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

Need for an ‘Inertial Frame’  

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

Celestial and Terrestrial Inertial Frames  

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

Absolute Motion and Absolute Rest? 

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

Abandon of the Principle of Relativity in UET

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

Assumption of Continued Existence  

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

Why is acceleration so difficult?

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

• …………………….………………..●

0   ←                        c*                   → c

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

The Inertial Ratchet 

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

The Systems Axiom  

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

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

        Another statement of the principle would be :

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

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

The spaceship on the way to the moon

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

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

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

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

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

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

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

Co-ordinate systems

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

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

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

Redundancy of two spatial axes 

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

Expected form of Transformation formulae 

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

x′ = Ax – Bt + C₁        t′ = Cx – Dt + C₂  

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

Upper Limit the speed of light  

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

x_c′ /t_c′ = (x_c – vt_c)/t_c = (x_c /t_c ) – (vt_c)/t_c = c – v 

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

Stopping the limit being exceeded  

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

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

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

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

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

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

       x^’/t^’ = γ (x – vt)/γ(t – vx/c²)         =   (x – vt)/(t – vx/c²)      =   c²(x – vt)/ (c²t – vx)   
                                                                  = c²x – c²vt)/(c²t – vx)         = c²x – cv(ct)/ (c(ct) – vx)

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

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

The Einstein/Lorentz Transformations

But the Transformations

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

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

γ = c /√(c² – v²) = 1/√1 –(v²/c²)

we get some extra spin-offs.

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

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

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

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

The Time Transformation in UET

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

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

Principle of Constant Occupied Area and its consequences 

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

Problems with the time transformation  

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

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

The 1/cos φ parts cancel out so we get

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

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

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

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

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

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

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

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

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

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

Ultimate Values for the Space/Time Rectangle

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

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

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

               = (1/√1 – n²/m²)   > 1

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

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

The dimensions of the ‘ultimate’ Space/Time Rectangle should, however, be calculated, not according to the γ formula which breaks down at or near the upper limit, but simply according to the fixed dimensions of the original rest’ capsule s₀ by t₀. If there are c* simultaneous ultimate events, and this is the absolute limit, the events occupy (or could occupy) the whole of one dimension of the ‘rest capsule’ which has a fixed volume of s₀³ where s_o­ is a maximum. Now each ‘event kernel’, the exact region where an ultimate event has occurrence has a small but non-zero size which may be noted as s_u³. Since there are, in the extreme case, c* = (c –1) ultimate events, or emplacements for possible ultimate events, we have s_o = c s_u.
What of time? To keep the overall area constant, time has to be stretched to a maximum, or, rather, the gap between two successive ultimate events is the largest possible. Since s_v t_v = s₀ t₀ = constant , this means that t_max, or as it is usually written t_u , = c* t_0. .
        This maximum time dilation should be observable when a ‘particle’ (event-chain) approaches the upper limit. Moreover, no matter what, this maximum time-length can never be exceeded ─ we will not get the ludicrous scenario of the ‘time interval ’ between two screams of someone falling into a black hole becoming infinite.

Deductions and Predictions  

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

Beyond the speed limit? 

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

NOTES

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

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

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

 Note 4   The argument goes something like this.

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

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

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

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