A statistical basis for special relativity.

The Fitzgerald-Lorentz contraction.

Light was identified as an electro-magnetic wave. A medium, for that wave to travel thru-out the universe, was supposed. This was called the ether. It was assumed to be a sort of universal ocean with an absolute motion, in Newtonian universe of absolute space and time. The Michelson-Morley experiment sought to detect the velocity of the ether in relation to a split beam of light, back and forth in perpendicular directions.

If and when one of the perpendicular directions happened to be aligned parallel to the ether "stream," as in wind or water, this would cause the light beam maximum ether drag. With the ether velocity for a tail wind, the beam would make most progress. But the return journey, into a head wind, would more than offset this.

The perpendicular beam, with only an ether cross-wind to contend with in both directions, would under-go the least loss of time.

The Michelson-Morley experiment, repeated in all directions, showed no difference in light speed between the split beams.
The two calculations were not confirmed. But Fitzgerald and Lorentz independently equated them, to conform to experiment, with a "contraction factor." This phrase owed to a supposition that the ether had a contracting effect on motion into it, as a ball flattens when kicked into the wind.

Later, Einstein abandoned the idea of an ether, with an absolute velocity in an absolute space and time. Instead, locally differing space and time measurements, related by the Lorentz transformations, were taken at their face values.

We can show how Fitzgerald and Lorentz equated the Michelson-Morley calculations. The distance the split light beams travel is the same, so their time taken can be compared. Let that distance simply be length, l. The velocity of light is usually given by c. The ether velocity, v, is equated to earth velocity, like a bank on which we are moving relative to an ether "stream."

One split light beam gets a head- and tail-stream trip. Their respective times, given as distance divided by their head and tail stream velocities, are l/(c-v) and l/(c+v). This compares with a time, unaffected by the ether, for light to travel the same distance, which is: l/c, or, counting both ways, 2l/c.

The supposed maximum time, due to maximum ether drag was:

l/(c-v) + l/(c+v) = 2lc/(cē-vē).

The cross-stream trip is the same velocity each way and over-all the least affected by supposed ether drag. The chapter about the Michelson-Morley experiment, using Pythagoras theorem, shows the cross-wind time, taken in both directions (and so multiplying by two) to be:


To make the calculated differences in times equal, as the Michelson-Morley experiment showed them to be, the maximum time, 2lc/(cē-vē), had to be multiplied by a "contraction factor," F = (1-vē/cē)^1/2 = {(cē-vē)/cē}^1/2.

Then, F x 2lc/(cē-vē) = 2l/(cē-vē)^1/2, making the two times equal.

This contraction factor only becomes important at velocities that are a significant fraction of that of light. It was incorporated in the Lorentz transformations, of observers different space and time measurements, to make them consistent with each other. Einstein rested his special relativity on such local times and lengths having equal validity. Thus, he ruled out an absolute velocity, as of a supposed "ether."

The "contraction factor" of averages.

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The Michelson-Morley calculation for least ether drag, with an ether 'cross-wind' or 'cross-stream' on a light beam ( which is the same value as the Fitzgerald-Lorentz contraction ) looks like a geometric mean, especially if it is put in the form:


The geometric mean is one possible calculation of an average. An average is the representative value in a range of values. If the geometric mean's two values in brackets are considered as the lower and upper limits of a range of values, then the geometric mean takes their representative value to be the square root of the lower limit times the upper limit. That is a range from 1-v/c to 1+v/c, or from (c-v)/c to (c+v)/c. Respectively, these are the ratios of tail-wind and head-wind speeds to the speed of light.

Another possible average, of these speeds, is the harmonic mean, which is the reciprocal of the arithmetic mean of the range reciprocals. In this example, { c/(c-v) + c/(c+v) }/2 = cē/(cē - vē ). The harmonic mean is its inverse: (cē - vē)/cē.

The contraction factor was introduced to multiply by the maximum ether drag time to derive the minimum ether drag time. It was an ad hoc hypothesis to cover for the Michelson-Morley experiment not confirming the existence of an ether drag on the speed of light, as light speed remained unchanged in all directions of observation.

The statistical basis of special relativity derives the contraction factor, F, as a by-product of the ratio of the harmonic mean velocity (under maximum ether drag) to the geometric mean velocity (under minimum ether drag):

{(cē - vē)/cē}/{(cē-vē)/cē}^1/2 = F.

The inverse of the contraction factor, 1/F, is the ratio of maximum to minimum "drag" times.

The contraction factor reduces the range of dispersed observation. The Michelson-Morley calculations, modified by the contraction factor, reduce the maximum to the minimum range of dispersion. A statistical interpretation, of the Michelson-Morley test for maximum and minimum 'ether drags', is that they are maximum and minimum ranges of dispersion.

The maximum ether 'drag' or dispersion was taken to be from end-limits (c-v)/c to (c+v)/c. Multiplying the two ratios measures how closely they approximate the value of unity they are more or less dispersed from.
In fact this value, (cē - vē)/cē, equals the harmonic mean.

The geometric mean of the end-limits might also be taken as a multiple of 'contracted' dispersion limits from {(c-v)/c}^1/2 to {(c+v)/c}^1/2.

( If these limits had their own, smaller, geometric mean, the harmonic mean would in turn be reduced, given that the contraction factor relates two means of observation. )

The geometric mean of the maximum dispersion's end-limits produces a smaller average than the harmonic mean of that range's limits, and is therefore capable of representing a narrower dispersion. This is equated to the maximum range's harmonic mean by the contraction factor.

If a different average applies between observers, this suggests they measure different ranges of dispersion, if their relative motion is light-proximate. This reminds of the varying statistical paths moved by quantum particles of light and matter, at the microscopic other end of the scale of things.

Constant light speed as a constant average.

In statistics, different averages are used, according to which represents better a given range. The geometric mean is more likely to be the closer average to a range of items that follows an exponential rate of growth, like population growth. Growth is gradual at first but becomes steep. The upshot of this is that an average of this kind of growth tends to dip below the average for a flat growth. Flat growth may have a higher average if it starts more steeply.

The arithmetic mean is the average that represents constant growth. So, the arithmetic mean may give a higher estimate than the geometric mean for the average of a given range. But the range, suitably measured by the arithmetic mean, remains on a steady slope, like an inclined plane. Exponential growth will over-take it eventually, like the difference between simple interest and compound interest on savings, tho the latter may start at a lower rate of interest.

The arithmetic mean, applied to the M-M calculations, would add the two range values, up and down stream velocities, and divide by two, to give a value of one:

{(c-v)/c + (c+v)/c}/2 = 1.

The arithmetic mean is one, whereby ether head- and tail-winds cancel each other out. This gives a statistical explanation for the constant velocity of light, as an averaging ( by arithmetic mean ) of the speed of light with respect to light-relative velocity.

The statistical formulation still works if the limit values, for the range of velocity ratios are inverted. That is if the time ratios become the velocity ratios and vice versa. Light speed and journey time both remain at unity, as the inverse of one is one. The roles of the harmonic mean and arithmetic mean are exchanged. The contraction factor is now a by-product of the ratio of the arithmetic mean to the geometric mean. And the harmonic mean cancels out relative velocity to leave constant light speed.

In that case of a velocity range between c/(c-v) and c/(c+v) the averages work out over unity, that is over the speed of light. In other words, these are the velocities of hypothetical faster-than-light particles or tachyons.

Albert Einsten's original role, for the constant speed of light, was as an axiom or basic postulate of special relativity, in the spirit of his scientific philosophy of empirical rationalism. His idea was to base his deductions on incontrovertible facts. (These deductions would derive implications of his theory, which could test its value. ) One of these basics was the solid backing of experimental evidence for light always being observed at the same velocity.

(Another undeniable basis, to special relativity, was that there is no privileged frame of reference. There is democratic equality between every point of view from which observers choose to take measurements of a given event.)

Einstein's scientific method is fine. No surprises there. But it would be an advantage to be able to go beyond just accepting the basic fact of constant light speed. The relation between light and its constant speed is not, on the face of it, a necessary relationship.

Conceivably, someone might construct a tight theory of why light, the medium of our observation, logicly implies speed constancy to our perception. However, present evidence does not compel such an assumption. Indeed, the physicist, Joao Magueijo, argues that light may have had a different speed early in the evolution of the universe. Light speed puts a limit on the speed of cause and effect connections. So, light, at its presently known speed, cannot explain the near uniformity of stellar matter in the universe without some expansion of space-time stupendous even by the standards of the Big Bang. Hence, the so called inflationary theories of the origin of the cosmos. Alan Guth, in The Inflationary Universe, cites, in the notes, some fifty different versions of inflation. Hence, the opportunity for an alternative theory.

Saying, that light speed may have changed once-over, is one thing. Paul Davies reported an experiment that reversed the effect of a medium on light waves, which normally makes their group velocity less than their phase velocity. When a stone is thrown in a pond, one crest appears faster than its group of waves.

But when a laser excited cold caesium, 'the peak of the wave pulse appears to exit the gas before it enters.' However, Heisenberg's uncertainty principle prevents a foton being picked out from the pulse as traveling faster than light.

The philosopher David Hume made the distinction between contingent relations and logicly necessary relations. A contingent relation is an observed association between facts. The association may be a general relationship. That is the association may always be observed without fail. According to Hume, that is no guarantee that the association must persist. There might be a time or place when the link is suspended.

A logical relationship is by definition always true: Socrates is a man. All men are mortal. Therefore, Socrates is mortal. The distinction is not absolutely clear cut. But this syllogism is basicly a logical statement, as the relation, between light and constant speed, is basicly an evidential statement.

Hume's distinction, between necessary and contingent relations, is the basis of the objection to constant light speed as an axiom to special relativity. The statistical basis of special relativity removes constant light speed as an axiom, to make it the derivation of an arithmetic (or possibly harmonic) mean velocity of a range of velocity observations. The arithmetic mean always cancels out whatever relative velocity with respect to light, to produce just the constant light speed.

The theory of equating relative observations becomes a theory of equating averaged observations (at significant approximations to the speed of light).

Michelson-Morley range and Lorentz range.

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The nineteenth century introduction of the contraction factor may not have explained anything new. But it provided the form for conversion of space and time measurements between observers.

This conversion is more generaly given in the Lorentz transformations. The Michelson-Morley calculations, in terms of maximum and minimum ether drag, just give the outer limits of possible co-ordinate changes between different observers.

Likewise, a statistical basis of special relativity has a more general Lorentz form to the Michelson-Morley form. Taking the maximum drag time as t' and the minimum drag time as t, the Michelson-Morley calculation can be considered as a special form of the Lorentz transformation of time measurements between observers: t' = t/F, where F is the contraction factor.

The actual Lorentz transformation is:

t' = ( t + xv/cē )/F = t( 1 + uv/cē )/F.

In the equation, distance x equals velocity u multiplied by time t. In the Michelson-Morley calculation, the light beam going in a cross-wind direction to the ether, therefore has a zero velocity with respect to it. When velocity, u, in the Lorentz time transformation equals zero, it reduces to the Michelson-Morley calculation.

Substituting in the Michelson-Morley values for F = {(cē - vē)/cē}/{square root(cē-vē)/c}, then:

(1/t)/(1/t') = {(cē+uv)/cē}{square root(cē-vē)/c}/{(cē - vē)/cē}.

In the denominator, the value for 1/t', is the harmonic mean of the range (c-v)/c to (c+v)/c, as for the M-M calculations. In the numerator, the value for 1/t, is modified from the M-M value, by a coefficient. If this numerator is a geometric mean, it has to be squared, and the square shown to be a multiple of two distinct end-limits to a range.

The following range answers this description:

{(cē+uv)/cē}{(c-v)/c} to {(cē+uv)/cē}{(c+v)/c}.

In the Lorentz transformations, there are usually the three distinct velocities, v, u' and u. The Lorentz transformation, of velocities between observers, u' = (u+v)/(1+uv/cē), can be substituted in the above time transformation:

(1/t)/(1/t') = {(cē+uv)/cē}/F = {(u+v)/u'}/F.

Because, u = 0, in the M-M calculation, the relative velocity, v, found in the contraction factor, equals u' or the velocity of the observer of time t'. Substitute u = 0 and u' = v in the right side of the above equation, reducing it to 1/F. This gives the Michelson-Morley special case of the Lorentz transformation of time.

The purpose here is to extend the Michelson-Morley range of velocities to a Lorentz range of velocities. The Michelson-Morley ratio of harmonic mean to geometric mean, equal to the contraction factor, ( as given above ) is compared to a Lorentz transformation ratio of those means equal to F.

The harmonic and geometric means are different averages of different ranges of dispersion. Here, the Lorentz form of the harmonic mean's range is the same as for the Michelson-Morley range. Taking the harmonic mean from the numerator of the ratio, just given, substitute u' for v, to get the range, (c-u')/c to (c+u')/c.

The range, averaged geometricly, is smaller than the range, averaged harmonicly.

The 'Lorentz' harmonic mean multiplied by the contraction factor equals the 'Lorentz' geometric mean of velocity dispersals ( that are the inverses of time dispersals ).


In 1905, Einstein's special theory of relativity was based on generalisations, that had simply to be accepted as facts. There was the constant speed of light (apparently verified by Michelson and Morley ) and the equality of all frames of reference, related by ( Lorentz ) transformations thru a 'contraction factor,' when observed motions reached a noticable fraction of light speed.
Einstein dispensed with the idea of the ether and its contracting effect, like the compression of a ball kicked into the wind. But the factor remained as a given fact of nature, in relativistic physics.

A statistical basis to special relativity derives both constant light speed and the contraction factor, in the relative motion between observers, as by-products from three different mathematical ways of averaging a dispersion of measurements. The contraction factor ( or its inverse ) is the ratio of the harmonic mean to the geometric mean velocities ( or inversely, the times ) significantly aproaching light speed.

In the Michelson-Morley case, the observer of the slowest ( most 'ether-dragged' or most dispersed ) velocity and therefore longest time, measures the harmonic mean velocity of the most dispersed range of velocities. The other observer of the least slowed velocity and least slowed-up time, compared to light speed and time, measures the geometric mean velocity of that range of velocities, contracting its dispersion.

But taking the arithmetic mean velocity of this range cancels out the velocity in the same direction and the opposite direction of a light beam, so that the arithmetic mean velocity is always a constant light speed.

This statistical formulation reduces light speed constancy from an axiom of the theoretical system to a derivation. Light speed constancy is explained as a statistical consequence rather than asserted as an essential condition of special relativity.

This is in keeping with quantum physics. One only has to quote from Richard Feyman's QED:

"It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, c. The amplitudes for these possibilities are very small compared to the contribution from speed c; in fact, they cancel out when light travels over long distances. However, when the distances are short...these other posibilities become vitally important and must be considered."

Previously, Feynman's popular book explained that light didnt always travel in straight lines, as shown in 'Feynman diagrams'. Forked lightning looks like Feynman diagrams written in the sky.

It may be that at speeds approximating to light speed, a statistical explanation of special relativity is that relative observers are measuring light following significantly different paths with different ranges of dispersion, which are, represented by different averages, related thru the contraction factor.
This contraction is not caused by a greater or lesser ether drag but is the measure of a greater or lesser statistical dispersion in times or velocities recorded between observers.

In the nineteenth century, physicists such as Clerk-Maxwell introduced statistics into the measurement of the behavior of gases, considered in terms of the mass motions of their molecules or atoms. Individual bodies large enough to be observed could have their motions tracked in a predictable manner. Motions of very small bodies could only be measured in the aggregate by statistics. In principle, the particles were believed to behave in as determinate a manner as the familiar scale of things. The classical mechanics of cause and effect was still deemed to apply.

With twentieth century quantum physics, most physicists gradually came to accept that statistics was more than a convenient second-best measurement of the small scale of things. Tho, Einstein, who helped to pioneer quantum mechanics' most radical concepts, never believed it. By the nineteen eighties, Alain Aspect's experiments favored 'quantum correlations' over the classical principle of local causes.

Yet, the belief has prevailed that there is a radical difference between Einstein's relativity theory and quantum physics. Relativity theory has been lumped with Newtonian physics as 'classical physics' of determinate relationships. This page, hopefully, shows how statistics may offer a more coherent conception of special relativity, which may relate to the statistics of quantum paths.

In QED, from 1985, Feynman admits straight away:

"Does this mean that physics, a science of great exactitude, has been reduced to calculating only the probability of an event, and not predicting exactly what will happen? Yes. That's a retreat, but that's the way it is: Nature permits us to calculate only probabilities. Yet science has not collapsed."

As late as 1960, in his monumental work, The Structure of Science, Ernest Nagel was conducting a rear-guard argument for determinism, even in the social sciences. The assumption has remained that classical mechanics including relativity is deterministic.

However much I may be mistaken, I hope the basic idea here is justified. The spirit of this page is known in law as a tu quoque, meaning "you also," namely that special relativity is also an implicitly statistical theory.


See references at the end of other relativity pages on this site. Also:
Graham Farmelo. Einstein's theory? It's all relative. The Sunday Telegraph, 6 April 2003.
Paul Davies. Light goes backwards in time. The Guardian. Science section, 20 july 2000.
Derek Gregory and Harold Ward. Statistics for business studies. 1967. Descriptions of the arithmetic, geometric and harmonic means in elementary statistics.
Albert Einstein. Ideas and Opinions. The article: What is the theory of relativity? 1919.

Richard Lung
9 october 2003

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