Archives for category: Minkowski

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

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

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

# •  •    •    ••

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

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

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

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

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

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

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

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

v =    ( tan θ)/ (tan β cos φ)

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

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

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

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

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

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

Parable of the Two Friends and Railway Stations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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