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?

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.