Archives for category: Time

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

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

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 × s0  × s= 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 sv and tv (subscript v for variable) such that sv × tv  = s0 × t0  = Ω = constant. Just conceivably, one could make s0  a minimum and t0 a maximum but this would result in a very strange universe indeed. In this model of UET, I take s0 as 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 RELATIVITY  HYPERBOLA DIAGRAMis 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  su    and tu  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 Relativity Circle Diagram tan sinthemselves 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) sand 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   =  s 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 su, 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* su , i.e. the relation s0 : su = c*: 1. But s0 and su are, by hypothesis, universal constants and so is c* . Furthermore, since by definition sv tv = s0 t0 = Ω = constant , t0 /tv = sv/s0 and, fitting in the ‘ultimate’ s value, we have t0 /tu = su/c* su    = 1 : c*. In the case of ‘time’, the ‘ultimate’ dimension tu 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 sv and tv . 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.

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The phenomenon of time dilation, though not noticeable in everyday circumstances, is not a mathematical trick but really exists. Corrections for time dilation are made regularly to keep the Global Positioning System from getting out of sync. The phenomenon becomes a good deal more comprehensible if we consider a network of ultimate events which does not change and spaces between then which can and do change. We are familiar with the notion that ‘time speeds up’ or ‘slows down’ when we are elated or anxious : the same, or very similar occurrences, play out differently according to our moods. Of course, it will be pointed out that the distances between events do not ‘actually change’ in such cases, only our perceptions. But essentially the same applies to ‘objective reality’. or would apply if our senses or instruments were accurate enough the selfsame events would slow down or speed up according to our viewpoint and relative state of motion.
RELATIVITY TIME DILATION DIAGRAMThis could easily be demonstrated by making a hinged ‘easel’ or double ladder which can be extended at will in one direction without altering the spacing in the other, lateral, direction. The ‘time dimension’ is down the page.The stars represent ultimate events, light flashes perhaps which are reflected back and forth in a mirror (though light flashes are made of trillions of ultimate events packed together). The slanting zig-zag line connects the ultimate events : they constitute an event-chain. By pulling the red right hand side of the double ladder outwards and extending it at the same time, we increase the difference between the ultimate events on this part of the ladder but do not increase the ‘lateral’ distance. These events would appear to an observer on the black upright plane as ‘stretched out’ and the angle we use represents the relative speed. As the angle approaches 90 degrees, i.e. the red section nearly becomes horizontal, the red part of the ladder has become enormously long. Setting different angles shows the extent of the time dilation for different relative speeds.
Note that in this diagram the corresponding space contraction is not shown since it is not this spatial dimension that is being contracted (though there will still be a contraction in the presumed direction of motion). We are to imagine someone flashing a torch across a spaceship and the light being reflected back. Any regular repeating event can be considered to be a ‘clock’.
What such a diagram does not show, however, is that, from the point of view of the red ladder, it is the other event-chain that will be stretched : the situation is reversible.

The idea for this diagram is not original : Stefan Wolfram has a similar more complicated set of diagrams on p. 524 of his great work A New Kind of Science. However, he makes the ‘event-lines’ continuous and does not use stars to mark ultimate events.  More elaborate models could actually be made and shown in science museums to demonstrate time dilation. There is, I think, nothing outrageous in the idea that the ‘distance between two events’ is variable : as stated we experience this ourselves all the time. What is shocking is the idea of the whole of Space/Time contracting and dilating. Ultimate events provide as it were the skeleton which shows up in an X-ray : distances between events are flesh that does not show up. There is ‘nothing’ between these events, nothing physical at any rate.        SH 

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Our civilization is, and always has been, object based, tactile, much more concerned with space than with time.  The main reason would seem to be that we inherit our science and mathematics from the Greeks and, for historic and perhaps also temperamental reasons, they hankered after the stability and permanence which, for most of their era, they did not possess (note 1). The Greeks excelled above all in two areas where change is not immediately apparent or was deliberately eliminated : namely astronomy and geometry. Archimedes formulated the basic laws of statics and hydrostatics but the world had to wait nearly another two thousand years before Galileo and Kepler gave us the laws of motion. This preoccupation with solid matter and with changelessness is by no means a universal attitude : Buddhism, for example, has been described as “essentially a long meditation on impermanence” while Taoism positively exults in the ebb and flow of natural processes and counsels us, if we want to achieve happiness, to ‘abandon ourselves to the flow’.
Even when movement did make its belated appearance in ‘classical’ (i.e. Renaissance to mid 19th century) physics the early scientists dealt primarily with ‘extended bodies’, ‘inertial systems’, ‘ideal’ gases, and viewed the whole of Nature as subject to unchanging mathematical laws laid down by the Creator. When thermodynamics in the nineteenth century gave physics the notion of ‘entropy’ there was general consternation since it seemed to show that the whole marvellous machine was running down and that the universe would ‘in time’ become uninhabitable as all temperatures became uniform. Then came Einstein and Relativity and the world of science was never the same again (note 2).
The difficulty that so many people, even some physicists, have in dealing with four dimensions instead of three only demonstrates the materialistic bias of our whole civilization. Obviously, if we are dealing with relatively persistent and unchanging entities such as rocks or lumps of metal, we can afford to neglect the time dimension since such substances  remain the same, or very nearly the same, from one day to the next, often from one century to another. But if we are talking about events, this is quite another matter.
Suppose a staccato event such as a pistol shot or a sudden flash of lightning. To locate this event spatially for the benefit of someone who was not present, it is sufficient to specify three, and no more than three, distances from a fixed point, say the corner of the room for the pistol shot. These three directions do not need to be at right angles to each other but in general this is the most convenient way of proceeding : an imaginary rectangle, extendable in all directions, with one corner baptised the ‘origin’ gives us a coordinate system. Alternatively, we can imagine a perfect sphere, make its centre the origin and precisely identify any spot by giving its distance from the centre and two angles. Latitude and longitude suffice to locate Waterloo if we mean the village in Belgium (or what is left of it), but this is not much use if we are referring to the battle. Human history is much more concerned with what happened than where such and such an event took place — what does it matter where Napoleon was when he decided to invade Russia?
The time dimension is, in reality, more fundamental than the spatial one, as Pearse correctly observed. One can (just about) imagine a world without extendable bodies, but not a world where nothing happens — one would not call it a world. All subjective events, wishes, fancies and so forth, though technically speaking localized ‘in our head’, have nothing to do with space : this is precisely why dreams and fantasies are attractive — they liberate us from the tyranny of spatial boundaries. This is a rather important point : it shows how the ‘time dimension’ is relatively independent of the spatial one, is dislocated from it, as it were, at least in our consciousness, if not in reality.

‘Time’ has in common parlance two very different meanings  which are frequently confused : it sometimes means Succession (before and after) and sometimes Duration (how long?). Of these two meanings, there can be no doubt that the first, succession, is the more important since the only way we can evaluate duration meaningfully is by measuring the interval between two prominent events, the first and last in a chain of events. On the other hand, succession has no connection to duration : event B succeeds event A whether the interim is scarcely perceptible or centuries long. A person’s ‘life’ spans the inerval between a first event, the moment of birth (or sometimes the moment of conception) and a last event, the moment of death. Today, the ‘times’ of these two events can be specified with great precision and there can be no doubt about which of the two events comes before the other and that they really are the first and the last (of this particular sequence anyway). The extent of a lifetime or other interval between two successive events is always measured by referring it to some repeated event or cycle which we have reason to believe is regular, for example the periodic reappearance of the sun (so many days), or its return to the same spot as judged by the shadow cast on a stone (so many years). Otherwise, we have man-made devices on the same principle, swinging pendulums, vibrating atoms and so on. One could in principle measure a time interval  with tolerable precision by recording one’s own heartbeats. In all such cases we have (1) a first and last event in a specific event chain, and (2) a (finite) number of events taking place in an adjacent event chain, or, in the case of heart beats, ‘within’ the very event chain we are measuring.
The Axioms of Ultimate Event Theory make this even more decisive. In a causally connected event chain — for the moment we assume that it is possible to recognize such a chain — there are always single events which we can take as ‘first’ and ‘last’. Every macroscopic event is made up of a finite number of ultimate events (Axiom of Finitude) and every ultimate event is precisely localized (Axiom of Locality). Moreover, in much the same way as we can, or could, measure a life-span in terms of heartbeats, we can measure the duration of any connected event chain by the number of (actual or possible) intermediate events. This number will be large but will not be infinite (Axiom of Finitude). Of course, in practice there may be serious difficulties in deciding whether such and such macroscopic events are causally connected or not, i.e. belong to the same chain, and the number of intermediary ultimate events will not, with our current technology at any rate, be ascertainable. But in principle this could be done. Such an event chain would be entirely determined both as to its constituent ultimate events and their number. (More will be said about this in a subsequent post.) We know athat, because of Relativity, there will be serious problems about relating this particular event chain to all the other event chains going on ‘at the same time’ but if we consider only a particular connected chain, and other chains occurring in its immediate vicinity, these problems do not arise.
Four, and seemingly no more than  four, specifications are required to localize an event exactly. However, this does not mean that the four ‘dimensions’ enter the arena ‘on the same footing’ : manifestly they do not. The three spatial dimensions are inextricably intertwined and how we label them, i.e. which we call length, breadth, depth, x, y or z is obviously neither here nor there — provided we keep to the same labels we have assigned them to a particular case. I conceive of ‘space’ as being ‘continuous’ in the sense of there not being any obvious breaks or barriers between specific spots which can receive ultimate events, even though (by the Axiom of Locality combined with the Axiom of Finitude) there is not an ‘infinite’ number of possible locations for events between any two given spots. The three spatial dimenions -dimensions are thus (1) arbitrarily labelled with respect to the three different directions; (2) are at right angles to each other (in the normal coordinate representation); and (3) are continuous in that they ‘run into each other’ without any apparent breaks. But none of this is true of the time dimension (which renders all this talk about the fourth dimension being compared to ‘adding a third dimension to a flat surface’ completely vacuous). Why is this not true? Because the time dimension has only one possible direction and this direction is (I believe) imposed on us — the so-called Arrow of Time. Secondly, the time dimension does not  ‘run into’ the three spatial dimensions but is very much out on its own, which is precisely why it was possible for so long to neglect it. Thirdly and most important of all, the time dimension is not continuous.
Suppose a set of objects in a particular neighbourhood at a particular moment, the contents of this room for example. A moment later, I see the same set of objects to all intents and purposes as they were before — though I  know there have been some slight changes at an invisible level. I cannot conceive for the life of me how this room can get from what it is at one moment to what it is at another, later moment, except by disappearing and reappearing. Strangely enough, few Western thinkers have addressed this problem since nearly all of them assume that it is ‘natural’ for things to keep on existing from one moment to the next and, moreover, to make the transition without any kind of an interim. Descartes is about the only Western philosopher I know who was worried by the question, and he resolved it, to his own satisfaction at least, by invoking the perpetual intervention of God who, at each and every moment, prevented the entire universe from disappearing without trace by recreating it anew — the so-called ‘Theory of Continuous Creation’ (Note 3).
Now, I believe that the movement from present to future is discontinuous, that “Being is shot through with nothingess” as Heidegger put it in a memorable phrase. Not only that, I believe that we have a certain instinctive awareness of this breakage in ‘the flow of time’. For Newton and countless others, ‘time’ is imaged as a stream, a fluid at any rate that ‘flows’ in a particular direction — an idea that goes back to Heraclitus. The classic Indian Buddhist simile is quite other : it is that of a lamp flashing on and off repeatedy (note 4) . I find this simile much more plausible, and indeed, I have never felt at ease with the whole concept of ‘continuity’, which is not a physical but a mathematical concept. We now know that all transfers of energy are not continuous, as they were once thought to be, but are subject to quantum laws (come in chunks). Fluids appear to be  continuous but in reality they are made up of molecules which we can even ‘see’, if only via an electron microscope. The last bastions of the continuous are ‘space’ and ‘time’ and, even here, some physicists are already suggesting that the ‘fabric of Space-Time’  might be ‘grainy’, as they put it.         S.H.

Notes :  (1) : During Plato’s lifetime Athens lost a long drawn out war with Sparta and was decimated by a plague. It is hardly surprising that Plato, and other men like him at the time, recoiled with horror from the human spectacle, taking refuge in a transcendent reality.

(2)  Paradoxically, the man who did more than anyone else to bring time to the centre of the stage, ended up by dispensing with it completely. During his last years, Einstein apparently came to the conclusion that everything takes place “in an eternal present”. He seems to have been serious about this, since he mentions this idea in a letter of condolence he wrote after hearing about the death of his undergraduate friend, Besso.

(3) Bergson “The world the mathematician deals with is a world that dies and is reborn at every instant, the world which Descartes was thinking about when he spoke of continuous creation” (Bergson, Creative Evolution p. 23-4). Stcherbatsky, who quotes this statement, remarks that “His [Descartes’] idea is quite Buddhistic and the Sanscrit translation sounds like a quotation from an Indian text” (Stcherbatsky, Buddhist Logic Vol. 1 Footnote p. 108).  The Buddhist theory is rather one of ‘Instantaneous Being’ since there is no Creator : the appearance and disappearance of dharma is a purely mechanical process, a sort of cosmic karma.

(4)    “The Buddhist theory of Universal Momentariness (ksanikatva’), converting the universe into a kind of cinema, meaintains that there is no other cause of destruction than origination, entities disappear as soon as they appear, the moment when the jar is broken a stroke of a hammer does noit differ in this respect from all preceding moments, since every moment a new or ‘other’ jar appears, constant desrtruction or renovation is inherent in every existence which is really a compact series of ever new moments.” Stcherbatsky, Buddhist Logic Vol. 2 Page 93 Footnote)