Archives for category: Special Relativity

The Two Postulates of Special Relativity

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

The Third Postulate 

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

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

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

Inertial frames in UET 

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

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

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

Upper Speed limit?  

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

Derivation of basic formulae in UET 

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

 

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

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

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

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

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Most schoolchildren these days who have got beyond GCSE, certainly those who study science, have heard of the Lorentz transformations which, in the theory of Special Relativity, replace the Galilean transformations. These ‘transformations’ ─ ‘adaptations’ would perhaps be a better term ─   enable us to plot the motion of a body using two different co-ordinate systems  (basically three lines at right angles to each other), and to convert the specifications of a body’s position within one system to their equivalent specifications in the other. Why do we need these transformations/adaptations? Because in everyday life, especially modern life,  we are forever switching from one event environment to another and often need to communicate our changing position to someone whose instantaneous position and general state of motion is different to ours. If I take a couple of steps in an aeroplane I have changed my location within the aeroplane by a few metres at most. But, from the point of view of a land-based control system at an airport, my position has changed very substantially indeed because the controller will have ‘added in’ the speed of the aeroplane. If we did not continually convert positions within one system to equivalent positions in another, there would be serious accidents all the time and modern life would be impossible. Even at the level of a band of hunters stalking a mammoth, quite sophisticated calculations of the prey’s movements relative to, say, certain prominent rocks or trees visible to all the hunters would have been necessary. Indeed, some ethnologists have even conjectured that it was this need to communicate vital information to colleagues while on a hunting expedition that gave rise to spoken language in the first place ─ “Go right, he’s behind that tree”, “He’s turning round, watch out!!” and so forth (Note 1.)

    Although they have a fancy name, ‘Galileian’ transformations (named after Galileo) are entirely commonsensical. If an object is moving steadily in a set direction while we remain at the same spot, we obviously have to take the object’s speed into account when, for example,  we take aim with a rifle or bow and arrow. Distances from a fixed point ─ a particular rock or tree, say ─  will be different  depending on whether we are using the animal’s changing ‘position system’ or our own static one. The object’s distance from a fixed point along the direction of travel is going to increase if the object is moving relative to it, but, if the object is moving at a constant speed this distance will increase in a completely  predictable manner. Our distance d from the fixed point, however, is going to stay the same if we are stationary. To work out the moving object’s changing distance from the fixed point we have to factor in the object’s speed v in order to predict where it will be in so many seconds or minutes and aim accordingly. What we do is to multiply the object’s speed, which we assume to be constant, by the anticipated lapse of time, not forgetting to add the original distance from the fixed point.  Mathematically d′ (d prime) = d + vt where v is (by hypothesis) constant, say 5 metres a second. If the original distance between the object and a certain landmark is 10 metres and the object is moving at a rate of 5 metres per second, the distance at the end of the fifth second will be 10 + 25 = 35 metres.

The situation becomes more complicated if we ourselves are moving relative to a fixed point, stalking our prey in a canoe for example, since we have to factor in our speed as well. Nonetheless, predators and even experienced human hunters are incredibly good at making these sort of complicated predictions, even taking into account a prey’s variable  speed and zig-zag path. Big cats and hunters with bow and arrow  became good at this sort of thing because their survival depended on such calculations : sheer speed and brute strength are, by themselves, not enough. Just as the science of geometry (literally ‘land-measurement’) had its unromantic origin in the accurate surveying of land for taxation purposes, kinematics (the study of objects in motion) almost certainly had its origins in hunting and warfare, especially  archery.

But if we ourselves are stationary whle the distance of our moving object changes along the direction of travel, i.e. we have to modify the x  coordinate, everything else remains the same. If there is a wall exactly parallel to the direction of travel, the object’s distance from the wall, provided he or it keeps on track, remains the same, also the object’s height above sea level if there are no bumps or hills on its path. As for time, ‘obviously’ a second is a second wherever you are or whatever you are doing. Newton himself stated his firm belief in ‘absolute time’ in his Principia though he conceded that actual methods or devices for measuring time might vary quite a lot for technical reasons ─ precise time-keeping on a ship, for example, is notoriously difficult  because of the erratic motion of the ship in bad weather and designing a timepiece that ‘kept time’ accurately on board ship was a huge challenge (see the book Longitude).

To sum up, in a simple case of an object’s motion in one direction at constant speed, the Galileian transformations remained the same except in the direction of motion. Mathematically, we have x′ = x ± vt,  y′ = y, z′ = z, t′ = t.

All this sounds so obvious that scarcely anyone gave much thought to the matter until the null result of the Michelsen-Morley experiment designed to detect the Earth’s movement through the surrounding ‘aether’. Lorentz seems to have developed his transformations in an essentially ad hoc manner in order to cope with the Michelsen-Morley experiment and other puzzling experimental results. The Lorentz transformations differ from the ‘normal’ Galileian ones  in two respects. Firstly, the adaptation required for distances along the presumed path of travel when changing from one coordinate system to the other, is somewhat more complicated. Secondly, the ‘time’ dimension needs tinkering with. In effect the Lorentz Transformations imply that ‘time’ does not run at the same rate in a stationary system as it does in a moving one ─ though Lorentz himself does not seem to have drawn this particular conclusion. The Irish physicist Fitzgerald did seriously suggest that “lengths were contracted in the direction of the Earth’s motion through the ether” but no one before Einstein seems to have thought that time ‘runs slower’ or ‘faster’ depending on one’s ‘state of motion’ and reference system. Since any regular sequence of events can be used as a time-keeping system, this in effect means that physical processes ‘speed up’ or ‘slow down’ depending on where you are standing and how you and the observed object are moving relative to each other. (Note 2).

Einstein, in his 1905 paper,- developed the exact same Lorentz formulae from first principles and always maintained that he did not at the time (sic) know of Lorentz’s work. What were Einstein’s assumptions? Only two.

 

  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.

It has since been pointed out that 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 3). Einstein also seems to have envisaged  ‘space’ and ‘time’ as being ‘continuous’ ─ certainly all physicists at the time  assumed this without question and the wave theory of electro-magnetism required it as its inventor, Maxwell, was well 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 become more prominent when the mathematician Minkowski (one of Einstein’s teachers) got to work on the problem and coined the well-known phrase ‘Space/Time continuum’. It is only quite  recently that one or two physicists  have dared to suggest that space and time might be  ‘grainy’ and, even then, very few have seriously thought about the serious consequences of such an assumption. Unless specifically told otherwise, physics students still tend to  assume that physical processes are continuous, proceed ‘without a break’ as it were.  Despite the photo-electric effect, quantum wave/particle duality, the complete victory of the digital computer over the analogue and all sorts of other phenomena that point in the direction of discreteness and discontinuity, most physicists  still think of ‘space’, ‘time’ and electro-magnetism as being ‘continuous’. And training in calculus methods, of course, only reinforces this essentially misguided model of physical reality.
(Note 4)

Coordinate Systems ─ who needs them?

Although inertia itself is undoubtedly a force to be reckoned with (and thus ‘real’) inertial ‘frames’, which play such a big role in Special Relativity, do not exist in Nature. Migrating birds navigate expertly across the surface of the globe without knowing anything at all about co-ordinate systems  ─ though they must have some sort of an inherited ‘neural positioning system’ of their own. Co-ordinate systems and similar mathematical devices are man-made systems, constructed for our own interest and convenience, and it was precisely this realisation that, along with other considerations, motivated Einstein himself at a slightly later date to try to formulate the ‘laws of physics’ in a way that did not depend on any particular reference frame, inertial or not. Practically speaking, at any rate in an advanced industrial society, we need the complicated and often tiresome paraphernalia of co-ordinate systems, inertial frames and transformation formulae, but this does not make any of them independently real. And this is worth saying because the tendency today is to give the student the impression that there is only one way to Special Relativity and other branches of physics, the co-ordinate way : the student is taught to distrust his or her ‘intuition’ and ‘common sense’ in favour of familiarity with complicated mathematical constructs which are in effect treated as being  ‘more real than reality’.
The problem is compounded by warnings about “sticking strictly to observables”. Heisenberg and Bohr are the main people responsible for this positivistic approach and it may be some consolation to those of us who find this kind of talk irritating and counter-productive to know that Einstein was of the same opinion. He in effect told Heisenberg that it made no sense to ‘keep strictly to observables’ since “It is the theory that decides what we can observe” ─ a pretty devastating retort. Certainly, it is theory that points the experimenter in a direction that otherwise he or she would most likely never have taken. It is ironical that the experimentalist Millikan was so outraged by Einstein’s ‘particles of light’ theory that he at once embarked on a series of experiments to prove that the hare-brained notions  of an obscure employee at the Swiss Patents Office were a lot of rubbish (Note 5) ─ and ended up by providing Einstein with valuable evidence in favour of  his (Einstein’s) theory.
Though I certainly hope that one day someone will undertake experiments that test some of my own guesses and predictions, I make no apology for the ‘qualitative’ approach I am currently taking. Formulae we already have, enough to sink a battleship, but, to make further progress, we may well need to ‘return to basics’ and think differently about well-known physical phenomena. I also believe it is desirable where possible to obtain a clear visual or tactile picture of what is most likely going on beneath the surface of phenomena. Most people, even pure mathematicians, work with a semi-conscious model of reality at the back of their minds anyway, so one might as well lay one’s hands on the table and admit the fact. Several of these mental picture-maps have proved to be extremely helpful and we would not be where we are today without the had to be abandoned in the end. Incredible though it sounds today, the majority of the scientific establishment at the beginning of the 20th century was deeply sceptical about the ‘reality’ of atoms  and elementary particles because their existence could, at the time, only be inferred, not observed directly. The Austrian physicist Mach (of Mach numbers fame) remained sceptical practically to the end of his life.

A Limiting Speed for all causal processes

During the years following Einstein’s two 1905 papers that laid out the basic ideas of the theory of Special Relativity ─ ‘special’ because it only applied to inertial systems and, in particular,  ignored gravity altogether ─ Einstein was criticised, and in a sense rightly, for the excessive importance he gave to light and electro-magnetism. Einstein himself conceded that “it is immaterial what physical process one chooses for a definition of time…..[provided] the process enables relations to be established between different places.” Nonetheless, the emphasis given to electro-magnetism (and for that matter to clocks and measuring rods) masks the true nature of the Special Theory : it is basically a theory about the propagation not of light but of causality. Time and time keeping come into the picture only because causality is sequential ─ the effect follows the cause and there is always a time lag between the two.
There is, in the literature of Relativity, a great deal of talk  about ‘sending messages across Space/Time’  and how one cannot do this at a speed greater than that of light (though now and again some reputable scientists raise their heads above the parapet and claim to have done just this). But why is the sending of a message important? To many people today, it is important because that is basically what everything is about, i.e. the transfer of ‘information’ ─ as if the universe had nothing better to do than to amass data and send it around at high speed over its own Internet. But this is no answer at all. The rate of transmission of ‘information’ is important because if I want something done at a place far away from where I am standing, I have in some way or other to ‘send a message’ so that someone or something will carry out my wishes. It is the resulting event and not the transfer of information than matters, not data and its dissemination but what actually happens. “The world is everything that has occurrence”, not “everything that can be computer-programmed” ─ to paraphrase Wittgenstein (Note 6).
It remains regrettable that Einstein did not emphasize more strongly the causal aspect of his theory. Minkowski upstaged Einstein by transferring the basic issue around which SR revolved away from the realm of physics to the realm of (semi-pure) mathematics. This development ─ of which Einstein originally disapproved ─ was in one sense fruitful since it led on to the General Theory of Relativity, but it also closed off other avenues along which Einstein’s thinking had been moving. Einstein in effect transferred his interest from particles (photo-electric effect, Brownian motion &c.) to fields and the field is, by definition, a continuous concept. There are no ‘holes’ or ‘gaps’ in the magnetic field surrounding a horseshoe or bar magnet, nor any ‘safe places to hide’ in the gravitational field near the centre of the Earth. Indeed, Einstein ended up by believing so much in  fields that he decided they were the only reality and attempted to develop a theory that there was, at bottom, only one all embracing, ‘Unified Field’.

Relevance to Ultimate Event Theory

 Why is all this significant in the context of the theory of ‘Ultimate Events’ I am trying to develop? Because, at one stage in his career, Einstein focussed his attention on precisely localized events in Space/Time and their causal interconnections and made the most important contribution to date to a scientific theory of causality by pointing out that the speed of light, if it was an upper limit to the propagation of causal influences of any nature whatsoever, divided the universe into distinct ‘causal zones’. The possible ‘causal range’ of an event was thus limited in a precise way and one thus had, in principle, a foolproof way of deciding whether events occurring at distant  places could or could not be, causally connected (Note 7).
        In a sense, I suppose what I am attempting to do, without at first realizing it and naturally without Einstein’s abilities,  is to turn the clock back to 1905 before Minkowski and Quantum Mechanics muddied the event waters. What we had then for a brief moment was  a world-view centred on pointlike events that had occurrence at precise positions on the Locality (Space/Time if you like) and a network of possible causal relations stretching out in every direction (except backwards in time). Light and electro-magnetic phenomena are not essential to this overall picture  which covers all physical phenomena and the idea that there is an ‘upper speed limit’ for all physical processes would apply even in a world where there was no such thing as electro-magnetic radiation, supposing this to be possible.
So, I asked myself what assumptions, within the general schema of Ultimate Event Theory, are required to derive the basic results of Special Relativity or their equivalents? Only two as far as I can see ─ apart from certain very general assumptions about the nature of ‘events’ and their localization, the equivalent of a crude molecular/atomic theory such as existed around 1900. The second postulate is tricky to formulate and will be dealt with separately but the first is straightforward enough : it is the Limiting Speed Principle. This states that there is a limiting ‘speed’ to the propagation of all causal influences ─ a fairly reasonable assumption I think. What is its basis? Well, I simply cannot imagine a ‘world’ where there was not such a principle : effects would be instantaneous with causes and everything would be happening at once. Such a state of affairs does apparently exist, at least exceptionally,  in sorcery but not usually in science (Note 8) Eddington once observed that one could decide simply from a priori assumptions that there  must, in every possible universe, be an upper speed limit to the transfer of energy or information ─ though he added that the actual value of  such a constant would have to be decided by experiment.  I entirely go along with this; so what effect does the Principle have on the ‘normal’ method of compounding velocities?

Addition of Velocities 

If I roll a marble  down the floor of a carriage in a train moving at a constant rate, it will have a certain speed that I naturally calculate relative to the ‘co-ordinate system’ of the carriage which for me is stationary ─ it might as well be held up at a station for all the difference it makes. However, for someone standing outside the train, say in a field as the train whizzes by, the marble has a much greater velocity. Calling the marble’s velocity relative to me and the train v1 and the velocity of the train v2, the marble’s velocity, viewed by a farmer or shepherd in a field is simply v3 = v1 + v2 . But the Earth itself is moving on an approximately straight path during a short interval of time at nearly constant speed, so if we add on the Earth’s orbital speed and then the speed of the solar system as it orbits the centre of the Milky Way galaxy, we soon arrive at quite stupendous speeds even for a slow-moving  marble.
The big question is : can a ‘linear’ combination of speeds end up by exceeding the upper limit, c, and, if so, does this matter? The answer to the first question is, “Yes” and to the second question, “That depends”.
In UET, a distinction is made between an unattainable upper limit c, and the highest attainable speed, c* . But this complication need not bother us for the moment and I shall follow established usage and treat c as being an all-round upper limit, whether attained or not, which is around  3 × 108 m/sec. So v, the speed of any object or process (excluding light, X-rays, microwaves and so on)  is by definition less than c, i.e. v < c . But suppose we ‘compound’ speeds as in the case of the marble in the train which is itself fixed to the moving earth &c. &c., can a combination of speeds, each less than c, exceed c ?  Obviously, yes, since, for example, c/2 + c/2 + c/2  = (3/2)c  > c.
        What about if we restrict ourselves to just two speeds ? The most instructive way to deal with this is to express v in terms of c, i.e. make it a ‘fraction’ of c, conceivably an improper fraction. So we have c/m + c/n where both m and n are > 1 . The ‘compounded’ speed is thus  c/m + c/n  =  c(1/m + 1/n)  m, n > 1
The smallest possible integral choice for m and n is 2. In such a case the sum of the two individual speeds, though both less than c, equals c since c(1/2  + 1/2)  = c. However, even this speed is not actually greater than c  and for any other choice of integers we will not even be able to equal c. For example,  c/2 + c/3 = (5/6) c < c . More generally,  c(1/m + 1/n) = c (m + n)/(mn)  and mn, the denominator > (m + n) unless m, n = 2 (both of them). This is actually an interesting result in UET though not in normal  physics.
What about if we let m and n be ‘improper fractions’, i.e. rational numbers > 1. We will still have individual speeds < c since in each case the denominator > 1. All we need is to find a case where both m and n  > 1 (and so c/m and c/n are each < c) but where (m + n)/(mn) > 1. This is not too difficult. For example, if v1 = c/(3/2)        v2 = c/(9/8)

v1 + v2  = c ((3/2) + (9/8)) )  = c ( (12 + 9)/8  = 42/27 = c (14/9) > c
                   (3/2) × (9/8)          27/16

        There are in fact any number of cases where we can find rational numbers (but not integers) that fit the case.

So the ‘obvious’ rule for adding velocities doesn’t work if we restrict all velocities to be less than c ─ or in the sole case of light equal to c. So how can we manipulate the rule for adding velocities to always keep under the upper limit of c ?  Examining the expression  c (m + n)/mn  it is clear that we must somehow negate the fateful influence of the dominator mn since it is this factor, increasing as it does so much faster than (m + n), that tips the balance. So what we need is a new factor  mn/f(mn)    which when  applied will get rid of the mn in c (m + n)/mn   and stop the whole  thing from exceeding c. The simplest such function is mn/(1 + mn) since, for all m, n > 1 this is greater than mn while at the same time ridding us of the mn in c (m + n)/mn    Also, and crucially,   the final result    

(m + n)  ×     mn        =  (m + n)    < 1           since the denominator is
mn         (1 + mn)        ( 1 + mn)

larger than the numerator for all m, n > 1. (If you don’t believe me try out a few values.)

So, if we make the rule for combining two velocities, each less than c,

v1 + v2 = (v1 + v2 )
1 + v1 v2
we shall always get a result less than c.
Whether this is the only rule that ‘does the trick’ I do not know ─ I would guess that it is not but it does seem to be the simplest such rule and the one that follows most naturally from the situation. Nonetheless, I am not altogether happy with this derivation (given in more detail in the post 48) because it is too mathematical. I would like to see this rule emerging from some inevitable physical, visualizable situation but currently I don’t see how to manage this (Note 9).
Anyway, we have now the first of Einstein’s new formulae of SR, the rule for the addition or compounding of velocities so that the resulting velocity never exceeds a certain limit. What is required now is a way of deriving, using the basic concepts of UET, the Einstein formulae for space contraction, time dilation and, more important of all, the celebrated E = mc2 equivalence of mass and energy equation.    SH 
Note 1. It is worth pointing out that, although the use of some sort of reference system to locate a moving object, must go back very far in time, the most natural way of doing this is not numerical/geometrical but topological. We do not say an object in a room is so many feet from a particular corner at ground level, so many feet above the ground and so on. We say the object is ‘on’ the table, ‘underneath’ the chair, ‘above the bookcase’, ‘alongside’ the fireplace and so on. These directions are ‘topological’ since topology is the branch of mathematics which deals in ‘nearness’ and ‘connectedness’ to the exclusion of metrical distance. Bohm is the first person to have pointed this out as far as I know (in a recorded debate with, I think, Price).

Note 2   From a Newtonian, or even classical scientific’ point of view, it was unthinkable that ‘the rate of physical processes should ‘speed up’ or ‘slow down’ depending on where you were standing and how you and the observed object were moving relative to each other. But it is not actually such a shocking idea from a non-scientific, ‘subjective’ standpoint. We are all familiar with how ‘experienced time’ speeds up or slows down according to mood, “A watched kettle never boils” and so on. It is only necessary to extend the range and validity of the basic principle. Explaining, or perhaps simply describing, the Mossbauer effect to someone the other day, I said, “The basic idea is that all processes at the top of a high building proceed at a faster rate than at the bottom” and she did not find this particularly startling.

Note 3  According to General Relativity ‘Space-Time’ is not homogeneous and isotropic in all directions but ‘blotchy’ and warped even when there are no massive objects in the immediate vicinity; also the velocity of light in free space is not strictly constant since a light ray deviates from a straight line (accelerates) when passing near a massive object such as the Sun.

Note 4   Even the few physicists who do entertain the idea that space and time might be ‘grainy’, do not usually go so far as to suggest that there are ‘gaps’ or ‘holes’ in the apparently dense fabric of Space/Time. However, see the excellent article Follow the Bouncing Universe by Martin Bojowald in the Scientific American booklet “Our Universe and Beyond”.
A few philosophers such as Plato and Heidegger have indeed suggested something along these lines, but without making much of the notion. Hinayana Buddhism, of course, takes it for granted that physical reality is ‘gapped’, since there are no ‘continuous’  entities whatsoever ─ except perhaps nirvana which is not a ‘normal’ entity to say the least.

Note 5  “The only person who took much notice of it [Einstein’s 1905 paper] was an American experimental physicist, Robert Millikan, who was so infuriated when he heard about it that he promptly set out to try to prove Einstein was wrong.” John & Mary Gribbin, Annus Mirabilis p. 85.   

Note 6 At one point in his astonishing career, Alexander the Great became convinced that Philotas, the son of Alexander’s most trusted general, Parmenio, was plotting to kill him. After a hurried ‘court-martial’ Macedonian-style Philotas was found guilty and ‘pierced with javelins’. What of Parmenio? It seemed to Alexander unwise to leave the old general alive, whether guilty or not, since he would automatically become a focus for further rebellions. But he was far away, in Media, so a message was sent for him to be put to death. The point is that it was not the sending of the message that mattered, it was Paremenio’s death, an event, that mattered.
The Romans set up an elaborate system of beacons over much of the Empire, thus using the speed of light for the transmission of information, and the English did the same at the time of the Spanish Armada. But even this system involved some delay, partly because of the finite speed of action and reaction within the human nervous system but also because of the nature of light itself. Most people at the time of the Armada still believed that the speed of light was ‘infinite’ but Galileo, for one, thought otherwise and tried to ‘time’ the delay involved in the transmission of light signals. He was not successful in this but he paved the way for more accurate estimates using advanced astronomical techniques. Einstein, in so many ways a man treading in the very footprints of his illustrious predecessor, realized how important this issue was since, if light had a fixed speed and light was the fastest ‘thing’ there was, this put an upper limit to the speed of propagation of all causal processes. And Einstein, like the man in the street, was a fervent believer in causality ─ so much so that this put him off Quantum Theory since the latter (arguably) violates the ‘laws’ of causality.
The question of whether or not a message could have been sent from one point to another ‘in Space/Time’ remains an incredibly important issue, and we have not heard the last of it,  since it not only concerns theoretical physics but, for example, criminal law : it is on this basis that we decide whether an alibi is valid, whether a Mafia chief  ‘had the time’ and the means to send orders by mobile phone and so on and so forth.

Note 7    ????

Note 8 One might reasonably wonder why no such principle had been formulated before. The answer is that no one seem to have envisaged that any physical process could happen at anything remotely approaching, even less exceeding, the speed of light. Newton was embarrassed by the fact that the operation of gravity was, in his theory, ‘instantaneous’ and it was precisely for this and related reasons that almost all continental scientists totally rejected the Law of Universal Attraction as being much too far-fetched. Contemporary physicists seem to think that gravitational effects, ripples in Space/Time and the like, propagate at the speed of light.

Note 9 How exactly Einstein himself arrived at this simple but absolutely crucial formula is not clear. His 1905 paper makes difficult reading today while subsequent ‘popular’ accounts by the great man are a little too clearcut. One suspects that Einstein knew exactly where he wanted to get to and fished around for a likely formula that would take him to the desired conclusion. This is the normal way in which physicists and mathematicians discover things, i.e. they work backwards from a conjectured result, not forwards, step by step, following a straight deductive path. Humans rarely discover anything important by applying painstaking logical procedures : we make ‘inspired guesses’, some of which eventually turn out to be the case and others not.

 

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

The Principle of Spatio-Temporal Continuity 

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

Limit to unitary re-appearance rate

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

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

Reappearance rates 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

Lower Limit of re-creation rate 

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

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

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

Re-appearance rates and collisions 

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

Reappearance rates 2

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

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

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

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

 

 

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

 

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

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

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

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

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

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

•   

•        

•    •    •    

•    •    •    •    

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

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

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

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

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

Relativity cos diagram

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

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

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

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

 

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

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

Relativity tangent diagram

  

   

 

 

 

 

 

 

 

 

 

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

Relativity Circle Diagram tan sin

 

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

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

     

 

 

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

Parable of the Two Friends and Railway Stations

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

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

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

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

 

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

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

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

 

 

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