Archives for category: ksana

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

 Beginnings  

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

Events instead of things  

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

Rejection of Infinity 

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

 Size and shape of Ultimate Events

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

Event Kernels and Event Capsules 

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

Length of a ksana 

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

Formation of Event-Chains and Event-clusters 

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

‘Rest Chains’

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

True rest?

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

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

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

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

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

 

Although, in modern physics,  many elementary particles are extremely short-lived, others such as protons are virtually immortal. But either way, a particle, while it does exist, is assumed to be continuously existing. And solid objects such as we see all around us like rocks and trees are also assumed to carry on being rocks and trees from start to finish even though they do undergo considerable changes in physical and chemical composition. What is out there is  always there when it’s out there, so to speak.
However, in Ultimate Event Theory (UET) the ‘natural’ tendency is for everything to flash in and out of existence and most ultimate events, the ‘atoms’ or elementary particles of  Eventrics,  disappear for ever leaving no trace and even with more precise instruments than we have at present, wouldshow up as a sort of faint permanent background ‘noise’, a ‘flicker of existence’. Certain ultimate events, those that have acquired persistence ─ we shall not for the moment ask how and why they acquire this property ─ are able to bring about, i.e. cause, their own re-appearance and eventually to constitute a repeating event-chain or causally bonded sequence. And some event-chains also have the capacity to bond to other event-chains, eventually  forming relatively persistent clusters that we know as matter.  All apparently solid objects are, according to the UET paradigm, conglomerates of repeating ultimate events that are bonded together ‘laterally’, i.e. within  the same ksana, and ‘vertically’, i.e. from one ksana to the next. And the cosmic glue is not gravity or any other of the four basic forces of contemporary physics but causality.

The Principle of Spatio/Temporal Continuity

Newtonian physics, likewise 18th and 19th century rationalism generally, assumes what I have referred to elsewhere as the Postulate of Spatio-temporal Continuity. This postulate or principle, though rarely explicitly  stated in philosophic or scientific works,  is actually one of the most important of the ideas associated with the Enlightenment and thus with the entire subsequent intellectual development of Western society (Note 1). In its simplest form, the principle says that an event occurring here, at a particular spot in Space-Time (to use the traditional term), cannot have an effect there, at a spot some distance away without having effects at all (or at least most or some) intermediate spots. The original event, as it were, sets up a chain reaction and a frequent image used is that of a whole row of upright dominoes falling over one after the other once the first has been pushed over. This is essentially how Newtonian physics views the action of a force on a body or system of bodies, whether the force in question is a contact force (push/pull) or a force acting at a distance like gravity ─ though in the latter case Newton was unable to provide a mechanical model of how such a force could be transmitted across apparently empty space.
As we envisage things today, a blow affects a solid object by making the intermolecular distances of the surface atoms contract a little and they pass on this effect to neighbouring atoms which in turn affect nearby objects they are in contact with or exert an increased pressure on the atmosphere, and so on. Moreover, although this aspect of the question is glossed over in Newtonian (and even modern) physics, each transmission of the original impulse  ‘takes time’ : the re-action is never instantaneous (except possibly in the case of gravity) but comes ‘a moment later’, more precisely at least one ksana later. This whole issue will be discussed in more detail later, but, within the context of the present discussion, the point to bear in mind is that,  according to Newtonian physics and rationalistic thought generally, there can be no leap-frogging with space and time. Indeed, it was because of the Principle of Spatio-temporal Continuity that most European scientists rejected out of hand Newton’s theory of universal attraction since, as Newton admitted, there seemed to be no way that a solid body such as   the Earth could affect another solid body such as the Moon thousands  of kilometres without affecting the empty space between. Even as late as the mid 19th century, Maxwell valiantly attempted to give a mechanical explanation of his own theory of electro-magnetism, and he did this essentially because of the widespread rock-hard belief in the principle of spatio-temporal continuity.

So, do I propose to take the principle over into UET? No, except possibly in special situations. If I did take over the principle, it would mean that certain regions of the Locality would soon get hopelessly clogged up with colliding event-chains. Indeed, if all the possible positions in between two spots where ultimate events belonging to the same chain had occurrence were occupied, event-chains would behave as if they were solid objects and one might as well just stick to normal physics. A further, and more serious, problem is that, if all event-chains were composed of events that repeated at every successive ksana, one would expect event-chains with the same ‘speed’ (space/time ratio with respect to some ‘stationary’ event-chain) to behave in the same way when confronted with an obstacle. Manifestly, this does not happen since, for example, photon event-chains behave very differently from neutrino event-chains even though both propagate at the same, or very similar, speeds.
One of the main reasons for elaborating a theory of events in the first place was my deep-rooted conviction ─ intuition if you like ─ that physical reality is discontinuous. I do not believe there is, or can be, such a thing as continuous motion, though there is and probably always will be succession and thus change since, even if nothing else is happening, one ksana is perpetually being replaced by another, different, one ─ “the moving finger writes, and, having writ, moves on” (Rubaiyat of Omar Khayyam). Moreover, this movement is far from smooth : ‘time’ is not a river that flows at a steady rate as (Newton envisaged it) but a succession of ‘moments’, beads of different sizes threaded together to make a chain and with minute gaps between the beads which allow the thread that holds them together to become momentarily visible.
If, then, one abandons the postulate of Spatio-temporal Continuity, it becomes perfectly feasible for members of an event-chain to ‘miss out’ intermediate positions and so there most definitely can be ‘leap-frogging’ with space and time. Not only are apparently continuous phenomena discontinuous but one suspects that they have very different staccato rhythms.

‘Atomic’ Event Capsule model

 At this point it is appropriate to review the basic model.
I envisage each ultimate event as having occurrence at a particular spot on the Locality, a spot of negligible but not zero extent. Such spots, which receive (or can receive) ultimate events are the ‘kernels’ of much larger ‘event-capsules’ which are themselves stacked together in a three-dimensional lattice. I do not conceive of there being any appreciable gaps between neighbouring co-existing event-capsules : at any rate, if there are gaps they would seem to be very small and of no significance, essentially just demarcation lines. According to the present theory these spatial ‘event-capsules’ within which all ultimate events have occurrence cannot be extended or enlarged  ─ but they can be compressed. There is, nonetheless,  a limit to how far they can be squeezed because the kernels, the spots where ultimate events can and do occur, are incompressible.
I believe that time, that is to say succession, definitely exists; in consequence, not only ultimate events but the space capsules themselves, or rather the spots on the Locality where there could be ultimate events, appear and disappear just like everything else. The lattice framework, as it were, flicks on and off and it is ‘on’ for the duration of a ksana, the ultimate time interval (Note 2). When we have a ‘rest event-chain’ ─ and every event-chain is ‘at rest’ with respect to itself and an imaginary observer moving on or with it ─ the ksanas follow each other in close succession, i.e. are as nearly continuous as an intrinsically  discontinuous process can be.
According to the theory, the ‘size’ or ‘extent’ of a ksana cannot be reduced  ─ otherwise there would be little point in introducing the concept of a minimal temporal interval and we would be involved in infinite regress, the very thing which I intend to avoid at all costs. However, the distance between ksanas can, so it is suggested, be extended, or, more precisely, the distance between the successive kernels of the event capsules, where the ultimate events occur, can be extended. That is, there are gaps between events. As is explained in other posts, in UET the ‘Space/Time region’ occupied by the successive members of an event-chain remains the same irrespective of ‘states of motion’ or other distinguishing features. But the dimensions themselves can and do change. If the space-capsules contract, the time dimension must expand and this can only mean that the gaps between ksanas widen (since the extent of an ‘occupied’ ksana is cnstant. The more the space capsules contract, the more the gaps must increase (Note 3).  But, as with everything else in UET, there is a limiting value since the space capsules cannot contract beyond the spatial limits of the incompressible kernels. Note that this ‘Constant Region Principle’ only applies to causally related regions of space ─ roughly what students of SR view as ‘light cones’.

The third parameter of motion

 In traditional physics, when considering an object or body ‘in motion’, we essentially only need to specify two variables : spatial position and time. Considerations of momentum and so forth is only required because it affects future positions at future moments, and aids prediction. To specify an object’s ‘position in space’, it is customary in scientific work to relate the object’s position to an imaginary spot called the Origin where three mutually perpendicular axes meet. To specify the object’s position ‘in time’ we must show or deduce how many ‘units of time’ have elapsed since a chosen start position when t = 0. Essentially, there are only two parameters required, ‘space’ and ‘time’ : the fact that the first parameter requires (at least) three values is not, in the present context, significant.
Now, in UET we likewis need to specify an event’s position with regard to ‘space’ and ‘time’. I envisage the Event Locality at any ‘given moment’ as being composed of an indefinitely extendable set of ‘grid-positions’. Each ‘moment’ has the same duration and, if we label a particular ksana 0 (or 1) we can attach a (whole) number to an event subsequent to what happened when t = 0 (or rather k = 0). As anyone who has a little familiarity with the ideas of Special Relativity knows, the concept of an  ‘absolute present’ valid right across the universe is problematical to say the least. Nonetheless, we can talk of events occurring ‘at the same time’ locally, i.e. during or at the same ksana. (The question of how these different  ‘time zones’ interlock will be left aside for the moment.)
Just as in normal physics we can represent the trajectory of an ‘object’ by using three axes with the y axis representing time and, due to lack of space and dimension, we often squash the three spatial dimensions down to two, or, more simply still, use a single ‘space’ axis, x (Note 4). In normal physics the trajectory of an object moving with constant speed will be represented by a continuous vertical straight line and an object moving at constant non-zero speed relative to an object considered to be stationary will be represented by a slanting but nonetheless still straight line. Accelerated motion produces a ‘curve’ that is not straight. All this essentially carries over into UET except that, strictly, there should be no continuous lines at all but only dots that, if joined up, would form lines. Nonetheless, because the size of a ksana is so small relative to our very crude senses, it is usually acceptable to represent an ‘object’s’ trajectory as a continuous line. What is straight in normal physics will be straight in UET. But there is a third variable of motion in UET which has no equivalent in normal physics, namely an event’s re-appearance rhythm.
        Fairly early on, I came up against what seemed to be an insuperable difficulty with my nascent model of physical reality. In UET I make a distinction between an attainable ‘speed limit’ for an event-chain and an upper unatttainable limit, noting the first c * and the second c. This allows me to attribute a small mass ─ mass has not yet been defined in UET but this will come ─  to such ‘objects’ as photons. However, this distinction is not significant in the context of the present discussion and I shall  use the usual symbol c for either case. Now, it is notorious that different elementary particles (ultimate event chains) which apparently have the same (or very nearly identical) speeds do not behave in the same way when confronted with obstacles (large dense event clusters) that lie on their path. Whereas it is comparatively easy to block visible light and not all that difficult to block or at least muffle much more energetic gamma rays, it is almost impossible to stop a neutrino in its path, so much so that they are virtually undetectable. Incredible though it sounds, “about 400 billion neutrinos from the Sun pass through us every second” (Close, Particle Physics) but even state of the art detectors deep in the earth have a hard  job  detecting a single passing neutrino. Yet neutrinos travel at or close to the speed of light. So how is it that photons are so easy to block and neutrinos almost impossible to detect?
The answer, according to matter-based physics, is that the neutrino is not only very small and very fast moving but “does not feel any of the four physical Reappearance rates 2forces except to some extent the weak force”. But I want to see if I can derive an explanation without departing from the basic principles and concepts of Ultimate Event Theory. The problem in UET is not why the repeating event-pattern we label a neutrino passes through matter so easily ─ this is exactly what I would expect ─ but rather how and why it behaves so  differently from certain other elementary event-chains. Any ‘particle’, provided it is small enough and moves rapidly, is likely, according to the basic ideas of UET, to ‘pass through’ an obstacle just so long as the obstacle is not too large and not too dense. In UET, intervening spatial positions are simply skipped and anything that happens to be occupying these intermediate spatial positions will not in any way ‘notice’ the passing of the more rapidly moving ‘object’. On this count, however, two ‘particles’ moving at roughly the same speed (relative to the obstacle) should either both pass through an  obstacle or both collide with it.
But, as I eventually realized, this argument is only valid if the re-appearance rates of the two ‘particles’ are assumed to be the same. ‘Speed’ is nothing but a space/time ratio, so many spatial positions against so many ksanas. A particular event-chain has, say, a ‘space/time ratio’ of 8 grid-points per ksana. This means that the next event in the chain will have occurrence at the very next ksana exactly eight grid-spaces along relative to some regularly repeating event-chain considered to be stationary. On this count, it would seem impossible to have fractional rates and every ‘re-appearance rate’ would be a whole number : there would be no equivalent in UET of a speed of, say, 4/7 metres per second since grid-spaces are indivisible.
However, I eventually realized that it was not one of my original assumptions that an event in a chain must repeat (or give rise to a different event) at each and every ksana. This at once made fractional rates possible even though the basic units of space and time are, in UET, indivisible. A ‘particle’ with a rate of 4/7 s0 /t0 could, for example, make a re-appearance four times out of every seven ksanas ─ and there are any number of ways that a ‘particle’ could have the same flat rate while not having the same re-appearance rhythm. 

Limit to unitary re-appearance rate

It is by no means obvious that it is legitimate to treat ‘space’ and ‘time’ equivalently as dimensions of a single entity known as ‘Space/Time’. A ‘distance’ in time is not just a distance in space transferred to a different axis and much of the confusion in contemporary physics comes from a failure to accept, or at the very least confront, this fact. One reason why the dimensions are not equivalent is that, although a spatial dimension such as length remains the same if we now add on width, the entire spatial complex must disappear if it is to give rise to a similar one at the succeeding moment in time ─ you cannot simply ‘add’ on another dimension to what is already there.
However, for the the time being I will follow accepted wisdom in treating a time distance on the same footing as a space distance. If this is so, it would seem that, in the case of an event-chain held together by causality, the causal influence emanating from the ‘kernel’ of one event capsule, and which brings about the selfsame event (or a different one) a ksana later in an equivalent spatial position, must traverse at least the ‘width’ or diameter of a space capsule, noted s0 (if the capsule is at rest). Why? Because if it does not at least get to the extremity of the first spatial capsule, a distance of ½ s0  and then get to the ‘kernel’ of the following one, nothing at all will happen and the event-chain will terminate abruptly.
This means that the ‘reappearance rate’ of an event in an event-chain must at least be 1/1 in absolute units, i.e. 1 s0 /t0 , one grid-space per ksana. Can it be greater than this? Could it, for example, be  2, 3 or 5 grid-spacesper ksana? Seemingly not. For if and when the ultimate event re-appears, say  5 ksanas later, the original causal impulse will have covered a distance of 5 s0   ( s0 being the diameter or spatial dimension of each capsule) and would have taken 5 ksanas to do  this. And so the space/time displacement rate would be the same (but not in this case the actual inter-event distances).
It is only the unitary rate, the distance/time ratio taken over a single ksana, that cannot be less (or more) than one grid-space per ksana : any fractional (but not irrational) re-appearance rate is perfectly conceivable provided it is spread out over several ksanas.  A re-appearance rate of m/n s0/t0  simply means that the ultimate event in question re-appears in an equivalent spatial position on the Locality m times every n ksanas where m/n ≤ 1. And there are all sorts of different ways in which this rate be achieved. For example, a re-appearance rate of 3/5 s0/t0 could be a repeating pattern such as

 

   ™˜™™™™™™™™™™™™™™™™™™™™™™Reappearance rates 1

 

 

 

 

 

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

Lower Limit of re-creation rate 

Since, by definition, everything in UET is finite, there must be a maximum number of possible consecutive non-reappearances. For example, if we set the limit at, say, 20 blank lines, or 200 03 2000, this would mean that, each time this was observed, we could conclude that the event-chain had terminated. This is the UET equivalent  of the Principle of Spatio-Temporal Continuity and effectively excludes phenomena such as an ultimate event in an event-chain making its re-appearance a century later than its first appearance. This limit would have to be estimated on the  basis of experiments since I do not see how a specific value can be derived from theoretical considerations alone. It is tempting to estimate that this value would involve c* or a multiple of c* but this is only a wild guess ─ Nature does not always favour elegance and simplicity.
Such a rule would limit how ‘stretched out’ an event-chain can be temporally and, in reality , there may not after all be a hard and fast general rule  : the maximal extent of the gap could decline exponentially or in accordance with some other function. That is, an abnormally long gap followed by the re-appearance of an event, would decrease the possible upper limit slightly in much the same way as chance associations increase the likelihood of an event-chain forming in the first place. If, say, there was an original limit of a  gap of 20 ksanas, whenever the re-appearance rate had a gap of 19, the limit would be reduced to 19 and so on.
It is important to be clear that we are not talking about the phenomenon of ‘time dilation’ which concerns only the interval between one ksana and the next according to a particular viewpoint. Here, we simply have an event-chain ‘at rest’ and which is not displacing itself laterally at all, at any rate not from the viewpoint we have adopted.

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

Since Galileo, and subsequently Einstein, it has become customary in physics to distinguish, not between rest and motion, but rather between unaccelerated motion and  accelerated motion. And the category of ‘unaccelerated motion’ includes all possible constant straight-line speeds including zero (rest). It seems, then,  that there is no true distinction to be made between ‘rest’ and motion just so long as the latter is motion in a straight line at a constant displacement rate. This ‘relativisation’ of  motion in effect means that an ‘inertial system’ or a particle at rest within an inertial system does not really have a specific velocity at all, since any estimated velocity is as ‘true’ as any other. So, seemingly, ‘velocity’ is not a property of a single body but only of a system of at least two bodies. This is, in a sense, rather odd since there can be no doubt that a ‘change of velocity’, an acceleration, really is a feature of a single body (or is it?).
So what to conclude? One could say that ‘acceleration’ has ‘higher reality status’ than simple velocity since it does not depend on a reference point outside the system. ‘Velocity’ is a ‘reality of second order’ whereas acceleration is a ‘reality of first order’. But once again there is a difference between normal physics and UET physics in this respect. Although the distinction between unaccelerated and accelerated motion is taken over into UET (re-baptised ‘regular’ and ‘irregular’ motion), there is in Ultimate Event Theory a new kind of ‘velocity’ that has nothing to do with any other body whatsoever, namely the event-chain’s re-appearance rate.
When one has spent some time studying Relativity one ends up wondering whether after all “everything is relative” and that  the universe is evaporating away even as we look it leaving nothing but a trail of unintelligible mathematical formulae. In Quantum Mechanics (as Heisenberg envisaged it anyway) the properties of a particular ‘body’ involve the properties of all the other bodies in the universe, so that there remain very few, if any, intrinsic properties that a body or system can possess. However, in UET, there is a reality safety net. For there are at least two  things that are not relative, since they pertain to the event-chain or event-conglomerate itself whether it is alone in the universe or embedded in the dense network of intersecting event-chains we view as matter. These two things are (1) the number of ultimate events in a given portion of an event-chain and (2) the re-appearance rate of events in the chain. These two features are intrinsic to every chain and have nothing to do with velocity or varying viewpoints or anything else.  To be continued SH

Note 1   This principle (Spatio-temporal Continuity) innocuous  though it may sound, has also had  extremely important social and political implications since, amongst other things, it led to the repeal of laws against witchcraft in the ‘advanced’ countries. For example, the new Legislative Assembly in France shortly after the revolution specifically abolished all penalties for ‘imaginary’ crimes and that included witchcraft. Why was witchcraft considered to be an ‘imaginary crime’? Essentially because it  violated the Principle of Spatio-Temporal Continuity. The French revolutionaries who drew the statue of Reason through the streets of Paris and made Her their goddess, considered it impossible to cause someone’s death miles away simply by thinking ill of them or saying Abracadabra. Whether the accused ‘confessed’ to having brought about someone’s death in this way, or even sincerely believed it, was irrelevant : no one had the power to disobey the Principle of Spatio-Temporal Continuity. The Principle got somewhat muddied  when science had to deal with electro-magnetism ─ Does an impulse travel through all possible intermediary positions in an electro-magnetic field? ─ but it was still very much in force in 1905 when Einstein formulated the Theory of Special Relativity. For Einstein deduced from his basic assumptions that one could not ‘send a message’ faster than the speed of light and that, in consequence,  this limited the speed of propagation of causality. If I am too far away from someone else I simply cannot cause this person’s death at that particular time and that is that. The Principle ran into trouble, of course,  with the advent of Quantum Mechanics but it remains deeply entrenched in our way of thinking about the world which is why alibis are so important in law, to take but one example. And it is precisely because Quantum Mechanics appears to violate the principle that QM is so worrisome and the chief reason why some of the scientists who helped to develop the theory such as Einstein himself, and even Schrodinger, were never happy with  it. As Einstein put it, Quantum Mechanics involved “spooky action at a distance” ─ exactly the same objection that the Cartesians had made to Newton. 

Note 2  Ideally, we would have a lighted three-dimensional framework flashing on and off and mark the successive appearances of the ‘object’ as, say, a red point of light comes on periodically when the lighted framework comes on.

Note 3 In principle, in the case of extremely high speed event-chains, these gaps should be detectable even today though the fact that such high speeds are involved makes direct observation difficult. 

Note 4 This is not how we specify an object’s position in ordinary conversation. As Bohm pertinently pointed out, we in effect speak in the language of topology rather than the language of co-ordinate geometry. We say such and such an object is ‘under’, ‘over’, ‘near’, ‘to the right of’ &c. some other well-known  prominent object, a Church or mountain when outside, a bookcase or fireplace when in a room.
Not only do coordinates not exist in Nature, they do not come at all naturally to us, even today. Why is this? Chiefly, I suspect because they are not only cumbersome but practically useless to a nomadic, hunting/food gathering life style and we humans spent at least 96% of our existence as hunter/gatherers. Exact measurement only becomes essential when human beings start to manufacture complicated objects and even then many craftsmen and engineers used ‘rules of thumb’ and ‘rough estimates’ well into the 19th century.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

                                          =   c

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

Relativity from Ultimate Event Theory?

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

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

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

♦        ♦       ♦       ♦       ♦    ………

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

The ‘Space-Time’ Capsule

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

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

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

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

The Rest Postulate

This says that

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

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

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

Diverging Regular Event-chains

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

tan θr  = vr  = 1, 2, 3, 4……   ­­

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

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

Which distance or time interval to choose?

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

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

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

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

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

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

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

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

To be continued  SH  18 July 2013

 

A ksana is the minimal temporal interval : within the space of a ksana one and only one ultimate event can have occurrence. There can thus be no change whatsoever within the space of a ksana — everything is at rest.
In Ultimate Event Theory every ultimate event is conceived to fill a single spot on the Locality (K0) and every such spot has the same extent, a ‘spatial’ extent which includes (at least) three dimensions and a single temporal dimension. A ksana is  the temporal interval between the ‘end’ of one ultimate event and the ‘end’ of the next one. Since there can be nothing smaller than an ultimate event, it does not make too much sense to speak of ‘ends’, or ‘beginnings’ or ‘middles’ of ultimate events, or their emplacements, but, practically speaking, it is impossible to avoid using such words. Certainly the extent of the spot occupied by an ultimate event is not zero.
The ksana is, however, considerably more extensive than the ‘vertical’ dimension of the spot occupied by an ultimate event. Physical reality is, in Ultimate Event theory, a ‘gapped’ reality and, just as an atom is apparently mainly empty space, a ksana is mainly empty time (if the term is allowed)..  Thus, when evaluating temporal intervals the ‘temporal extent’ of the ultimate events that have occurrence within this interval can, to a first approximation, be neglected. As to the actual value of a ksana in terms of seconds or nanoseconds, this remains to be determined by experiment but certainly the extent of a ksana must be at least as small as the Planck scale, 6.626 × 10–34 seconds.
A ‘full’ event-chain is a succession of bonded ultimate events within where it would not be possible to fit in any more ultimate events. So if we label the successive ultimate events of a ‘full’ event-chain, 0, 1, 2, 3……N  there will be as many ksanas in this temporal interval as there are ultimate events.
Suppose we have a full event-chain which, in its simplest form, may be just a single ultimate event repeated identically at or during each successive ksana. Such an event-chain can be imaged as a column of dots where each dot represents an ultimate event and the space in between the dots represents the gap between successive ultimate events of the chain. Thus , using the standard spacing of 2.5  this computer we have

Now, although the ‘space’ occupied by all ultimate events is fixed and  an absolute quantity (true for ‘all inertial and non-inertial frames’ if you like), the spacing between the spots where ultimate events can occur both ‘laterally’ — laterally is to be understood as including all three normal spatial dimensions — and vertically, i.e. in the temporal direction, is not  constant but variable. So, although the spots where ultimate events can occur have fixed (minuscule) dimensions, the ‘grid-distance’, the distance between the closest spots which have occurrence within the same ksana,  and so  does the temporal distance between successive ultimate events of a full event-chain. So the ksana varies in extent.  However, there is, by hypothesis,  a minimum value for both the grid-distance and the ksana. The minimal value of both is attained whenever we have a completely isolated event-chain. In practice, there is no such event-chain any more than, in traditional physics, there is a body that is completely isolated  from all other bodies in the universe. However, these minimal values can be considered to be attained for event-chains that are sufficiently ‘far away’ from all other chains. And, more significantly, these minimal values apply whenever we have a full regular event-chain considered in isolation from its event environment.
The most important point, that cannot be too strongly emphasized, is that although the number of ultimate events in an event-chain, or any continuous section of an event-chain, is absolute, the interval between successive events varies from one chain to another, though remaining constant within a single event-chain (providing it is regular). Unless stated otherwise, by ‘ksana’ I mean the interval between successive ultimate events in a ‘static’ or isolated regular event-chain. This need not cause any more trouble than the concept of intervals of time in Special Relativity where ‘time’ is understood to mean ‘proper time’, the ‘time’ of a system at rest, unless a contrary indication is given.
Thus, the ‘vertical’  spacing of events in different chains can and does differ and the minimal value will be represented by the smallest spacing available on the computer I am using. I could, for example, increase the spacing from the standard spacing to

•           or to                     •

•                                     •

moment’, is not an absolute. However, unless stated otherwise, by ‘ksana’ we are to understand the duration of a ksana within a ‘static’ or isolated regular event-chain. This should not cause any more trouble than the concept of ‘time’ in Special Relativity where ‘time’ is understood to mean ‘proper time’, the ‘time’ of a system at rest, unless a contrary indication is given. However, the ‘vertical’  spacing of events in different chains can and does differ. I could, for example, increase the spacing from the standard spacing to

•           or to                     •

•                                        •

•                                       •

S.H. 11/7/13