These scenarious escape from the particular situation of the Michelson-Morley experiment (in the previous web page about null and non-null results) and consider the Interval more generally as a statistical equation of an average value to a range of values.
For example, eqn. 1:
I² = t²(c² - u²) = t'²(c² - u'²) = I'².
The only way you can tell this is a statistical Interval, as distinct from the conventional Interval, is by the indexed Interval (squared), I'².
On the previous web page, I considered a modified Michelson-Morley
experiment, in which one of the split light beams was not reflected. In
effect, this meant instead of a two-way beam with respect to earth motion,
represented by (c-u)(c+u), the two factors were the same sign to represent
one-way relative motion.
Bearing this in mind, from a statistical point of view, as will be duly
shown, eqn. 1 could become a new eqn. 2:
I² = t²(c² - u²) = t'²(c - u')².
Expanding eqn.2:
I² = t²(c² - u²) = t'²c² + t²u'² - 2cu't'².
Expressing eqn. 2 in terms of I':
I² = t²(c² - u²) = I'² - 2t²u'² - 2cu't'²
= I'² - 2x't'(u' - c) = I'² + 2xt'(c - u').
Also t'(c - u'){(c + u')/(c - u')}^1/2 = t'(c² - u'²)^1/2 = I'.
Therefore eqn. 3:
I'² + I'.2x{(c + u')/(c - u')}^1/2 - I² = 0.
This is a quadratic equation that can be solved for I' with the well-known formula, eqn. 4:
I' = (-2x{(c + u')/(c - u')}^1/2 ± {4x²(c + u')/(c - u') + 4I²}^1/2)/2
= -x{(c + u')/(c - u')}^1/2 ± {x²(c + u')/(c - u') + I²}^1/2.
The purpose of a statistical version of the Interval is to find the Interval, I' as an average of the Interval, I, as a range of values being averaged.
The limits of a range are given in the positive and negative solutions of the quadratic equation. Just solve for I' as the geometric mean of that range.
Geometric mean I' is symbolised as [I']. Eqn. 5:
[I']² = x²(c + u')/(c - u') - {x²(c + u')/(c - u') + I²} = -I².
Therefore, as the root of a square can be positive or negative, eqn. 2 solves as eqn. 6:
[I'] = ±iI.
Notice that taking the geometric mean cancels the coefficient of the first order term in I'. So, almost the same result would have come directly from eqn. 1. As a special case of a quadratic equation, the variable, I', is solved in terms of the constant, I. Again treating I' as an average of a range I, then eqn. 1 would solve as eqn. 7:
[I'] = ±I.
Notice that equation 7 contains the conventional Interval, that is where I' = I, as a special case.
To get the same answer for eqn. 1, as eqn. 2, only requires a change of
sign, so that: -I² = I'². In physics, both "signatures," as they are called,
are used, so that either sequence of subtraction, between (ct)² and x², is
allowed as a definition of the Interval.
But it goes beyond convention to equate the two signatures. It would be
illogical from the deterministic point of view, especially the view that
Special Relativity was invented to save the determinism of classical
physics.
As has just been shown, a statistical interpretation, of equating the two conventions, makes sense. Moreover, as both conventions are allowed as the Interval, their equation must be, in some sense, still a version of the Interval, even if it is a statistical version that makes statistical sense.
A mathematicly more complete expression would combine equations 6 and 7 to give a complex (statistical) Interval, ±I ±iI, equated to [I'], with some slight change of symbolism, say a change of brackets as: {I'}, to show it was now a complex number.
To get some intuition of what a statistical Interval might mean, it might be helpful to re-consider the special case, from my previous page. This considered two local observers with the same local times. This reduced a statistical Interval from in terms of I or I' to simply in terms of u and u', two locally observed velocities in relation to an event. This is because the same local times cancel and with them the light speed on both sides of the Interval for both observers.
It is then possible to substitute u and u', in the I and I' results for the statistical Interval above. Eqn. 6 becomes eqn. 6a: [u'] = ±iu. And eqn. 7 becomes eqn. 7a: [u'] = ±u.
In the conventional Interval, the observers are inter-changable, without
changing the value of the Interval. The conventional Interval has the
symmetry of uniform velocity in a straight line. Thus, the Interval measure
stays unchanged, as the common space-time measure, for all observers with
different local space and time measures.
Measurement inter-changes, which do not affect the result, are known as
symmetry transformations.
Inter-changable observation carries over to a statistical Interval. It does not matter which observer's (statisticly) local Interval is chosen to solve for a geometric mean. Nor, for the restricted case of simultaneous local times, does it matter which observer's local velocity is chosen to be solved as the geometric mean velocity of the other observer's values as velocity limit values.
The geometric mean of two polar limits is either of the other two polar limits, at right angles to them. The average values are inter-changable with the limit values, or the representative values with the represented values. This might be called, perhaps, a symmetry of statistical representation. (Tho, this need not mean the same representation, at least directionly.)
A statistical Interval appears to show velocities which oscillate
perpendicularly between their respective polar limits. Where the local times
are the same, the magnitude, tho not the directions, of the velocities must
be the same.
The Michelson-Morley experiment is like an oscillation of velocities, with
time and distance held equal.
When observers' local times differ, their respective velocity limits may
also differ, tho the distance dimension of the Interval itself remains
constant in magnitude. A statistical Interval allows for differences in
directions of the Interval between local observers. That is to say the form
of the statistical Interval appears to give different observers the Interval
measure in perpendicular oscillation with respect to each other.
This is reminiscent of electro-magnetic waves, which are perpendicular.
A statistical Interval, as I imagine it, is a symmetrical oscillation of observers' localised Intervals. Either local observer's measures can be considered as an average of the other observer's local measurements as limits.
An analogy of a statistical Interval with electro-magnetism could compare two range limits with polarity. Magnetism is bipolar and there is also such a thing as an electric dipole (so Ive heard). Likewise, the geometric mean as one average, of two limits, might be characterised as monopolar. But an electro-magnetic oscillation is not symmetric in that there are no known magnetic monopoles. In terms of an analogy with a statistical Interval, this would be as if one of the observers could not average the observations of the other, tho the latter could.
Another analogy with democratic representation might illustrate the differences between a statistical Interval and electro-magnetism. Consider a representative democracy that consists of a parliament and people, that is representatives and represented. The representatives are the monopolar averages of opinion that cover a bi-polar political spread from Left to Right. If only parliament consists of the representatives, only parliament is monopolar, so to speak, while the people are limited to bi-polarity. Tho, the representatives may be positive or negative, meaning government or opposition. Parliament is analgous to the electric and the People analgous to the magnetic.
Suppose the People are also allowed to be representative and Parliament in turn becomes represented. This can happen when politicians cannot agree (say, are split or polarised between Left and Right) and leave the decision to the public in a referendum. In that case, with a combination of direct democracy, and representative democracy, there is a symmetry, of representation and represented, between People and Parliament. And this is analgous to a statistical Interval, as distinct from electro-magnetism.
Take the equation 8:
I'² + I² = 0.
This is reminiscent of the form of the simple harmonic wave equation, where,
say, I'² compares to an acceleration variable, and I² compares to a constant.
The two terms are equal and opposite. That would be like a weight pulling
against a spring forced into oscillations.
It can be written four ways:
I'² + (+I)² = 0. So, in terms of two range limits, I' = (-I)(+I).
I'² - (+iI)² = 0. So, I' = (-I)(+I).
I'² + (-I)² = 0. So, I' = (-iI)(+iI).
I'² - (-iI)² = 0. So, I' = (-iI)(+iI).
These four different values of I produce two pairs of directionly
different range limits, which average, respectively, as: [I'] = iI; iI; -iI;
-iI.
Solving the first, of these four, gives the straightforward solution: I' =
±I. But the concern here is a statistical solution. The geometric mean of the
limits, plus or minus I, average in terms of another observer's localised
Interval, I', as the square root of their multiple. So, GM of I expressed in
terms of I' is symbolised as, say: [I'] = (-I.I)^1/2 = iI.
A similar four-fold treatment of eqn.1 gives, respectively: [I'] = I; -I; -I; I.
Likewise distinguishing between I' and iI' produce different results. Whether
they are positive or negative makes no difference but we tabulate all 16
results anyway, in table 1.
| limits. | [I'] | [iI'] | [-I'] | [-iI'] |
| I | I | iI | I | iI |
| iI | -I | iI | -I | iI |
| -I | -I | -iI | -I | -iI |
| -iI | I | -iI | I | -iI |
Notice that the last two columns repeat the first two. Then the main diagonals of table 1 repeat the sequence of complex numbers associated with the x,y co-ordinates of a circular function. That is, in full: I,0; 0,iI; -I,0; 0,-iI. To put it another way, their directions are like the four points of the compass: east, north, west, south, respectively.
The left-down full diagonal gives this series in that order. The right-down diagonal gives the series starting with the second term, iI. If the column of limits are all negated (to their opposite limits) then the left-down diagonal starts the series with the third term, -I, and the right-down diagonal starts with the fourth term, -iI.
Negating the values of these 16 table entries, in effect, flipping them all thru 180 degrees, reminds of a two dimensional wave consisting of a membrane of reversing "dunes" where hills and hollows change places. Taking the special case, where I reduces to u, because time is held constant, this might be analgous to a solution to a 2-D Laplace equation, where time is also held constant. That equation consists of two acceleration terms, maybe analgous to I'² and I².
Table 1 applies as well to quadratic equation 2 or (a simplified quadratic) equation 1, perhaps analgous to an ordinary wave equation, a simple harmonic wave. This makes sense, in that the Interval relates to the kind of oscillations in the Michelson-Morley experiment, where a light beam is reflected back and forth relatively to earth motion. And that harmonic equation is the same form as the conventional Interval itself. The Interval is eqn. 9:
I² = t²(c² - u²) = t'²(c² - u'²) =
I² = (tc)² - x² = (tc)² - x'².
The second version of the Interval merely multiplies time by velocity to put the equation in terms of distances. This is the same form as the simpler special case of assuming times, t and t' to be equal, so that the Interval is expressed merely as velocities. But what is done formally in terms just of velocities, could be done equally well in terms of distances.
The values of table 1 are the same for equations 8 and 2, the latter including a first order term in I, which cancels in the working of a geometric mean of the positive and negative solutions of the quadratic formula. This term is analgous to the first order term that physicly damps a simple harmonic wave's full oscillation (given by a second-order term, being a square of I) to successively decrease the crests and trofs of the wave.
For taking the geometric mean, the only difference between eqn. 2 and 8 is that in the former, the first order term's coefficient, c, is explicit but cancels, whereas in the latter, the first order term coefficient is already zero. In eqn. 2, the range is in terms of the limits, positive and negative {c² - I²}^1/2 about c. But in eqn. 8, the limits are effectively just positive and negative I, after cancelings of c.
When the geometric mean is taken of the positive and negative solutions of a quadratic equation, any first order term is automaticly canceled in the working. That is why it does not matter whether the Interval is kept in its conventional form, analgous to a simple harmonic wave form, or whether it is extended to a quadratic equation, analgous to a damped wave form, for the purpose of constructing table 1.
It would be possible to impose the conventional form of the Interval on eqn. 2. such that, say:
(c - I')² = (c² - Y²). Any velocity in the Interval is implicitly a vector or a possible composite of other velocities, such that, for example: Y² = (2I'c - I'²). Note the significance of this is that velocities +Y and -Y, considered as the limits of a statistical range of velocities, are really the composite limits +(2I'c - I'²)^1/2 and -(2I'c - I'²)^1/2. Composite, that is, with a damping term, 2I'c. (Some waves may be amplified rather than damped, so I suppose their equations may have an amplifying term.)
The point about these particular substitutions in the Interval, that is including the damping term to the simple wave form of the conventional Interval, is that the Interval considered as a geometric mean velocity (or more generally, a geometric mean distance) remains directionly the same.
This reminds of the mathemeatical jargon of "eigenvectors" and eigenvalues". "Eigen" is German for "the same". This owes to German mathematicians, who discovered that equations were more easily solved if they could be put in terms of vectors, whose directions remained the same, acting as a fixed co-ordinate system to make comparisons possible. A vector consists of both a direction and a magnitude. The eigenvalues bring-in the magnitudes of the eigenvectors.
In this way, the Interval may be considered as a geometric mean that remains directionly the same (like an eigenvector) whether the limits that it averages are "simple harmonic" like the conventional Interval form or include a "damping" term. The statistical limits are akin to eigenvalues for more or less damped or undamped wave magnitudes.
One of my early web pages is a commentary on W W Sawyer's popular explanation of then Modern Mathematics. I was trying, with only partial success, to get a better understanding. His book was eventually helpful in a way that the textbooks were not. But there was no smooth transition in understanding for me. Sawyer did leave holes in his explanation, which he seemed to expect the bright pupils to be able to fill in.
So, I never really got a proper grasp of the basics of the subject. But one of the requirements for converting variables into their eigenvectors, was that they must not have repeated roots. My basic approach to a statistical treatment of Relativity equations is to consider the Fitzgerald-Lorentz contraction or more generally the Minkowski Interval in terms of observed velocities, of an event, as light velocity plus or minus some locally observed velocity. These are only repeated roots, as far as the velocity's magnitude is concerned, not its direction, which opposite signs show as opposite directions.
Actually, it is possible to reproduce, from a damped wave form of the Interval, the very form given by Sawyer as an example, namely the Fibonacci series, graphed with its axes conveniently transformed. Tho Sawyer does not say so, this transformation is along the axis of the Fibonacci series that brings out its shape as an approximation to a damped wave. (This is discussed on an earlier web page.)
To repeat this result in the context of a "damped" Interval, on this occasion, allowing for different local times between two observers, O and O', and including the damping term, tcx' for observer O', in eqn. 10:
I² = (tc)² - x² = (t'c)² + t'cx' - x'².
It follows:
x'² - t'cx' - x² = 0.
Using the quadratic formula:
x' = {t'c ±({-t'c}² + 4x²)^1/2}/2.
A particular case gets the algebraic expression for the Fibonacci series:
let x = t'c. This means that observer, O, is measuring a light speed event,
unless the local time, t, of O is greater than the time, t', of observer O'.
Observer, O, would be measuring a faster-than-light or tachyonic event
(theoreticly possible in Quantum ElectroDynamics for sub-atomicly extreme
short ranges) if time t was less than time t'.
In the particular case:
x'/t'c = {1 ±(5)^1/2}/2 ~ 1.62 or -.62.
These are the approximate numbers associated, to ever closer approximation, with ratios of successive terms of the Fibonacci series.
Another version of this result could be achieved with a more heavily damped term, -3tcx', instead of just tcx'.
Then:
x'/t'c = {-3 ±(5)^1/2}/2.
Therefore: x'/t'c + 2 = {1 ±(5)^1/2}/2.
In the standard texts, quadratic equations are classed, to put it in physical terms, by the degree of damping. That is to say by the weight of the damping term. Given the quadratic equation 11:
ax² + bx + c = 0.
And the formula for solving it, (12):
x = {-b ±{(b² - 4ac}^1/2}/2a.
Then it is often useful to classify quadratic equations as to whether: b² > 4ac; b² = 4ac; b² < 4ac. For instance, when first and second order terms, in so-called second-order linear differential equations, can be related thru an exponential rate of change, they may be compared to this algebraic form, the so-called auxiliary quadratic equation. Then the weight of the damping term given by the magnitude of the coefficient, b, signifies: heavy; critical; light damping, respectively.
In my treatment, on this page, for what it is worth, the threefold distinction makes no difference.
It turns out, that a statistical treatment of a quadratic equation has matching terms for a second order linear equation. The second derivative, giving an ultimate ratio of change in a change, is matched by a geometric mean. The coefficient of a first order derivative, giving an ultimate ratio of change, is matched by the arithmetic mean. And the constant (no change) or zero-order derivative is matched by a range limit.
Turning the jargon around, this would make, a geometric mean a second order mean or average; an arithmetic mean a first order average; and the range limit a zero-order average.
Take a modified form of equation 11, namely the homogenous equation 13:
I'² - (b/a)II' - (c/a)I² = 0.
This could solve statisticly in terms of either of the localised Intervals of observer O' or O. Solving for I', this implies treating I' as the variable and I as the constant, tho it could equally well be the other way round. Compare with the nearly similar form of eqn. 3 and its solution in eqn. 4: if I' is the variable, then a statistical nomenclature would solve for I' in terms of the geometric mean. The coefficient, (b/2a)I, would in effect be an arithmetic mean. And (c/a)I² would be a positive or negative amplitude about the arithmetic mean as an equilibrium (or level between crests and trofs).
This would remain true whatever change in the mid-term coefficient, that is whatever the arithmetic mean is set at. If the mid term coefficient or arithmetic mean is zero, then there is no mid-term and the amplitude (a magnitude in the positive or negative direction, like crest and trof) is found directly as plus or minus the square root of (c/a)I². That is: ±I(c/a)^1/2.
If the arithmetic mean is non-zero, then the amplitude remains the same,
tho the quadratic formula always modifies the positve and negative limits to
cancel the non-zero value of the arithmetic mean, when the geometric mean is
found.
One might say that the amplitude is conserved thru changes or translations in
the arithmetic mean and its equal and opposite limits. A change that leaves
something (the amplitude) the same is called a symmetry transformation.
I started this (and other) pages with the Interval, considered as a geometric mean. But I have gone on to consider the geometric mean in terms of only one of the observers' measurements of distance (or more particularly velocity).
The Interval is a measure, in the dimensions of distance, common to all observers. (It is often said to be a four-dimensional measure of space-time.) Physicists use the jargon "global invariance." I'm not sure physicists use the term for a common measure like the Interval. But that is my translation of the term for my present purposes.
The corresponding term, physicists use, is a "local invariance." And I
adapt that term to a relativistic local observer's geometric mean Interval,
that is the average Interval representing a range of Intervals within the
limits of another observer's measurements.
The Interval shows that observers give different space and time measures
from their different co-ordinate systems, which are merely out of phase with
each other. Their only difference is an angle of rotation between the
different co-ordinate systems, about a common origin. (This is shown on my
web page explaining the Minkowski Interval.)
The jargon describes the Interval as space-time rotational symmetry.
The Interval formula gives a global invariance of distance measures, in
the sense of the same distance measure, of a given event, by every local
observer, tho their local space and time measures differ in themselves.
The Interval is also, I have claimed, a geometric mean distance. It might be
called a global geometric mean distance.
The Interval, as a global invariance, is distinct from a local invariance, in terms of local geometric mean Intervals, introduced on this page. (I have to emphasise this, if only not to confuse the reader into thinking that I am just relating already accepted physics. This is not the case. For better or worse, this is my own idea of localised geometric mean Intervals as a local invariance in special relativity.)
The Interval, I, as global geometric mean is statisticly reduced to local
geometric means, when one observer is assigned a local Interval in terms of
limits of a range, and the other observer is thereby assigned another local
Interval to be solved in terms of the average of those limits.
Thru these statistical manipulations, the magnitude of the local Intervals
remains the same as the original global Interval but the directions do not:
they become localised.
That is to say the latter local Interval is the variable to be solved in terms of the former local Interval as the constant in a quadratic equation. The quadratic equation solves in terms of two roots, positive or negative. These roots could be treated as limits of a range, like a wave oscillation, that can be averaged in terms of a geometric mean.
The conventional Interval effectively defines each observer's local measurements of some object's speed and direction with and against the speed of light - like an oscillation. (Multiplying by each observer's local time measure gives distance dimensions.)
The conventional Interval is like a quadratic equation without a first order term. Even if the Interval were modified to include such a term, it would not make any difference to a geometric mean local Interval solution. There would be no problem in so modifying the Interval, because local distances are vectors or composites. (It might be more technicly correct to speak of vector distances as displacements, just as it is more correct to rename speeds as velocities, when a direction, as well as a magnitude, is implied.)
The first order term is analgous to a damping term in a wave equation that reduces the amplitude of the wave. The damping term would not change the geometric mean of the oscillation, it would just change the range limits of the oscillation. It correspondingly changes the value, from zero, of what the limits range about, and so brings back the same measurement or restores the symmetry.
If there is a local invariance of geometric mean Interval, in special
relativity, what does it mean?
The Interval's global invariance of geometric mean Interval distance (or
displacement), as I have characterised it, is despite any rotations of local
co-ordinate systems of measurement. The change of angle, or phase, between
these local co-ordinate systems, leaves the conventional Interval the same or
invariant. A change or transformation, that leaves something the same, is
called a symmetry transformation.
The Interval is globally rotation-invariant or phase-invariant. The Interval
is the same measure, it has symmetry, despite turning round the local
observers' different space and time co-ordinate systems - like a sphere looks
the same from turning about its center.
How then is a local geometric mean Interval displacement invariant?
It can remain unchanged with respect to amplitude. The amplitude is the
height of a wave or corresponding depth of a trof. Such a wave can be plotted
from the rotation of a circle, whose radius is the same length as the
amplitude.
So, I suggest that just as special relativity is globally rotation, or phase,
invariant, it might also be considered as locally amplitude, or radius,
invariant.
This seems logical, because if one changing property of a wave, phase, leaves a given law unchanged, why should not the other property, amplitude? In other words, if one property of a circle, rotation, may leave some law invariant, why should not the other property, radius?
A symmetry operation is one that leaves something it works on, unchanged. It displays an invariance in the world, which implies conservation of something. (Noether's theorem derives conservation laws from symmetry.)
This page does not attempt to survey that subject. I merely advance with some diffidence, my own ideas on this in relation to my discussion of special relativity in terms of global and local invariance, as I have adapted these terms.
My starting point is the Minkowski Interval illustrated by the Michelson-Morley experiment. The Interval is treated as a geometric mean of each of the reflected journeys of the split light beam. As explained before, the reflected light beam, with and against earth motion, is in relative acceleration to it.
This implies, in principle, Einstein's equivalence principle of acceleration to gravity. For the discussion to be relevant, the gravitational effect on the light beam has to be experimently observable.
A light beam has constant speed, so it will not be slowed down by moving away from a massive body (planet or star) or speeded-up by moving with the gravitational mass. But that constant light speed is a product of wavelength times frequency. Moving away will slow the frequency and lengthen the wavelength, which means an observable red-shift of the wave-length.
A good analogy is the difference between Mr and Mrs Lincoln - "the long and the short of it" - walking together. Tall Abe Lincoln walking with long slow strides uses less energy than short Mrs Lincoln moving with short quick strides to keep at the same velocity.
As energy is equivalent to mass, so the light beam moving with the star's gravitational attraction, gains energy and therefore mass, tho light has no rest mass. Therefore, it also gains momentum, which is mass times velocity, tho light has constant velocity.
Therefore, when the light is reflected with respect to moving with or away from a massive enough body, there must be a change in momentum.
Besides Special Relativity's combined conservation law of mass-energy, it is standard knowledge, to use the jargon, that space translation symmetry implies conservation of linear momentum. There is also rotation symmetry for angular momentum and time symmetry for conservation of energy. As mentioned, the Interval is based on a space-time rotation symmetry.
I'm not going to try to unravel all those conservation laws and their symmetries, in the context of this discussion. But maybe I can make a limited point without going too far wrong. It seems to me that the Minkowski Interval, as interpreted in terms of the Michelson-Morley experiment, at least implies a conservation of change in linear momentum. This is a second order change, a change in velocity times mass, implying an acceleration.
The only reason this second order change is not identified with Newtonian force, defined as mass times acceleration, is that mass, or its energy equivalent (derived by special relativity), does not remain constant, but increases, for objects approaching light speed.
My argument is that the Interval is effectively a geometric mean, the
appropriate average for acceleration of the reflected light beam, in its
two-way journey with and against great mass of planet or star.
Suppose, however, that change in linear momentum is not conserved in this
Michelson-Morley version of the Interval. Then it would be permissible to use
an average other than the geometric mean, which does "conserve" the
acceleration and the change in momentum to the reflected light beam.
This is precisely what Michelson and Morley did in their original calculation of the two-way journey of the Earth-longitudinal light beam. They used the arithmetic mean, which does not conserve the acceleration and therefore (in principle if not measurable) change in momentum of the beam. The arithmetic mean is only for measuring constant change or velocity, not change in velocity, induced by the reflection.
But, in a previous web page (Lorentz transformation non-solutions of the Michelson-Morley calculation), I showed that the Michelson-Morley calculation with the arithmetic mean is inconsistent with the equations of special relativity. Therefore, it follows that the equations of special relativity depend for their consistency on conservation of change in linear momentum.
The Interval, invariant for all space-time rotations or phases, of local co-ordinate systems, may be considered as globally conserving a change in linear momentum, as well as its mentioned relevance to other conservation laws.
In contrast to the Interval as a global geometric mean, a local geometric
mean Interval is invariant with respect to changes in radius or amplitude of
another observer's local Interval measures.
So, as well as rotation or phase symmetry for a globally conserved law, as of
a change in linear momentum, there may be a radius or amplitude symmetry for
a locally conserved law.
I talked about a geometric mean "conserving" acceleration and an
arithmetic mean not doing so. This may be a questionable use of the term
conservation as it is meant in physics. It might be argued that a better way
of describing this is as a logical question of the right and wrong use of
averages for a given type of distribution.
Tho, I know from experience of the logic of voting choice that the right
logic does "conserve" votes (from being wasted) in an obviously meaningful
sense of the word.
Another short-coming of this page is that it only goes outside Special Relativity in terms of a change in the direction of the Interval and not a change in its magnitude. (Tho, velocities contained in the Interval might also change in magnitude with compensating changes in local times.)
Wholly inadequately, I know, I have sought before to convey the Interval in terms of a binomial expansion of the arithmetic mean and the geometric mean, so that the first term is pure AM and the last, pure GM, with a graduation of more or less one to the other, in the intervening terms of the expansion. These gradations of averages represent more or less flat to curved geometries. The Minkowki flat space-time geometry comes near the start of the series. Geometries of extreme curvature would be represented by more or less purely geometric means.
This web page does not pretend to be an authoritative account of symmetry and conservation law. Victor J Stenger, in The Comprehensible Cosmos, recently (2006) supplies that very well. I thought I understood some of the earlier part of this popular exposition fairly well at the time but had long since forgotten even reading it.
Heinz Pagels' two popularisations, regardable as modern classics: Perfect Symmetry (1982); The Cosmic Code (1985).
Richard Feynman: The Character of Physical Laws. (1965).
Richard Lung.
19, 20 & 24 December 2009.