{ 1 - (u'.v)/c}{ 1 + (u'.v)/c})/( { 1 - i(u.v)/c}{ 1 + i(u.v)/c}).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's lower bound ( value ) from its upper bound.
From the page Lorentz Transformations of Geometric Means, take equation (5) ( renamed (1) here ) for the ratio of two observers' times, t/t', with regard to an event, and equation (12) ( renamed (2) here ) for those observers' distance ratios, x/x', with regard to the same event.
(1):
t/t' = ({ 1 - (u'.v)/c}{ 1 + (u'.v)/c})/( { 1 - i(u.v)/c}{ 1 + i(u.v)/c}).
(2):
x/x' = {( 1 - { v/u'})( 1 + {v/u'})}/ {( 1 + i{ v/u} )( 1 - i{v/u} )}.
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 (1) gives the dispersion in terms of the difference. That is equation (3):
{ 1 + (u'.v)/c} - { 1 - (u'.v)/c} = 2u'.v)/c.
The dispersion with respect to t' is: 2iu.v)/c.
When the velocities are small compared to light speed, the dispersion are too small to be noticed.
The dispersions with respect to x and x' are, respectively:
2{ v/u'} and 2i{ v/u}.
For a special case of the Lorentz transformations, one, of the two observers, measures his velocity, u, as zero. This makes the other observer's velocity, u' equal to v, as a measured difference in velocities between the two observers. We could substitute these velocity values in equation (1). But we get the same result using the shorter, familiar form of the Lorentz transformation for time, which may be written as equation (4):
t/t' = ( 1 - u'.v/c² )/ ( 1 -
v²/c² ).
Given u = 0 and u' = v, we have equation (5):
t/t' = ( 1 - v²/c² )/1.
The right side numerator ( corresponding to t, on the left of equation (5) ) is the so-called "contraction factor," being a contracting factor on one of the observer's measurements, namely time t', in this example, in order that it might correspond with other observer's measure of time, t. That is the point of the transformation.
Here, the effect ( as would have been seen more clearly using equation (1) ) 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 average of. 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 dispersion about the contraction factor, as geometric mean, is given by equation (6):
{( 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, the statistical view-point of this page is that special relativity is precisely the condition of statistical dispersion becoming apparent. As velocity v approaches light speed c, the contraction factor approaches zero, and therefore a dispersion value of one about observed geometric mean time, t.
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.
We took the total dispersion about a geometric mean, when it is a Lorentz transformation. This implies an addition, of the dispersion above the geometric mean, to the dispersion below the geometric mean. One can also subtract the upper from the lower dispersion.
The upper dispersion, for t, the numerator in equation (1), is:
{ 1 + (u'.v)/c} - ({ 1 - (u'.v)/c}{ 1 + (u'.v)/c}).
The lower dispersion is:
({ 1 - (u'.v)/c}{ 1 + (u'.v)/c}) - { 1 - (u'.v)/c}.
Subtracting the upper and lower dispersions, gives the dispersion difference, with respect to geometric mean time t:
2(1 - {1 - (u'.v/c²)}).
Similar results can be obtained for dispersion differences with respect to
t', x and x'.
The difference between the upper and lower dispersions only becomes
significant for relativistic velocities approaching light speed. For low
velocities, only a tiny fraction of light speed, the difference between the
upper and lower dispersions becomes zero. An average is of a more or less
dispersed distribution. When no dispersion is apparent, as in classical
physics for low velocities, it is not apparent that observers' measures of
distances and times are averages at all.
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 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 Minkowski's 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 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's 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: I have rather clumsily given the middle velocity a capital "V"
to avoid confusion with the relative velocity v in the Lorentz
transformations. The velocities u and u' used here are the same as the u and
u' velocities in the 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 sphere's 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 (7):
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's 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.
Minkowski's Interval, I, another distance term, is thus (8):
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 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, which is here the radius, r ( = st = s't' ) sometimes called the radius vector.
Thus (9):
I² = (ct)² - r²t² = (ct')² - r'²t'².
This can be re-stated in the form of a geometric mean (10):
I = {( ct - rt )( ct + rt )}
= {( ct' - r't' )( ct' + r't
)}
Or (10b):
I/c = {( t - rt/c )( t + rt/c
)} = {( t' - r't'/c )( t' +
r't/c )}
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 (11):
( ct + rt ) - ( ct - rt ) = 2rt.
Or:
( 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.
However, reversing the signature of the Interval changes the dispersion from 2rt to 2ct:
( rt + ct ) - ( rt - ct ) = 2ct.
One might say that 2rt is a spatial dispersion, as r can be a vector of velocity in three dimensions. Whereas, ct is conventionly assigned as the temporal dimension, and so here 2ct might be characterised as a temporal dispersion.
Moreover, the Interval, expressed in (10b) 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.
As for the Lorentz transformations, so 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) = 2(ct - {(ct)² - r²t²}.
Here again when the distance rt becomes of the relatively small classical order of value, the dispersion difference tends to zero.
(If the Interval's signature is reversed, the dispersion difference becomes:
( rt + ct ) - I - { I - ( rt - ct )} = 2(rt - I) = 2(rt - {(rt)² - c²t²}.
In this case a relatively low velocity, r, leaves a dispersion difference seemingly of about -2ict. This imaginary number signifies an operation for a certain amount of rotation in a circle. And the Interval can be expressed as a circular function. There is more about this on my web page about Mach's principle etc.)
Special relativity may register, in disguised form, the statistics implicit but unmeasurable in classical mechanics. This contradicts the still current view that special relativity belongs to a classical physics, where statistics was only considered a second best approximation.
Finally, why bother to take a dispersion difference as well as a dispersion sum total? The two options may correspond to relative motion, respectively moving in the same direction or in the opposite 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 to below average dispersions can apply to any observer's velocity, distinct from any other observers' measures.
It may be that this relative motion, as simple additions or subtractions of any given observer's dispersions, is a statistical re-invention of Galileo's relative motion as simple additions and subtractions of different observers' velocities.
Richard Lung.
13 december 2005; revised, 24 & 25 may2007.