The Two Postulates of Special Relativity

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

The Third Postulate 

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

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

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

Inertial frames in UET 

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

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

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

Upper Speed limit?  

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

Derivation of basic formulae in UET 

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


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

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

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

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