I'd better mention again where my ideas come from. I once studied social science, including scientific method, which I saw could be used to determine voting method, one of those limited, precise problems that scientific method does well. When I was young, I also noticed, in several ways, that special relativity was akin to an electoral method. But not till I was already old, could I give a fairly consistent analogy.
Anyway, in the course of the demoralising attempts at comparisons, I decided that I would have to put special relativity ( SR ) in statistical terms. Because, an election counts a distribution of choice. And I was thinking that SR could be considered as a distribution of choices that all observers are in an equal position to make of measuring an event.
At that stage, I didnt see the deeper senses in which elections and special relativity are intrinsicly statistical procedures. As to SR, I noticed that the contraction factor has the same form as a geometric mean, and to cut a long story short, it turned out that the Lorentz transformations and the Interval can be reformulated as geometric means. In particular, the Interval can be understood as the commonly observed geometric mean.
The geometric mean is a sort of average, a typical or representative item of a range of items or values, that forms a geometric series. And it is also possible to measure deviations from that average. On a previous web page, I call them dispersion formulas, which were applicable to the Lorentz transformations for time and space, and to the Interval. ( They also apply to the dynamic as well as the kinematic versions of SR. )
The interesting result of all these very simple dispersion formulas was that they showed for situations of classical mechanics with low velocities, that there is no significant dispersion about the geometric mean. In other words, if your geometric mean has no apparent range of values to represent, it is not apparent that it is a geometric mean. And that is why classical mechanics appears to be about uniquely determined values. In reality, it is implicitly statistical, the statistics only becoming apparent for high energy physics.
SR postulates a constant maximum speed of light. But a statistical interpretation has a new slant on this. A geometric mean is suitable for measuring a range of values subject to diminishing returns, such as energising increases in the velocity of a massive object. But light, has no rest mass never being at rest, and is not subject to such diminishing returns. So, you would not consider an average speed of light from the point of view of a geometric mean but the more familiar average known as the arithmetic mean.
And the arithmetic mean has the property, which the geometric mean does not have, that for relative velocities equally above and below light speed, they exactly cancel. In other words, light speed considered as an arithmetic mean is a constant speed. So, this statistical interpretation of light speed gives the same result as SR but as an average speed without necessarily postulating that light always moves at the same rate.
This statistical interpretation is consistent with Quantum Electro-dynamics. According to Feynman's popular lectures, on QED, at extremely short ranges, light can move at slower or faster rates but that these very soon average out to the generally observed constant speed.
If SR can be explicitly reformulated as a statistical theory, then so might general relativity in a comparable way. Once again, relative observations might translate into statistical ranges of observations. And, at a first guess, classical gravity would involve the reduction to a mean without a significant range, which therefore was not apparent as a statistical average.
In the nineteenth century, Laplace produced his treatise to estimate ranges of error in astronomy and so get nearer to some definitive measurement. Likewise, the Gaussian curve is also called the error curve. Clerk-Maxwell's statistical theory of molecular motion in the mass was designed as a large scale approximation of molecular collisions, assumed, on their microscopic scale, to obey Newton's laws. Einstein belonged to this tradition of assuming Newton's laws were a definitive kind of law, tho in need of revision. Whereas statistical laws were supposed to be second best approximations. Hence, his famous debates, with Niels Bohr, challenged (unsuccessfully) that statistics gives a complete explanation of quantum mechanics.
Two of Einstein's three famous papers of 1905 are avowedly statistical: the study of Brownian motion, explained by random molecular motion, and the quantum statistics of the photo-electric effect. So, it is especially ironic that his third paper on Special Relativity also seems to have an underlying statistical explanation.
On previous web pages, I have described the Fitzgerald-Lorentz contraction factor as a geometric mean. More than that, it is a geometric mean of a range, which, if averaged by the arithmetic mean, would make the speed of light appear constant.
The contraction factor is the square root of the following: one minus the ratio of relative velocity, v, squared, over light speed, c, squared. Or:
( 1 - v²/c² )^1/2.
( The Google notation for a square root as a fractional power is: ^1/2. )
The contraction factor can be considered a geometric mean, because the terms in the brackets factorise to: ( 1 - v/c)( 1 + v/c ). These factors may represent a range of v/c below and above one. The contraction factor may be defined as the geometric mean velocity, a representative average of this range of velocities.
Suppose you take the arithmetic mean of the two factors of the contraction factor. That means adding (c-v)/c to (c+v)/c, which gives 2c/c, and dividing by two. The arithmetic mean is c/c. Essentially, the contraction factor involves a positive and negative range of velocity, v, about light velocity, c, such that their arithmetic mean is c, whatever the value of v. This makes a constant of c, even if it were a kind of average.
The situation reminds of that mentioned above, in QED, of variations in light speed, at extreme short ranges, that average out over distance to the perceived constant speed.
When the velocity of any normally massive object significantly approaches light speed, its path describes a geometric series, of diminishing returns of increased speed for increased energy of propulsion. That is represented by the contraction factor as a geometric mean which shows that the velocity can never quite reach that of light.
The contraction factor as a geometric mean, with arithmetic mean implications, seems to lend it the quality of an average of averages, when expanded by the binomial theorem, as shown in later sections.
( Appendix 1 speculates wildly on light speed as an average. )
Galileo's relativity principle derived from the experience of someone at rest on a moving boat relative to the bank. In the heyday of the train, we experienced this in the surprise we felt, when sitting in a railway carriage, at the start of a journey, that the station platform appeared to lurch away in the opposite direction to the way we were going.
The point of this odd feeling is that there is no way we can decide whether the train or the platform is the one that is really moving or really at rest. Neither situation is a privileged frame of reference.
The contraction factor was first introduced to explain the unexpected
result of the Michelson-Morley experiment that light always measured at the
same speed.
Consider two observers moving uniformly, at a significant fraction of light
speed, in opposite directions, like passing trains. "Einstein proposed ...no
measurement could determine which train was stationary and which was moving.
That being the case, the equations of electricity and magnetism would have to
appear the same on the two trains, and thus the speed of light must also be
the same." ( Robert Laughlin, "A Different Universe." )
The contraction factor weighted the observers' distance and time measurements of an event, to bring the Michelson-Morley calculation into line with evidence of light always measuring the same speed. This factor was the first adjustment of classical mechanics to what was to become the theory of special relativity.
My web page, Statistical prediction of the Michelson-Morley experiment, gave a geometric mean time (and an analgous geometric mean mass). Following on from that page's approach, the geometric mean velocity is the square root of the product of light speed, c, minus a given velocity, v ( if only of a supposed absolute motion of "the universal ether" ) and of light speed, c, plus velocity, v. That is the geometric mean velocity is: {(c-v)(c+v)}^1/2 = ( c² - v² )^1/2.
This geometric mean velocity averaged the up and down "ether stream" journeys. In practise the Earth's velocity was used supposedly as the "bank" moving relative to that stream. But the point of the above-mentioned page was that the same geometric mean time was found for the cross-stream journeys as for the up and down stream journeys.
Similarly, the same geometric mean velocity can be found for the cross-stream journeys, as for the up and down stream journeys. The cross-stream mean is calculated using Pythagoras' theorem that the triangle hypotenuse side squared equals the sum of the other two squared sides. The Michelson-Morley experiment sends part of a light beam at right angles or cross-ways to the up and down stream direction. Mirrors send both beams back on their return journey.
With regard to the cross journey, a light beam, sent perpendicular to the flow of the stream, is imagined to drift, like a boat, with the flow. This drift across stream is in effect the hypotenuse of a triangle traversed with light speed, c. By the time the beam has reached the far "bank", it is no longer directly opposite its point of origin on the near "bank". The far bank has moved on by a distance covered thru the earth's velocity, v.
The third side of the triangle is the perpendicular distance across stream. The velocity with which the other two sides were covered can tell us, by Pythagoras' theorem, the velocity with which the perpendicular crossing is covered. This is the square root of c squared minus v squared, or ( c² - v² )^1/2.
This perpendicular velocity is the same both ways back and forth. It is
its own geometric mean velocity, as multiplying the same two values and
taking their square root brings us back to where we started.
But this geometric mean velocity for the perpendicular crossing is the same
as for the up and down stream crossing.
The contraction factor is sometimes refered to in inverted form, as on my previous-mentioned page. Here, we may refer to the contraction factor in terms of the geometric mean velocity divided by the light velocity, c. This just turns the form into a ratio, which sometimes may be a more convenient way to do any algebraic working: {( c² - v² )^1/2}/c = (1 - v²/c² )^1/2.
The contraction factor, as a geometric mean, is only the most summary of geometric means. If special relativity is really amenable to a statistical treatment, then one might expect a fuller version of the geometric mean to apply. The contraction factor is in effect a geometric mean that works by multiplying the two end points of a range of values and taking their square root.
Tho I started out with the contraction factor as geometric mean, it turned out that a similar form, that can be found in the Interval, is more central to my argument. And this should be borne in mind, as I shall shift from the contraction factor to it. (It is in terms of a vector, of the three dimensions of space, subtracted from the so-called fourth dimension, that occurs in the formula for the Interval.)
A geometric mean may be taken not only of the extreme values but also of any number of inter-mediate values. Whatever the number of such values, they are all multiplied together. Then a root of the multiple gives the geometric mean. For two values multiplied, the root taken is the square root. For three items, the cubic root of the multiple gives the geometric mean. The number of the root is the number of values (on the range) multiplied.
The geometric mean of four values in a range multiplies them and takes the quartic root, or takes the multiple to the power of one quarter.
Instead of assuming a two-valued range of velocities, say, from ( 1 - v/c ) to ( 1 + v/c ), we assume a four-valued range of velocities.
Where f and g are new velocity terms, let the four-valued range of velocities from least to greatest be: ( 1 - f/c ), ( 1 - g/c ), ( 1 + g/c ), ( 1 + f/c ). Their geometric mean is their multiple to the power of one-quarter ( using again Google's notation for powers ), or:
{( 1 - f/c )( 1 - g/c )( 1 + g/c )( 1 + f/c )}^1/4.
This relates to a form like the contraction factor (but not necessarily
the same): ( 1 - r²/c² )^1/2. Why might we do this?
Taking a geometric mean of more values may give a more detailed result. Nevertheless, when we only calculate with the two end values, we still hope the result is not far out. And in equating the simple end-valued geometric mean to the geometric mean of a fuller range of values, we may assume that the simpler calculation happens to be in accord with fuller calculations.
Hence, equation (1):
{( 1 - f/c )( 1 - g/c )( 1 + g/c )( 1 + f/c )}^1/4 = ( 1 - r²/c² )^1/2.
Raising both sides of the equation by the power of four, we get (2):
( 1 - f²/c² )( 1 - g²/c² ) = ( 1 - r²/c² )².
Therefore (3),
1 - ( f²/c² + g²/c² ) + f²g²/c²c² = 1 - 2r²/c² + r²r²/c²c².
Cancel the ones and multiply thru by c². Then (4):
f²g²/c² - f² - g² = r²r²/c² - 2r².
Suppose r² = fg. Then, r = ( fg )^1/2.
Thus, for velocities significantly approaching light speed, the velocity,
r, is the geometric mean velocity of observers' velocities, considered as a
velocity range from f to g.
For this to hold, then the square of the velocity, r, must also equal an arithmetic mean of the squares of the observers' velocities, f and g.
That is:
r² = ( f² + g² )/2.
An average, that is a geometric mean, with an arithmetic mean element in
it, expands into a distribution of terms that are themselves averages,
including the geometric mean in one term and an element of arithmetic mean in
another term. The contraction factor type geometric mean is an average of
averages.
Expansions involving more than two observers' velocities show a refinement of
this basic feature.
Taking the geometric mean of a range of three observed velocities, u, v and w would involve three multiplied factors. Two new range values, ( 1 - h/c ) and ( 1 + h/c ), would be multiplied by those on the right side of equation (1). This time the geometric mean of the six multiplied values would require a root to the power of one-sixth, for equation (5):
{( 1 - f/c )( 1 - g/c )( 1 - h/c )( 1 + h/c )( 1 + g/c )( 1 + f/c )}^1/6 = ( 1 - r²/c² )^1/2.
This simplifies to (6):
( 1 - f²/c² )( 1 - g²/c² )( 1 - h²/c² ) = ( 1 - r²/c² )^3.
Using the binomial theorem to expand both sides, (7):
1 - ( f² + g² + h² )/c² + ( f²g² + f²h² + g²h² )/c^4 - ( f²g²h² )/c^6
= 1 - 3(r²/c²) + 3(r^4/c^4) - r^6/c^6.
As for the example of two observers, for the three observers' velocities, we assume the terms, on each side of the equation, correspond. Taking from the last term on each side, r^6 = ( f²g²h² ). So, r = ( fgh )^1/3. The velocity, r, is the geometric mean of the three observed velocities.
That assumes the other terms correspond. So, 3(r²/c²) = ( f² + g² + h²
)/c². Here, the square of the velocity, r, is the arithmetic mean of the
squares of the three observed velocities.
It seems significant that the first term, on either side, namely one, is the
arithmetic mean of the two factors that each observer contributes, such as (
1 - f/c ) and ( 1 + f/c ), in the case of observer's velocity measure u.
Likewise, for the other two observers' velocities v and w.
The term corresponding to 3(r^4/c^4), that is ( f²g² + f²h² + g²h² )/c^4, is the closest to the simple geometric mean but it still has some residual arithmetic averaging. Here, the velocity, r, to the fourth power, r^4, is the arithmetic mean of alternately paired multiples of the three observed velocities squared.
Anyone looking at 3(r²/c²) = ( f² + g² + h² )/c² might think that it
involves Pythagoras' theorem in three dimensions of space, with f, g and h,
being velocities in the direction of x, y and z co-ordinates. And r² appears
the hypotenuse squared, moving in a sphere, like the radius vector outcome of
a tug of war between the three velocities at right angles to each other.
(The coefficient 3, not normally there, and constant c² doesnt essentially
alter the relationship.)
( The term "geometric" mean as a kind of average of a "geometric" series is not to be confused with "geometric" as in the geometry of space that yields the likes of Pythagoras' theorem, with its use of an arithmetic sum of squares. )
The three di-mensions, literally twice-measured, can be considered as six "mensions", the six factors on the left side of equation (5), which measure three pairs of values each equally above and below a central value of unity.
These mensions are multiplied together thru all their logicly possible combinations, which fall into classes of unequal numbers of members, thus forming a distribution given by the binomial theorem, as in equation (7). Then, the second term (especially if we multiply out the constant denominator c²) becomes a logicly possible geometry structure, namely Euclid's geometry deriving Pythagoras' theorem.
The first and second terms, combined, extend Euclid's geometry to Minkowski's Interval. That is to go from a three-dimensional geometry of space to the four dimensional geometry of space-time. The first term in the series is identified with the fourth so-called temporal dimension in the Interval connected by a negative sign to the other three dimensions, which are positive in relation to each other.
The first term in the series is unity, which c is sometimes set at, and it becomes c², when that value is multiplied out of the denominator of the second term.
The feature of the Interval is that it is a space-time measure, for light-approaching velocities, on which all observers agree, despite their differing results when their space and time measures are not treated in the Interval's unitary manner.
The difference of the left side of equation (7), from the conventional Interval, is that it only has the first two terms and these are multiplied by an observer's time variable, t. This observer's three velocity co-ordinates are not necessarily the same, in the conventional Interval, usually labeled, u, v, w. The right side of the conventional Interval mirrors the left, except the velocity and time variables are indexed to indicate another observer's differing time and velocity values.
Let one observer's local time, squared, be: t². As the Interval squared, I² is constant for two or more observers, then a second observer must have a local time, t', and differing velocities, u', v', w'. The (Pythagorean) vector of these three velocities may be labeled velocity r', corresponding to a like vector r, for u, v, w. Then (8):
I²/c² = t²{1 - (r²/c²)} = t'²{1 - (r'²/c²)}.
Or:
I² = t²{c² - (u² + v² + w²)} = t'²{c² - u'² + v'² + w'²)}.
At first, I assumed that the velocity terms of the Interval were contained in the binomial expansion of the contraction factor as a geometric mean of more than just two terms, an upper and lower term. Later, I realised the contraction factor (or its square) was only a particular case of the appropriate factor here. The key factor is in the Interval itself, namely a function of the fourth dimension minus the the vector of the three dimensions of space.
This, namely {1 - (r²/c²)}, looks like the square of the contraction factor and in a particular case can be so. In the standard Interval, r is the hypotenuse given by Pythagoras' theorem applying in three dimensions of space, u, v, w. But I have also related r to define the upper and lower limits of range that has an average velocity. This average can be estimated, more fully, using three velocities, f, g, h, to each mark more or less upper and lower limits to the range.
There is nothing special about using three velocities, f, g, h, to mark out a range of velocities. One could use any number. It just so happens that these three can be used to approximate the velocities, u, v, w, on the three dimensions of space. To get the standard Interval, using, u, v, w, to approximate to the Interval-like binomial expansion, using f, g, h, the two last terms of the expansion have to be small enough to be ignored.
Looking at this composite term from equation (7), ( f²g² + f²h² + g²h²
)/c^4, we note that if only one of the velocities, f, g, h, is significant
compared to light speed, c, then none of the three sub-terms in the brackets
can be more than of the order of 1/c². In that case, we assume that this
whole term can be ignored.
A similar argument applies to the fourth term.
Thus, we can treat the standard Interval in two different ways. The
conventional form allows r to be considered as a 3-D vector of u, v, w.
A statistical way is for r to be part of a velocity form, of which the
contraction factor is a particular case, which can be treated as an average
over a fuller range of velocities, f, g, h, than is encompassed by r
alone.
Arithmetic can illustrate how velocities u, v, w and f, g, h can approximate each other. Indeed, it is easy to make two pairs the same, with the third pair approximating each other more closely, the slower the velocities compared to light speed. For instance, if f, g, h, are 1/2, 1/3, and 1/4 of light speed, 1, then u, v, w are 1/2, 1/4, and 1/4. If f, g, h, are 1/2, 1/9, 1/10, then u, v, w are 1/2, (1/150)^1/2 and 1/10. If f, g, h, are 1/2, 1/99, and 1/100, then u, v, w, are 1/2, (17/330000)^1/2, and 1/100.
The velocities are only completely equal when they are all zero, which is a trivial result. The conventional Interval and the statistical form are equal when velocity r is the same in both versions. Suppose that velocity r in the statistical version just is the 3-D vector of f, g and h, so that these three are actually the same as u, v and w. Now we re-label statistical version r as R. In other words,
( 1 - u²/c² )( 1 - v²/c² )( 1 - w²/c² ) = ( 1 - R²/c² )^3.
This is essentially the same as equation (6) except that we now consider f, g, h, in a geometric series, in terms of the relativist observation co-ordinates u, v , w. This adjustment has the interesting result that the standard Interval becomes an approximation, found in the first two terms of the binomial expansion. Hence equation (9):
I²/c² = t²{1 - ( u² + v² + w² )/c² + ( u²v² + u²w² + v²w² )/c^4 - ( u²v²w² )/c^6}
= t'²{1 - ( u'² + v'² + w'² )/c² + ( u'²v'² + u'²w'² + v'²w'² )/c^4 - ( u'²v'²w'² )/c^6}.
We've divided thru by the light speed squared in equation (9), unlike equation (8). Allowing for that, equation (8) looks like a statistical approximation to the first two terms, in the curly brackets, of equation (9).
The famous equation, E = mc^2, is essentially a binomial expansion of the contraction factor, for two locally observed masses of a body, taken to the first two terms.
Mass is proportional to time in the formulas of special relativity. The Interval, which has dimensions of distance, has been comparably derived but by an inverse expansion. It may be that the conventional Interval is an approximation of a statistical expansion, just as the famous energy equation is.
The binomial distribution here is a distribution of logicly possible combinations of mensions. These form kinds of geometrical structure or "geometries" such as Euclid's geometry. Also, another term plus the Euclidean term formed Minkowski's Interval, a further geometry. The distribution clusters about the geometry with the average number of mensions. And its structure is about half arithmetic mean and half geometric mean, considered as an average as well as a geometry.
Different geometries can be considered as different averages. And the binomial theorem, on which the expansion of mensions is based, can be considered as an average of averages, which is also a geometry of geometries. In other words, geometries may be considered as averages, which can be expanded from second order averages or second order geometries.
Special relativity postulates no privileged frames of reference within the context of Minkowski space-time. But Minkowski geometry is itself to some extent a confining frame of reference to observers only in relative velocities to each other, excluding consideration of accelerated reference frames. Minkowski's Interval is a privileged reference frame, in Special Relativity, which only works for the conditions, it exacts for observers, of uniform motion in a straight line of flat space-time.
But Minkowski's Interval can be treated not only in terms of the uniform motion implied in Euclid's geometry but also in terms of non-uniform motion implied in the expansion of a geometric mean, in which Euclidean uniform motion is observed but as an approximation.
This may mean that the conventional Minkowski Interval is an approximation of some statistical expansion, that gives both an Euclidean treatment for uniform motion and a non-Euclidean or geometric mean treatment for non-uniform or accelerated motion.
This treatment may also imply no privileged classes of geometry frames of reference, (Euclid's, Minkowski's, and presumably others), as different geometries (different classes of reference frame) may appear as terms in a random distribution represented by, and expanded from, a geometry of geometries, as an average of averages.
But suppose the binomial distribution that derives the Interval in its first two terms has its conditions changed. Suppose we multiply the left side of equation (7), by the stupendous amount of c^4, for equation (10):
c^4 - c²( u² + v² + w² ) + ( u²v² + u²w² + v²w² ) - ( u²v²w² )/c².
Suppose also that ( u² + v² + w² ) is as close as we like to c². This still allows a great deal of possible variation in the three velocities individually. But it effectively cancels out the first two terms, making zero the Interval, as we know it. This still leaves the subsequent two terms which might be considered as a subsequent Interval.
One of these terms, ( u²v² + u²w² + v²w² ) might be relabeled, if f = uv, g = uw, h = vw, as f² + g² + h², which may be given as equal to vector k². Thus the second two terms may be rendered:
k² - ( f²g²h² )/c².
Multiply this thru by a time, T, squared and we have a new Interval (squared), (I²c²) for two observers, given that the observations of the second observer are indexed. This is equation (11):
I²c² = T²{k² - ( u²v²w² )/c²} = T'²{k'² - ( u'²v'²w'² )/c²}.
We note that the velocity, k, is a vector of velocity pairs multiplied, whereas uvw is a triple multiple. Therefore, the triple multiple divided by the double multiple gives a result of the order of a single velocity. As such, we treat this quotient, uvw/k, as a new velocity, say V. Then equation (11) becomes equation (12):
I²c² = T²k² (1 - {(u²v²w²)/k²}/c²) = T'²k'²(1 - {(u'²v'²w'²)/k'²}/c²).
As V = uvw/k, then k = uvw/V. And Tk = uvwT/V = uvw/A where A stands for acceleration. Re-writing equation (12) as (13):
I²c² = {(uvw)²/A²}(1 - {V²}/c²) = {(u'v'w')²/A'²}(1 - {V'²}/c²).
We note the near resemblance of equations (12) and (13) to the conventional Interval, as in equation (8). Equation (13) may be refered to as an "acceleration Interval", where two observers in relative acceleration can transform each other's co-ordinates to agree their observations of a given event.
This would still be a relativistic equation with the restriction that: uvw = u'v'w', allowing both terms to be canceled. Thus:
{1/A²}(1 - {V²}/c²) = {1/A'²}(1 - {V'²}/c²).
Or (14):
A/(1 - {V²}/c²)^1/2 = A'/(1 - {V'²}/c²)^1/2.
I dont want to try to re-invent General Relativity, which is far beyond me, and which seems right on its own terms. The above formulations may be wrong or misconceived. In particular, when I treated Special Relativity velocities in terms of a geometric series, instead of Pythagoras' theorem, this may have been too simplistic to be valid.
But I suggest that this statistical approach to an acceleration Interval might have the advantage of applying finite mathematics to problems in General Relativity whose laws break down when its equations produce infinities, encountered as singularities in its solutions, such as at the origin of the Big Bang and at the destination of Black Holes.
Instead of a singularity, an infinitely dense point of zero spatial dimensions, statistics may come up with alternative formulations. When the conventional Interval was derived in the first two terms of the binomial distribution, the first term was comparable to the so-called time or fourth dimension, as distinct from the three spatial dimensions of the second term. (Remember, it is velocities we are dealing with on the four dimensions of space-time, not the space and time variables themselves, called, say: x, y, z and t.)
Consider the later two terms of this distribution ( as in equation (7). The third term was in effect treated as another spatial term, leaving the final term, in terms of uvw, to be compared to the temporal or fourth dimension. Suppose this term represents one temporal dimension. Yet, this final term, in uvw, is like a synthetic dimension of three spaces on one ("temporal") dimension. Contrast the conventional Interval's second (Pythagorean) term in the distribution as three analytic dimensions of three distinct velocities.
This final, uvw, term is a function of a geometric mean. This is an
average of a series associated with curvature, the graphic description in
geometry of acceleration or deceleration, characteristic of a geometric
series. General Relativity, dealing with observers in accelerated reference
frames, uses the geometry of curvature.
The third term in the distribution, before the geometric mean term, is the
next closest to a geometric mean, but is not merely a multiple of all the
velocities. Different velocities are partly multiplied and partly added.
So, the binomial distribution of terms progresses from arithmetic mean to
geometric mean, with more or less one or the other mixed in, as the series
progresses. When the binomial theorem is expanded, with more than three
velocities, further inter-mediate terms are introduced, suggesting a more and
more refined distinction between the original contrast of a conventional
Interval of flat space-time and an acceleration Interval of curved
space-time. Tho, at the same time, the expansion, in more and more
velocities, seems to be producing any number of dimensions, indeed an
infinite number in the normal distribution's extreme of the binomial
distribution.
It might be possible to treat these many dimensions as sub-dimensions to three or a few dimensions. Above, we labeled uv = f, etc, which might be similar in principle. This may be another advantage of finite or discrete mathematics over continuous mathematics.
Note that, in equation (13), it is possible for one observer's local acceleration to be zero while the other observer's local acceleration is not zero. This may be relevant to Einstein's principle of equivalence. This is between an observer at rest under gravity (or its attractive force of gravitational mass) and an observer of that "rest" as acceleration, which is the measure of inertial mass.
Classical mechanics is not the master science, it was once thought to be, not even with regard to the progress of mechanics itself. But it does seem a bit surprising that special relativity does not follow classical mechanics, in that the concept of acceleration follows in a straight-forward manner from space, time and velocity.
Special relativity deals in space, time and velocity, but acceleration
seems to come out of no-where, on the basis of a principle of equivalence.
The next section is an attempt to relate this founding principle of general
relativity to special relativity.
Michelson and Morley's calculation didnt agree with their experimental result of equal times for the split light beams on the two journeys. This was despite certain "ether wind resistance" considerations with respect to a back and forth journey directly into the wind and a back and forth journey across the wind.
From Einstein's point of view, there was no universal ether wind but just two differing local times for the two observations of the split light beam at right angles to itself. In terms of the Lorentz transformations, that worked out as meaning that the observation of the cross-wind beam has an observer's velocity of zero (say u = 0). Whereas the observation of the light beam with a head and tail (ether) wind has an observer's velocity equal to the relative velocity between the two observations ( say u' = v).
In terms of Galileo's relativity principle, the relative motion of two observers' velocities is straight-forward addition or subtraction. That wont do for the Michelson-Morley experiment, even when one of the observers measures zero velocity, and the other observer's velocity is the same as the relative velocities between the two observers.
But the point may be that the Michelson and Morley calculation was based on Galileo's relativity principle when it should have been based on Galileo's force principle, which became Newton's second law of motion. Light is a force.
Einstein's photo-electric effect demonstrates this. Light of shorter wave-lengths has more energy and acts like faster bullets which are powerful enough to knock electrons out of the surface of a metal. Only a few of them are enough to do this trick, whereas it doesnt matter how much of the lower energy light is played on the metal, its photons dont have the energy to dislodge the electrons.
Force is mass multiplied by acceleration, or more generally force is change of momentum over (change of) time. Change of momentum leaves the door open to special relativity recognising a possible change of mass as well as change of velocity. Einstein and Infeld, in The Evolution of Physics, say that Galileo's great innovation, that opened the door to modern physics, was to recognise that force is not a function of motion but of change in motion, or acceleration.
When air-craft approach the speed of sound, they encounter a sound barrier. When bodies approach the speed of light they encounter a light barrier. This becomes an ever increasing deterrent force. Therefore, when motion significantly approaches light speed, it should be measured not in terms of Galileo's relative motion, by adding (or from the other point of view, subracting) observers' velocities with respect to each other, but in terms of Galileo's law of force, in this case light force, which is measured not in terms of velocity but change in velocity, which is acceleration or deceleration.
For a steadily increasing input of energy into a body's motion approaching light speed, there is a deceleration in the extra velocity the body achieves. The energy input increases the body's mass, which in theory would have to become infinite before the body reached light speed.
The Michelson-Morley calculation takes an average time for a light beam's back and forth journey but it takes the wrong average, an arithmetic mean, which deals in terms appropriate to the adding and subtracting of velocities. The average should be the geometric mean dealing with change in velocity, in this case deceleration with respect to a body significantly approaching light speed.
The Michelson-Morley experiment, with the averaging calculation based on Galileo's acceleration principle of force, means the geometric mean applies, instead of Galileo's velocity principle of relative motion applying an arithmetic mean. In that case, we can apply Einstein's principle of equivalence to the experiment.
Using his famous thought experiment of the accelerating lift in outer space, the man in the lift feels his feet hit floor as if rooted there under the force of gravity. A chain to the roof is really accelerating him away. A beam of light comes thru the window. To an outside observer, it is going in a straight line. To the man in the lift, he theoreticly sees the light beam slightly dip from its entrance. He assumes it too feels the force of gravity bending it down. But to the out-sider, it's just a case of the lift accelerating so fast that it is leaving the entering light beam behind, rather as it would leave the man in the lift further behind if he were not caught by the floor.
Hence, Einstein derived his principle of the equivalence of acceleration and gravity and predicted that light would bend under strong gravitational attraction. We may predict an analgous situation with regard to the Michelson-Morley experiment's light beam split at right angles. The cross-wind or cross-stream light beam observation, for zero velocity, may be compared to the outside observer of Einstein's lift. The cross-stream journey was calculated by Michelson and Morley to take the same time. They used the arithmetic mean to average the time taken both ways.
In fact, the same result is obtained by using the geometric mean as an average, precisely because both trips do take the same time, and their average must be equal to both times. But this means that the cross journey time, and therefore the cross-journey velocity is compatible with an arithmetic series, which implies a constant velocity or linear progress like an air-craft moving steadily across the sky in a straight line, perhaps marked by a vapor trail. This is the uniform motion in a straight line, to which special relativity has been deemed to apply for all observers to observe the same laws of physics. So, the cross-journey light beam may move in a straight line, as has previously been assumed.
The inside observer sees the light bend, because the lift, which is his reference frame, has lifted somewhat before the beam reaches the other side of the lift. The lift's acceleration effect, of appearing to curve the light beam, may be compared to the acceleration implicit in a geometric series of velocities, we have made apparent, in geometric averaging the Michelson-Morley experiment's up and down stream journey.
For the prediction to conform to the experimental times of the light beams being equal, then the geometric mean is the average that must be taken of the up and down stream journeys. Thus, this beam should appear curved, because acceleration is equivalent to gravity in its effect on light, which has mass and is therefore subject to gravitational attraction. And the reference frame of the man inside the lift is comparable to the reference frame of the observation of the up and down stream (or head and tail wind) light beam journeys.
Suppose, when the bending light beam reaches the other side of Einstein's lift, that it (or part of it) is reflected and the acceleration instantly reversed, so that the light beam repeats its path in the other direction, forming a loop. Thus, reflecting the beam, for a head and tail wind journey, implies a reverse gravitational effect, in the Michelson-Morley experiment.
The thought experiment may differ from the M-M experiment but it seems reasonable to compare Einstein's lift with the head-wind part of the M-M experiment's head and tail wind journey. Einstein's lift in reverse would compare with the tail wind part of the head and tail wind journey. The outside observer, to Einstein's lift in reverse, sees the same straight line of light passing thru the lift only this time in the opposite direction. That is like the cross-stream beam in the M-M experiment.
The comparison, between experiment and thought experiment, is not exact and looks to involve a difference like that between transverse and longitudinal waves. Take a string tied to a post or wall and hold the other end. Jerking the string end up and down will produce up and down waves along the string. These are called transverse waves. The more energy put into the vertical shaking, the more wave vibrations formed along the string. Since the string is fixed at both ends, the vibrations tend to form a definite number of up and down vibrating loops (refered to as harmonics). Just one loop is called the fundamental. This is like the reversed lift scenario, where the observer under-goes a single up and down oscillation with regard to the light beam.
Longitudinal waves are produced when the holder of the string pushes and pulls the string along the line of the string. This looks more like the Michelson-Morley situation of the head and tail wind journey, when the wind pushes you back or forward. (I avoid the analogy, up and down stream, here to avoid any suggestion of vertical action.)
The more energeticly oscillatory the situations, such as those in Einstein's lift or Michelson-Morley experiment, the more oscillations to the gravitational wave.
More complicated oscillations presumably could result in further standing waves in the harmonic series. But to the observer in the lift, these light beam oscillations imply gravitational waves rather than reverse accelerations. He might assume he had been bobbed up and down on a ripple in the fabric of space-time caused by a super-nova explosion.To the outside observer, tho, this is not a gravitational ripple but just the lift being yanked up and down, with the light passing back and forth in a straight line.
Ive talked about light beams perhaps from reading the old popular accounts of physics. I am not well up on these matters but lasers have long since taken over as the light beams of choice in the M-M experiment. White light is light mixed at different wavelengths. Sound, at all different wavelengths, is noise. A laser coheres light into a beam all of one wavelength and phase.
Einstein's lift was only a theoretical argument. His prediction of its consequences was rather on an astronomical scale, namely that light from a star passing close by the sun would be bent in passing by gravitational attraction. If there is a "loop" formed in the head and tail wind journey of the M-M experiment, I guess this might be marked by some decoherence in the sustained bounce of a laser beam between head and tail wind journeys, provided, of course, that the scale or definition were sufficiently great for such a minute effect to be measurable.
With current instrumentation, gravitational waves have not been directly detected. A supernova in the local heavens (not too local we hope) is one prospect for detectable gravitational waves.
In any case, there seems to be no reason why GR should not have a statistical under-pinning. Special Relativity equates observers' measures in uniform motion, thru their respective linear co-ordinate systems. General Relativity equates observers' measures in accelerated motion, thru their respective curvilinear co-ordinate systems. (Albert Einstein discovered this extension from SR to GR in consequence of his "principle of equivalence".) If statistics can derive SR in those geometry terms, statistics should also be able to derive GR in its more sophisticated geometry terms - if with more difficulty!
We might take a glance at the nature of dimensions. In statistics, the normal distribution is a continuously bell-shaped curve version of the binomial distribution. These curves may be standardised for purposes of exact comparisons between them. And one of the standards used is to take the central position under the norm or highest point of the "bell," for any such curve, as zero. By convention, points to the right of zero, increase to +1, +2, +3 etc. Points to the left, decrease from -1, -2, -3 etc. In fact you have a range or dimension that measures indefinite increases or decreases. Zero is the implicit average of this dimension.
Then why not consider all dimensions as potential ranges about an average? Riemann's geometry was adapted to GR to measure curved space. There are spaces of positive or negative curvature, which might be considered statisticly as ranges of curvature about an average curvature.
Special relativity is the geometry of a space-time of zero curvature. This four-dimensional flatness, on average, characterises the physics of the observed universe. One might also say that mathematicly, SR is about the statistics of average space-time curvature. And, perhaps, GR is about the statistics of deviations, from that average, to positive and negative ranges of space-time curvature.
Euclid's geometry generating Pythagoras' theorem and by extension the Interval were statisticly derived from an expansion of the geometric mean, as terms in a series. Einstein's Relativity assumed that there is no privileged co-ordinate frame of reference for the observation of physical laws. This is to make each frame a random choice.
It is enough to assume that observers' frames of reference are random in
relation to each other. Relativity of observation frames becomes Randomness
of observation frames. And their relation to each other is expressed as an
over-all or representative average of their random sample of observation.
More than that, statistical derivation of relativity, such as in the Interval
or an acceleration Interval, has the benefit that no absolute framework of
space and time, and no one and only geometry, such as Euclid's, from
classical physics need be assumed.
A suitable statistics for the geometry of general relativity holds out the hope of compatibility with quantum theory. Under extremely small measurements, far below experimental scrutiny, at the Planck scale, space and time is likened to a sea that breaks up into a "quantum foam." According to physicists, space and time cease to be the basic concepts they are in classical mechanics and relativity.
Statistics does not have to assume a range of values in space and time, tho, of course, it can do. Statistics merely averages any range of values, tho these may derive geometry-like terms in a series of averages.
Mach's principle related the motions of bodies to the motions of all other bodies in the universe, rather than put them in some arbitrary context of any imposable space and time frame-work. Barbour and Bertotti derived Newton's laws from Mach's principle, which they also found was implicit in General Relativity.
Their process of "best matching" of bodies from one configuration to another may be likened to a statistical idea of finding the ranges of bodies' motions by putting successive motions of bodies in a series. There are "goodness of fit" tests in statistics, such as the chi-squared test, that show how well actual ranges of values match theoreticly expected values. In principle, this class of tests might be used to best match configuration changes. At any rate, Barbour says: "it is very convenient to measure how far each body moves by making a comparison with a certain average of all the bodies in the universe." ( A fuller quotation is in my full review of "The End Of Time." )
As the title of Barbour's book suggests, he is trying to replace time as a basic concept in classical as well as quantum mechanics. A note suggests what I take to be a use of topology to remove distance measurements, that define the traditional concept of space, in physical theory. This would be consistent with a statistical treatment that also does not need to assume space and time.
In so far as Barbour and Bertotti's work is a statistical treatment of General Relativity, it has the merit of showing the necessity of that treatment, without having pre-supposed that need with a program for reformulating GR in terms of statistics.
Mach's principle appears to be a statistical program under another name. And it shows the kinship of science and democracy. It is scientific because it insists on reference only to observables. It does not impose any outside reference, which anyone can impose, resulting in a dead-lock of prejudices. Progress in knowledge depends on shedding unfounded assumptions for which there is no basis in agreement from common experience. This is democratic progress because it depends on all points of view agreeing on the rules or laws, rather than an arbitrary rule, essentially a privileged anarchy, in imposing the conventions. Relativity is a democracy of observers all equally free to observe physics' laws. Mach's principle is a democracy of the observed physical phenomena, representatively measured, rather than externally measured.
It may be significant that the Lorentz addition of velocities for observed light speeds is, in effect, an arithmetic mean velocity of light. In common sense experience of low velocities, the ( Galilean ) relative motion, v, between two observers' measured velocities, u' and u, is u' = u + v.
This becomes, for motions significantly approaching light speed, c, the Lorentz version:
u' = ( u + v )/( 1 + uv/c²).
Consider that light speed is "the same regardless of the relative speed of the observer to the source." ( This quotes Bob Lerwill of the SR e-mail group. ) Thus u' = u = c. Then for the Lorentz velocity transformation to hold, v must also equal c. And the Lorentz transformation reduces to c = ( c + c )/( 1 + c²/c²) = 2c/2.Thus observed light speed velocity for u' is in effect light speed as the arithmetic mean velocity of two light speeds for u and v.
I would be surprised if this speculation were right, but it wouldnt take much to modify the Lorentz addition of velocities, for light as massless motion, into an arithmetic mean formula. You could just make the Quantum Electro-dynamic assumption, for extremely short distances, that light speeds have an equitable variation above and below the average speed that is the generally observed constant speed.
The arithmetic doesnt matter but for illustrative purposes, say light speed is unitary or c = 1. Suppose an equitable variation or arithmetic series variation in light speeds of 1 1/2 and 1/2. Obviously their arithmetic mean is one, or their sum, which is two, divided by two ( because there are two speeds to average ).
To get this result for the Lorentz addition of velocities, we have only to consider the denominator term of light speed squared, as - figuratively speaking - light speed "rectangled". In other words, c is subject to similar variation. In this example, the sides of the figurative rectangle may be one-and-one-half and one-half. In such cases, a modified Lorentz addition of varying light speed velocities would be an arithmetic mean light speed familiar as the constant, c.
Richard Lung.
27 September 2006, with slight changes, 28 sept. and 5 dec. '06.
Considerably revised 3 & 6 jan. 2007.