Archives for category: Time Dilation

Special Relativity from UET Postulates

March 9, 2014 //

As related in the previous post, Einstein, in his epoch-making 1905 paper, based his theory of Special Relativity on just two postulates,

1. The laws of physics take the same form in all inertial frames.
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.

I asked myself if I could derive the main results of the Special Theory, the Rule for the Addition of Velocities, Space Contraction, Time Dilation and the ‘Equivalence’ of Mass and Energy from UET postulates.
Instead of Einstein’s Postulate 2, the ‘absolute value of the speed of light’, I employ a more general but very similar principle, namely that there is a ‘limiting speed’ for the propagation of causal influences from one spot on the Locality to another. In the simplest case, that of an event-chain consisting of a single ultimate event that repeats at every ksana, this amounts to asking ourselves ‘how far’ the causal influence can travel ‘laterally’ from one ksana to the next. I see the Locality as a sort of grid extending indefinitely in all directions where each ‘grid-position’ or ‘lattice-point’ can receive one, and only one, ultimate event (this is one of the original Axioms, the Axiom of Exclusion). At each ksana the entire previous spatial set-up is deftly replaced by a new, more or less identical one. So, supposing we can locate the ‘same’ spot, i.e. the ‘spot’ which replaces the one where the ultimate event had occurrence at the last ksana, is there a limit to how far to the left (or right) of this spot the ultimate event can re-occur? Yes, there is. Why? Well, I simply cannot conceive of there being no limit to how far spatially an ‘effect’ ─ in this case the ‘effect’ is a repetition of the original event ─ can be from its cause. This would be a holographic nightmare where anything that happens here affects, or at least could affect, what happens somewhere billions of light years away. One or two physicists, notably Heisenberg, have suggested something of the sort but, for my part, I cannot seriously contemplate such a state of affairs. Moreover, experience seems to confirm that there is indeed a ‘speed limit’ for all causal processes, the limit we refer to by the name of c.
However, this ‘upper speed limit’ has a somewhat different and sharper meaning in Ultimate Event Theory than it does in matter-based physics because c (actually c*) is an integer and corresponds to a specific number of adjacent ‘grid-positions’ on the Locality existing at or during a single ksana. It is a distance rather than a speed and even this is not quite right : it is a ‘distance’ estimated not in terms of ‘lengths’ but only in terms of the number of the quantity of intermediary ultimate events that could conceivably be crammed into this interval.
In UET a distinction is made between an attainable limiting number of grid-positions to right (or left) denoted c* and the lowest unattainable limit, c, though this finicky distinction in many cases can be neglected. But the basic schema is this. A ‘causal influence’, to be effective, must not only be able to at least traverse the distance between one ksana and the next ‘vertically’ (otherwise nothing would happen) but must also stretch out ‘laterally’ i.e. ‘traverse’ or rather ‘leap over’ a particular number of grid-positions. There is an upper limit to the number of positions that can be ‘traversed’, namely c*, an integer. This number, which is very great but not infinite ─ actual infinity is completely banished from UET ─ defines the universe we (think we) live in since it puts a limit to the operation of causality (as Einstein clearly recognized), and without causality there can, as far as I am concerned, be nothing worth calling a universe. Quite conceivably, the value of this constant c i(or c*) is very different in other universes, supposing they exist, but we are concerned only with this ‘universe’ (massive causally connected more or less identically repeating event-cluster).
So far, so good. This sounds a rather odd way of putting things, but we are still pretty close to Special Relativity as it is commonly taught. What of Einstein’s other principle? Well, firstly, I don’t much care for the mention of “laws of physics”, a concept which Einstein along with practically every other modern scientist inherited from Newton and which harks back to a theistic world-view whereby God, the supreme law-giver, formulated a collection of ‘laws’ that everything must from the moment of Creation obey ─ everything material at any rate. My concern is with what actually happens whether or not what happens is ‘lawful’ or not. Nonetheless, there do seem to be certain very general principles that apply across the board and which may, somewhat misleadingly, be classed as laws. So I shall leave this question aside for the moment.
The UET Principle that replaces Einstein’s First Principle (“that the laws of physics are the same in all inertial frames”) is rather tricky to formulate but, if the reader is patient and broad-minded enough, he or she should get a good idea of what I have in mind. As a first formulation, it goes something like this:

The occupied region between two or more successive causally related positions on the Locality is invariant.

         This requires a little elucidation. To start with, what do I understand by ‘occupied region’? At least to a first approximation, I view the Locality (the ‘place’ where ultimate events can and do have occurrence) as a sort of three-dimensional lattice extending in all directions which flashes on and off rhythmically. It would seem that extremely few ‘grid-spots’ ever get occupied at all, and even less spots ever become the seats of repeating events, i.e. the location of the first event of an event-chain. The ‘Event Locality’ of UET, like the Space/Time of matter-based physics, is a very sparsely populated place.
Now, suppose that an elementary event-chain has formed but is marooned in an empty region of the Locality. In such a case, it makes no sense to speak of ‘lateral displacement’ : each event follows its predecessor and re-appears at the ‘same’ ─ i.e. ‘equivalent’ ─ spot. Since there are no landmark events and every grid-space looks like every other, we can call such an event-chain ‘stationary’. This is the default case, the ‘inertial’ case to use the usual term.
We concentrate for the moment on just two events, one the clone of the other re-appearing at the ‘same spot’ a ksana later. These two events in effect define an ‘Event Capsule’ extending from the centre (called ‘kernel’ in UET) of the previous grid-space to the centre of the current one and span a temporal interval of one ksana. Strictly speaking, this ‘Event Capsule’ has two parts, one half belonging to the previous ksana and the other to the second ksana, but, at this stage, there is no more than a thin demarcation line separating the two extremities of the successive ksanas. Nonetheless, it would be quite wrong (from the point of view of UET) to think of this ‘Event Capsule’ and the whole underlying ‘spatial/temporal’ set-up as being ‘continuous’. There is no such thing as a ‘Space/Time Continuum’ as Minkowski understood the term. ‘Time’ is not a dimension like ‘depth’ which can seamlessly be added on to ‘length’ or ‘width’ : there is a fundamental opposition between the spatial and temporal aspect of things that no physical theory or mathematical artifice can completely abolish. In the UET model, the demarcations between the ‘spatial’ parts of adjacent Event Capsules do not widen, they remain simple boundaries, but the demarcations between successive ksanas widen enormously, i.e. there are gaps in the ‘fabric’ of time. To be sure there must be ‘something’ underneath which persists and stops everything collapsing, but this underlying ‘substratum’ has no physical properties whatsoever, no ‘identity’, which is why it is often referred to, not inaccurately, both in Buddhism and sometimes even in modern physics, as ‘nothing’.
To return to the ‘Constant Region Postulate’. The elementary ‘occupied region’ may be conceived as a ‘Capsule’ having the dimensions s₀ × s_{0 ×}s₀= s₀³ for the spatial extent and t₀ for time, i.e. a region of extent s₀³ × t₀ . These dimensions are fixed once and for all and, in the simplest UET model, s₀ is a maximum and t₀ is a minimum. Restricting ourselves for simplicity to a single spatial dimension and a single temporal dimension, we thus have an ‘Event Rectangle’ of s₀ by t₀ .
        For anything of interest to happen, we need more than one event-chain and, in particular, we need at least three ultimate events, one of which is to serve as a sort of landmark for the remaining pair. It is only by referring to this hypothetical or actual third event, occurring as it does at a particular spot independently of the event-pair, that we can meaningfully talk of the ‘movement’ to left or right of the second ultimate event in the pair with relation to the first. Alternatively, one could imagine an ultimate event giving rise to two events, one occurring ‘at the same spot’ and the other so many grid-spaces to the right (or left). In either case, we have an enormously expanded ‘Event Capsule’ spatially speaking compared to the original one. The Principle of the Constancy of the Area of the Occupied Region asserts that this ‘expanded’ Event Capsule which we can imagine as a ‘Space/Time rectangle’ (rather than Space/Time parallelipod), always has the ‘same’ area.
How can this be possible? Quite simply by making the spatial and temporal ‘dimensions’ inversely proportional to each other. As I have detailed in previous posts, we have in effect a ‘Space/Time Rectangle’ of sides s_v and t_v (subscript v for variable) such that s_v× t_v = s₀× t₀ = Ω = constant. Just conceivably, one could make s₀ a minimum and t₀a maximum but this would result in a very strange universe indeed. In this model of UET, I take s₀as a maximum and t₀ as a minimum. These dimensions are those of the archetypal ‘stationary’ or ‘inertial’ Event Capsule, one far removed from the possible influence of any other event-chains. I do not see how the ‘mixed ratio’ s₀ : t₀ can be determined on the basis of any fundamental physical or logical considerations, so this ratio just ‘happens to be’ what it is in the universe we (think we) live in. This ratio, along with the determination of c which is a number (positive integer), are the most important constants in UET and different values would give rise to very different universes. In UET s₀/t₀ is often envisaged in geometrical terms : tan β = s₀/t₀ = constant.    s₀  and   t₀   also have minimum and maximum values respectively, noted as s_u and t_u respectively, the subscript u standing for ‘ultimate’. We thus have a hyperbola but one constrained within limits so that there is no risk of ‘infinite’ values.

What is ‘speed’? Speed is not one of the basic SI units. The three SI mechanical units are the metre, the standard of length, the kilogram, the standard of mass, and the second, the standard of time. (The remaining four units are the ampere, kelvin, candela and mole). Speed is a secondary entity, being the ratio of space to time, metre to second. For a long time, since Galileo in fact, physicists have recognized the ‘relational’ nature of speed, or rather velocity (which is a ‘vector’ quantity, speed + direction). To talk meaningfully about a body’s speed you need to refer it to some other body, preferably a body that is, or appears to be, fixed (Note 1). This makes speed a rather insubstantial sort of entity, a will-o’-the-wisp, at any rate compared to ‘weight’, ‘impact’, ‘position’, ‘pain’ and so forth. The difficulty is compounded by the fact that we almost always consider ourselves to be ‘at rest’ : it is the countryside we see and experience whizzing by us when seated in a train. It requires a tremendous effort of imagination to see things from ‘the other object’s point of view’. Even a sudden jolt, an acceleration, is registered as a temporary annoyance that is soon replaced by the same self-centred ‘state of rest’. Highly complex and contrived set-ups like roller-coasters and other fairground machines are required to give us the sensation of ‘acceleration’ or ‘irregular movement’, a sensation we find thrilling precisely because it is so inhabitual. Basically, we think of ourselves as more or less permanently at rest, even when we know we are moving around. In UET everything actually is at rest for the space of a single ksana, it does not just appear to be and everything that happens occurs ‘at’ or ‘within’ a ksana (the elementary temporal interval).
I propose to take things further ─ not in terms of personal experience but physical theory. As stated, there is in UET no such thing as ‘continuous motion’, only succession ─ a succession of stills. An event takes place here, then a ksana or more later, another event, its replica perhaps, takes place there. What matters is what occurs and the number and order of the events that occur, everything else is secondary. This means not only that ultimate events do not move around ─ they simply have occurrence where they do have occurrence ─ but also that the distances between the events are in a sense ‘neither here nor there’, to use the remarkably apt everyday expression. In UET v signifies a certain number of grid-spaces to right or left of a fixed point, a shift that gets repeated every ksana (or in more complex cases with respect to more than one ksana). In the case of a truncated event-chain consisting of just two successive events, v is the same as d, the ‘lateral displacement’ of event 2 with respect to the position of event 1 on the Locality (more correctly, the ‘equivalent’ of such a position a ksana later). Now, although the actual number of ‘grid-positions’ to right or left of an identifiable spot on the Locality is fixed, and continues to be the same if we are dealing with a ‘regular’ event-chain, the distance between the centres (‘kernels’) of adjacent spots is not fixed but can take any number (sic) of permissible values ranging from 0 to c* according to the circumstances. The ‘distance’ from one spot to another can thus be reckoned in a variety of legitimate ways ─ though the choice is not ‘infinite’. The force of the Constancy of the Occupied Region Principle is that, no matter how these intra-event distances are measured or experienced, the overall ‘area’ remains the same and is equal to that of the ‘default’ case, that of a ‘stationary’ Event Capsule (or in the more extended case a succession of such capsules).
This is a very different conception from that which usually prevails within Special Relativity as it is understood and taught today. Discussing the question of the ‘true’ speed of a particular object whose speed is different according to what co-ordinate system you use, the popular writer on mathematics, Martin Gardner, famously wrote, “There no truth of the matter”. Although I understand what he meant, this is not how I would put it. Rather, all permissible ‘speeds’, i.e. all integral values of v, are “the truth of the matter”. And this does not lead us into a hopeless morass of uncertainty where “everything is relative” because, in contrast to ‘normal’ Special Relativity, there is in UET always a fixed framework of ultimate events whose number within a certain region of the Locality and whose individual ‘size’ never changes. How we evaluate the distances between them, or more precisely between the spots where they can and do occur, is an entirely secondary matter (though often one of great interest to us humans).

Space contraction and Time dilation

In most books on Relativity, one has hardly begun before being launched into what is pretty straightforward stuff for someone at undergraduate level but what is, for the layman, a completely indigestible mass of algebra. This is a pity because the actual physical principle at work, though it took the genius of Einstein to detect its presence, is actually extreme simple and can much more conveniently be presented geometrically rather than, as usual today, algebraically. As far as I am concerned, space contraction and time dilation are facts of existence that have been shown to be true in any number of experiments : we do not notice them because the effects are very small at our perceptual level. Although it is probably impossible to completely avoid talking about ‘points of view’ and ‘relative states of motion’ and so forth, I shall try to reduce such talk to a minimum. It makes a lot more sense to forget about hypothetical ‘observers’ (who most of the time do not and could not possibly exist) and instead envisage length contraction and time dilation as actual mechanisms which ‘kick in’ automatically much as the centrifugal governor on Watt’s steam-engine kicks in to regulate the supply of heat and the consequent rate of expansion of the piston. See things like this and keep at the back of your mind a skeletal framework of ultimate events and you won’t have too much trouble with the concepts of space contraction and time dilation. After all why should the distances between events have to stay the same? It is like only being allowed to take photographs from a standing position. These distances don’t need to stay the same provided the overall area or extent of the ‘occupied region’ remains constant since it is this, and the causally connected events within it, that really matters.
Take v to represent a certain number of grid-spaces in one direction which repeats; for our simple truncated event-chain of just two events it is d , the ‘distance’ between two spots. d is itself conceived as a multiple of the ‘intra-event distance’, that between the ‘kernels’ of any two adjacent ‘grid-positions’ in a particular direction. For any specific case, i.e. a given value of d or v, this ‘inter-possible-event’ distance does not change, and the specific extent of the kernel, where every ultimate event has occurrence if it does have occurrence, never changes ever. There is, as it were, a certain amount of ‘pulpy’, ‘squishy’ material (cf. cytoplasm in a cell) which surrounds the ‘kernel’ and which is, as it were, compressible. This for the ‘spatial’ part of the ‘Event Capsule’. The ‘temporal’ part, however, has no pulp but is ‘stretchy’, or rather the interval between ksanas is.
If the Constant Region Postulate is to work, we have somehow to arrange things that, for a given value of v or d, the spatial and temporal distances sort themselves out so that the overall area nonetheless remains the same. How to do this? The following geometrical diagram illustrates one way of doing this by using the simple formula tan θ = v/c = sin φ . Here v is an integral number of grid-positions ─ the more complex case where v is a rational number will be considered in due course ─ and c is the lowest unattainable limit of grid-positions (in effect (c* + 1) ).
Do these contractions and dilations ‘actually exist’ or are they just mathematical toys? As far as I am concerned, the ‘universe’ or whatever else you want to call what is out there, does exist and such simultaneous contractions and expansions likewise. Put it like this. The dimensions of loci (spots where ultimate events could in principle have occurrence) in a completely empty region of the Locality do not expand and contract because there is no ‘reason’ for them to do so : the default dimensions suffice. Even when we have two spots occupied by independent, i.e. completely disconnected, ultimate events nothing happens : the ‘distances’ remain the ordinary stationary ones. HOWEVER, as soon as there are causal links between events at different spots, or even the possibility of such links, the network tightens up, as it were, and one can imagine causal tendrils stretching out in different directions like the tentacles of an octopus. These filaments or tendrils can and do cause contractions and expansions of the lattice ─ though there are definite elastic limits. More precisely, the greater the value of v, the more grid-spaces the causal influence ‘misses out’ and the more tilted the original rectangle becomes in order to preserve the same overall area.
We are for the moment only considering a single ‘Event Capsule’ but, in the case of a ‘regular event-chain’ with constant v ─ the equivalent of ‘constant straight-line motion’ in matter-based physics ─ we have a causally connected sequence of more or less identical ‘Event Capsules’ each tilted from the default position as much as, but no more than, the last (since v is constant for this event-chain).
This simple schema will take us quite a long way. If we compare the ‘tilted’ spatial dimension to the horizontal one, calling the latter d and the former d′ we find from the diagram that d′ cos φ = d and likewise that t′ = t/cos φ . Don’t bother about the numerical values : they can be worked out by calculator later.
These are essentially the relations that give rise to the Lorentz Transformations but, rather than state these formulae and get involved in the whole business of convertible co-ordinate systems, it is better for the moment to stay with the basic idea and its geometrical representation. The quantity noted cos φ which depends on v and c , and only on v and c, crops up a lot in Special Relativity. Using the Pythagorean Formula for the case of a right-angled triangle with hypotenuse of unit length, we have

(1 cos φ)² + (1 sin φ)² = 1² or cos² φ + sin² φ = 1
Since sin φ is set at v/c we have
cos² φ = 1– sin² φ = 1 – (v/c)² cos φ = √(1 – (v/c)²

More often than not, this quantity (√(1 – (v²/c²) (referred to as 1/γ in the literature) is transferred over to the other side so we get the formula

d′ = (1/cos φ) d = d /( √(1 – (v²/c²)) = γ d

Viewed as an angle, or rather the reciprocal of the cosine of an angle, the ubiquitous γ of Special Relativity is considerably less frightening.

A Problem
It would appear that there is going to be a problem as d, or in the case of a repeating ‘rate’, v, approaches the limit c. Indeed, it was for this reason that I originally made a distinction between an attainable distance (attainable in one ksana), c*, and an unattainable one, c. Unfortunately, this does not eliminate all the difficulties but discussion of this important point will be left to another post. For the moment we confine ourselves to ‘distances’ that range from 0 to c* and to integral values of d (or v).

Importance of the constant c*

Now, it must be clearly understood that all sorts of ‘relations’ ─   perhaps correlations is an apter term ─ ‘exist’ between arbitrarily distant spots on the Locality (distant either spatially or temporally or both) but we are only concerned with spots that are either occupied by causally connected ultimate events, or could conceivably be so occupied. For event-chains with a 1/1 ‘reappearance rhythm’ i.e. one event per ksana, the relation tan θ = v/c = sin φ (v < c) applies (see diagram) and this means that grid-spots beyond the point labelled c (and indeed c itself) lie ‘outside’ the causal ‘Event Capsule’ Anything that I am about to deduce, or propose, about such an ‘Event Capsule’ in consequence does not apply to such points and the region containing them. Causality operates only within the confines of single ‘Event Capsules’ of fixed maximum size, and, by extension, connected chains of similar ‘Event Capsules’.
Within the bounds of the ‘Event Capsule’ the Principle of Constant Area applies. Any way of distinguishing or separating the spots where ultimate events can occur is acceptable, provided the setting is appropriate to the requirements of the situation. Distances are in this respect no more significant than, say, colours, because they do not affect what really matters : the number of ultimate events (or number of possible emplacements of ultimate events) between two chosen spots on the Locality, and the order of such events.
Now, suppose an ultimate event can simultaneously produce a clone just underneath the original spot, and also a clone as far as possible to the right. (I doubt whether this could actually happen but it is a revealing way of making a certain point.)
What is the least shift to the right or left? Zero. In such a case we have the default case, a ‘stationary’ event-chain, or a pair belonging to such a chain. The occupied area, however, is not zero : it is the minimal s₀³ t₀. The setting v = 0 in the formula d′ = (1/cos φ) d makes γ = 1/√(1 – (0²/c²) = 1 so there is no difference between d′ and d. (But it is not the formula that dictates the size of the occupied region, as physicists tend to think : it is the underlying reality that validates the formula.)
For any value of d, or, in the case of repetition of the same lateral distance at each ksana, any value of v, we tilt the rectangle by the appropriate amount, or fit this value into the formula. For v = 10 grid-spaces for example, we will have a tilted Space/Time Rectangle with one side (10 cos φ) s₀and the other side                 (1/10 cos φ) t₀ where sin φ = 10/c   so cos φ = √1 – (10/c)² This is an equally valid space/time setting because the overall area is
         (10 cos φ) s₀ ×   (1/10 cos φ) t₀   = s₀ t₀

We can legitimately apply any integral value of v < c and we will get a setting which keeps the overall area constant. However, this is done at a cost : the distance between the centres of the spatial element of the event capsules shrink while the temporal distances expand. The default distance s₀ has been shrunk to s₀ cos φ, a somewhat smaller intra-event distance, and the default temporal interval t₀ has been stretched to t₀ /cos φ , a somewhat greater distance. Remark, however, that sticking to integral values of d or v means that cos φ does not, as in ‘normal’ physics, run through an ‘infinite’ gamut of values ─ and even when we consider the more complex case, taking reappearance rhythms into account, v is never, strictly never, irrational.
What is the greatest possible lateral distance? Is there one? Yes, by Postulate 2 there is and this maximal number of grid-points is labelled c*. This is a large but finite number and is, in the case of integral values of v, equal to c – 1. In other words, a grid-space c spaces to the left or right is just out of causal range and everything beyond likewise (Note 2).

Dimensions of the Elementary Space Capsule

I repeat the two basic postulates of Ultimate Event Theory that are in some sense equivalent to Einstein’s two postulates. They are

1. The mixed Space/Time volume/area of the occupied parallelipod/rectangle remains constant in all circumstances

2. There is an upper limit to the lateral displacement of a causally connected event relative to its predecessor in the previous ksana

        Now, suppose we have an ultimate event that simultaneously produces a clone at the very next ksana in an equivalent spot AND another clone at the furthest possible grid-point c*. Even, taking things to a ridiculous extreme to make a point, suppose that a clone event is produced at every possible emplacement in between as well. Now, by the Principle of the Constancy of the Occupied Region, the entire occupied line of events in the second ksana can either have the ‘normal’ spacing between events which is that of the ‘rest’ distance between kernels, s₀, or, alternatively, we may view the entire line as being squeezed into the dimensions of a single ‘rest’ capsule, a dimension s₀ in each of three spatial directions (only one of which concerns us). In the latter case, the ‘intra-event’ spacing will have shrunk to zero ─ though the precise region occupied by an ultimate event remains the same. Since intra-event distancing is really of no importance, either of these two opposed treatments are ‘valid’.
What follows is rather interesting: we have the spatial dimension of a single ‘rest’ Event Capsule in terms of s_u, the dimension of the kernel. Since, in this extreme case, we have c* events squashed inside a lateral dimension of s₀, this means that
s₀ = c* s_u , i.e. the relation s₀ : s_u = c*: 1. But s₀ and s_u are, by hypothesis, universal constants and so is c* . Furthermore, since by definition s_v t_v = s₀ t₀ = Ω = constant , t₀ /t_v = s_v/s₀ and, fitting in the ‘ultimate’ s value, we have t₀ /t_u = s_u/c* s_u    = 1 : c*. In the case of ‘time’, the ‘ultimate’ dimension t_u is a maximum since (by hypothesis) t₀ is a minimum. c* is a measure of the extent of the elementary Event Capsule and this is why it is so important.
In UET everything is, during the space of a single ksana, at rest and in effect problems of motion in normal matter-based physics become problems of statics in UET ─ in effect I am picking up the lead given by the ancient Greek physicists for whom statics was all and infinity non-existent. Anticipating the discussion of mass in UET, or its equivalent, this interpretation ‘explains’ the tremendously increased resistance of a body to (relative) acceleration : something that Bucherer and others have demonstrated experimentally. This resistance is not the result of some arbitrary “You mustn’t go faster than light” law : it is the resistance of a region on the Locality of fixed extent to being crammed full to bursting with ultimate events. And it does not matter if the emplacements inside a single Event Capsule are not actually filled : these emplacements, the ‘kernels’, cannot be compressed whether occupied or not. But an event occurring at the maximum number of places to the right, is going to put the ‘Occupied Region’ under extreme pressure to say the least. In another post I will also speculate as to what happens if c* is exceeded supposing this to be possible.      SH    9/3/14

Notes:

Note 1 Zeno of Elea noted the ‘relativity of speed’ about two and a half thousand years before Einstein. In his “Paradox of the Chariot”, the least known of his paradoxes, Zeno asks what is the ‘true’ speed of a chariot engaged in a chariot race. A particular chariot has one speed with respect to its nearest competitor, another compared to the slowest chariot, and a completely different one again relative to the spectators. Zeno concluded that “there was no true speed” ─ I would say, “no single true speed”.

Note 2 The observant reader will have noticed that when evaluating sin φ = v/c and thus, by implication, cos φ as well, I have used the ‘unattainable’ limit c while restricting v to the values 0 to c*, thus stopping 1/cos φ from becoming infinite. Unfortunately, this finicky distinction, which makes actual numerical calculations much more complicated, does not entirely eliminate the problem as v goes to c, but this important issue will be left aside for the moment to be discussed in detail in a separate post.
If we allow only integral values of v ranging from 0 to c* = (c – 1), the final tilted Casual Rectangle has a ludicrously short ‘spatial side’ and a ridiculously long ‘temporal side’ (which means there is an enormous gap between ksanas). We have in effect

tan θ = (c–1)/c (i.e. the angle is nearly 45 degrees or π/4)
and γ = 1/√1 – (c–1)²/c² = c/√c² – (c–1)² = c/√(2c –1)
Now, 2c – 1 is very close to 2c so γ ≈ √c/2

I am undecided as to whether any particular physical importance should be given to this value ─ possibly experiment will decide the issue one day.
In the event of v taking rational values (which requires a re-appearance rhythm other than 1/1), we get even more outrageous ‘lengths’ for s_v and t_v . In principle, such an enormous gap between ksanas, viewed from a vantage-point outside the speeding event-chain, should become detectable by delicate instruments and would thus, by implication, allow us to get approximate values for c and c* in terms of the ‘absolute units’ s₀ and t₀ . This sort of experiment, which I have no doubt will be carried out in this century, would be the equivalent in UET of the famous Millikan ‘oil-drop’ series of experiments that gave us the first good value of e, the basic unit of charge.

Categories Buddhism, Einstein, Greeks, Special Relativity, Time, Time Dilation, Time dilation, Uncategorized

Special Relativity and Ultimate Event Theory

July 18, 2013 //

Almost everyone schoolboy these days has heard of the Lorentz transformations which replace the Galileian transformations in Special Relativity. They are basically a means of dealing with the relative motion of two bodies with respect to two orthogonal co-ordinate systems. Lorentz first developed them in an ad hoc manner somewhat out of desperation in order to ‘explain’ the null result of the Michelson-Morley experiment and other puzzling experimental results. Einstein, in his 1905 paper, developed them from first principles and always maintained that he did not at the time even know of Lorentz’s work. What were Einstein’s assumptions?

1. The laws of physics take the same form in all inertial frames.
2. The speed of light in free space has the same value for all observers in inertial frames irrespective of the relative motion of the source and the observer.

As has since been pointed out, Einstein did, in fact, assume rather more than this. For one thing, he assumed that ‘free space’ is homogeneous and isotropic (the same in all directions) (Note 1). A further assumption that Einstein seems to have made is that ‘space’ and ‘time’ are continuous ─ certainly all physicists at the time assumed this without question and the wave theory of ele tro-magnetism required it as Maxwell was aware. However, the continuity postulate does not seem to have played much of a part in the derivation of the equations of Special Relativity though it did stop Einstein’s successors from thinking in rather different ways about ‘Space/Time’. Despite everything that has happened and the success of Quantum Mechanics and the photo-electric effect and all the rest of it, practically all students of physics think of ‘space’, ‘time’ and electro-magnetism as being ‘continuous’, rather than made up of discrete bits especially since Calculus is primarily concerned with ‘continuous functions’. Since nothing in the physical world is continuous, Calculus is in the main a false model of reality.

Inertial frames, which play such a big role in Special Relativity, as it is currently taught, do not exist in Nature : they are entirely man-made. It was essentially this realisation that motivated Einstein’s decision to try to formulate physics in a way that did not depend on any particular co-ordinate system whatsoever. Einstein assumed relativity and the constancy of the speed of light and independently deduced the Lorentz transformations. This post would be far too long if I went into the details of Special Relativity (I have done this elsewhere) but, for the sake of the general reader, a brief summary can and should be given. Those who are familiar with Special Relativity can skip this section.

The Lorentz/Einstein Transformations Ordinary people sometimes find it useful, and physicists find it indispensable, to locate an object inside a real or imaginary three dimensional box. Then, if one corner of the imaginary box (e.g. room of house, railway carriage &c.) is taken as the Origin, the spot to which everything else is related, we can pinpoint an object by giving its distance from the corner/Origin, either directly or by giving the distance in terms of three directions. That is, we say the object is so many spaces to the right on the ground, so many spaces on the ground at right angles to this, and so many spaces upwards. These are the three co-ordinate axes x, y and z. (They do not actually need to be at right angles but usually they are and we will assume this.)

Also, if we are locating an event rather than an object, we will need a fourth specification, a ‘time’ co-ordinate telling us when such and such an event happened. For example, if a balloon floating around the room at a particular time, to pinpoint the event, it would not be sufficient to give its three spatial co-ordinates, we would need to give the precise time as well. Despite all the hoo-ha, there is nothing in the least strange or paradoxical about us living in a ‘four-dimensional universe’. Of course, we do done : the only slight problem is that the so-called fourth dimension, time, is rather different from the other three. For one thing, it seems to only have one possible direction instead of two; also the three ‘spatial’ directions are much more intimately connected to each other than they are to the ‘time’ dimension. A single unit serves for the first three, say the metre, but for the fourth we need a completely different unit, the second, and we cannot ‘see’ or ‘touch’ a second whereas we can see and touch a metre rod or ruler.
Now, suppose we have a second ‘box’ moving within the original box and moving in a single direction at a constant speed. We select the x axis for the direction of motion. Now, an event inside the smaller box, say a pistol shot, also takes place within the larger box : one could imagine a man firing from inside the carriage of a train while it has not yet left the station. If we take the corner of the railway carriage to be the origin, clearly the distance from where the shot was fired to the railway carriage origin will be different from the distance from where the buffers train are. In other words, relative to the railway carriage origin, the distance is less than the distance to the buffers. How much less? Well, that depends on the speed of the train as it pulls out. The difference will be the distance the train has covered since it pulled out. If the train pulls out at constant speed 20 metres/second metres/second and there has been a lapse of, say, 4 seconds, the distance will be 80 metres. More generally, the difference will be vt where t starts at 0 and is counted in seconds. So, supposing relative to the buffers, the distance is x, relative to the railway carriage the distance is v – xt a rather lesser distance.
Everything else, however, remains the same. The time is the same in the railway carriage as what is marked on the station clock. And, if there is only displacement in one dimension, the other co-ordinates don’t change : the shot is fired from a metre above ground level for example in both systems and so many spaces in from the near side in both systems. This all seems commonsensical and, putting this in formal mathematical language, we have the Galilean Transformations so-called

x^’ = x – vt y^’ = y z^’ – z t^’= t

All well and good and nobody before the dawn of the 20^th century gave much more thought to the matter. Newton was somewhat puzzled as to whether there was such a thing as ‘absolute distance’ and ‘absolute time’, hence ‘absolute motion’, and though he believed in all three, he accepted that, in practice, we always had to deal with relative quantities, including speed.
If we consider sound in a fluid medium such as air or water, the ‘speed’ at which the disturbance propagates differs markedly depending on whether you are yourself stationary with respect to the medium or in motion, in a motor-boat for example. Even if you are blind, or close your eyes, you can tell whether a police car is moving towards or away from you by the pitch of the siren, the well-known Doppler effect. The speed of sound is not absolute but depends on the relative motion of the source and the observer. There is something a little unsettling in the idea that an object does not have a single ‘speed’ full stop, but rather a variety of speeds depending on where you are and how you are yourself moving. However, this is not too troublesome.
What about light? In the latter 19^th century it was viewed as a disturbance rather like sound that was propagated in an invisible medium, and so it also should have a variable speed depending on one’s own state of motion with respect to this background, the ether. However, no differences could be detected. Various methods were suggested, essentially to make the figures come right, but Einstein cut the Gordian knot once and for all and introduced as an axiom (basic assumption) that the speed of light in a vacuum (‘free empty space’) was fixed and completely independent of the observer’s state of motion. In other words, c, the speed of light, was the same in all co-ordinate systems (provided they were moving at a relative constant speed to each other). This sounded crazy and brought about a completely different set of ‘transformations’, known as the Lorentz Transformations although Einstein derived them independently from his own first principles. This derivation is given by Einstein himself in the Appendix to his ‘popular’ book “Relativity : The Special and General Theory”, a book which I heartily recommend. Whereas physicists today look down on books which are intelligible to the general reader, Einstein himself who was not a brilliant student at university (he got the lowest physics pass of his year) and was, unlike Newton, not a particularly gifted pure mathematician, took the writing of accessible ‘popular’ books extremely seriously. Einstein is the author of the staggering put-down, “If you cannot state an issue clearly and simply, you probably don’t understand it”.
If we use the Galileian Tranformations and set v = c , the speed of light (or any form of electro-magnetism) in a vacuum, we have x = ct or with x in metres and t in seconds, x = 3 × 10⁸metres (approximately) when t = 1 second. Transferring to the other co-ordinate system which is moving at v metres/sec relative to the first, we have x^’ = x – vt and, since t^’ is the same as t, when dividing we obtain for x^’/t^’ , (x – vt)/t = ((x/t) – v) = (c – v), a somewhat smaller speed than c. This is exactly what we would expect if dealing with a phenomenon such as sound in a fluid medium. However, Einstein’s postulate is that, when dealing with light, the ratio distance/time is constant in all inertial frames, i.e. in all real or imaginary ‘boxes’ moving in a single direction with a constant difference in their speeds.

One might doubt whether it is possible to produce ‘transformations’ that do keep c the same for different frames. But it is. We need not bother about the y and z co-ordinates because they are most likely going to stay the same ─ we can arrange to set both at zero if we are just considering an object moving along in one direction. However, the x and t equations are radically changed. In particular, it is not possible to set t = t^’, meaning that ‘time’ itself (whatever that means) will be modified when we switch to the other frame. The equations are

x^’ = γ (x – vt) t^’ = γ(t – vx/c²) where γ = (1/(1 – v²/c²)^1/2)

The reader unused to mathematics will find them forbidding and they are indeed rather tiresome to handle though one gets used to them. If you take the ratio If x^’ /t^’ you will find ─ unless you make a slip ─ that, using the Lorentz Transformations you eventually obtain c as desired.

We have x = ct or t = x/c and the Lorentz Transformations

x^’ = γ (x – vt) t^’ = γ(t – vx/c²) where γ = (1/(1 – v²/c²)^1/2)

Then x^’/t^’ = γ (x – vt) = (x – vt) = c²(x – vt)
γ (t – vx/c²) (t – vx/c²) (c²t – vx)

= c²(x – vt) = c²x – cv(ct)
(c²t – vx) (c(ct) – vx)

= c²x – cvx) = (cx)(c – v)
(cx – vx) x(c – v)

= c

The amazing thing that this is true for any value of v ─ provided it is less than c ─ so it applies to any sort of system moving relative to the original ‘box’, as long as the relative motion is constant and in a straight line. It is true for v = 0 , i.e. the two boxes are not moving relatively to each other : in such a case the complicated Lorentz Transformations reduce to x^’ = x t^’ = t and so on.
The Lorentz/Einstein Transformations have several interesting and revealing properties. Though complicated, they do not contain terms in x² or t² or higher powers : they are, in mathematical parlance, ‘linear’. This is what we want for systems moving at a steady pace relatively to each other : squares and higher powers rapidly produce erratic increases and a curved trajectory on a space/time graph. Secondly, if v is very small compared to c, the ratio v/c which appears throughout the formulae is negligible since c is so enormous. For normal speeds we do not need to bother about these terms and the Galileian formulae give satisfactory results.
Finally, and this is possibly the most important feature : the Lorentz/Einstein Transformations are ‘symmetric’. That is, if you work backwards, starting with the ‘primed’ frame and x^’ and t^’, and convert to the original frame, you end up with a mirror image of the formulae with a single difference, a change of sign in the x^’to x formula denoting motion in the opposite direction (since this time it is the original frame that is moving away). Poincaré was the first to notice this and could have beaten Einstein to the finishing line by enunciating the Principle of Relativity several years earlier ─ but for some reason he didn’t, or couldn’t, make the conceptual leap that Einstein made. The point is that each way of looking at the motion is equally valid, or so Einstein believed, whether we envisage the countryside as moving towards us when we are in the train, or the train moving relative to the static countryside.

Relativity from Ultimate Event Theory?

Einstein assumed relativity and the constancy of the speed and deduced the Lorentz Transformations : I shall proceed somewhat in the opposite direction and attempt to derive certain well-known features of Special Relativity from basic assumptions of Ultimate Event Theory (UET). What assumptions?

To start with, the Event Number Postulate which says that
Between any two events in an event-chain there are a fixed number of ultimate events.
And (to recap basic definitions) an ultimate event is an event that cannot be further decomposed — this is why it is called ultimate.
Thus, if the ultimate events in a chain, or subsection of a chain, are numbered 0, 1, 2, 3…….n there are n intervals. And if the event-chain is ‘regular’, sort of equivalent of an intertial system, the ‘distance’ between any two successive events stays the same. By convention, we can treat the ‘time’ dimension as vertical — though, of course, this is no more than a useful convention. The ‘vertical’ distance between the first and last ultimate events of a regular event-chain thus has the value n × ‘vertical’ spacing, or n × t. Note that whereas the number n¸ indicating the quantity of ultimate events and intervals, is fixed in a particular case, t turns out to be a parameter which, however, has a minimum ‘rest’ value noted t₀. This minimal ‘rest’ value is (by hypothesis) the same for all regular event-chains.
♦

♦

…. Likewise, between any two ‘contemporary’ i.e. non-successive, ultimate events there are a fixed number of spots where ultimate events could have (had) occurrence. If there are two or more neighbouring contemporary ultimate events bonded together we speak of an event-conglomerate and, if this conglomerate repeats or gives rise to another conglomerate of the same size, we have a ‘conglomerate event-chain’. (But normally we will just speak of an event-chain).
A conglomerate is termed ‘tight’, and the region it occupies within a single ksana (the minimal temporal interval) is ‘full’ if one could not fit in any more ultimate events (because there are no available spots). And, if all the contemporary ultimate events are aligned, i.e. have a single ‘direction’, and are labelled 0, 1, 2, 3…….n , then, there are likewise n ‘lateral’ intervals along a single line.

♦ ♦ ♦ ♦ ♦ ………

If the event-conglomerate is ‘regular’, the distance between any two neighbouring events will be the same and, for n events has the value n × ‘lateral’ inter-event spacing, or n × s. Although s, the spacing between contemporary ultimate events must obviously always be greater than the spot occupied by an ultimate event, for all normal circumstances it does not have a minimum. It has, however, a maximum value s₀ .

The ‘Space-Time’ Capsule

Each ultimate event is thus enclosed in a four-dimensional ‘space-time capsule’ much, much larger than itself — but not so large that it can accommodate another ultimate event. This ‘space-time capsule’ has the mixed dimension s³t.
In practice, when dealing with ‘straight-line’ motion, it is only necessary to consider a single spatial dimension which can be set as the x axis. The other two dimensions remain unaffected by the motion and retain the ‘rest’ value, s_0. Thus we only need to be concerned with the ‘space-time’ rectangle st.
We now introduce the Constant Size Postulate

The extent, or size, of the ‘space-time capsule’ within which an ultimate event can have occurrence (and within which only one can have occurrence) is absolute. This size is completely unaffected by the nature of the ultimate events and their interactions with each other.

           We are talking about the dimensions of the ‘container’ of an ultimate event. The actual region occupied by an ultimate event, while being non-zero, is extremely small compared to the dimensions of the container and may for most purposes be considered negligible, much as we generally do not count the mass of an electron when calculating an atom’s mass. Just as an atom is mainly empty space, a space time capsule is mainly empty ‘space-time’, if the expression is allowed.
Note that the postulate does not state that the ‘shape’ of the container remains constant, or that the two ‘spatial’ and ‘temporal’ dimensions should individually remain constant. It is the extent of the space-time parallelipod’ s³t which remains the same or, in the case of the rectangle it is the product st ,that is fixed, not s and t individually.  All quantities have minimum and maximum values, so let the minimum temporal interval be named t₀ and, conversely, let s₀ be the maximum value of s. Thus the quantity s₀ t₀, the ‘area’ of the space-time rectangle, is fixed once and for all even though the temporal and spatial lengths can, and do, vary enormously. We have, in effect a hyperbola where xy = constant but with the difference that the hyperbola is traced out by a series of dots (is not continuous) and does not extend indefinitely in any direction (Note 3).
         This quantity s₀t₀ is an extremely important constant, perhaps the most important of all. I would guess that different values of s₀t₀   would lead to very different universes. The quantity is mixed so it is tacitly assumed that there is a common unit. What this common unit is, is not clear : it can only be based on the dimensions of an ultimate event itself, or its precise emplacement (not its container capsule), since K₀ , the backdrop or Event Locality does not have a metric, is elastic, indeterminate in extent.
         Although one can, in imagination, associate or combine all sorts of events with each other, only events that are bonded sequentially constitute an event-chain, and only bonded contemporary events remain contemporary in successive ksanas. This ‘bonding’ is not a mathematical fiction but a very real force, indeed the most basic and most important force in the physical universe without which the latter would collapse at any and every moment — or rather at every ksana.
         Now, within a single ksana one and only one ultimate event can have occurrence. However, the ‘length’ of a ksana varies from one event-chain to another since, although the size of the emplacements where the ultimate events occur is (by hypothesis) fixed, the spacing is not fixed, is indeterminate though the same in similar conditions (Note 5). The length of a ksana has a minimum and this minimal length is attained only when an event-chain is at rest, i.e. when it is considered without reference to any other event-chain. This is the equivalent of a ‘proper interval’ in Relativity. So t is a parameter with minimal value t₀. It is not clear what the maximum value is though there must be one.
         The inter-space distance s does not have a minimum, or not one that is, in normal conditions ever attained — this minimum would be the exact ‘width’ of the emplacement of an ultimate event, an extremely small distance. It transpires that the inter-space distance s is at a maximum in a rest-chain taking the value s₀. I am not absolutely sure whether this needs to be stated as an assumption or whether it can be derived later from the assumptions already made.)

Thus, the ‘space-time’ paralleliped s³t has the value (s₀)³t₀ , an absolute value.

The Rest Postulate

This says that

Every event-chain is at rest with respect to the Event Locality K₀ and may be considered to be ‘stationary’.

Why this postulate and what does it mean? We all have experience of objects immersed in a fluid medium and there can also be events, such as sounds, located in this medium. Now, from experience, it is possible to distinguish between an object ‘at rest’ in a fluid medium such as the ocean and ‘in motion’ relative to this medium. And similarly there will be a clear difference between a series of siren calls or other sounds emitted from a ship in a calm sea, and the same sequence of sounds when the ship is in motion. Essentially, I envisage ultimate events as, in some sense, immersed in an invisible omnipresent ‘medium’, K₀, — indeed I envisage ultimate events as being localized disturbances of K₀. (But if you don’t like this analogy, you can simply conceive of ultimate events having occurrence on an ‘Event Locality’ whose purpose is simply to allow ultimate events to have occurrence and to keep them separate from one another.) The Rest Postulate simply means that, on the analogy with objects in a fluid medium, there is no friction or drag associated with chains of ultimate events and the medium in or on which they have occurrence. This is basically what Einstein meant when he said that “the ether does not have a physical existence but it does have a geometric existence”.

What’s the point of this constant if no one knows what it is? Firstly, it by no means follows that this constant s₀t₀ is unknowable since we can work backwards from experiments using more usual units such as metres and seconds, giving at least an approximate value. I am convinced that the value of s₀t₀ will be determined experimentally within the next twenty years, though probably not in my lifetime unfortunately. But even if it cannot be accurately determined, it can still function as a reference point. Galileo was not able to determine the speed of light even approximately with the apparatus at his disposal (though he tried) but this did not stop him stating that this speed was finite and using the limit in his theories without knowing what it was.

Diverging Regular Event-chains

Imagine a whole series of event-chains with the same reappearance rate which diverge from neighbouring spots — ideally which fork off from a single spot. Now, if all of them are regular with the same reappearance rate, the nth member of Event-chain E₀ will be ‘contemporaneous’ with the nth members of all the other chains, i.e. they will have occurrence within the same ksana. Imagine them spaced out so that each nth ultimate event of each chain is as close as possible to the neighbouring chains. Thus, we imagine E₀ as a vertical column of dots (not a continuous vertical line) and E₁ a slanting line next to it, then E₂ and so on. The first event of each of these chains (not counting the original event common to all) will thus be displaced by a single ‘grid-space’ and there will be no room for any events to have occurrence in between. The ‘speed’ or displacement distance of each event-chain relative to the first (vertical one) is thus lateral distance in successive ksanas/vertical distance in successive ksanas. For a ‘regular’ event-chain the ‘slant’ or speed remains the same and is tan θ = 1 s/t₀ , 2 s/t₀ and so on where, if θ is the slant angle,

tan θ_r = v_r = 1, 2, 3, 4……

“What,” asked Zeno of Elea “is the speed of a particular chariot in a chariot race?”  Clearly, this depends on what your reference body is. We usually take the stadium as the reference body but the charioteer himself perceives the spectators as moving towards or away from him and he is much more concerned about his speed relative to that of his nearest competitor than to his speed relative to arena. We have grown used to the idea that ‘speed’ is relative, counter-intuitive though it appears at first.
But ‘distance’ is a man-made convenience as well : it is not an ‘absolute’ feature of reality. People were extremely put out by the idea that lengths and time intervals could be ‘relative’ when the concept was first proposed but scientists have ‘relatively’ got used to the idea. But everything seems to be slipping away — is there anything at all that is absolute, anything at all that is real? Ultimate Event Theory evolved from my attempts to ponder this question.
The answer is, as far as I am concerned, yes. To start with, there are such things as events and there is a Locality where events occur. Most people would go along with that. But it is also one of the postulates of UET that every macroscopic ‘event’ is composed of a specific number of ultimate events which cannot be further decomposed. Also, it is postulated that certain ultimate events are strongly bonded together into event-chains temporally and event-conglomerates laterally. There is a bonding force, causality.
Also, associated with every event chain is its Event Number, the number of ultimate events between the first event A and the last Z. This number is not relative but absolute. Unlike speed, it does not change as the event-chain is perceived in relation to different bodies or frames of reference. Every ultimate event is precisely localised and there are only a certain number of ultimate events that can be interposed between two events both ‘laterally’ (spatially) and ‘vertically’ (temporally). Finally, the size of the ‘space-time capsule’ is fixed once and for all. And there is also a maximum ‘space/time displacement ratio’ for all event-chains.
This is quite a lot of absolutes. But the distance between ultimate events is a variable since, although the dimensions of each ultimate event are fixed, the spacing is not fixed though it will remain the same within a so-called ‘regular’ event-chain.
It is important to realize that the ‘time’ dimension, the temporal interval measured in ksanas, is not connected up to any of the three spatial dimensions whereas each of the three spatial dimensions is connected directly to the other two. It is customary to take the time dimension as vertical and there is a temptation to think of t, time, being ‘in the same direction’ as the z axis in a normal co-ordinate system. But this is not so : the time dimension is not in any spatial direction but is conceived as being orthogonal (at right angles) to the whole lot. To be closer to reality, instead of having a printed diagram on the page, i.e. in two dimensions, we should have a three dimensional optical set-up which flashes on and off at rhythmic intervals and the trajectory of a ‘particle’ (repeating event-chain) would be presented as a repeating pinpoint of light in a different colour.
Supposing we have a repeating regular event-chain consisting for simplicity of just one ultimate event. We [resent it as a column of dots, i.e. conceive of it as vertical though it is not. The dots are numbered 0, 1, 2….    and the vertical spacing does not change (since this is a regular event-chain) and is set at  t₀since this is a ‘rest chain’. Similar regular event-chains can then be presented as slanting lines to the right (or left) regularly spaced along the x axis. The slant of the line represents the ‘speed’. Unlike the treatment in Calculus and conventional physics, increasing v does not ‘pass through a continuous set of values’, it can only move up by a single ‘lateral’ space each time. The speeds of the different event-chains are thus 0s/t₀ (= 0) ; 1s/t₀ ;
2s/t₀ ;  3s/t₀ ; 4s/t₀ ;……  n s/t₀ and so on right up to c s/t₀ .  But to what do we relate the spacing s ? To the ‘vertical’ event-chain or to slanting one? We must relate s to the event-chain under consideration so that its value depends on v so v = v s_v    The ratio s/t₀ is thus a mixed ratio s_v/t₀ .   t_v  gives the intervals between successive events in the ‘moving’ event-chains and the number of these intervals does not increase because there are only a fixed number of events in any event-chain evaluated in any way. These temporal intervals thus undoubtedly increase because the hypotenuse gets larger. What about the spacing along the horizontal ? Does it also increase? Stay the same? If we now introduce the Constant Size Postulate which says that the product s_v t_v = s₀ t₀    we find that   s_v decreases with increasing v since t_v certainly increases. There is thus an inverse ratio and one consequence of this is that the mixed ratio s_v/t₀ = s₀/t_vand we get symmetry. This leads to relativity whereas any other relation does not and we would have to specify which regular event-chain ‘really’ is the vertical one. One can legitimately ask which is the ‘real’ spatial distance between neighbouring events? The answer is that every distance is real and not just real for a particular observer. Most phenomena are not observed at all but they still occur and the distances between these events are real : we as it were take our pick, or more usually one distance is imposed on us.

Now the real pay off is that each of these regular event-chains with different speeds v is an equally valid description of the same event-chain. Each of these varying descriptions is true even though the time intervals and distances vary. This is possible because the important thing, what really matters, does not change : in any event-chain the number and order of the individual events is fixed once and for all although the distances and times are not. Rosser, in his excellent book Introductory Relativity, when discussing such issues gives the useful analogy of a gamer of tennis being played on a cruise liner in calm weather. The game would proceed much as on land, and if in a covered court, exactly as on land. And yet the ‘speed’ of the ball varies depending on whether you are a traveller on the boat or someone watching with a telescope from another boat or from land. The ‘real’ speed doesn’t actually matter, or, as I prefer to put it, is indeterminate though fixed within a particular inertial frame (event system). Taking this one step further, not just the relative speed but the spacing between the events of a regular event-chain ‘doesn’t matter’ because the constituent events are the same and appear in the same order. It is interesting that. on this interpretation, a certain indeterminacy with regard to distance is already making its appearance before Quantum Theory has even been mentioned.

Which distance or time interval to choose?

Since, apparently, the situation between regular event-chains is symmetric (or between inertial systems if you like) one might legitimately wonder how there ever could be any observed discrepancy since any set of measurements a hypothetical observer can make within his own frame (repeating event system) will be entirely consistent and unambiguous. In Ultimate Event Theory, the situation is, in a sense, worse since I am saying that, whether or not there is or can be an observer present, the time-distance set-up is ‘indeterminate’ — though the number and order of events in the chain is not. Any old ‘speed’ will do provided it is less than the limiting value c. So this would seem to make the issue quite academic and there would be no need to learn about Relativity. The answer is that this would indeed be the case if we as observers and measurers or simply inhabitants of an event-environment could move from one ‘frame’ to another effortlessly and make our observations how and where we choose. But we can’t : we are stuck in our repeating event-environment constituted by the Earth and are at rest within it, at least when making our observations. We are stuck with the distance and time units of the laboratory/Earth event-chain and cannot make observations using the units of the electron event-chain (except in imagination). Our set of observations is fully a part of our system and the units are imposed on us. And this does make a difference, a discernible, observable difference when dealing with certain fast-moving objects.
Take the µ-meson. µ-mesons are produced by cosmic rays in the upper reaches of the atmosphere and are normally extremely short-lived, about 2.2 × 10^–6sec. This is the (average) ‘proper’ time, i.e. when the µ-meson is at rest — in my terms it would be N × t₀ ksanas. Now, these mesons would, using this t value, hardly go more than 660 metres even if they were falling with the speed of light (Note 4). But a substantial portion actually reach sea level which seems impossible. Now, we have two systems, the meson event-chain which flashes on and off N times whatever N is before terminating, i.e. not reappearing. Its own ‘units’ are t₀ and s₀ since it is certainly at rest with itself. For the meson, the Earth and the lower atmosphere is rushing up with something approaching the limiting speed towards it. We are inside the Earth system and use Earth units : we cannot make observations within the meson. The time intervals of the meson’s existence are, from our rest perspective, distended : there are exactly the same number of ksanas for us as for the meson but, from our point of view, the meson is in motion and each ‘motion’ ksana is longer, in this case much much longer. It thus ‘lives’ longer, if by living longer we mean having a longer time span in a vague absolute way, rather than having more ‘moments of existence’. The meson’s ksana is worth, say, eight of our ksanas. But the first and last ultimate event of the meson’s existence are events in both ‘frames’, in ours as well as its. And if we suppose that each time it flashed into existence there was a (slightly delayed) flash in our event-chain, the flashes would be much more spaced out and so would be the effects. So we would ‘observe’, say, a duration of, say, eight of ‘our’ ksanas between consecutive flashings instead of one. And the spatial distance between flashes would also be evaluated in our system of metres and kilometres : this is imposed on us since we cannot measure what is going on within the meson event-chain. The meson actually would travel a good deal further in our system — not ‘would appear to travel farther’. Calculations show that it is well within the meson’s capacity to reach sea level (see full discussion in Rosser, Introductory Relativity pp. 71-3).
What about if we envisaged things from the perspective of the meson? Supposing, just supposing, we could transfer to the meson event-chain or its immediate environment and could remember what things were like in the world outside, the familiar Earth event-frame. We would notice nothing strange about ‘time’, the intervals between ultimate events, or the brain’s merging of them, would not surprise us at all. We would consider ourselves to be at rest. What about if we looked out of the window at the Earth’s atmosphere speeding by? Not only would we recognize that there was relative motion but, supposing there were clear enough landmarks (skymarks rather), the distances between these marks would appear to be far closer than expected — in effect there would be a double or triple sense of motion since our perception of motion is itself based on estimates of distance. As the books say, the Earth and its atmosphere would be ‘Lorentz contracted’. There would be exactly the same number of ultimate events in the meson’s trajectory, temporarily our trajectory also. The first and last event of the meson’s lifetime would be separated by the same number of temporal intervals and if these first and last events left marks on the outside system, these marks would also be separated by exactly the same number of spatial intervals. Only these spatial intervals — distances — would be smaller. This would very definitely be observed : it is as if we were looking out at the countryside on a familiar journey in a train fantastically speeded up. We would still consider ourselves at rest but what we saw out of the window would be ludicrously foreshortened and for this reason we would conclude that we were travelling a good deal faster than on our habitual journey. I do not think there would be any obvious way to recognize the time dilation of the outside system.

One is often tempted to think that the time dilation and the spatial contraction cancel each other out so all this talk of relativity is purely academic since any discrepancies should cancel out. This would indeed be the case if we were able to make our observations inside the event-chain we are observing, but we make the measurements (or perceptions) in a single frame. Although it is the meson event-chain that is dictating what is happening, both the time and spatial distance observations are made in our system. It is indeed only because of this that there is so much talk about ‘observers’ in Special Relativity. The point is not that some intelligent being observes something because usually he or she doesn’t : the point is that the fact of observation, i.e. the interaction with another system seriously confuses the issue. The ‘rest-motion’ situation is symmetrical but the ‘observing’ situation is not symmetrical, nor can it be in such circumstances.

This raises an important point. In Ultimate Event Theory, as in Relativity, the situation is ‘kinematically’ symmetrical. But is it causally symmetrical? Although Einstein stressed that c was a limit to the “transfer of causality” he was more concerned with light and electro-magnetism than causality. UET is concerned above all with causality — I have not mentioned the speed of light yet and don’t need to. In situations of this type, it is essential to clearly identify the primary causal chain. This is obviously the meson : we observe it, or rather we pick up indications of its flashings into and out of existence. The observations we make, or simply perceptions, are dependent on the meson, they do not by themselves constitute a causal chain. So it looks at first sight as if we have a fundamental asymmetry : the meson event-chain is the controlling one and the Earth/observer event chain is essentially passive. This is how things first appeared to me. But on reflection I am not so sure. In many (all?) cases of ‘observation’ there is interaction with the system being observed and it is inevitably going to be affected by this even if it has no senses or observing apparatus of its own. One could thus argue that there is causal symmetry after all, at least in some cases. There is thus a kind of ‘uncertainty principle’ due to the interaction of two systems latent in Relativity before even Quantum Mechanics had been formulated. This issue and the related one of the limiting speed of transmission of causality will be dealt with in the subsequent post.

Sebastian Hayes 26 July
Note 1. And in point of fact, if General Relativity is to be believed, ‘free space’ is not strictly homogeneous even when empty of matter and neither is the speed of light strictly constant since light rays deviate from a straight path in the neighbourhood of massive bodies.

Note 2 For those people like me who wish to believe in the reality of K₀ — rather than seeing it as a mere mathematical convenience like a co-ordinate system — the lack of any ‘friction’ between the medium or backdrop and the events or foreground would, I think. be quite unobjectionable, even ‘obvious’, likewise the entire lack of any ‘normal’ metrical properties such as distance. The ‘backdrop’, that which lies ‘behind’ material reality though in some sense permeating it, is not physical and hence is not obliged to possess familiar properties such as a shape, a metric, a fixed distance between two points and so on. Nevertheless, this backdrop is not completely devoid of properties : it does have the capacity to receive (or produce) ultimate events and to keep them separate which involves a rudimentary type of ‘geometry’ (or topology). Later, as we shall see, it would seem that it is affected by the material points on it, so that this ‘geometry’, or ‘topology’, is changed, and so, in turn, affects the subsequent patterning of events. And so it goes on in a vicious or creative circler, or rather spiral.
The relation between K₀, the underlying substratum or omnipresent medium, and the network of ultimate events we call the physical universe, K₁ is somewhat analogous to the distinction between nirvana and samsara in Hinayana Buddhism. Nirvana is completely still and is totally non-metrical, indeed non-everything (except non-real), whereas samsara is turbulence and is the world of measure and distancing. It is alma de-peruda, the ‘domain of separation’, as the Zohar puts it. The physical world is ruled by causality, or karma, whereas nirvana is precisely the extinction of karma, the end of causality and the end of measurement.

Note 3 The ‘Space-time hyperbola’ , as stated, does not extend indefinitely either along the ‘space’ axis s (equivalent of x) or indefinitely upwards along the ‘time’ axis (equivalent of y). — at any rate for the purposes of then present discussion. The variable t has a minimum t₀ and the variable s a maximum s₀ which one suspects is very much greater than t_c . Since there is an upper limit to the speed of propagation of a causal influence, c , there will in practice be no values of t greater than t_c and no s values smaller than s_c . It thus seems appropriate to start marking off the s axis at the smallest value s_c = s₀/ c which can function as the basic unit of distance. Then s₀ is equal to c of these ‘units’. We thus have a hyperbola something like this — except that the curve should consist of a string of separate dots which, for convenience I have run together.

Note 4 See Rosser, Introductory Relativity pp. 70-73. Incidentally, I cannot recommend too highly this book.

Note 5 I have not completely decided whether it is the ‘containers’ of ultimate events that are elastic, indeterminate, or the ‘space’ between the containers (which have the ultimate events inside them)’. I am inclined to think that there really are temporal gaps not just between ultimate events themselves but even between their containers, whereas this is probably not so in the case of spatial proximity. This may be one of the reasons, perhaps even the principal reason, why ‘time’ is felt to be a very different ‘dimension’. Intuitively, or rather experientially, we ‘feel’ time to be different from space and all the talk about the ‘Space/Time continuum’ — a very misleading phrase — is not enough to dispel this feeling.

To be continued SH 18 July 2013

Categories Einstein, ksana, Rosser, Special Relativity, Special Relativity, Time Dilation, Uncategorized

Time Dilation and Space Contraction in Ultimate Event Theory

December 7, 2012 //

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

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

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

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

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

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

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

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

Categories Einstein, Ether, Minkowski, Special Relativity, Time Dilation