{(ct - rt)(ct + rt)}^1/2 =
{(ct' - r't')(ct' + r't)}^1/2Lorentz transformations of geometric means.
The Interval as the commonly observed geometric mean.
By simple algebra, the Lorentz transformations can be re-stated as transformations between geometric means of time and of space over related ranges of velocities.
Take the Lorentz transformations of time. These are two complementary formulas by which two observers transform or translate their local times (t' and t), which will differ significantly with respect to an observed event at velocities significantly approaching the speed of light.
The observers respective velocities are u' and u. In classical physics of the Galilean relativity principle, their relative velocity, v, is simple addition or subtraction of their two velocities. Coming to measure velocities of a significant fraction of light speed, a revised formula was needed to accomodate the fact that no combination of a velocity with light speed could ever exceed light speed, c.
If light speed was set at unit distance traveled per period of time, (c = 1) then a light beam carrier itself traveling at one-quarter that distance per time period, could no-how effect a combined speed of one and one-quarter, or any speed in excess of light.
The Lorentz transformations of velocities are usually derived from combining the transformations for space and for time. First, the time transformations, numbering the first equation, (1):
t = t'(1 - u'.v/c²)/F
where F is the so-called contraction factor:
(1 - v²/c²)^1/2.
Conversely, equation (2):
t' = t(1 + u.v/c²)/F.
The times, t' and t, can be shown as geometric means, that transform into each other, by combining their converse equations. An average implies a range of items or values that it is the average of. Therefore, the trick is to express the Lorentz forms as such ranges.
Expressing the times, t and t', as a ratio means that the contraction factor, F, in both their equations, can cancel. Hence (3):
t/t' = t'(1 - u'.v/c²)/t(1 + u.v/c²).
Therefore (4),
t²/t'² = (1 - u'.v/c² )/(1 + u.v/c²)
= {1 - [(u'.v)^1/2]/c}{1 + [(u'.v)^1/2]/c}/{1 - i[(u.v)^1/2]/c}{1 + i[(u.v)^1/2]/c}.
The numerator and the denominator of the equation have both been factorised
in terms of one plus and one minus the ratio of [(u'.v)^1/2]/c and
i[(u.v)^1/2]/c, respectively.
In the latter case, i means the square root of minus one, to make possible the
factorisation of the denominator.
If we were taking an arithmetic mean of these two ranges plus or minus one,
then their average would be estimated in both cases to be just one. This
assumes a constant ("arithmetic") increase in values from the given lowest to
highest terms in the range. The arithmetic mean is calculated by adding these
two terms and dividing by two, resulting in an answer of one.
This taking the arithmetic mean also works for the denominator even tho, i, the
square root of minus one is involved, because the two imaginary terms cancel
each other out.
In this instance, the arithmetic mean is not a suitable average to take,
because we are not dealing with constant increases in velocity. Instead, the
Lorentz transformations are adapted to take into account geometric
distributions, one way or another.
The cumulative decrease in the rate of velocity increase of an object,
corresponds to its cumulative increase in mass, as light speed is approached. A
geometric mean measures the average distribution of high velocities as they
approach light speed.
The geometric mean can be found by multiplying the end values of a distribution range and taking their square root. In equation (4), of t²/t'², the numerator and denominator are both multiplications of end values: the two factors in curly brackets are the lowest and the highest velocities. Hence, the square root of the numerator gives a geometric mean, as does the square root of the denominator. But these respectively, are equal to t and t', which are, therefore, geometric mean times.
That is (5):
t/t' = [(1 - u'.v/c²)/(1 + u.v/c²)]^1/2.
Similarly, for Lorentz transformation of geometric mean distances.
(6):
x = t'( u' - v )/F and x' = t( u + v)/F.
(7):
x/x' = t'(u' - v)/t(u + v) = ut/u't' =
(8):
{(u' - v)/(u + v)}[(1 + u.v/c²)/(1 - u'.v/c²)]^1/2.
Notice that the geometric means of the times, t and t', are inverted for the distances, x and x'. For example, the geometric mean that appeared in the numerator, corresponding to t, in the ratio t/t', now appears in the denominator corresponding to x', in the ratio x/x'. Tho, the geometric means of distances, x and x' (as times multiplied by velocities) also have the coefficent factors (u'-v) and (u+v).
It is possible to get rid of these coefficient factors in the ratio x/x' for an equivalent form, as follows.
From the ratios of the Lorentz time transformations and velocity transformations, (9):
t²/t'² = u'(u' - v)/u(u + v).
Therefore (10):
t/t' = {u'(u' - v)/u(u + v)}^1/2.
And (11):
x/x' = ut/u't' = [{(u' - v)u}/{(u + v)u'}^1/2.
= [{(u' - v)/u'}/{(u + v)/u}]^1/2
= [{(1 - v/u'}/{(1 + v/u}]^1/2
(12):
= {(1 - {v/u'}^1/2)(1 + {v/u'}^1/2)}/ {(1 + i{v/u}^1/2)(1 - i{v/u}^1/2)}.
The above expression for x/x' is a simple ratio of a geometric mean in the
numerator and in the denominator.
Thus, a ratio of velocities, u/u', could be expressed as the x/x' ratio of
distance geometric means, divided by the ratio of time geometric means, t/t'.
That is, u/u' equals equation (12) divided by equation (5).
When the Lorentz transformations were treated as ratios of one to another observers values of time (or distance) this canceled the contraction factor found in both observers versions of the Lorentz transformation for time (or distance). But the contraction factor itself has the form of a geometric mean. This first alerted me to the possibility of a statistical basis for special relativity.
This procedure "simply" gives the Lorentz transformations as of geometric means determined by their observers related but different ranges of observation of the same event. The ratio shows these observers ranges as geometric series distributions.
The contraction factor, F, comes into our present reckoning, when a special case for the Lorentz transformations of time is considered. That is when u' = v and therefore u = 0.
From equation (1), t = t'(1 - u'.v/c²)/F. When u' = v, t = t'( 1 - v²/c² )/F. Therefore, t = t'F, because F is the square root of (1 - v²/c²). Taking the ratio t/t' = F/1, if t is taken to be F as geometric mean, then the value of t' = 1.
For equation (2), t' = t(1 + u.v/c²)/F, reduces to t' = t(1 + 0)/F when u = 0. Then the ratio of equations (1) to (2) for t/t' can no longer be expressed as the ratios of two geometric means, but as only one geometric mean, in the familiar guise of the contraction factor.
If the Lorentz transformations are geometric means, then they must be means to distributions. A mean represents a dispersion of values about it. Each Lorentz transformation can be re-written in terms of the lower and upper bounds of a dispersion, by which the mean or average is calculated.
Most simply, subtract the geometric mean lower bound (value) from its upper bound. This is equivalent to adding the dispersion above the average, to the dispersion below the average, for the total dispersion.
Take equation (4) for the ratio of two observers times, t/t', with regard to an event, and equation (12) for those observers distance ratios, x/x', with regard to the same event.
There are four dispersion calculations to be made for the two observers
times and distances.
With respect to the geometric mean time t, the numerator of equation (4) gives the dispersion in terms of the difference between the factors. That is equation (13):
{ 1 + [(u'.v)^1/2]/c} - { 1 - [(u'.v)^1/2]/c} = 2[(u'.v)^1/2]/c.
Similarly, the dispersion with respect to t' is: 2i[(u.v)^1/2]/c.
When the velocities are small compared to light speed, the dispersions are too small to be noticed.
Not so easy to interpret are the dispersions with respect to x and x', respectively:
2(v/u')^1/2 and 2i(v/u)^1/2.
All four Lorentz transformations, for the respective time and distance measures, of two observers in constant relative motion, have the contraction factor in their denominators. For small relative velocities, v, this factor approaches unity, which doesn't affect the above dispersions calculated, from the numerators.
For a special case of the Lorentz transformations, one, of the two observers, measures his velocity, u, as zero. This makes the other observers velocity, u' equal to v, as a measured difference in velocities between the two observers.
Taking equation (1):
t/t' = (1 - u'.v/c²)/(1 -v²/c²)^1/2.
Given u = 0 and u' = v, we have equation (14):
t/t' = {(1 - v²/c²)^1/2}/1
= (1 - v²/c²)^1/2.
This is the so-called "contraction factor," being a contracting factor on one of the observers measurements, namely time t', in this example, in order that it might correspond with other observer measure of time, t. That is the point of the transformation.
Here, the effect is to reduce a Lorentz transformation of geometric mean times to the contraction factor as geometric mean, in the numerator, and to unity, in the denominator.
The denominator no longer is a mean, because a mean must have a range, which it is the mean or average of. It is no more than a zero range mean. However, the contraction factor is interpretable as a geometric mean, considered as the square root of a lower bound (1 - v/c) multiplied by an upper bound (1 + v/c).
The addition of dispersions to the contraction factor, as geometric mean, is most simply given by subtracting the upper from the lower range bound, in equation (15):
{(1 + v/c) - (1 - v/c) = 2v/c.
This dispersion is regulated by the velocity, v, that observers measure relative to each other, and the speed of light, c. In classical mechanics, v was very small compared to c, so there is no apparent dispersion. It is still there in classical physics but it appeared not to be, because it was too small to be measured.
On this reasoning, classical physics is not more definite and precise than statistics, as was assumed. Rather it is a less than precise approximation to statistical measurement. (As is well known, this is the quantum physics view that gradually prevailed in the twentieth century.)
Neither does special relativity belong to some supposedly ultra-statistical precision of classical physics. Rather, my statistical view-point is that special relativity is precisely the condition of statistical dispersion becoming apparent.
Much the same considerations apply to mass, as time. For a steady increase in speed of a body, significantly towards light speed, the mass of the body increases geometricly. The Lorentz transformations of energy and momentum are similar in form to those for time and distance and can be likewise re-formulated in terms of geometric mean transformations.
For a steady increase in energy applied to a body approaching light speed, the velocity only takes on a geometricly decreasing increase. One such event should only represent one such geometric distribution of velocities. Why isnt its measurement represented by only one geometric mean, instead of all the geometric means related by the Lorentz transformations?
This is a statistical re-working of the original question, that Einstein asked of Special Relativity in 1905. Then, the question was why do observers have different space and time measurements of a same event in high energy physics? The answer came in 1911 with the Minkowski Interval. Observers do have a common measurement of the event if it is treated as one combined four-dimensional space-time event.
The Interval is the Pythagoras theorem extended from three dimensions of
space to four dimensions of space-time. As is familiar from school, the theorem
in two spatial dimensions may be given as r² = x² + y². That is the square of a
triangle hypotenuse equals the sum of the squares of the right-angled sides. It
also holds in three spatial dimensions as r² = x² + y² + z². The spatial
dimension of distance equals velocity multiplied by time: ut. So, the theorem
can be re-stated as: (st)² = (ut)² + (vt)² + (wt)².
(Please note: The velocity, v, here is not the relative velocity, v, in the
Lorentz transformations. Tho, the velocities u and u' used here may be
considered the same as the u and u' velocities in the Lorentz transformations
without the extra two spatial dimensions taken into account.)
Pythagoras theorem has the property that the hypotenuse of a triangle, considered, say, as the constant radius, r, of a circle, can swing round the circle, so that the other sides of a right-angled triangle it forms, the x and y axes of a graph, change their lengths, while it stays the same. The same applies for a radius, r, moving in a spheres three dimensions, measured by x, y and z axes.
This means that the square of the radius equals the sum of the squares of its three trigonometric sides not just in one set of co-ordinate positions but in any set of such positions that any observer may arbitrarily take up with respect to the sphere. (The sphere, being symmetrical, gives no indication where the x, y and z co-ordinates should be. And where the radius happens to be corresponds equally arbitrarily to any given event.)
Thus, Pythagoras theorem not only holds for r, with respect to x, y, z, but also with respect to these co-ordinates swung round some angle (say angle Q) from the center or origin of the sphere. The new co-ordinate positions in the sphere may be denoted by indices, x', y', z', so that (17):
r² = x² + y² + z² = x'² + y'² + z'²
= (ut)² + (vt)² + (wt)² = (u't')² + (v't')² + (w't')².
We are now almost at Minkowski Interval, needing just a fourth term to these three. Tradition describes this as time supplying a fourth dimension. But this fourth term, when one looks at it, is actually another distance variable, being the velocity of light, c, multiplied by the time, t or t', depending on the co-ordinate frame of reference. The peculiarity of the fourth term is that the squares of ct or ct' are not added to the other three terms but subtracted.
Usually left as a square of its value, Minkowski Interval, I, another distance term, is thus (18):
I² = (ct)² - (ut)² + (vt)² + (wt)² = (ct')² - (u't')² + (v't')² +
(w't')².
(As mass is proportional to time, in Special Relativity, an analgous form for the Interval is derived in terms of energy and momentum.)
From the point of view of a statistical interpretation of special relativity, this unexpected negative is providential, because it enables the Interval to be simply re-stated as a geometric mean. This is simply seen by refering the three spatial dimensions to their combined equivalent in magnitude and direction, that is to their vector, sometimes called three-vector, which is here the radius, r ( = st = s't') sometimes called the radius vector.
Thus (19):
I² = (ct)² - r²t² = (ct')² - r'²t'².
This can be re-stated in the form of a geometric mean (20):
I = {(ct - rt)(ct + rt)}^1/2 =
{(ct' - r't')(ct' + r't)}^1/2
Or (20b):
(I/c) = {(t - rt/c)(t +
rt/c)}^1/2 = {(t' - r't'/c)(t' +
r't/c)}^1/2
This answers our question about any given observed geometric distribution of velocities or masses having the same geometric mean for different observational frames of reference (in uniform velocity with each other, as far as special relativity is concerned). The Interval provides that commonly observed geometric mean.
The dispersion about the geometric mean Interval, as for the Lorentz transformations, is a difference of an upper bound from a lower bound of dispersion, namely equation (21):
(ct + rt) - (ct - rt) = 2rt, for the dispersion to Interval, I
Or, for the dispersion to I/c:
(t + rt/c) - (t - rt/c) = 2rt/c.
The situation in classical mechanics is that a low velocity and time, or a short distance, with respect to light creates no observable dispersion.
The Interval, expressed as I/c, or a fraction of light speed, is itself negligible at low values. Thus, the Interval, conceived as a unitary space-time concept, does not appear in classical physics.
With the Interval, a difference between upper and lower dispersions, about the geometric mean, also can be taken.
The dispersion above the Interval, I, is:
(ct + rt) - I.
The dispersion below the Interval is:
I - (ct - rt).
Subtracting the two dispersions:
(ct + rt) - I - {I - (ct - rt)} = 2(ct - I).
In terms of dispersion about I/c, this difference, of upper and lower limits, means that when velocities become small compared to light speed, the Interval, I, approaches ct and the dispersal difference approaches zero.
Why bother to take a dispersion difference as well as a dispersion sum total? The two options may correspond to relative motion, effectively adding velocities by moving in contrary directions or subtracting velocities moving in the same direction.
This is a change from the conventional meaning of relative motion as between different observers local velocities. The relative motion, of above average dispersions, adds observer velocities to light speed, while below average dispersions subtract those observer velocities from light speed.
It may be that Galilean relative motion, as simple additions or subtractions of velocities is statisticly re-invented in simple additions and subtractions of motion dispersions.
Richard Lung.
13 december 2005;
revised, may 2007;
revised & corrected, june 2015.