Archives for category: Einstein

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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Space-Time

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

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

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

Succession

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

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

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

 Difference between Space and Time

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

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

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

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

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

Breaks in Time

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

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

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

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

Mosaic Time

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

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

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

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

 

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

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

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

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

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

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

 

 

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)

The Two Postulates of Special Relativity

 If you do not make some assumptions, you can never get started either in physics or mathematics ─  or for that matter in any area of research or endeavour. As stated in the previous post, Galileo kick-started a vast intellectual revolution with his originally rather innocuous suggestion that a man locked up in the  windowless cabin of a ship would not be able to tell whether the ship was in the harbour or proceeding at a steady pace in a straight line on a calm sea (presumably rowed by galley-slaves). Galileo does not seem to have been particularly interested in the topic of inertia as such and only introduced it into his Dialogue Concerning the Two Chief World Systems to meet the obvious objection, ”If the Earth is moving round the Sun, why don’t we register this movement?” In effect, Galileo’s answer was that neither do we necessarily register certain differences of motion here on Earth such as the difference between being ‘at rest’ in the harbour and being rowed at a steady pace on a calm sea. According to Galileo, the behaviour of physical objects inside the cabin would be exactly the same whether the ship was at rest or in constant straight-line motion.
Newton made a good deal more of the principle since it appears as his 1st Law of Motion and provides him with an extremely useful definition of ‘force’, namely something that disturbs this supposedly ‘natural’ state, that of rest or constant straight-line motion. Newton was nonetheless somewhat unhappy about Galileo’s principle because he felt that there ought to be some way of distinguishing between ‘absolute’ rest and constant straight-line motion. However, no mechanical experiment was actually able to decisively distinguish between the two states, either in Newton’s time or in later epochs. At the end of the 19th century, most physicists thought that an optical experiment, provided it was refined enough, ought to be able to distinguish between the two states and the failure of Michelsen and Morley to do so caused a crisis in the physical sciences.
This takes us to 1905 and to Einstein, then a ‘Technical Expert III Class’ in the Zurich Patent Office. Einstein subsequently claimed that the famous null result of the Michelsen-Morley experiment played very little role in what came to be known as the Special Theory of Relativity ─ special because it only applied to ‘inertial frames’ and ignored gravity completely. Einstein does briefly allude to “the unsuccessful attempt to discover any motion of the earth relative to the ‘light medium’ ” on the first page of his 1905 article but seems to be much more impressed by various experiments in electricity and magnetism, some of which he may have conducted himself as a student. In any case, Einstein from the beginning makes ‘relativity’ a matter of principle (rather than a conclusion based on data) though he does state that various ‘examples’ relating to electro-magnetism “suggest that the phenomena of electro-dynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest”.
Thus, in contradistinction to the various other physicists of the time who were anxious to find ingenious explanations for the null result of the Michelsen-Morley experiment, and in contrast to Newton himself who had misgivings on the subject, Einstein makes the ‘Principle of Relativity’ into a postulate  and one to which he is clearly strongly attracted. He immediately adds a second postulate, that “light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body”. Einstein claims that “these two postulates suffice for the attainment of a simple and consistent theory of the electro-dynamics of moving bodies”  

The Third Postulate 

 But do they? Are these two postulates in fact enough? We all take for granted a number of things and debate would be impossible if we had at every moment to state everything we assume to be the case, since this would include the notion that there is such a thing as a physical universe, that there is a ‘person’ who is writing these lines and so on and so forth. Einstein clearly takes on board a certain number of physical assumptions which practically everyone shared at the time, for example that there was such a thing as wave motion, such a thing as a rigid ‘body’, that physics was deterministic, that Maxwell’s equations were essentially correct and so on.
There is, however, one extra principle that is not completely obvious and which does play an important role in the derivation of Einstein’s results. This is the principle of the ‘homogeneity and isotropy of space and time’ as it is rather portentously stated in physics textbooks. Roughly what this means is that any ‘place’ and any ‘time’ is as good as another for carrying out observations or doing experiments. If ‘space’ were not homogeneous, an experiment carried out at a particular spot would not necessarily give the same results as one carried out at another spot (even if the temperature, pressure &c . were identical), nor would an experiment carried out today necessarily give the same result as an identical experiment carried out tomorrow. As for ‘isotropy’ it means “the same in all directions” and is put in to rule out the possibility of our being at the centre of a finite universe ─ for in such a case although each section of ‘space’ might be more or less the same our special position would affect what we saw and how far we saw.
The ‘homogeneity of space and time’ is by no means obvious : indeed, it is astonishing that scientists today feel able to talk confidently about what is happening, or has happened, in places no human being will ever be able to visit (such as distant galaxies). Even the principle is not strictly true ! In General Relativity ‘space’ is not a ‘neutral backdrop’ but is warped and deformed in the neighbourhood of massive bodies, so, in this sense, one ‘spot’ is not the same as another. And one ‘moment’ is not equivalent to another in Quantum Mechanics since exactly the same conditions can (and indeed sometimes must) give rise to different results.
But we can safely ignore such sophistications for the moment. The assumption of the ‘homogeneity of space’ enters implicitly into Einstein’s line of argument at certain points. It is essential that, for example, when he is talking of the velocity of one system relative to another inertial system that the situation is perfectly reversible and symmetric : there is no ‘up and down’, no ‘left and right’ and so forth in space. Whether we consider spaceship A to be moving away from spaceship B at constant velocity, or whether we consider it is spaceship B that is moving away from spaceship A is simply a matter of human convenience ─ and essentially comes down to where the observer, real or imagined, is positioned. This ‘equivalence’ is absolutely essential to Einstein’s thinking and that of his followers. The obstinate refusal to give preferential treatment to any ‘place’, ‘time’ or direction was subsequently extended to a refusal to give preferential treatment to any ‘frame’ and ultimately led on to the rejection (or radical redefinition of) the very concept of an ‘inertial frame’.
In his 1905 paper, Einstein does briefly allude to the homogeneity assumption since he says that “the equations [of motion] must be linear on account of the properties of homogeneity which we attribute to space and time” (Note 2).
Einstein also implicitly appeals to the principle of the conservation of energy in his 1905 paper and explicitly in the subsequent ‘popular’ book “Relativity, the Special and the General Theory”. Here, he writes, “The principle of [special] relativity requires that the law of the conservation of energy should hold not only with reference to a coordinate system K, but also with respect to every coordinate system K′ which is in a state of uniform motion of translation relative to K, or, briefly, relative to every ‘Galileian’ system.”
One could, of course, argue that belief in the conservation of energy was covered by Einstein’s blanket proposition that “the laws of physics take the same form in all inertial frames”. However, at the time very few people realized the full implications of the ‘law’ of the conservation of energy (which was only about fifty years old at the time anyway) so it is certainly worth singling it out for special consideration.

Concepts and Principles inherited from ‘classical’ and 19th century physics  

Since I am now irretrievably embarked on the reckless voyage towards a radically different physical theory, I have had to re-examine the basic concepts of matter-based physics and see what I can (and cannot) incorporate into UET while making only minor changes.
For a start, I am quite happy with the Newtonian concept of a ‘body’ which, redefined in UET terms, simply becomes a massive repeating event-cluster. And I have even less of a problem with the idea of the ‘homogeneity’ and general ‘neutrality’ of ‘Space/Time’ (in Special Relativity). The equivalent of the hybrid ‘Space/Time’ in UET is the Event Locality and it is assumed to be more or less the same everywhere and not to have any observable ‘effects’ on repeating event-clusters ─ e.g. it does not offer any resistance to their progress through or on it. So at least one of Einstein’s basic assumptions, the ‘homogeneity and isotropy of Space/Time’ carries over readily enough into Ultimate Event Theory.
So far so good. What of ‘inertial frames’? Newtonian mechanics considers a frame to be ‘inertial’ if a body inside it either stays put or continues on a straight path at constant speed. No force is required for this and Newton specifically defines a force as an external influence that causes a body to deviate from this ‘natural’ state. An inertial frame is not the same thing as a stationary frame, or rather one perceived as being so. Every ‘observer’  tends to consider him or herself ‘at rest’ firmly anchored to a stationary frame of reference which is why, for example, we still talk about the ‘rising’ and the ‘setting’ of the sun.
So, is it possible to decide whether we are ‘really’ at rest? It is, in many cases, possible to decide that we are not in a state of rest or constant straight-line motion even though at first sight it would seem that we are.  A rotating frame is not an inertial frame and within such a frame Newton’s laws of motion do not hold ─ to make them apply we have to add in so-called ‘fictitious’ forces, centrifugal, Coriolis and so on. Over a short period of time we might ─ and almost always do ─ consider the Earth to be an inertial frame but experiments like Foucault’s Pendulum (on show at the Science Museum, London and elsewhere) demonstrate that the Earth is not an inertial frame since there is, apparently, a force making a free-swinging pendulum move in an arc relative to the floor. Since we have not given the pendulum a push in any direction and can neglect varying air pressure and suchlike effects on a heavy object such as a pendulum, the pendulum should stay put relative to the floor and us. Since it does not stay put, either Newton’s Laws are wrong or what appears at first sight to be an inertial frame, i.e. the Science Museum and the Earth to which it is attached, is not in fact an inertial frame.
But this case is untypical : generally it is not at all easy to decide whether a ‘frame’ is inertial or not. In any case, a building attached to the Earth, even supposing the latter were not rotating on its axis, was, according to Einstein post-1905, not a true inertial frame. For Einstein decided that what had previously been thought of as an ‘inertial frame’ in the sense of it being a ‘force-free frame’ was not in fact inertial. Stand in a room with an apple in your hand and let go of the apple. What happens? It does not stay suspended in mid-air as by rights it ought to according to Newton’s 1st Law, nor for that matter does it fall to the ground at a constant speed. Photographs of astronauts in orbit in conditions that are to all intents and purposes  force-free frames for brief periods of time, or the experiences of parachutists falling from a balloon at great height, have given us a better idea of what a ‘true’ inertial frame is like. A ‘true’ inertial frame is what Einstein called a ‘freely-falling frame’ and in such a frame if you let go of an apple it stays at the same height as you relative to the Earth (Note 3).

Inertial frames in UET 

So, what is the equivalent of an inertial frame in UET? We require at least two ‘entities’, an enveloping structure which is more or less rigid and seemingly permanent, and something inside it which is free to move about. The simplest ‘inertial frame’ ─ and ultimate the only true one in UET ─ is actually the ‘event-capsule’ itself, though I have only recently realized this. Each ultimate event is conceived as being confined inside a certain region that I call an event-capsule. This capsule is ‘flexible’ in shape and form but has a maximum and a minimum size ─ everything in UET has a maximum and a minimum. There are, by hypothesis, c* possible emplacements for an ultimate event ‘inside’ this capsule, though only one emplacement can be occupied at any one ksana.         Why is this the equivalent of an ‘inertial frame’? Because, by hypothesis, nothing can change during the ‘space’ of a ksana so the ultimate event (the equivalent of our apple) has to stay where it is and that is that. Also, although the shape of the surrounding capsule can and sometimes does vary from ksana to ksana its shape, volume and so on does not and cannot vary between the limits of a single ksana.  Thus the image, the schema. It certainly fits all the requirements of an ‘inertial system’ though it is an extremely reduced one, to say the least.
Since nothing lasts in UET (except the Event Locality itself), each ephemeral ‘inertial frame’ either disappears or, if part of an event-chain, re-appears at the next ksana. And if we have a number of event-chains in sync with each other and spatially close, we can easily construct the equivalent of a solid framework which itself contains a smaller repeating event-cluster. However, we very soon run into exactly the same problem as crops up in General Relativity. If repeating massive event-clusters deform the local Event Locality and have observable effects on neighbouring event-chains, any such smaller cluster will change in some way, most likely by changing its overall shape. We can in fact make change of shape a criterion for something not being an ‘inertial’ event frame, with the conclusion that a ‘true’ inertial event-frame, or indeed event-chain, can only exist if it is completely remote from all other clusters.
It transpires that an inertial event-frame, or event-chain, i.e. one where the shape of the capsule and/or the position of the ultimate event inside it do not change, is unrealizable in practice ─ and would certainly be unobservable because any observation would ruin its isolation. There are thus no true inertial event-the frames that last for more than a single ksana, whereas every event-capsule functions as the equivalent of a ‘true’ inertial frame (or ‘freely-falling frame’).
Although you will find this point glossed over in physics textbooks, exactly the same situation applies within General Relativity. To use the terminology of matter-based physics, gravitational fields are not homogeneous ─ certainly that surrounding the Earth is not ─ and even Einstein’s ‘falling workman + lunch-box’ is subject to gravitational forces that are continually changing, to what are known as ‘tidal forces’. The ‘pull’ of gravity on the falling workman’s head will be slightly more than that on his feet, and his body will contract a little widthwise because he is not being pulled straight down but towards the centre of the Earth. As one commentator, Fock,  puts it:

“The equivalence of accelerations and gravitational fields is entirely local, i.e. refers to a single point in space (more exactly to the spatial neighbourhood of the points on a time-like world line.)
(…) One can so transform the equations of motion of a mass point in a gravitational field that in this new system they will have the appearance of a free mass point. Thus a gravitational field can, so to speak, be replaced, or rather imitated by a field of acceleration. Owing to the equality of inertial and gravitational mass such a transformation is the same for any value of the mass of the particle. But it will succeed in its purpose only in an infinitesimal region of space” (Note 3)

         So, really all I am doing in UET is replacing the vague concepts of ‘point’ (which comes from Euclid) and ‘infinitesimal region’ (which comes from Newton and Leibnitz) by the precise image of an ‘event-capsule’.
There are, as far as I can ascertain, no such things as homogeneous gravitational fields : they are useful constructs like the idea of an ‘ideal’ gas and no more. Moreover, the normal physical/mathematical presentation even today involves us in the same sophistries as the infinitesimal calculus : at a certain height above the Earth the gravitational field, though ‘continually changing’, for all that is given a specific value (otherwise we could say nothing of any significance). Any logically coherent theory inevitably ends up with a schema similar to that of Ultimate Event Theory, namely that, within a sufficiently small region there is no change at all, while at different  levels we have  different values for some property such as pressure or gravitational potential. In other words the non-existent continuum of calculus breaks up into a discontinuum of adjacent self-contained regions. We associate a different value of some property with each region but within this region nothing changes. This is what physicists and engineers in effect do, and have to do,  ─ in which case why not lay your hands on the table and dispense with all this continuum nonsense, the lumber of a bygone era?

Upper Speed limit?  

Einstein developed his special theory within the context of electro-magnetism ─ the title of the famous 1905 paper is On the electrodynamics of moving bodies. Light, or rather electro-magnetic radiation, is given a privileged place amongst physical phenomena and the speed of light becomes a universal constant. Einstein is doing two things at once. He is first of all proposing, or rather assuming, that there is an upper limit to the speed of propagation of  all particles/radiation and, secondly, he is assuring us that electro-magnetism actually propagates at exactly this limiting speed. In other words c is not an asymptote ─ a quantity that one can approach closer and closer but never actually attains ─ but a reality.
Now the first assumption ─ that there is a limiting speed for all particles/radiation ─ is entirely reasonable and I cannot myself imagine a universe where this would not be the case. However, the second part, that light actually propagates at this speed, though it sounds at first sight innocuous enough, leads him, and all the physicists who follow him, into deep trouble.  Einstein in effect has his cake and eats it too. He states, “we shall find in what follows that the velocity of light in our theory plays the part, physically, of an infinitely great velocity” (section 4 of the paper). And yet ‘something’, namely light, apparently attains this ‘infinitely great velocity’.
In a later section, he derives an expression for the ‘energy of motion’ of an electron, namely  W =  mc2{(1 – v2/c2)1/2 – 1} and notes that “when v = c, W becomes infinite”. We thus seemingly have to conclude that a photon, or for that matter any other particle that attains c, must be massless. As it happens, photons do have mass in certain circumstances since, in General Relativity, light rays can be bent in the vicinity of massive bodies ─ the bending of starlight observed during a solar eclipse was the first confirmation of Einstein’s later theory. Physics textbooks, realizing there is a problem here, glibly say that photons do have ‘gravitational mass’ but not the inertial variety ─ even though, from the point of view of GR, the two are ‘equivalent’.
Now, conceptually all this is a wretched muddle. An ‘object’ without any mass at all would have strictly no resistance to any attempt to change its state of rest or constant straight line motion, so it is hard to see how it could be anything at all for more than a single instant. In UET terms, such an entity  would lack ‘persistence’, would not be able to maintain itself for more than a single ksana.
Of course, a good deal of this hinges on the strictly mathematical issue of what sense we are to give to division by zero. Whenever v actually is equal to c, the ubiquitous tag known as γ = 1/√1 – (v2/c2)  goes to 1/0 which in the bad old days was actually equated to infinity ─ and many physicists even today speak of a particle’s mass ‘going to infinity’ as v goes to c.
As a matter of fact, this situation can be very easily remedied. We simply prohibit v from attaining c for any particle/radiation and envisage c as an unattainable speed limit ─ the least of such upper limits. Moreover, since everything is ‘quantized’ in UET, this is much easier to do than in continuum physics. We interpret v as a certain number of emplacements for ultimate events in a single spatial direction which are ‘covered’ or ‘skipped’ from one ksana to the next. If c is unattainable and we are dealing in ‘absolute’ units, this means v can be at most (c – 1) which I note as c* (Note 4).
Unfortunately, as any mathematician reading this will see at once, this stratagem makes the usual formulae of SR much more difficult to derive : in effect one has perpetually to deal in inequalities rather than equalities. Though Einstein originally used a rather more tortuous method, he subsequently realized ─ and said so in a footnote to a later edition ─ that the simplest way to derive the Lorentz transformations is to employ the postulate of the ‘absolute’ speed of light in all inertial frames and then express this in two different coordinate systems. We thus have x2 + y2 + z2 = c2t2  in one frame and (x′)2 + (y′)2 + (z′)2 = c2 (t′)2   in the other. Using the Lorentz transformations        x′ = γ(x – vt)   y′ = y   z′ = z    t′ = γ(t –vx/c2)   you will find that this comes out right ─ provided you don’t make a slip ! It can be shown that this is the only solution given the assumptions, or alternatively one can, with some labour, derive these relations by assuming that the transformations are linear. (No one these days bothers much with the derivation since we know that the formulae work.)

Derivation of basic formulae in UET 

There is, dreadful to admit, a great deal wrong with the Special Theory of Relativity ─ despite it being one of the most successful and revolutionary ideas in the history of science. I have mentioned the trouble with c and massless particles, but this is not all. Far too much importance is given to one particular phenomenon (light) and to the traditional way of modelling such phenomena. Coordinate systems are entirely man-made inventions : Nature does not bother with them and seems to cope pretty well considering. As Einstein himself subsequently felt about his theory, it very soon got highjacked by pure mathematicians and removed as far as possible from the plane of reality.
So how would I propose to establish the formulae of SR or something similar? All I can give at present is a very rough plan of campaign. One should certainly not start with coordinate systems or even with velocity as such but with ‘mass’, which certainly for me is not a mathematical fiction but a reality. The equivalent of mass in UET is ‘persistence’. If an event repeats and forms an event-chain, it has persistence, if not not. This is the most basic property of an event-chain and is inherent to it, i.e. does not necessarily involve any other event-chain.  But everything to do with ‘motion’, ‘acceleration’ and so forth is a property of a system of at least two event-chains and there is,  by hypothesis confirmed by experience, a limit to how much a system of two event-chains can expand spatially, so to speak, from one ksana to the next. The ‘persistence’ of each event-chain in the system (as viewed by the other) increases with each expansion and strongly resists further expansion; moreover, this increase is not linear. (We all know how easy it is to go from 5 to 10 mph and how difficult to go from 90 to 100 mph.)
Now, I do not know if it is possible to derive a precise mathematical function on the basis of this and the  current assumptions of Ultimate Event Theory : hopefully it will eventually be possible. But what we can say right now  is that a function of the form p /cos φ   where cos φ = √1 – (v2/c2and  0 ≤ v ≤ c  has desirable properties when confronted with experience. That is, when v = 0 we have just the basic ‘persistence’ which is never lost. As one would expect the ‘persistence’ increases very slowly at first while it rises precipitously as v approaches c (but never attains it). The reason for the complications of the squares and the square root in (√1 – (v2/c2is something that must emerge from the initial assumptions and conclusions drawn therefrom. Once we have established a likely formula for increasing persistence (aka mass) most of the other formulae of SR can be derived employing basic mechanical principles. It should not be necessary to even mention light or electro-magnetism. However, all this is for another day.       SH 

 

Note 2  (page 44 The Principle of Relativity A collection of original papers Dover edition). The point is that we must, according to Einstein, have equations of motion of the type x′ = Ax + Bt, x = Cx′ + Dt′ where A, B, C, D are constants ─ or at least ‘parametric constants’ involving the relative speed, v. If ‘space/time’ were non-homogeneous, for example ‘patchy’ like the atmosphere or viscous like treacle, so-called linear equations would not work, nor would situations necessarily be ‘reversible’.

Note 3 Apparently Einstein got the idea of a ‘freely-falling frame’ (which became a cornerstone of General relativity) one morning when he was travelling to work and passed by a large building under construction. He wondered what a workman on the scaffolding of the building would feel if he fell off and let go of his hammer and lunch-pack as he fell. Einstein later said that it was “the happiest thought of my life”.

Note 3   The quotation is from Fock’s book Space, Time and Gravitation. It is given in Rosser, Introductory Relativity  p. 263

 Note 4 The ‘speed’, i.e. the ‘lateral’ ratio of emplacements/ksana, for any event-chain with a 1/1 appearance rhythm (one event per ksana), thus has an attainable upper limit of 1/√1 – ((c–1)2 /c2)  = c/√(2c – 1) ≈ √(c/2) . Note that this is in ‘absolute’ limits, not metres per second!

In classical mechanics, the ‘natural’ state of a body is to be at rest or, if in motion, to continue in a straight line at constant speed indefinitely. This is Newton’s 1st Law. Any deviation from this state is to be attributed to the action of a force and, for Newton, all forces were external since Newton knew nothing of  chemical bonding or nuclear reactions.
Similarly, since Newton did not have at his disposal the notion of the field  (which was only invented by Faraday in the 19th century), he assumed that all forces were to be attributed to the action of other bodies : there were contact forces (such as those due to collisions) or distant forces such as those due to gravitational attraction. And for anything to happen at all one needed at least two bodies and the situation was supposed to be symmetric : if A affects B, then B affects A to exactly the same extent ─ Newton’s 3rd Law.
The important point here is that, in the Newtonian scheme, no body is an island but always “part of the main” (to paraphrase Donne) and this ‘main’, because attraction was universal and instantaneous, turns out to be the entire universe. In the Newtonian schema, every atom is enmeshed in a complex net of forces stretching out in every direction and from which there is no hope of escape (Note 1)

Need for an ‘Inertial Frame’  

So, could, in the Newtonian scheme, an entirely isolated body be said to have any properties at all apart from occupying a certain position in space? As Bishop Berkeley observed at the time (Note 2), to speak of a completely isolated body being in ‘motion’ or ‘having a velocity’ is meaningless : we require at least one other body with which we can compare the first body’s changes of position over time. And, in like manner, the ‘second’ body  requires the first.
But a ‘two-body system’ where each body is moving relative to the other is not much of an advance on a single body if we want to work out the successive positions of either, or both, of these bodies, especially if they are circling round each other. What is required is a rigid framework which  encloses our ‘test’ body and which does not itself move around appreciably while the ‘test’ body is free to move inside it. Hence the idea of an ‘inertial frame’  : an absolutely  indispensable concept without which physics would never have developed very far.

Celestial and Terrestrial Inertial Frames  

On the astronomical scale, the required framework was supposed to be provided by the ‘fixed stars’ ─ even though it was already known by Newton’s time that the stars were not completely fixed in their positions relative to each other. But compared to the Earth the stars provided a good enough backdrop.
What of terrestrial frames? Galileo’s ‘inertial frame’ was the windowless  cabin of a ship conceived to be either at rest or moving at a constant speed on a calm sea (presumably rowed by galley-slaves). Today, we have much better ‘inertial frames’, cars, trains, ocean liners, aircraft, spaceships and so on ─ indeed it is remarkable that Galileo and his contemporaries were able to conceive of the idea of an ‘inertial frame’ at all since methods of transport at the time were so jerky.
Of course, not all physical objects are situated inside recognizable ‘inertial frames’, but, if need be, we simply imagine a frame, usually either the classic Cartesian box frame or a spherical ‘frame’ like that of an idealized Earth (without flattening at the poles). If we take one corner of the box as the fixed origin, or the centre of the Earth, we can fix the position of any small body relative to the ‘origin’ using at most three ‘specifications’, i.e. co-ordinates.
Galileo was not particularly interested in inertial frames as such and only introduced the windowless cabin ‘thought experiment’ to meet the standard objection to the heliocentric theory, “If the Earth is moving round the Sun, why do we not register this movement?”  Galileo replied, in effect, that neither do we necessarily register motion here on Earth provided this motion is more or less constant and in a straight line. He challenges a traveller, shut up in the windowless cabin of a ship, to decide whether the ship is at the dock or travelling at constant speed over a calm sea. Galileo argues that no experiment undertaken inside the cabin, would enable the voyager to come to a final decision on the matter. We ourselves know how difficult it sometimes is, when in a train for example, to decide without looking out of the window whether we are in motion (relative to the platform) or are still at the station.
This question of distinguishing between inertial frames has had enormous importance in the history of physics since it ultimately gave rise to Einstein’s Theory of Special Relativity. 19th century physicists, although accepting that no mechanical experiment would be able to distinguish between Galileo’s two situations, reasoned that there ought nonetheless to be a foolproof method of distinguishing between rest and constant straight-line motion by way of optical experiments. The Michelsen-Morley experiment was designed to detect the (very nearly) constant straight-line motion of the Earth through the all-pervading ether. The famous null result caused a crisis in theoretical physics which was only resolved by Einstein. He made it an axiom (assumption) of his theory that no experiment that ever would or could distinguish, from the inside, between different inertial frames. (More precisely, what Einstein assumed  was that “the laws of physics take the same form in all inertial frames”. If identical bodies in similar physical conditions were observed to behave differently in different inertial frames, then this would show either that Einstein’s assumption was wrong or that there were no universally valid ‘laws of physics’.

Absolute Motion and Absolute Rest? 

Newton himself was reluctant to accept what Galileo’s Principle of Relativity implied : namely that there was no such thing as ‘absolute’ motion, or for that matter ‘absolute’ rest, only motion or rest relative to some agreed body or point in space. Instinctively  Newton felt that there ought to be some way of distinguishing between ‘absolute’ and ‘relative’ motion, and consequently between constant straight line motion and rest. But he conceded that, practically speaking, he could not see how this could be achieved ─ “the parts of space cannot be seen or distinguished from one another by our senses, therefore in their stead we use sensible measures of them” (Principia Motte’s translation p. 8).
Newton did, however, point out that we can make an ‘absolute’ distinction when speaking of rotational movement — “There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion” (Newton, Principia p. 11) As evidence for this Newton used the ‘bucket and rope’ experiment.
If a bucket of water is suspended on the end of a twisted rope and we leave the rope to untwist, the water climbs up the sides of the bucket which it would not otherwise do. This was, to Newton, an example of ‘absolute rotational movement’ within an ‘absolute’ frame ultimately provided by the fixed stars. In this case, the situation was not symmetrical : one could, by observing the torsion of the rope, conclude that it was the bucket, and not the stars, that was rotating (Note 4).
In much the same manner, one might well expect there to be some inertial frame that was ‘truly at rest’ and against which the motions of all other inertial frames could be judged.

Abandon of the Principle of Relativity in UET

 What is the comparable situation in Ultimate Event Theory? After agonising over this question and related issues for the best part of a year, I have finally taken the plunge and decided to discard one of the most firmly established and fruitful principles in the whole of physical science. So there we are : the Rubicon is crossed.
In UET, the equivalents (sic) of inertial frames are not generally equivalent. In principle at least,  it should be  possible to distinguish between a ‘truly stationary’ event-chain and a ‘non-stationary’ one, as also between event-chains which have different constant displacement rates. Indeed, I aim to propose an axiom which in effect says just this. At present, contemporary experimental methods most likely do not allow one to make such fine distinctions, but this situation may change during this century, and indeed I predict that it will.
Dispensing with the Principle of Special Relativity does not mean we have to abandon all the formulae and predictions based upon it. There exists now a substantial body of evidence that ‘verify’ the formulae Einstein originally deduced on the basis of his particular assumptions, and in the last resort this evidence is the justification of the formulae, not the other way round. It is, for example, a matter of empirical fact that it is  not possible to accelerate a body beyond a certain well-defined limit, and that the closer one gets to this limit, the more difficult it is to accelerate the body.

Assumption of Continued Existence  

In ‘classical’ mechanics and physics generally, it is taken for granted that, once in existence, a ‘solid body’ carries on existing more or less in the same shape and form. Even rocks and mountains get worn down in the end, of course, but their constituent ‘bodies’, namely their atoms, last far longer. For Newton and his contemporaries atoms were indestructible, just as they were for the originators of the atomic theory, Democritus and Epicurus. Although twentieth-century discoveries have overturned this rash assumption ─ most elementary particles are very short-lived indeed ─ there remain plenty of small ‘bodies’ all around and inside us, that we are assured have been in existence for millions, sometimes even billions, of years. The idea that “once in existence an object tends to carry on existing indefinitely” is so deeply ingrained in Western thought that it has  rarely been seriously questioned.
In Western thought but not Eastern. Two of the principal Indian and Chinese systems, Buddhism and Taoism, on the contrary emphasize the transience and ephemerality of all physical (and mental) phenomena. According to Buddhism nothing lasts for more than an instant and even solid objects such as rocks and corpuscles are flickering in and out of existence even as we look at them. It is significant that the ancient Chinese equivalent of the (under normal conditions)  unchanging elements in the Periodic Table are the shifting configurations of the Y Ching, the Book of Changes.
Now, in UET the ‘natural’ state is for every ultimate event to appear and disappear for ever. If an ultimate event reappears and keeps on doing so, thus inaugurating an event-chain, this can only be due to a ‘force’ ─ I have thought of calling it ‘existence-force’. Most ultimate events never become subject to ‘existence force’ ─ never acquire ‘existence energy’ if you like ─  but, once they do acquire this capacity to repeat, they generally retain it for a considerable length of  time. Once an event-chain is established, then, no extra force, inner or outer, is required for it to ‘keep on existing’ : on the contrary effort is required to terminate an event-chain, i.e. to stop the ultimate event or event-cluster repeating. And this is a very important fact.

Why is acceleration so difficult?

Why is it so difficult to make a particular object ‘go faster’? And why, the faster an object is already ‘moving’, is it all the more difficult to make it go faster still?
This state of affairs might appear ‘obvious’, but I do not  believe that it is. Take a bath with a little water in it. Does it become more difficult to add one more teaspoonful as the bath fills up? No, it does not. And even when the bath is full, you can still merrily carry on adding water, though in this case some of the water will spill onto the floor. On the other hand, it is extremely difficult to get a lot of personal objects to fit into a travelling case : we have to fold clothes carefully so that they lie flat, arrange solid objects so they fit together neatly and so on.
What is the difference between the two sets of examples ─  the bath and the travelling case or trunk? It is, of course, simply a matter of the available space. In the case of an ‘open’ container, such as a bath, there is more or less unlimited space; in the case of a trunk, the available space is seriously limited.
Now apply this to ultimate events (which are the equivalent of ‘elementary particles’ in UET). Taking as our starting point the spot where an event occurs at a particular ksana, there is seemingly a built in limitation to how far away the next event in the event-chain can occur. If there is to be only one  ‘next event’, it can only occur in a single spatial direction relative to its predecessor ─ as opposed to all three directions at once. And there is an upper limit to the possible ‘lateral distance’ between successive events if  they are members of an event-chain. This is so because the ‘range’ of the causal connection is finite ─ everything in UET except the Event Locality itself is finite. There may conceivably be relations of some sort between two events that are separated by more than c emplacements at successive ksanas, but, in UET as in the theory of Special Relativity, such relations cannot be causal, at any rate as the term is normally understood (Note 5).
In UET everything is static though one static set-up is constantly being replaced by another. ‘Motion’ in UET simply means the replacement of one ultimate event or event-cluster by another event or event-cluster. Instead of particles in perpetual motion, we must think rather in terms of evanescent point-like ultimate events encased in ‘event containers’. In the proposed schema for UET, each ultimate event has its own particular ‘event-capsule’ of variable dimensions. If we label the  boundary positions in any one spatial direction 0 and c , we can say that there are c* = (c – 1) possible emplacements for ultimate events inside a single capsule in a single direction. (This excludes the two boundary positions.) But only one of these positions or ‘event-pits’ can be occupied at a single ksana (moment).

  • …………………….………………..●

0   ←                        c*                   → c

Now, there is seemingly, also a limit on how far the very next  ultimate event in  an event-chain can be displaced in a single direction. This is a matter of experience and observation though it would be difficult to imagine a ‘world’ in which there was not a limit of some sort. If there was no such limit, something that I do here, wherever ‘here’ is, could have immediate consequences at some arbitrarily distant spot in the universe. The speed of the transmission of causality would be ‘infinite’. This is scarcely conceivable and, in any case, for the purposes of UET, one can  simply rule out any such possibility by invoking the ‘Anti-Infinity Postulate’. Eddington,  rightly in my view, argued that one could decide for strictly a priori reasons that there must be a ‘speed limit’ for the transmission of energy (or information) in any universe, though one could not for a priori reasons decide exactly what this limit must be.
There is, then, a permanent constraint on all event-chains without exception : successive ultimate events cannot be more than c* positions apart in any one direction. And if we have two event-chains where the distance between successive events of each of the two chains regularly increases by, say, d positions (where d < c*) at every ksana, there is a further constraint on this dual system, namely that the greatest possible subsequent increase is (c* – d) emplacements. (Note that I am speaking of event positions or emplacements not distances in the ‘metric’ sense.)
In effect, looming over and above each individual ‘event-capsule’ with its ultimate event, there is a sort of ghostly potential event-container which dictates how far the next ultimate event of an event-chain can be relative to its previous position. If we label the boundary positions of this ‘macroscopic’ event-container 0 and c , we can say that this creature has the  capacity to accommodate c* ultimate events in any one spatial direction but no more. It is, in effect, a scaled-up version of an individual event-capsule since, in the case of an individual event-capsule, there are exactly the same number of possible emplacements for an ultimate event ─ but only one position can be occupied at a time. This parallelism turns out to be extremely significant in UET.
In matter-based physics, we say that a ‘body’ cannot go any faster than c metres per second. The equivalent statement in UET would be : “It is not possible to fit more than c* ultimate events into a causal event container. Once this container is ‘full’, there is no room for any more events and that is that. This question of available space, and the increasing difficulty of cramming events into it, is the crucial issue in UET from which all sorts of  consequences follow. As this available space becomes curtailed, the system as a whole becomes subject to increasing pressure and strongly resists any further constriction. To speak in mathematical terms, any supposed ‘event-packing function’  ─ the equivalent of the acceleration function ─ would not be linear, would start off almost as a straight line but would rise precipitously as v gets nearer and  nearer the maximum possible value c* = (c – 1).

The Inertial Ratchet 

This picture of an event-container and ultimate events inside it is, of course, not quite right. If we are considering an event-chain where each constituent event is ‘laterally displaced’ at each successive ksana, all the intermediate possible emplacements ─ spots where ultimate events could in principle have occurred ─ are not actually occupied. But it is as if they were. There is no way of going back to the previous state of affairs ─ except by applying a completely new force. Galileo’s notion of inertia should not be interpreted negatively, i.e. as showing our personal incapacity to distinguish ‘inertial frames’, but realistically as a sort of ‘valve’ or  ‘space-time ratchet’ which stops an event-chain reverting to its previous occurrence pattern. Not only can the ‘moving finger’ of Omar Khayyam not be lured back to “cancel half a line” but it must inexorably keep on writing at the same rate. If, then, for some reason, an event-chain A suddenly increases its lateral distance from event-chain B by d emplacements at every ksana, it must seemingly keep increasing its distance by this precise amount of d event-emplacements indefinitely.
This property of maintaining a constant ‘speed’ without extra effort is an astonishing and extremely important fact about physical reality which has been glossed over because of the exclusive concentration on the technicalities of how one might  actually be able to distinguish between one ‘inertial frame’ and another. Galileo, foreshadowed by the great medieval thinker Oresme, realized that it is not the distance between two bodies (event-clusters) that is important, but the increase in the distance. Why should this be? Because, as far as we know, there is no built in restriction on how far two event-chains can be apart. But the doctrine of the equivalence of all inertial frames means that conditions within any one of the inertial frames remain exactly the same whether or not the two frames (repeating event-clusters) are right alongside each other or are moving apart at a fantastic speed provided this speed is constant ksana by ksana (and less than the upper limit).
But can one really believe this? One can ─ or I can ─ understand only too well why people (including Newton) were disinclined to accept Galileo’s Principle of Relativity and subsequently at first even more disinclined to accept Einstein’s more extended version. Only repeated experiments of increasing precision made some such acceptance mandatory.

The Systems Axiom  

Let us examine the reasons for this reluctance. It has been argued that velocity has little if any meaning if we are speaking of a completely isolated body, aka event-chain. We thus require at least two bodies that then form a dual system. And although, if we are confined to the point of view of one ‘inertial frame’ (which  we naturally consider to be at rest), we will attribute a certain ‘velocity’ to the other inertial frame (if it seems to be getting further away at each successive moment), this ‘velocity’ really belongs to the dual system ─ and not to either of the components of the system to the exclusion of the other. Very well. Considering the dual system, can we say that a situation where this system is expanding by d emplacements per ksana is ‘equivalent’ to a situation where it is expanding by 2d emplacements per ksana, or for that matter by zero emplacements per ksana? Or even by c* emplacements per ksana? Clearly, the situations or configurations of the dual system are not equivalent, cannot possibly be. However, we are asked to accept that the situations in each of the two components of the system are indistinguishable. Is this reasonable? No.
Why is it not reasonable? Because one would expect conditions of the system as a whole to have repercussions of some sort on all parts of the system : indeed, it is hardly conceivable that it could be otherwise. Certainly in most physical contexts this is what we find. If two bodies are linked by gravitational attraction, this systems situation is detectable in either one of the two (or more) bodies ─ provided we have sensitive enough instruments, of course. Similarly, for components of an electrical circuit. Indeed, one could argue that Newton’s 3rd Law makes something of the kind not just possible but obligatory.
This can be presented as an axiom :

If a system as a whole is subject to certain constraints, then so are all parts of the system and in a similar manner.

        Another statement of the principle would be :

If two or more configurations of a dual event-chain system are distinguishable when considering the system as a whole, then the configurations of each of the two or more components of the system must also be distinguishable considered individually. 

        Now, this axiom is incompatible with the Principle of Relativity, or at any rate what the Principle is taken to imply, namely that there is no way, from the inside, of distinguishing between inertial frames. Let us take a practical example.

The spaceship on the way to the moon

 Once a spaceship bound for the Moon has got sufficiently outside the Earth’s gravitational grip, the rocket motors are turned off. Neglecting minuscule perturbations from other planets, comets and so on, the rocket carries on at the same speed relative to the Earth and in more or less exactly the same direction : it does not drop back to what it was before. To reproduce previous conditions, say when the rocket was momentarily stationary relative to the Earth, it would be necessary to start other motors firing, i.e. to introduce a new force.
Now, so we are assured, astronauts devoid of radio contact, windows and so on, would not be able to tell whether the rocket or spaceship  was ‘in motion’ relative to the Earth or motionless. The dual system ‘rocket/Earth’ is very definitely not  the same in the two contexts. And the same goes for Galileo’s ‘port/ship’ system. If the ship is being rowed by galley-slaves on a calm sea, the distance between the repeating event-clusters we call ‘port’ and those we call ‘ship’ is increasing at every ksana, and so is the distance between the  rocket on its way to the Moon and the Earth.
If now we apply the ‘Systems under Constraint Axiom’, we must conclude that there is, at least  in principle, some way of distinguishing between the two situations. Why so? Because the constraints on the system are not the same. In the case of a stationary dual system, there is the constraint that, at the subsequent ksana, the distance between the two components can be at most c* units. If the system is already expanding by d units of distance, then there is the more stringent constraint that no increase greater than c* – d units of distance is possible. And in the case when the system is expanding at the maximum possible rate, c* positions per ksana, no further increase is possible at all : the constraint becomes a total ban.
So, according to the ‘Systems under Constraint Axiom’, since the system as a whole is under constraint, each component of the dual system is also under constraint and this constraint should in principle be observable. How can this be? Well, first of all, I need to work out a schema that allows such a distinction to be observable in principle; subsequently, it is for experiments to detect such a distinction, or some eventual consequence of such a distinction.
This is why, as stated earlier, I have eventually come to the unwelcome conclusion that the schemas of Ultimate Event Theory and Relativity diverge : they are not ‘homologous’ as mathematicians might put it.    This topic will be pursued in subsequent posts.    SH  1/8/14

 

Note 1  One is reminded at once of the human (pseudo)individual enmeshed in the web of karma. Except that, according to the Buddha, there is hope of escape.

Note 2   “Up, down, right’ left, all directions and places are based on some relation and it is necessary to suppose another body distant from the moving one ….. so that motion is relative in its nature, [and] it cannot be understood until the bodies are given in relation to which it [a particular body] exists, or generally there cannot be any relation, if there are no terms to be related.  Therefore, if we suppose that everything is annihilated except for one globe, it would be impossible to imagine any movement of that globe.”                              Bishop Berkeley, quoted Sciama

 Note 3  “All things are placed in time as to order of succession and in space as to order of succession. It is from their essence or nature that they are places; and that the primary places of things should be movable is absurd. (…) But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures for them. For from the positions and distances of things from any body considered as immovable, we define all places; and then with respect to such places, we estimate all motions….. And so, instead of absolute places and motions, we use relative ones.”                Newton, Principia ‘Scholium’ p.8 Motte’s translation  

Note 4  The late 19th century Austrian physicist Mach argued that the two descriptions, Earth rotating, Heavens fixed and Heavens rotating, Earth fixed, were equally valid and this was also Einstein’s view.

Note 5   Entangled photons and other particles do give rise to event correlations that far exceed c, but asuch associations of distant events are not considered to be causal in the normal sense. The issue bothered Einstein so much that he never accepted Quantum Mechanics in its then current form, nor would he have been any happier with it in its present form.

 

Reputedly, Einstein got the first inklings of his theory of Relativity at the age of sixteen when he asked himself what would happen if one were travelling at the speed of light (Note 1). This is exactly the sort of question an inquisitive adolescent asks himself and doubtless one which, at a slightly later date, the undergraduate coterie composed of Einstein, Besso, Grossmann and Mileva Maric (Einstein’s girl-friend and eventually first wife) debated earnestly in smoke-filled cafés in Zurich. Most people ‘grow up’ and put aside childish things, but Einstein not only continued puzzling over the apparent paradoxes involved in motion at or near the upper limit, but fleshed out his ideas into a testable mathematical theory.
Einstein ‘deduces’ the Lorentz Transformations ─ apparently without being familiar with Lorentz’s work ─ in his 1905 paper On the Electro-dynamics of Moving Particles but the paper makes rather heavy reading today. Later, in Appendix I to the ‘popular’ book Relativity, the Special and the General Theory (1916), he gives a much simpler derivation. These days, hardly anyone bothers about the reasoning involved : the Lorentz Transformations and related formulae must be right, or not very far from the truth, since otherwise a whole bunch of things that we know do work (like nuclear power stations) wouldn’t work. One doubts whether Einstein derived the formulae in the approved step by step logical fashion : he knew where he wanted to get to and worked backwards covering his tracks. This is the usual way important scientific and even mathematical discoveries are made : the rigorous deductive presentation is strictly “for the gallery”, or rather, for the editors of learned journals.
So where did Einstein want to get to? To the second of only two postulates on which the entire Theory of Special Relativity was based, namely that the observed speed of light in a vacuum is always the same irrespective of the relative movement of the source. Using the paraphernalia of co-ordinate Systems, this means that we must in some way make x′/t′ = c when x/t = c, with c constant.

Co-ordinate systems

To pinpoint a particular event we need to be able to answer the double question Where? and When? Traditionally we use a so-called co-ordinate system which, in the simplest case, may be conceived as composed of three directions emanating from a single point ─ an indefinitely expandable room with the bottom left-hand corner fixed as the ‘Origin’. An event occurring at a given time within this room can be located precisely by stating the distances along three axes at right angles to each other, the ‘length’, ‘depth’ and ‘height’ or in mathematics the x, y, z axes. But to specify not just the place but the time at which an event occurred we need a fourth co-ordinate t. A printed page is two-dimensional so we need to cheat a bit, retaining just a single spatial dimension and a single time dimension (Note 2). This simplification is less drastic than it may appear, since we are more often than not only concerned with a single spatial dimension.
A series of causally bonded events can be represented as a collection of dots on a Space/Time graph and if the relative displacement to the left (or right) at each ksana is the same, we get a straight line when we join up the dots. If the space/time displacement ratio is not constant, we do not get a straight line but some sort of ‘curve’. A ‘curve’ shows that we are dealing with an ‘irregular’ event-chain where the causal impulse that produces each next event fluctuates in some way, or the event-chain in question is subject to the influence of a second event-chain (or both possibilities at once). All this is familiar enough : it is the distinction between ‘rest or constant straight-line motion’ and ‘accelerated’ motion.
Einstein’s First Postulate of Special Relativity, that “the laws of
physics take the same form in all inertial frames” implies that a description of a series of events in terms of one particular coordinate system is ‘just as valid’ as a description of the same events in a different coordinate system, provided the second system is conceived as moving with constant velocity v relative to the first (which for simplicity we can consider to be stationary). All we need to do is make sure we ‘translate’ the specifications in one coordinate system into specifications in the second system in an intelligible and consistent manner (Note 3).

The occurrence, or not, of an event does not depend on coordinate systems

It is absolutely essential to grasp (or rather accept) that the occurrence or not of an event, either in UET or in matter-based physics, has absolutely nothing at all to do with coordinate systems or such like man-made constructions. An event either occurs or it does not occur, and if it does occur, it occurs at a specific spot ─ these are all part of the preliminary assumptions of UET. In principle, much the same applies even in ‘traditional’ physics ─ leaving aside for the moment complications introduced by Quantum Mechanics ─ though textbooks tend to be rather evasive on this point.
Whatever coordinate system or other localization method we are using, we need to make sure that we always home in on the same exact spot where such and such an event occurred, or could have occurred. A ‘moving body’ is really a succession of point-like events, but we humans don’t perceive it as such; Einstein’s First Postulate means that, amongst other things, the trajectory or ‘event-line’ of such a (pseudo) body should be the same no matter what localization method we employ. Transferring from one system to another is not too difficult post-Descartes and a very useful trick is to choose two coordinate systems, one ‘moving’ with respect to the other, in such a way that the ‘direction of motion’ of the ‘body’ we are interested in coincides with the x (and the x′) axis (Note 3)

Redundancy of two spatial axes 

What about the other two spatial axes? They are treated as irrelevant since, by hypothesis, nothing is going on in those directions and so we can simply make y′ = y   z′ = z. This just leaves x′ and t′ to bother about.
‘Motion’ is a ‘function of time’ : at different moments ‘in time’ a ‘moving’ body will be at different places. Moreover, in the general case, motion is only a function of time. We do not, for example, usually need to consider things like temperature or air pressure when we are studying motion : if they make a difference this will show up in the equations anyway. On the other hand,since we are dealing with the general case of steady straight-line motion, we need a ‘parametric constant’ v. By ’parametric constant’ I mean a quantity that in any particular set-up does not change but which can and does change in a different set-up. Such a quantity is somewhat in-between a true variable and a true constant : logically speaking the notion is quite subtle but, surprisingly, we find it natural enough which suggests that the brain uses a similar procedure ‘unconsciously’.

Expected form of Transformation formulae 

Mathematically, the above brief discussion means that, algebraically, we would expect the ‘transformed formulae’ x′ and t′ to be expressions in x and t only apart from certain constants and the ‘parametric constant’ v. Moreover, without going into too much detail for the moment (Note 4), we expect both x′ and t′ to be ‘linear’ functions of x and t, i.e. they will not contain powers higher than the first. So we are looking for expressions such as

x′ = Ax – Bt + C1        t′ = Cx – Dt + C 

where A, B, C, D are either pure constants or involve v.
        If we make the two origins coincide at a certain moment in time, we can eliminate the two constants C1 and C2 which makes things easier still.
Now, the simplest ‘transformation’ is simply to ‘add on’ (or ‘take away’) the distance the ‘moving’ frame ∑′ has travelled in t seconds, namely a distance of vt metres (or some other unit of length). If the particular spot where an event occurs is, say, d metres from the origin of ∑ and the event keeps on occurring at the ‘same’ spot this distance d from the origin of the ‘stationary’ frame ∑ remains the same. But if ∑′ is moving at a rate of v metres/second this original distance d will decrease in ∑′ by such and such an amount every second. This gives x′ = x – vt where in effect we are making a = 1, B = v   C1 = 0 in the equation x′ = Ax – Bt + C1  .
We leave aside the t′ co-ordinate for the moment.

Upper Limit the speed of light  

Now, according to traditional electro-magnetic theory light (and all other electro-magnetic radiation such as radio waves, X-rays and so on) travels at a constant speed c. So there are particular values for x and t when x/t = c. Call one pair xc and tc .
So, we have xc /tc = c. According to Einstein’s Second Postulate, whenever we get xc /tc = c we must get xc′ /tc′ = c .  Now, this is impossible using the Galilean Transformations  x′ = x – vt    t′ = t            becausewe obtain for specific values xc and tc (where xc /tc = c)

xc′ /tc′ = (xc – vtc)/tc = (xc /tc ) – (vtc)/tc = c – v 

But xc′ /tc does not equal (c – v) except when except when v = 0. And we don’t need all the paraphernalia of Galilean or Lorentz Transformations to conclude that when there is no relative movement between two co-ordinate systems (expandable boxes) the distances from the corner to where the event occurred are going to be the same.
Obvious though this sounds today, it led Einstein and one or two other people at the time, to seriously entertain the possibility that both time and distance change when there is relative motion between the systems that Einstein calls ‘inertial frames’. So ‘time is not always the same time’ and ‘distance is not always the same distance’ as Newton for one certainly believed.

Stopping the limit being exceeded  

So are there any possible ways of tampering with the equations but keeping the general pattern

x′ = Ax – Bt               t′ = Cx – Dt

while, miraculously, obtaining x′ /t′ = c whenever x/t = c ?
The answer is, yes.
If we use the Transformation Formulae

x′ = γ (x – vt)      y′ = y       z′ = z     t′ = γ(t – vx/c2)

where γ is a constant independent of x and t, it all comes out right.
The proof, or rather verification of the claim, is as follows. Remember that we are concerned for the moment only with those values of x and t for which the distance/time ratio is c ─ for example when x is a little less than 108 metres and t is one second.

x/t = c makes x = ct   and t = x/c for these values of x and t and we can legitimately replace any of the above by their equivalents.
So, assuming these Transformation Rules, and x/t = c

       x/t = γ (x – vt)/γ(t – vx/c2        =   (x – vt)/(t – vx/c2)      =   c2(x – vt)/ (c2t – vx)   
                                                                  = c2x – c2vt)/(c2t – vx)         = c2x – cv(ct)/ (c(ct) – vx)

                        = c2x – cvx)/  (cx – vx)            (using ct = x)

                                                                   = (cx)(c – v)/x(c – v)
=   c
                                                                 = x/t

The Einstein/Lorentz Transformations

But the Transformations

x′ = γ (x – vt)      y′ = y       z′ = z     t′ = γ(t – vx/c2)

would work for any constant value of γ provided it is independent of x and t. If, now, we make

γ = c /√(c2 – v2) = 1/√1 –(v2/c2)

we get some extra spin-offs.

The Einstein/Lorentz Transformations then have the following four very desirable features:

1. Whenever x/t = c   x′ /t′ = c or, as it is more frequently expressed n textbooks, whenever

        (x2 + y2 + z2)/t2 = c2           (x′2 + y′2 + z′2)/t′2 = c2

This is in line with Einstein’s 2nd Special Relativity Postulate.
2. They are linear in x and t, i.e. they don’t involve more complicated functions such as t2, sin t, ex and so forth.
3. They reduce to the normal Galilean Transformations as v/c → 0 , thus explaining why we don’t need them when the speeds we are dealing with are much smaller than c.
4.     The factor γ = 1/√1 –(v2/c2) ‘blows up’ as v gets closer and closer to c which Einstein interprets as demonstrating that no ‘ordinary’ particle or causal physical process can actually attain c, the only exception being light itself. c thus functions as an upper limit to the transfer of energy and/or information (Note 4).

The Time Transformation in UET

So, how does this go over into UET? Not very well, actually. I have had more trouble over this point than any single aspect of the theory since I first sketched it out some thirty-five years ago.

To start with, I am not so much concerned in transforming from one coordinate system to another ─ essential though it is to be able to do this for practical purposes. l wish rather to concentrate attention on individual ‘Event Capsules’ and causally connected sequences of (more or less) identical Event Capsules, i.e. event-chains. Coordinate systems don’t exist in Nature but I believe that ‘event capsules’, or something very much like them, do.
I also prefer a more visual, geometric approach (while recognizing its limitations) and thus opt for expressions like
sin φ = v/c and cos φ = √1 – sin 2 φ) which makes γ = 1/cos φ. But this is not a big deal : certain more approachable textbooks on Relativity do employ diagrams and use trigonometrical aids.
Now, in principle, I should be able to derive the basic formulae of Special Relativity from strictly UET premises. Can I actually do this and if not, why not?

Principle of Constant Occupied Area and its consequences 

The equivalent of Einstein’s first postulate (“that the laws of physics take the same form in all inertial frames”) is the so-called Principle of the Constancy of the ‘Occupied Area’.
In UET, though there is change, there is no such thing as continuous motion. This means that dynamical problems tend to reduce to problems of statics. Two successive causally bonded ultimate events are envisaged as constituting ‘Event Capsules’ (more strictly ‘Transitional Event Capsules’) and Einstein’s ‘Principle of Relativity’ in effect means that the actual dimensions of a specific Transitional Event Capsule are unimportant provided it contains, or could contain, the same event or events. The ‘occupied area’ of the Event Locality remains the same because the spatial and temporal dimensions are inversely proportional : roughly, if space contracts, time must expand.
Any description of an Event Capsule which makes its spatial ‘spread’ v spaces and its temporal dimension one ksana is acceptable irrespective of the value of v which in the simplest case considered here must be a positive integer. We may equally well consider a pair of causally related ultimate events as stretching out over v individual grid-spaces, or alternatively as being confined to a single grid-space of dimension 0 (but the ‘length’ of a ksana will be different in the two descriptions). In the extreme case of the upper attainable limit of c* positions per ksana, we may either consider that we have a very extended spatial ‘spread’ of c* gridpositions with the distance between the emplacements contracted, or that we are dealing with a single Event-capsule of spatial dimension s0. The ‘occupied area’ will be the same because the time-length adapts accordingly, sv tv = constant.
This is very close to the normal Special Relativity treatment which analyzes relative motion in terms of fictitious ‘observers’. Were an observer travelling ‘on a light beam’, he or she would consider himself to be at rest and not notice anything untoward. But for an observer in a ‘stationary’ system, stationary relative to the light beam, things would be rather different and the spatial dimension of, say, a spaceship travelling near the speed of light, would appear contracted in the direction of motion (Note 5).
In UET, I attempt where possible to dispense with ‘observers’. I take a more realistic approach whereby the quantity and order of ultimate events within a particular region of the Event Locality is fixed, is ‘absolute’ if you like and not relative. Any way of adjusting the spatial and temporal inter-event distances is acceptable so long as the extent of the overall region, which reduced for simplicity to a Space/Time ‘Rectangle’, remains the same.
There is also one description of the total occupied region that takes precedence over all the others, namely the ‘Space/Time volume’ when v = 0 . This ‘default volume’ s03 t0 is what the occupied region actually is when there are no other event-chains in the neighbourhood : it is the ‘proper’ Space/Time volume because intrinsic to the specific events and the region they occupy. Such a realistic approach is not possible in physics as it is taught at present because in the latter the key notions of ‘ultimate event’ and ‘occurrence’, which are absolute notions, are absent. Einstein was very close to this viewpoint at one stage but then Minkowski muddled everything with his introduction of the idea of a ‘Space/Time Continuum’.
Now, when we have this same ‘rest capsule’ in an event environment that is not empty, not only are there alternative descriptions of the extent of the occupied region available but, rather more significantly, but there is interference from nearby event-chains (provided they are within causal range). This interference, the real possibility of ‘alternative viewpoints’, ma infests itself as a sort of tension within the ‘rest capsule’, a tension that depends on v. In the extreme case where v = c* , the tension within the single event capsule is the greatest possible : the capsule is, as it were, full to bursting. This is the UET reason why ‘particles’ forcefully resist any attempt to increase their speed relative to a stationary frame when we approach the upper limit ─ and we know that they do this because of numerous experiments (Note 6).

Problems with the time transformation  

In my treatment there is not too much trouble with the spatial part. In effect, in UET, d′ cos φ = (d – vt)  where v < c which is just another way of saying x′ = γ(x – vt) (Diagram). And, as in ‘normal’ physics, I am happy to take on board the ‘Independence of Axes Postulate’ (at least for the moment), i.e. y′ = y, z′ = z also.
So far, so good. Where I make things difficult for myself is by insisting that no actual event-chain has a ‘Space/Time Displacement Ratio’ of c exactly. The original reason for doing this was to avoid having to assume that ‘light’, or its equivalent, is ‘massless’, on the face of it a nonsense since I can’t really see how anything can be anything at all without having some mass (and anyway we know photons have gravitational mass since they are bent when they pass close to massive bodies).
There is thus a difference in UET between the lowest unattainable Space/Time Displacement Ratio, labelled c, and the highest attainable ratio which I label c*. For the simplest case, that of event-chains with a 1/1 Reappearance Rate, i.e. one ultimate event at each successive ksana, this would make
c* = (c –1) in ‘ultimate units’ ─ but for more complex cases c* can get a good deal closer to c.
I take time dilation to be a fact of life, or rather of the Event Locality. The extent of the time dilation depends entirely on the ratio of v to c which is a rational number (proper fraction or zero). The angle sin–1 v/c in effect gives both the space and time ‘settings’ for all systems of event-chains that have a constant displacement ratio of v relative to each other, i.e. a ratio of v grid spaces per ksana or, in ‘absolute units’, v s0 /t0 .
Note that the denominator in v/c is c and not c*. This is deliberate. Since v never actually attains c, the denominator in γ = 1/√1 – v2/c2 never risks becoming ‘infinite’, or, if you like, is always defined. And tan φ does not attain unity, i.e. φ < π/4 for all the cases we are considering.
The dual requirement in UET is to keep what I call the area of the ‘Space/Time Event Rectangle’ constant for all legitimate values of v, while not allowing the ratio of the sides to exceed c*.
The basic equation is sv tv = s0/t0 (the subscript v stands for ‘variable’) where s0 and t0 are constants, the spatial and temporal dimensions of a ‘rest event capsule’, i.e. when v = 0. (I am, of course, not considering the two other spatial dimensions but they will also be s0 for a ‘rest capsule’.)
To keep the overall area constant, we have to have the spatial and temporal ‘sides’ inversely proportional to each other. I define s0 to be a maximum and t0 to be a minimum. If we make tan β = s0/t0 , the ‘angle’ tan–1 s0/t0 is also a universal constant and crucial in defining any particular ‘universe’ (massive interrelated event cluster). I do not see how the value of tan β can be deduced a priori : its value in our part of the Locality will have to be determined by experiment.
For a given setting v/c   (0 < v < c) the basic rectangle will deform, the spatial dimension contracting and the temporal expanding, but keeping the overall area constant. If we have sv = s0 cos φ and tv = t0/cos φ where sin φ = v/c this does the trick.
And, it has been argued in previous posts, the distance d of a spot from a repeating event when everything is at rest, will contract according to the formula d′ cos φ = (d – vt).
So the obvious way to keep the overall area constant in all eventualities is to make t′ = t/cos φ . So what happens to the ratio of one side to the other which, remember, must not ever exceed c*?

We have d′/t′ = (1/cos φ) (d – vt)
                                     (1/cos φ) t

The 1/cos φ parts cancel out so we get

        d′/t′ = (d – vt)/t  = d/t – (vt)/t = d/t – v 

This is OK, since if d/t ≤ c* , d′/t′ which is less then d/t will also not exceed the upper limit of c*.

However, we are not out of the woods yet. ‘Time’ is quantized in UET since there is no such thing as a ‘fraction’ of the ‘rest length’ of a ksana, t0 , a constant fixed once and for all (for any one ‘universe’). So everything has to be taken to the nearest integral value. Any distance d, the number of grid-spaces from a ‘stationary’ event-chain to a particular spot on the Locality, will not be ‘within causal range’ within the ‘space’ of a single ksana if it is c spaces away, or any number > c. So for such distances t must at least be 2 since no value of v, in effect no space contraction, is feasible in such a case. Of course, one could establish some sort of connection, or rather correlation, between arbitrarily distant spots on the Locality, but any such correlation would be entirely abstract if d ≥ c, in particular it cannot be a causal connection.

To find how many ksanas will be required for an event-chain with lateral displacement v spaces per ksana, we divide

d by v and make t the first integer ≥ d/v . For example, if we are looking at a spot c/2 spaces away, and an event-chain has a lateral displacement rate of c/4 s0/t0 only (c/2)/(c/4) = 2 ksanas will be required. In this case an event will occupy the exact spot but this will not normally happen. For v = (c/3) , straight division gives (c/2)/(c/3) = 3/2 which is not an acceptable value for t. We thus have to take the first integer > d/v which in this case is 2.
It is important to realize that, in UET, you cannot just have any old values for v, d and t : all such values are integral for the simplest event-chains which have a 1/1 Re-appearance Rate, and even those that do not, those which allow rational values of v, e.g. 7 spaces every 10 ksanas, never produce irrational values for v while d and t always remain integral.
Things become tricky when we consider a spot exactly c* or c spaces to the right. The distances are perfectly valid since the spots exist and we know the values we require for t in advance, namely a maximum value of 1 for the spot c* (corresponding to v = c*) and a value of 2 for the spot c since it is unattainable within a single ksana.
I have spent endless time messing around with the formula for t and the best I could come up with was the following,

   t′ = (1/cos φ) (t – d/c) where t is the first integer > d/c and v < c

This works in most but not all cases. For example, what if d = c*? Since c*/c < 1 , we have t = 1 as required ─ the causal impulse reaches its goal in a single ksana. However, if we fit this into the formula for d′ we get

d′ = (1/cos φ) (d – vt) = (1/cos φ) (c* – c*) = 0 

        Now d′ cos φ is never zero : this would imply that the distances between possible distinct ultimate events is zero ─ so they would all be merged together, as it were. However, we can make some sense of this formula if we take it to mean that the distances between the emplacements, shrunk to their minimum size, is zero. And this minimum size is not zero but su (the subscript u for ‘ultimate’) and su = s0/c* a small but finite and hopefully one day measurable extent. This value su is the ‘ultimate’ spatial dimension : nothing smaller than this exists, or can exist, at any rate in our ‘universe’ (independent sub-region of the Locality).
It must be borne in mind that the area of the ‘occupied region’ continually adjusts itself, keeping always to integral ‘ultimate’ values much in the way that the pitch of an organ pipe or other blown instrument adjusts to keep the number of nodes of the sound waves integral ─ you cannot have √2 of a pipe length, for example. In extreme cases, such as c* we are obliged perforce to jettison the formulae based on the angle φ and simply state the spatial and temporal distances that we know must be right. If an ultimate event ‘goes the whole distance’ c* in a single bound, using its maximum possible displacement rate, we may imagine the original ‘rest capsule’ as being ‘filled to bursting’. Were there an ultimate event simultaneously at every possible emplacement, the ultimate events would be jammed forcibly together and, since each of them occupies a dimension su , and there are c* of them, we have no inter-event spatial distance left and no room for a possible extra event.
All this doubtless seems tortuous and needlessly complicated to the layman, while being utterly pointless to the traditional physicist who just aplies the SR formulae. But, to me, it is not a waste of space and time (sic) because this is actually the first important issue where I find UET diverges from Special Relativity. Why so? Because, as several readers will have noted, I do not end up with exactly the Einstein/Lorentz Transformation for time. In the latter we have, using my notation,
t′ = (1/cos φ)(t – (sin φ)d/c) or, as you will find it written in textbooks t′ = γ(t – (vx/c2))     γ = 1/√1 – v2/c2

        The extra factor of v/c = sin φ is required if we wish to make d′/t′ exactly equal to c whenever d/t = c. This extra factor is redundantin my treatment since I am stipulating in advance that v < c and I only need to guarantee that d′/t′ ≤ c when d/t ≤ c not that d′/t′ = c exactly. Whether any experiment will differentiate the two treatments at such a precise level remains to be seen.

Ultimate Values for the Space/Time Rectangle

As I cannot state too often, everything in UET is quantized and the basic ‘quanta’ of space and time have fixed extent that hopefully we will soon be able to determine experimentally within certain limits.
So is there a final shape for the ‘Space/Time Rectangle’, one that cannot be exceeded if we are dealing with a causal process? Yes. As stipulated, for a 1/1 reappearance rate (one ultimate event per ksana), this makes c* , the greatest lateral displacement rate, equal to (c – 1). For any such event-chain, we end up with a grotesquely deformed ‘Space/Time Rectangle’ with a minimal spatial length and a maximum ‘time-length’. If we plug the value v = (c–1) into the formula for γ we obtain
1/√(1 – (c–1)2/c2   = c/√c2 – (c–1)2 = c/√(2c –1) ≈ c/√2c = √c/2

I do not know whether this value has any physical significance or utility, but I note it anyway.
If an ultimate event only reappears once every m ksanas, while it displaces itself mc* at each appearance, it will have the same ‘Space/Time Displacement Rate’ of mc*/mc* = m/m = 1 but it will miss out many more positions. And if has a displacement rate of n/m this makes γ =

 

1/√(1 – ((n/m)c)2/c2   = c/√c2 – (n/m)2 c2 = c/√c2(1 –(n/m)2)

               = (1/√1 – n2/m2)   > 1

There may, thus, in UET be all kinds of event-chains that have the same displacement rate (though different reappearance rates) and a great variety of possible event-chains with a great many gaps between successive events (n/m < 1) that come close to the maximum attainable value.

On the other hand, reappearance rates cannot be stretched indefinitely ─ nothing is infinite in UET ─ there will be a maximum possible gap between two successive appearances, probably one involving c or c*.

The dimensions of the ‘ultimate’ Space/Time Rectangle should, however, be calculated, not according to the γ formula which breaks down at or near the upper limit, but simply according to the fixed dimensions of the original rest’ capsule s0 by t0. If there are c* simultaneous ultimate events, and this is the absolute limit, the events occupy (or could occupy) the whole of one dimension of the ‘rest capsule’ which has a fixed volume of s03 where sis a maximum. Now each ‘event kernel’, the exact region where an ultimate event has occurrence has a small but non-zero size which may be noted as su3. Since there are, in the extreme case, c* = (c –1) ultimate events, or emplacements for possible ultimate events, we have so = c su.
What of time? To keep the overall area constant, time has to be stretched to a maximum, or, rather, the gap between two successive ultimate events is the largest possible. Since sv tv = s0 t0 = constant , this means that tmax, or as it is usually written tu , = c* t0. .
        This maximum time dilation should be observable when a ‘particle’ (event-chain) approaches the upper limit. Moreover, no matter what, this maximum time-length can never be exceeded ─ we will not get the ludicrous scenario of the ‘time interval ’ between two screams of someone falling into a black hole becoming infinite.

Deductions and Predictions  

The UET treatment differs from that of SR on several points. The most important being that Different event-chains can have the same maximum displacement ratio. They will differ amongst themselves by the number of spots on the Locality that are missed out, as it were.
A corollary of the above is that different event-chains with the same displacement ratio can have very different ‘penetration’ when confronted with massive event-clusters that block their path. This is the suggested explanation for the known fact that neutrinos are almost impossible to stop, thus to detect : they must have a very ‘open’ reappearance rate while also a displacement rate comparable to that of light (and possibly exceeding it). There seems no reason a priori to assume that the electro-magnetic event-chain is the only event-chain to attain the maximum and it seems to me quite possible that the neutrino, or some other particle, chain is ‘faster’. In UET it might just be possible to “send messages faster than the speed of light” but, certainly, it would not be possible to send them faster than the maximum possible displacement rate for all causal event-chains. There is so much fuss about ‘sending messages’ through empty space only because messaging is a causal process.

Beyond the speed limit? 

What would/could happen if we carried on accelerating a ‘particle’ (event-chain) beyond the maximum, just supposing for a moment that this is technically feasible? According to Einstein, this is not possible because the mass of a particle “becomes infinite” as it approaches the limit ─ though photons regularly travel at this speed. I can only guess what would happen within the context of UET but my guess is that the event-chain in question would simply terminate : to all intents and purposes the ‘particle’ would simply “disappear into thin air” without leaving a trace. There would be no Space/Time explosion, nothing dramatic like that. Would there be any compensatory creation of new particles (event-chains)? One could argue that a certain amount of ‘existence energy’ has been ‘released’ and thus has become available for other purposes. I, however, think it more likely that there would be no new production of events : the chain would terminate and that is that. This is, of course, the worst heresy since it is contrary to the dogma of the Conservation of Mass/Energy.         To accelerate anything to this enormous speed (relative to a stationary event-chain) is likely to be a rare event, so the possibility of the violation of energy conservation would not make too much difference in the short run. I think it very likely that the ‘disappearance’ of event-chains has already been observed in CERN and other accelerators, and has either been dismissed as “experimental error” or not mentioned because too controversial. If and when the possibility of the sudden termination of event-chains, i.e. sudden apparent disappearance of particles without any observed consequence, becomes theoretically acceptable, you will probably see several experimentalists publishing results stashed away in bottom drawers at present, results that show just this happening.

NOTES

 Note 1 “He [Einstein] asked himself what would be the consequences of his being able to move with the speed of light. This question, innocent as it appears, eventually brought him into conflicts and contradiction of enormous depth within the foundations of physics.”      Jeremy Bernstein, Einstein

Note 2 I have suggested elsewhere that what we really need is a three-dimensional lattice which flashes on and off rhythmically and a a spurt of differently coloured light marking an event, and ‘over time’ an event-chain.

 Note 3 This sort of problem comes up all the time in physics and was probably the motivation for Descartes invention of the coordinate system (or something very much like it) in the first place. The Greeks did not have a coordinate system : it would be interesting to know what sort of ‘localization system’ they used in daily life to locate, say, a particular house in Alexandria.

 Note 4   The argument goes something like this.

Suppose that the transformation formulae were not linear, for example that x′ = Ax – Bt2
        Then x′ = 0 marks the position of the origin of the ‘moving’ system ∑′ which gives
0 = Ax – Bt2   or x = (B/A) t2

However, we have already decided we are dealing with a case of constant relative motion of the two systems so x/t ought to be some constant, probably involving v, but not t. But in the above x/t = (B/A)t and (B/A)t is not a constant function (since t appears on the R.H.S.) and any graph of x against t will not be a straight line ─ it will, in the above case, be a parabola. Thus contradiction.
Incidentally, it does not matter whether we write
x′ = Ax + Bt   or x′ = Ax – Bt since we can make B positive or negative as required by the situation. All that matters is the relative motion of the two systems irrespective of direction.    

 Note 5 There is some doubt about what a hypothetical observer would actually ‘see’ (a fortiori what his brain would tell him he is ‘seeing’). According to current physics, photons from a ‘moving’ object leave that object at different times, and ones from parts further away will take longer to impinge on the eye than others. On would not see a ‘moving’ cube ‘face on’ but as if it were slightly rotated thus allowing a little of one side to be visible. The general conclusion seems to be the general case one would see the shape of a moving object rotated but otherwise undisturbed; the colour and brightness of the object would also change because of aberration and Doppler shift. See the discussion “The Visual Appearance of Moving Objects in Rosser, Introductory Relativity pp. 104-107.

 Note 6 For example the experiments carried out by Berozzi on electrons. They decisively show that (v/c)2 flattens off dramatically for kinetic energies approaching as v approaches c, whereas, for small speeds, the predictions of Einstein coincide with those of Newtonian Mechanics.

 

 

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.