Time-Dilation, the simplest possible derivation (Neo-Machian)

Let the intrinsic spectral period of an atom (that is, the peri0d of a particular atom relative to itself) be denoted by t, and let that period as it appears in the spectroscope of an observer be denoted by tR (called the relative time). The distance in metres  between that atom and the observer is then denoted by s.

Now, as we already know, in neo-Machian Normal Realism, this distance s in metres, divided by the constant c is  not a ‘velocity’ but simply a time in seconds, where c is a dimensional constant giving the ratio of measures in metres to those in seconds. Since these measures, t, tR and s/c are now all in the same units of seconds, they can be represented dimensionally in the orthogonal  (i.e., rectangular) way as depicted in Fig. 1  ]. where tis shown as the resultant, by Pythagoras,  of the two rectangular measures s/c and t  [i].

Fig. 1 here        The Pythagorean time-triangle          (ontrol-click to open)

Unlike the traditional scalar graph of a uniform motion, in Fig. 1 the length of the body’s motion vector tR, in the diagram is the geometrical resultant of the two components,, tR and s/c, viz.:  

 tR  =  √ (s/c)2 + t2          (1).

This simple, sixth form school-level formula gives precisely the same value of time-stretching (‘dilation’) as is given by  Einstein’s far more complicated formula for that same relativistic time-dilation. This can be seen if we substitute for the distance s in (1). the relative, or observed, velocity vtR and then simplify the formula by the ordinary rules of algebra. This produces the standard orthodox, Einsteinian equation:

 tR  = t/[[1 – (v2/c2)]     (2).                  

(Note that the direction of v is inconsequential, since squaring the term, whether v is negative or positive, makes the result uniformly positive. This means that the dilation is the same for a body moving away from the observer as coming back. This  explains the so-called ‘Twins’, or ‘Clock’ paradox, as being  due to the assymmetry in those relative motions due to the turnaround of the one twin, with all its force and visceral accompaniments, which makes him, on his return, the younger of  the two. This is what the young Viv Pope discovered soon after his short correspondence with Einstein (see above ). It was whilst on sick-leave following a climbing accident in his job as a telephone lineman.  Unable to cope with the standard, arcane version of Relativity,  he thought:  ‘If what one man discovers about nature is true, then others should be able to discover that same truth in different ways.’ (As it is said, ‘All the keys hang not at one man’s girdle.’)  The result was that while convalescing  in the garden one fine day, he had what may be called a ‘Eureka’ experience. Using no more than a notebook , a pencil,  a ruler and set of compasses, he had discovered the same relativistic space-time relation that Einstein had discovered, but in the simplest, purely geometrical way. This was from plain Pythagoras directly to the relativistic time-formula , bypassing all the historical rigmarole connected with the standard Einsteinian version. This unique derivation of relativistic time-dilation is shown in Fig. 1.

Why is this derivation ‘unique’? It is because it makes no  reference to the so-called ‘speed of light in vacuo’ (Einstein’s Second Axiom of his Theory of Relativity). Only lately, in the journal New Scientist, is it reported that physicists are adually coming to that same conclusion, that Einstein’s Second Postuate is redundant and that the theory is much simpler without it (see article, ‘Shedding Light on Light’ (N.S., 1/11/08, pp. 28-31). Despite the fact that Pope’s books and papers have been widely circulated on ne Web, no mention was made of the fact that this discovery was, arguably. first made by a non-academic telephone lineman in 1954.

This discovery was later concurred in a correspondence with Herman Bondi, who reported his own independent discovery of that same truth (the redundancy of Einstein’s Second Postulate) in a book he wrote in 1964, entitled Assumption and Myth in Physical Theory  (Oxford University Press). Now, regardless of who might claim precedence for this discovery of a much simpler, purely geometrical approach to  relativity, what is certain is that, prior to POAMS, no-one has followed-up the full logical,  physical and philosophical implications of getting rid of Einstein’s Second Postulate.  Bondi told Pope that for him it was sufficient that this discovery could be used as a teaching aid for students, and that he would leave it to Pope, with his blessing, to follow-up any philosophical developments.

What are those ‘further implications’?  They are to create a revolutionary paradigm-shift from Einsteinian, into Machian relativity, for only in Mach’s Positivist approach to physics  is there no such concept as that of light as something travelling in any form whatsoever. In Mach’s approach, light is simply what you see, either directly or instrumentally, begging no question as to  how and by what means,  it ‘travels  in  space’. The trouble with this paradigm shift in thinking about light is that it bypasses, or wipes-out,  practically the whole complicated history of theories that students have to study as to how and by what means light travels between the object and the observer. In the Machian approach, the object is as it is observed in the clearest possible instances of direct observati0nal and/or instrumental perception. The history of this Machian approach, called Phenomenalism, goes right back to George Berkeley and David Hume, as later corrected and developed by Immanuel Kant – and of course, Mach.  The most up-to-date development of that phenomenalist legacy is called  Normal Realism, as described on this website.

It needs to be recognised, here, that this phenomenalist approach to physical reality has been detested and scorned  by mainline physicists from its very inception in the works of George Berkeley and David Hume. This is scarcely surprising since Berkeley’s ‘dictum esse is percipi  (‘To be is to be perceived’) seems to subjectify the whole of physical reality, making everything a product of one’s own mind, as if in a dream, the abortive position known as Solipsism. The corrective to this absurdity was supplied by Mach as developed in the up-to-date version of Neo-Machian Physics called Normal Realism.

Something else which has not been recognised, neither by Einstein nor Bohr,  is that the time-formula (2) is fundamentally quantised (see below, the Balmer Quantum Bar-code).

The Balmer Quantum Barcode

Something which physicists  seem not to have realised so far, is that the relativistic time formula is automatically quantised.  Neither Einstein  nor Bohr were aware of this direct logical connection between the two theories, Relatvity and Quantum Theory, otherwise they would not have remained so chronically at loggerheads over the notorious ‘EPR (Einstein-Podolski -Rosen) Paradox’.

For instance, here, again, is that relativistic time-formula:

                                  tR = t/√ 1 – (v2/c2)                 (2).     

The lower limit of this formula is, of course, when v = 0, at which stage tR = t , where  tR is some  multiple of t – in this case, one. Now any number can be expressed as a fraction n/N., where n is the series of integers in the numerator, and N is the series of integers in the denominator. Recall that in this formula, t  is the bottom rung of the scale of values, which is the intrinsic spectral period of the simplest atom – namely, an atom of hydrogen with N = 1, throughout and n the running series of natural integers making up the hydrogen spectrum.  (Needless to say, the t  on both sides of the  equation cancels.)

                        (n2/N2) = 1/ [1 –(v2/c2)]                (3).

Now v2 in this formula = e/m as in the standard formula for potential energy (i.e.,  twice the kinetic energy, so (3) becomes:

                         n2/N2  =  1 / [1 – (e/mc2)]            (4).

Rearranging this formula we have

                              e = mc2(1 – N2/n2),                (5) .      

Note that for n and N both equal to one, formula (5) is the standard mass-energy relation,

                                        e = mc2                             (5a)

 This  famous formula is now seen to be a direct logical consequence of the Machian, or phenomenalist approach to physics,  in which those spectral lines of the various atoms constitute the ultimate observational sense-data (pace Mach) from which, in the context of  the whole panoply of observational data, we obtain our knowledge of basic microphysical phenomena.

  Substituting, then, the symbol for mc2 in (5)  produces

                                        e = E (1 – N2/n2),         (6).

where E is the (asymptotic) upper limit of the equation, as the scale of values n approaches infinity

Now, as we know, dividing energy by Planck’s constant h expresses frequency ƒ  so, dividing (3) by h produces

                                ƒ = ƒlimit. (1 – N2/n2)            (7).

Already we see the resemblance between this formula (7) and the Balmer formula. Including the intermediate values for the integers n then produces:     

                    ƒ   =  N2ƒlim.[(1/n12) – (1/n22)]        (8) ,

where n1 and n2 are, respectively, the fixed and running terms of the various series.  

This  matches, in purely formal terms, the standard Balmer-Rydberg formulae for the series of all hydrogen-like (ionized) spectra of ordinal number N.  The empirical frequency-limit of that standard  formula is, of course, cR, where c is the usual space-time constant, and R the so-called Rydberg constant. Fitting this bit of empirical information into the numbers template, formula (8) produces:

                    ƒ   =  N2cR [(1/n12) – (1/n22)]          (9).

This is precisely the Balmer-Rydberg formula which these experimenters discovered by empirical trial and error in their studies of spectra. Needless to say, these formulae are quantised in the way Mach would surely have approved. It cannot be mere coincidence, as some have suggested, that the formula for relativistic time-dilation and that of the discrete lines in the spectrum are connected in this directly transformable way. This is by no means implausible,  since the inverse of time is frequency. The discovery of this direct connection between time and spectral frequency might therefore be expected to occur, sooner or later. 

However, be that as it may, physical science has had  to take its own wandering course, milestoned with the names and deeds of the deservedly admired Great Pioneers of Physics. Unfortunately, Nature cares nothing for those theoretical meanderings that we cherish in the name of ‘Physics’, and which our students have to follow like the trails of ants.


[i] Why ‘orthogonal’, says someone. It is because the only way in which measures can be geometrically projected without encroaching on one-another’s domains is at right-angles to one another.