if it does have occurrence, never changes ever. There is, as it were, a certain amount of ‘pulpy’, ‘squishy’ material (cf. cytoplasm in a cell) which surrounds the ‘kernel’ and which is, as it were, compressible. This for the ‘spatial’ part of the ‘Event Capsule’. The ‘temporal’ part, however, has no pulp but is ‘stretchy’, or rather the interval between ksanas is.
If the Constant Region Postulate is to work, we have somehow to arrange things that, for a given value of v or d, the spatial and temporal distances sort Relativity Circle Diagram tan sinthemselves out so that the overall area nonetheless remains the same. How to do this? The following geometrical diagram illustrates one way of doing this by using the simple formula tan θ = v/c  =  sin φ . Here v is an integral number of grid-positions ─ the more complex case where v is a rational number will be considered in due course ─ and c is the lowest unattainable limit of grid-positions (in effect (c* + 1) ).
Do these contractions and dilations ‘actually exist’ or are they just mathematical toys? As far as I am concerned, the ‘universe’ or whatever else you want to call what is out there, does exist and such simultaneous contractions and expansions likewise. Put it like this. The dimensions of loci (spots where ultimate events could in principle have occurrence) in a completely empty region of the Locality do not expand and contract because there is no ‘reason’ for them to do so : the default dimensions suffice. Even when we have two spots occupied by independent, i.e. completely disconnected,  ultimate events nothing happens : the ‘distances’ remain the ordinary stationary ones. HOWEVER, as soon as there are causal links between events at different spots, or even the possibility of such links, the network tightens up, as it were, and one can imagine causal tendrils stretching out in different directions like the tentacles of an octopus. These filaments or tendrils can and do cause contractions and expansions of the lattice ─ though there are definite elastic limits. More precisely, the greater the value of v, the more grid-spaces the causal influence ‘misses out’ and the more tilted the original rectangle becomes in order to preserve the same overall area.
We are for the moment only considering a single ‘Event Capsule’ but, in the case of a ‘regular event-chain’ with constant v ─ the equivalent of ‘constant straight-line motion’ in matter-based physics ─ we have  a causally connected sequence of more or less identical ‘Event Capsules’ each tilted from the default position as much as, but no more than, the last (since v is constant for this event-chain).
This simple schema will take us quite a long way. If we compare the ‘tilted’ spatial dimension to the horizontal one, calling the latter d and the former d′ we find from the diagram that d′ cos φ = d and likewise that t′ = t/cos φ . Don’t bother about the numerical values : they can be worked out  by calculator later.
These are essentially the relations that give rise to the Lorentz Transformations but, rather than state these formulae and get involved in the whole business of convertible co-ordinate systems, it is better for the moment to stay with the basic idea and its geometrical representation. The quantity noted cos φ which depends on  v and c , and only on v and c, crops up a lot in Special Relativity. Using the Pythagorean Formula for the case of a right-angled triangle with hypotenuse of unit length, we have

(1 cos φ)2 + (1 sin φ)2 = 12  or cos2 φ + sin2 φ = 1
        Since sin φ is set at v/c we have
        cos2 φ  = 1– sin2 φ   = 1 – (v/c)2       cos φ = √(1 – (v/c)2

         More often than not, this quantity  (√(1 – (v2/c2)  (referred to as 1/γ in the literature) is transferred over to the other side so we get the formula

         d′ = (1/cos φ) d   =     d /( √(1 – (v2/c2))      =  γ d

Viewed as an angle, or rather the reciprocal of the cosine of an angle, the ubiquitous γ of Special Relativity is considerably less frightening.

A Problem
It would appear that there is going to be a problem as d, or in the case of a repeating ‘rate’, v, approaches the limit c. Indeed, it was for this reason that I originally made a distinction between an attainable distance (attainable in one ksana), c*, and an unattainable one, c. Unfortunately, this does not eliminate all the difficulties but discussion of this important point will  be left to another post. For the moment we confine ourselves to ‘distances’ that range from 0 to c* and to integral values of d (or v).

Importance of the constant c* 

Now, it must be clearly understood that all sorts of ‘relations’ ─   perhaps correlations is an apter term ─ ‘exist’ between arbitrarily distant spots on the Locality (distant either spatially or  temporally or both) but we are only concerned with spots that are either occupied by causally connected ultimate events, or could conceivably be so occupied. For event-chains with a 1/1 ‘reappearance rhythm’  i.e. one event per ksana, the relation tan θ = v/c = sin φ (v < c) applies (see diagram) and this means that grid-spots beyond the point labelled c (and indeed c itself) lie ‘outside’ the causal ‘Event Capsule’ Anything that I am about to deduce, or propose, about such an ‘Event Capsule’ in consequence does not apply to such points and the region containing them. Causality operates only within the confines of single ‘Event Capsules’ of fixed maximum size, and, by extension, connected chains of similar ‘Event Capsules’.
Within the bounds of the ‘Event Capsule’ the Principle of Constant Area applies. Any way of distinguishing or separating the spots where ultimate events can occur is acceptable, provided the setting is appropriate to the requirements of the situation. Distances are in this respect no more significant than, say, colours, because they do not affect what really matters : the number of ultimate events (or number of possible emplacements of ultimate events) between two chosen spots on the Locality, and the order of such events.
Now, suppose an ultimate event can simultaneously produce a  clone just underneath the original spot,  and  also a clone as far as possible to the right. (I doubt whether this could actually happen but it is a revealing way of making a certain point.)
What is the least shift to the right or left? Zero. In such a case we have the default case, a ‘stationary’ event-chain, or a pair belonging to such a chain. The occupied area, however, is not zero : it is the minimal s03 t0 . The setting v = 0 in the formula d′ = (1/cos φ) d makes γ = 1/√(1 – (02/c2) = 1 so there is no difference between d′ and d. (But it is not the formula that dictates the size of the occupied region, as physicists tend to think : it is the underlying reality that validates the formula.)
For any value of d, or, in the case of repetition of the same lateral distance at each ksana, any value of v, we tilt the rectangle by the appropriate amount, or fit this value into the formula. For v = 10 grid-spaces for example, we will have a tilted Space/Time Rectangle with one side (10 cos φ) sand the other side                 (1/10 cos φ) t0 where sin φ = 10/c   so cos φ = √1 – (10/c)2  This is an equally valid space/time setting because the overall area is
         (10 cos φ) s0    ×   (1/10 cos φ) t0   =  s t0      

We can legitimately apply any integral value of v < c and we will get a setting which keeps the overall area constant. However, this is done at a cost : the distance between the centres of the spatial element of the event capsules shrink while the temporal distances expand. The default distance s0 has been shrunk to s0 cos φ, a somewhat smaller intra-event distance, and the default temporal interval t0 has been stretched to t0 /cos φ , a somewhat greater distance. Remark, however, that sticking to integral values of d or v means that cos φ does not, as in ‘normal’ physics, run through an ‘infinite’ gamut of values ─ and even when we consider the more complex case, taking reappearance rhythms into account, v is never, strictly never, irrational.
What is the greatest possible lateral distance? Is there one? Yes, by Postulate 2 there is and this maximal number of grid-points is labelled c*. This is a large but finite number and is, in the case of integral values of v, equal to c – 1. In other words, a grid-space c spaces to the left or right is just out of causal range and everything beyond likewise (Note 2).

Dimensions of the Elementary Space Capsule

I repeat the two basic postulates of Ultimate Event Theory that are in some sense equivalent to Einstein’s two postulates. They are

1. The mixed Space/Time volume/area of the occupied parallelipod/rectangle remains constant in all circumstances

 2. There is an upper limit to the lateral displacement of a causally connected event relative to its predecessor in the previous ksana

        Now, suppose we have an ultimate event that simultaneously produces a clone at the very next ksana in an equivalent spot AND another clone at the furthest possible grid-point c*. Even, taking things to a ridiculous extreme to make a point, suppose that a clone event is produced at every possible emplacement in between as well. Now, by the Principle of the Constancy of the Occupied Region, the entire occupied line of events in the second ksana can either have the ‘normal’ spacing between events which is that of the ‘rest’ distance between kernels, s0, or, alternatively, we may view the entire line as being squeezed into the dimensions of a single ‘rest’ capsule, a dimension s0 in each of three spatial directions (only one of which concerns us). In the latter case, the ‘intra-event’ spacing will have shrunk to zero ─ though the precise region occupied by an ultimate event remains the same. Since intra-event distancing is really of no importance, either of these two opposed treatments are ‘valid’.
What follows is rather interesting: we have the spatial dimension of a single ‘rest’ Event Capsule in terms of su, the dimension of the kernel. Since, in this extreme case, we have c* events squashed inside a lateral dimension of s0, this means that
s0 = c* su , i.e. the relation s0 : su = c*: 1. But s0 and su are, by hypothesis, universal constants and so is c* . Furthermore, since by definition sv tv = s0 t0 = Ω = constant , t0 /tv = sv/s0 and, fitting in the ‘ultimate’ s value, we have t0 /tu = su/c* su    = 1 : c*. In the case of ‘time’, the ‘ultimate’ dimension tu is a maximum since (by hypothesis) t0 is a minimum. c* is a measure of the extent of the elementary Event Capsule and this is why it is so important.
In UET everything is, during the space of a single ksana, at rest and in effect problems of motion in normal matter-based physics become problems of statics in UET ─ in effect I am picking up the lead given by the ancient Greek physicists for whom statics was all and infinity non-existent. Anticipating the discussion of mass in UET, or its equivalent, this interpretation ‘explains’ the tremendously increased resistance of a body to (relative) acceleration : something that Bucherer and others have demonstrated experimentally. This resistance is not the result of some arbitrary “You mustn’t go faster than light” law : it is the resistance of a region on the Locality of fixed extent to being crammed full to bursting with ultimate events. And it does not matter if the emplacements inside a single Event Capsule are not actually filled : these emplacements, the ‘kernels’, cannot be compressed whether occupied or not. But an event occurring at the maximum number of places to the right, is going to put the ‘Occupied Region’ under extreme pressure to say the least. In another post I will also speculate as to what happens if c* is exceeded supposing this to be possible.      SH    9/3/14

Notes:

Note 1  Zeno of Elea noted the ‘relativity of speed’ about two and a half thousand years before Einstein. In his “Paradox of the Chariot”, the least known of his paradoxes, Zeno asks what is the ‘true’ speed of a chariot engaged in a chariot race. A particular chariot has one speed with respect to its nearest competitor, another compared to the slowest chariot, and a completely different one again relative to the spectators. Zeno concluded that “there was no true speed” ─ I would say, “no single true speed”.

Note 2  The observant reader will have noticed that when evaluating sin φ = v/c and thus, by implication, cos φ as well, I have used the ‘unattainable’ limit c while restricting v to the values 0 to c*, thus stopping 1/cos φ from becoming infinite. Unfortunately, this finicky distinction, which makes actual numerical calculations much more complicated,  does not entirely eliminate the problem as v goes to c, but this important issue will be left aside for the moment to be discussed in detail in a separate post.
If we allow only integral values of v ranging from 0 to c* = (c – 1), the final tilted Casual Rectangle has  a ludicrously short ‘spatial side’ and a ridiculously long ‘temporal side’ (which means there is an enormous gap between ksanas). We have in effect

tan θ = (c–1)/c  (i.e. the angle is nearly 45 degrees or π/4)
and γ = 1/√1 – (c–1)2/c2 =  c/√c2 – (c–1)2 = c/√(2c –1)
Now, 2c – 1 is very close to 2c  so     γ  ≈ √c/2   

I am undecided as to whether any particular physical importance should be given to this value ─ possibly experiment will decide the issue one day.
In the event of v taking rational values (which requires a re-appearance rhythm other than 1/1), we get even more outrageous ‘lengths’  for sv and tv . In principle, such an enormous gap between ksanas, viewed from a vantage-point outside the speeding event-chain, should become detectable by delicate instruments and would thus, by implication, allow us to get approximate values for c and c* in terms of the ‘absolute units’ s0 and t0 . This sort of experiment, which I have no doubt will be carried out in this century, would be the equivalent in UET of the famous Millikan ‘oil-drop’ series of experiments that gave us the first good value of e, the basic unit of charge.

What is time? It is succession. Succession of what? Of events ─ events that take place ‘one after the other’. If there is no succession of events, either nothing happens at all or everything that can happen occurs in an endless, eternal present.
We measure time by referring to two (or more) events which are easily recognizable, themselves having  negligible extension ‘in time’  and which repeat in a fashion we have reason to believe does not change appreciably. Tick-tock. The sounds are sharp, easily recognizable and the old-fashioned pendulum swings fairly regularly ─ though, of course, certain crystals vibrate a good deal more regularly. If we do  not have two  easily recognizable, repeating,  ‘marker events’ which signify the beginning and end of an ‘interval of time’, ‘duration’ is vague and subjective. This shows that duration is a secondary notion compared to succession. Without succession, no duration ─ or at least no accurately measurable duration. One can argue interminably about how long the interval between two events ‘really’ is, or was, but the events themselves either happen/happened, or they don’t/didn’t happen. Occurrence and succession are primary features of physical reality, duration a secondary property.
As I see it, physics ought by rights to be based chiefly on notions of occurrence, succession and causality since all these ‘things’ are primary ─ causality perhaps a shade less fundamental than the other two (since there can be occurrence without causality as in the case of ‘random’ events). In Ultimate Event Theory, the occurrence of an event is absolute in the sense that its occurrence has nothing whatsoever to do with anyone’s location, state of mind, state of motion in relation to other objects, and so on. Also,  since all events are (by hypothesis) made up of a finite number of ultimate events, every event-chain A-B whose first event is A and last event B has an event-number which is a positive integer. But the distances between the successive ultimate events composing the chain are secondary features : they have minimum and maximum values but otherwise are flexible and are legitimately evaluated quite differently according to one’s viewpoint and ‘state of motion’ (Note 1).

Space/Time Event-Capsules and ‘Objects’ 

In the preceding posts, it was hypothesized that the region occupied by the ‘Space-Time Rectangle’ of a single event-chain  or event-complex is constant. In the simplest case of a single repeating ultimate event, we have a repeating unitary four-dimensional  ‘Event Capsule’ successively occupying a region s0 × s0 × s0 × t0 = s03 t0. We, of course, do not ‘see’, or otherwise register, each individual event capsule but run several of them together much as the eye/brain runs together the separate stills that compose a ‘motion’ picture. We eventually become aware of an ‘event block’ which is nonetheless (according to UET) composed of a set of identical unitary event capsules each of them occupying a region s03 t0. This ‘occupied region’ is thus  d s03 × t t0  where d and t are integers. If all the available positions within this region are occupied by repeating ultimate events, we have a repeating  Event Conglomerate of volume d3 × t  in ‘absolute’ units. For simplicity, we shall only consider one of the spatial dimensions and confine ourselves to the Space/Time Event Rectangle of d × t with d and t in ‘absolute’ units. This is the equivalent in UET of a ‘solid object’.
Of course, it is most unlikely that all neighbouring. spatial positions would be occupied by ultimate events which, taken together, constitute Relativity Unitary Capsule Emptythe equivalent of an  ‘object’ or body : there will, almost certainly, be sizeable gaps just as there are ‘holes’ in an apparently solid body. Nonetheless, at least in imagination, we can connect up any two positions on the Locality and construct a ‘Space/Time’ Event Region which, in the simplified case, reduces to a Space/Time Event Rectangle composed of unitary Space/Time Event Rectangles.

 

 

In the following simplified diagram each line represents a section of the Event Locality at one particular ksana (‘moment in time’) and each square a grid-position.

ksana 0  ……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
ksana 1   ……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐……..
ksana 2  ……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
ksana 3  ……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
ksana 4 ……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………

Any of these little squares symbolizing two-dimensional grid positions could, in theory, receive an ultimate event and any two arbitrarily selected grid-positions could be connected up, for example those marked in black

……..⅐∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐⅐⅐⅐⅐⅐⅐⅐∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………

If we treat these grid-positions as extremities of an ‘occupied region’ equivalent to a perfectly dense ‘object’ occupying the entire rectangle, we have

……..⅐∎∎∎∎∎∎∎∎∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐∎∎∎∎∎∎∎∎∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐∎∎∎∎∎∎∎∎∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………

An object in UET is, then,  an identically repeating event conglomerate that only persists because the individual ultimate events are powerfully bonded both ‘laterally’, within a single ksana, and, more significantly, ‘vertically’, from one ksana to the next. If ultimate events are not bonded ‘laterally’, they do not constitute an ‘object’ : their proximity is entirely coincidental and if one or more of these ultimate events for some reason ceased to reappear, this would have no consequences on the others. And if the ultimate events are not bonded ‘vertically’, i.e. lack the property called persistence, the entire conglomerate does not repeat : it simply disappears without a trace.
Now, since everything in UET is finite (except the extent of the Locality itself), there must be a limit to the possible extent of ‘lateral bonding’ : that is, an ‘object’ cannot exceed certain dimensions at any one ksana. There is presumably also a limit to the number of times any ultimate event can repeat though, to judge by the length of time the present universe has existed and certain elementary particles such as protons, this limit must be inconceivably great. Much more important practically is the ‘lateral displacement limit’. For example, consider two occupied grid positions at successive ksanas

……..⅐∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………

Could these two ultimate events be causally connected, i.e. part of an event-chain? It is quite irrelevant that there are a large number of blank spaces between the two occupied positions since, in UET, it is not necessary for an event chain to fill the intervening space ─ a completely dense repeating event conglomerate is almost certainly a rarity and even perhaps  impossible.
The only problem is thus the extent of the lateral displacement from one  ksana to the next since this has a maximum possible value, traditionally noted as c but  in UET noted as c*. This gives us a test as to whether the two events occupying two squares are, or could conceivably be, causally connected. If the lateral distance covered in one ksana exceeds a certain number of spaces, namely c*,  the events are not causally connected. Of course, it is essentially a matter of convenience, or of viewpoint, which of two regular event chains is considered to be ‘vertical’ and which ‘slanting’, but this does not mean that lateral displacement of events from one ksana to another does not occur. We can imagine the original ultimate event repeating in an exactly equivalent spatial position at the very next ksana, and simultaneously producing a  clone’ of itself so many spaces to the right or left.

……..⅐∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………

If the distance is too great, we can confidently conclude that the red ultimate event has not been produced by the black one since the lateral distance exceeds the reach of a causal impulse emanating from the point of its occurrence (Note 3). All this is, of course, well known to students of Relativity, but it is important to recognize how naturally and inevitably this result (and all that follows from it) arises once we have accepted once and for all that there must, for a priori reasons, be a spatial limit to the transmission of a causal impulse. Someone in another place and time could have (and possibly did) hit upon such a conclusion  purely from first principles centuries before Einstein was even born (Note 4 Galileo…..)and when there would have been no way of carrying out appropriate experimental tests just as there was, in  Newton’s own day,  no way of showing that two small objects suspended in a room and free to move actually attract each other.

Causality and the unitary Space/Time Event Capsule

In normal physics, it is assumed that the ‘natural’ state of a body is to carry on existing more or less in the same state from one moment to the next, and if the  body does change, a fortiori disappear, we conclude that an external (or sometimes internal) force is at work. An ‘object’ does not ‘cause itself to happen’ as it were : the very idea sounds absurd. However, In UET, this is precisely what does prevail since the ‘natural state’ of everything is to appear once and then disappear for ever. The reappearance of an ultimate event more or less in the same position on the Locality is just as much the result of causality as the sudden appearance of a completely new, and, in general, different, event regarded as the ‘effect’. The continued existence of anything is ‘self-caused’ (Note 5 Sheldrake).
So, before examining repeating batches and conglomerates of ultimate events, we should examine exactly what is involved in the reappearance of a single ultimate event since without this happening there will be no repeating conglomerates, no apparently solid objects, no universe, nothing at all except a sort of “buzzing. blooming confusion” (Piaget) brought about by ephemeral random events emerging from the Event Origin and at once disappearing back into it.
For anything to last at all, certain conditions must be met. The first and most important is that the causal influence which brings about the repetition of the ultimate event must at least be able to traverse the distance from the centre of one Event Capsule to another. For, just as the  nucleus of an atom does not extend to the outer reaches of the atom but on the contrary is marooned in a comparatively vast empty area, an ultimate event in UET is conceived as occupying a minute ‘kernel’ at the centre of a Space/Time Event Capsule. When such an ultimate event is isolated or part of an event-chain ‘at rest’, the dimensions of all such capsules are fixed and are always the same. These ‘rest’ dimensions are the ‘true’ dimensions of the capsule and are s0 for the spatial dimension and t0 for the ‘time’ dimension.
‘Distance’, both spatial and temporal, is not absolute in UET but can be legitimately ‘measured’ (or, better, experienced) in different ways — though there are nonetheless, as for everything in UET, minimal and maximal values. What is ‘absolute’ is firstly, the number of ultimate events in an event-chain (or portion thereof) and, secondly, the total extent of the ‘occupied region’ on the Locality. In the simplest possible case which is what we are considering here, we have just a single ultimate event which repeats in the ‘same’ position a ksana later. To accurately model what I believe goes on in reality, it would be necessary not only to have a three-dimensional grid or lattice-frame extending as far as the eye can see in all three directions but also for it to consist of lines of lights which are switched on and off regularly. An ultimate event would then be represented by, say, a red flash inside a rectangular ‘box’ consisting of lines of little lights arranged in series. The whole framework of ‘fairy lights’ representing the Event Locality is then switched on and off, and each time the framework is switched on, a red flash appears inside a lower ‘box’. In the case of two appearances, the ‘vertical displacement’ of the red light would scarcely be noticeable, but if we keep on switching the whole framework on and off and having the red light appear slightly below where it was previously, we obtain a rough  idea of a ‘stationary’ event-chain. To represent ‘lateral displacement’, which is generally what is understood by ‘movement’, we would need to have a second flashing coloured light displacing itself regularly in a slanting straight line relative to the red one. By speeding up the flashings, we would get an impression of ‘continuous movement’ even though nothing is moving at all, merely flashing on and off.
We have then, as it were, arrays of 3- dimensional boxes representing the Event Capsules at a particular ksana, and a  red light, symbolizing an ultimate event, which appears inside first one box and then another below it at the very next ksana.Relativity Rows of capsules

For a ‘rest’ event-chain which we are assuming, the event capsules have fixed ‘rest’ values for both spatial and temporal distances. If we neglect two of the three spatial dimensions we thus have a Space-Time Rectangle of extent s0  by t0.  

 

Minimum and Maximum Displacement Rates 

Imagine then an ultimate event occurring  at a certain spot on the Locality, or rather at the kernel (centre) of an empty grid-space. It remains there for the space of one ksana, disappears, and reappears (or not) at an equivalent spot at the next ksana. How far has the causal influence travelled? If the ultimate event is conceived as occurring at the very centre of a spatial cube of dimension s0

˜Relativity Single event capsule

So the causal impulse must cover very nearly ½ s0 relative to the first grid-space and another ½ s0 relative to the second ksana, in order to  be able to ‘recreate’ a clone of the original ultimate event. The total spatial distance covered is thus (very nearly) ½ s0  + ½ s0  =  1s0 and this distance has been covered  within the ‘space’ of a single ksana. The ultimate event’s ‘vertical displacement rate’ is thus 1s0 /t0 , one grid-space per ksana. And if it keeps re-appearing regularly in the same way, it will keep  the same ‘vertical’ displacement rate — ‘vertical’ because it is not displacing itself either to the left or the right at each ksana relative to where it was previously.
So what happens if the causal influence is not strong enough to traverse such a distance? In such a case, the ultimate event does not re-appear and that is the end of the matter. (I am assuming that the occurrence has taken place at a sparsely populated region of the Locality so that the ultimate event is effectively isolated and not subject to any influence from other event-chains.) Whether or not a causal influence that fails to ‘go the distance’ subsequently completes its work in subsequent ksanas need not concern us for the moment : the point is that during (or ‘at’) the following ksana the ultimate event either does or does not reappear, an open/shut case.
A displacement rate of 1s0 /t0 is thus the minimum (vertical) displacement rate possible. Moreover, the dimensions s0 and t0 are fixed and thus their product, the rectangular ‘area’  s0 × t0 also. It is a postulate of UET that this rectangular area (and equivalent 4-dimensional region) remains constant though the ‘length’ of the ‘sides’ can change, i.e.  for variable ‘sides’  sv and tv, sv tv = s0  t0  = Ω, a constant, and so tvtv /t0 = s0 /sv . It would seem that, in our universe at any rate, s0 is a maximum which makes t0 a minimum. So the vertical displacement rate is maximum spatial unit distance/ minimum temporal unit distance. Note that, since to is a minimum, there is no possible change that can occur anywhere within a smaller interval of time ─ ‘time’ is not infinitely divisible. Also, since there are limits to everything, the minimum spatial distance, which can be noted su (for s ultimate), will be paired off with the maximum ‘temporal length’ of a ksana, tmax and the other extreme ratio will be smin/tmax  = su/tmax where su  ×  tmax  = so × t0 since the area of the rectangle stays the same.  
        But we do not need to pay attention to this for the moment since the sides are not at present going to change. Given that the rate 1s0 /t0  is the least possible, what about the maximum possible rate? Rather surprisingly, this also turns out to be 1s0 /t0 ! For suppose a more powerful causal impulse is able to carry itself over two or more event capsules and recreate the selfsame ultimate event two, four, or a hundred ksanas later. Even if it could do this, the rate would still not exceed 1s0 /t0 since, if the ultimate event were re-created four ksanas later, the causal impulse would have traversed a spatial distance of 4 s0 ­and taken exactly four ksanas to do it.

Relation of vertical displacement rate to ‘speed’ or lateral displacement rate  

In ‘normal’ physics ─ and normal conversation ─ we completely neglect what I term ‘vertical displacement rate’ since we do not conceive of ‘bodies’ appearing and re-appearing. What we are passionately concerned about is what we refer to as ‘speed’, and this in  UET is what I call ‘relative lateral displacement rate’ ─ ‘lateral’ because the ‘time’ axis is usually imagined as being ‘vertical’. Now, it is today (almost) universally recognized that there is a maximum ‘speed’ for all bodies, namely c which is roughly 3 × 108 metres/sec in macroscopic unts. In UET, since everything has a limit, there is a maximum lateral displacement rate for the transmission of causality whether or not light and electro-magnetic radiation actually do travel at exactly this rate. This ‘relative lateral displacement rate’ covers not only cases of ‘cause and effect’ in the normal sense, but the special kind of causality involved when an ultimate event reinvents itself so many spaces to the right or left of its spatial position at the previous ksana. We can thus imagine an ultimate event being re-created (1) in the same position, i.e. zero spaces to the right or left;  (2) one space to the right (or left) of the original position, (3) two spaces to the right, and so on up to, but not exceeding, c* spaces to the right where c* is a positive integer.

……..⅐∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐⅐⅐⅐⅐⅐⅐⅐∎⅐⅐⅐⅐⅐⅐⅐⅐⅐………

←  spaces   →

……..⅐∎⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐………
……..⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐⅐∎⅐⅐⅐⅐⅐………

←                     c* spaces           →

We have, in each case, a ‘Causal’ Space/Time Event Rectangle of dimensions  d spaces × 1 ksana  where d can attain but not exceed c*. Each of these positions could in principle receive an ultimate event and if all are filled (an unlikely occurrence) we will have at each ksana exactly d ultimate events not counting the original one (or d + 1 if we do count the first one). This number of possible ultimate events, brought about by a single ultimate event a ksana earlier, does not and cannot change but, in accordance with the principle of constant area, the distances between events (or rather their positions on the Locality) can and does change. We need not concern ourselves at the moment with the question of whether this change is ‘real’ or essentially subjective : the important  point is to get a clear visual image of the ‘sides’ of the Space/Time Event Rectangle contracting and expanding but nonetheless maintaining the same overall area. Some idea of this is given  by the following diagrams where the blobs represent ultimate events.

Limit of lengths of spatial and temporal ‘sides’ 

Can this process of contraction/expansion be continued indefinitely? In normal physics it can since, with some exceptions and occasional trepidation, contemporary physics still retains a key feature of Leibnitz’s and Newton’s Calculus, namely the usually unstated assumption that ‘space and time’ are ‘infinitely divisible’, incredible though this sounds (Note 2).  In UET on the other hand the spatial contraction has a clearcut limit which is the dimension of the ‘kernel’ of a Space/Time capsule. The precise region where an ultimate event can have occurrence, the equivalent of the nucleus of an atom, does not (by assumption) itself change in size and thus constitutes the ‘ultimate’ unit of distance in UET since everything that we see and hear or otherwise apprehend is made up of ultimate events and each of them (by hypothesis) occupies an equivalent 3-dimensional spot on the Locality whose ‘rest’ dimensions do not change.
Noting the ‘ultimate’ dimension as su, we deduce that there must be a numerical ratio between su and s0, the maximum dimension of a grid-space, which is also that of a ‘rest’ capsule, since everything in UET is basically related by whole number ratios to the complete exclusion of irrationals. So s0 = n su where n is a positive integer ─ i.e. one could in principle fit n ‘kernels’ lengthways and in the other two directions, if we assume a cubic shape, or, if the capsule is conceived as being spherical, we would have a diameter of n kernels.
Now, if the Event Rectangle keeps the same area, it must stay equal to its rest size of  1 s0 × 1 t0  and the spatial distances between ultimate events, or positions where ultimate events could occur, must contract while the temporal distances expand. At the limiting value of c* positions the spatial dimension has shrunk from being s0  to its minimum  size of su , i.e. each grid-position has shrunk to the size of the kernel and it cannot shrink any more since no smaller length exists or can exist. But there are precisely n multiples of su in s0  which can only mean that c* = n. And so the ratio of ‘kernel’, the smallest unit of length that can possibly exist, to the ‘rest’ size of the capsule is the same as the ratio of the ‘rest’ length of a grid-position to the maximum number of positions that can be displaced by a causal impulse within a single ksana. This unexpected result which flows from the basic assumptions of Ultimate Event Theory shows a pleasing symmetry since the constant c* , the limiting displacement rate (‘speed’), also shows up within the structure of the basic Event Capsule, the equivalent in UET of an ‘atom’.

Time Dilation 

When sv , the  variable length of the side of the Event Capsule, reaches its minimum value of su , tv is a maximum since, if  s0 ­is a maximum, t0  must be a minimum. Since  s0 × to  = sv  ×  tv , we have tmax /t0  = s0/su  = c*/1  so  tmax = c* × t0      As sv gets shorter, tv  increases  to keep the ‘occupied area’ constant.  When the spatial distances  d × s0  are large enough to be noticeable, (though the same thing applies if they are small) this gives rise the well-known phenomenon of ‘length contraction’ ─ well-known to students of physics, I mean. And this in turn means that the ‘time dimension’ gets extended to keep the overall area constant.

Relativity Expanding Contracting Rectangles

This is the same as the situation in Special Relativity. But, in UET, the Space/Time Rectangle does not end up with one side becoming ‘infinitely long’ and the other ‘infinitesimally short’ since v can essentially only take integral (or at most rational)  values and peaks at v = c* = (c – 1), i.e. one grid-space short of the ‘unattainable’ value c  ─ unattainable if we are considering a  Causal Space/Time Event Rectangle (Note 3).

Summary    The chief points arising from this discussion are :

(1) Every ultimate event has occurrence at the ‘kernel’ of a Space/Time Capsule of fixed ‘rest’ dimensions
s0 ×  s0 × s0 ×  s0 × t0  

(2) The ratio of the spatial dimension of the kernel su to the spatial dimension s0  is 1 : n ; 

(3) The least ‘re-appearance rate’ of an ultimate event is 1 s0 per ksana which also turns out to be the maximum rate;

(4) The limiting ‘lateral displacement rate’ of a causal impulse is set at c* s0 per ksana ;

(5) The volume of a causal ‘Space/Time Parallelipod’, reduced for simplicity to the area of a causal Space/Time Event Rectangle, is
constant and = s03 t0  or s0 t0  for a rectangle   ;

(6) The number of possible or actual ultimate events within a Casual Space/Time Parallelipod or Rectangle is constant and always a positive integer;

(7) The spatial and temporal distances between possible or actual ultimate events on or inside the Space/Time Parallelipod or Rectangle contract and expand in order to keep the overall volume/area constant;

(8) Spatial distances are submultiples (proper fractions)  of the basic ‘rest’ dimension s0  ;

(9) Temporal distances, i.e. the ‘duration’ of a ksana, the smallest interval of time, are multiples of the basic ‘rest’ temporal unit t0 ;

(10)  The maximum ‘lateral displacement rate’ is c* per ksana (c* a positive integer) where c* = n  

(11) There is a limiting value of the contracted spatial unit distance, namely su  which is the dimension of the kernel ; 

(12) There is a limiting value of the expanded unit temporal  distance, namely tmax =  c* t0   ;

All these features are derived from the basic postulates of Ultimate Event Theory. They differ from the usual features of Special Relativity in the following respects:

(1) Lengths cannot be indefinitely contracted, nor will they ever appear to be, and the same goes for time dilation. Popular books referring to someone falling into a Black Hole suggest that he or she will fall an ‘infinite distance’ in a single mini-second and his or her cry of despair will last for all eternity ─ this does not and cannot happen in UET. It should be possible one day soon to test these predictions (though not with respect to Black Holes): UET says that ‘space contraction’ and ‘time dilation’ will approach but never exceed limiting values, and indeed such observed limits would give us some idea of the dimensions of the Event Capsule.

(2) The actual speed (lateral displacement rate) of event-chains may attain, but not exceed, c* , which means they can be attributed a small but finite ‘mass’ ─ though ‘mass’ has not yet been properly defined in UET (see coming post). A rough definition would be something like the following.

mass :   capacity of an event-chain to resist any attempt to change its re-appearance rate, relative direction of ‘motion’, and a fortiori its very continuing re-appearance.”

(3)  It is in principle possible in UET for an event-chain to eventually exceed the maximum ‘causal displacement limit’ (roughly ‘speed’) but such an event-chain would immediately cease to repeat (having lost persistence or self-causation) and to all intents and purposes would ‘disappear into thin air’ leaving no trace. This would explain the sudden disappearances of ‘particles’ should this be observed. There would be no appreciable energy loss which conflicts with the doctrine of the Constancy of Mass/Energy (energy is as yet undefined in UET).

(To be continued in next Post)

NOTES

Note 1.  One should speak of ‘state of succession’ rather than ‘state of motion’ since continuous motion does not exist in UET (or in reality). But the phrase ‘state of succession’ seems strange even to me and ‘relative state of succession’ even stranger. All this goes to show how strongly we have been marked by the fallacious idea of continuous motion.

Note 2  Hume, the arch sceptic, memorably wrote “No priestly dogma invented on purpose to tame and subdue the rebellious reason of mankind ever shocked common sense more than the doctrine of  infinite divisibility with its consequences” (“Essay on Human Understanding”)

Note 3 We do not get the fantastic picture of someone falling into a Black Hole and being contracted down to nothing while his or her cry of despair lasts for all eternity. That there are definite limits to possible length contraction and time dilation is a proposition that is in principle verifiable — and I believe it will be verified during this century. And once we have approximate values for these limits, we may, by extrapolating backwards, obtain an idea of the dimensions of the basic Space/Time capsule, i.e.  s0 and  t0   .

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