SR non-solutions of the M-M calculation.
Transformations between affected and unaffected light beam times.
"The Minkowski Interval predicts the Michelson-Morley experiment" explained why the geometric mean was the suitable average of reflected light beam journeys with respect to earth motion, longitudinally and transversely.
Here, I hope to show that the more obvious choice of average, the arithmetic mean, used by Michelson and Morley, is mathematicly meaningless, when applied to the Lorentz transformation of Special Relativity. Likewise, to attempt application to the Minkowski Interval.
Previously, I gave proof positive that the geometric mean is the suitable calculation in the M-M experiment. Now to give proof negative, that the arithmetic mean is actually the wrong average to use for that experiment.
The Michelson-Morley experiment measures the times taken by a split light beam, reflected transversely and longitudinally with respect to earth motion. The original motivation of the experiment was to measure the differing drag on light waves transversely and longitudinally thru a supposed medium for those waves, in a universal ether. The earth motion stood-in for the ether motion, rather like a bank stands in relative motion to a stream.
The calculated time, for transverse motion of the reflected light beam with respect to earth motion, is the same whether the arithmetic mean or the geometric mean is used to average the time taken for both legs of the journey back and forth.
When the geometric mean is used, to average the back and forth journey of the reflected beam longitudinally with earth motion, it gives the same time as the reflected transverse journey. This confirms the result of the classic Michelson-Morley experiment. Their own calculation, using the arithmetic mean for the longitudinal journey, did not.
In terms of averaging by the arithmetic mean, the longitudinal and transverse motions are the simplest to measure and the most distinct, measuring maximum time difference.
The longitudinal and transverse distances of the light beam journeys are made the same for purposes of comparing the times taken. Therefore, both the transverse and longitudinal distance, d, are equal.
Time is measured as distance divided by velocity. As previously explained, the average transverse or "cross-stream" time, t, taken is given by simple geometry, using Pythagoras theorem, as distance, d, divided by a velocity, that is the square root of the squared light speed, c², minus the earth velocity squared, v².
That is, average transverse time, t = d/(c² - v²)^1/2.
The longitudinal time, t', is given over the same distance, d, covered by the part of the light beam, not split-off "across-stream," but reflected up and down "stream" against and with earth motion. This gives the beam velocity a head-stream (or head-wind) and tail-stream effect, respectively slowing and hastening its velocity.
The slowed time, under the "head-wind," is given by dividing distance, d, by the light speed minus earth velocity: d/(c - v). The tail-wind time is given by: d/(c + v).
Michelson and Morley used the arithmetic mean, adding the back and forth times and dividing by two, for the (evidently incorrect) average time, t', for the head and tail wind journey. This gives: t' = {2cd/(c² - v²)}/2 = cd/(c² - v²).
What I called the proof positive also implies a negation and refutation of the Michelson-Morley calculation. This is for the following reason. The transverse light reflection and the longitudinal light reflection are both with respect to earth velocity. But when you put the two different Michelson-Morley time calculations into the Minkowski Interval, I, equation, the answer only works for the transverse time side of the equation having a velocity zero, which is inconsistent with the earth velocity that both sides of the equation must have, to represent the Michelson-Morley experiment.
Hence the Interval is:
I² = t'²(c² - u'²) = t²(c² - u²).
Substituting in the Michelson-Morley values for the times, t' and t, and bearing in mind that both sides must have the earth velocity, v = u' = u, then:
{cd/(c² - v²)}²(c² - v²) = {d/(c² - v²)^1/2}²(c² - v²).
So:
c²/(c² - v²) = 1.
Or:
(1 - v²/c²)^1/2 = 1.
In this special case of applying the M-M calculation, the Interval reduces to the form of the contraction factor, because the arithmetically averaged longitudinal time is out by that amount. This error was the reason, in the first place, for the ad hoc "contraction factor," really a correction factor.
According to this (mis)calculation, v = 0 for the transverse beam journey. This is inconsistent with the beam journey being transverse to earth velocity, v. Some will say, as I once thought, that the transverse beam has zero relative motion to the earth. But that is inconsistent with having measured the transverse beam journey time in terms of actual earth velocity, v.
A similar result is found using the equivalent Lorentz transformations between the two times of two different local observers:
t' = t(1 + uv/c²)/(1 - v²/c²)^1/2.
The observer who locally measures the faster time, t, of the two times also
has a local velocity, u. This must be zero, so that the Fitzgerald-Lorentz
contraction factor, the square root factor in the denominator, relates time t'
to time t.
Therefore, like the Interval, the Lorentz transformation also makes the M-M
calculation inconsistent. And once again shows the two times out by the
"contraction" (correction) factor coefficient.
Hendrik Lorentz didnt start with the full Lorentz transformations, named after him. Historicly, the simple contraction factor is how Fitzgerald and Lorentz independently explained the discrepancy of the Michelson and Morley experimental result from their calculation. They came-up with this ad hoc hypothesis of a contraction factor, as if there was some physical effect of the supposed "universal ether wind" that balanced out the Michelson and Morley calculation.
A much simpler explanation is that Michelson and Morley should have used the
geometric mean, instead of the arithmetic mean, and then the times come out
equal.
The Lorentz transformations were built on the contraction factor, with its
implicit assumption that this was a physical correcting or accounting for the
M-M experiment rather than that the M-M calculation was mathematicly or
statisticly incorrect. This assumption prevented Lorentz and his successors
from seeing that his full transformation rendered the M-M calculation
inconsistent.
This explanation in terms of the two M-M experiment times, t and t', is admittedly simplistic.
A further time, to be taken into consideration, is the unaffected light beam journey time, T. This is simply distance, d, divided by the light speed, c, both ways, which gives the same value for the average time. That is: T = d/c.
The Minkowski Interval would give a different time for T, compared to identical light beam times for transverse time, t, and longitudinal time, t'. In fact, the Interval, I is the equal distance, d, of the light beams. And both have the same earth-relative velocities, v.
Hence,
I² = d² = t²{c² + (iv)²} = t'²(c - v)(c + v).
The factor with the imaginary earth velocity, iv, represents the transverse light beam traveling at 90 degrees to the earth. The two multiplied factors, on the left side, represent the longitudinal beam traveling, at speed, c, with and against earth velocity and therefore, relatively speaking, added and subtracted to and from earth velocity, v.
These Michelson-Morley experiment times would not be the same for a light beam traveling the same distance but not relative to earth velocity. In the scenario of the two boats, one goes back and forth or up and down stream. The other boat goes across from bank to bank and back.
The analogy, with a light beam not moving relative to earth velocity, would be for a third boat to be an amfibian that would not cross the stream but move the same distance, of the other two boats, on land, which is to say unaffected by the motion of the stream.
In that case the earth-relative velocity of the "amfibian" beam would be zero. In terms of the Interval, this beam taking time, T, would relate to either of the earth-relative beams, taking time, t, as:
I² = t²(c² - v²) = T²(c² - 0).
Therefore:
T = t(1 - v²/c²)^1/2.
In other words, the contraction factor would have to be applied to the earth-relative time in the Michelson-Morley experiment to arrive at a non-earth-relative time for light beam travel.
This contraction factor was used as an ad hoc means of getting the Michelson-Morley calculation to agree with experiment. Here, it is merely a consequence of applying the Minkowski Interval to earth-relative and non-relative travel times.
Also, the Lorentz transformation of times can be applied to relate the two times, earth-transverse time, t and earth-unaffected time, T.
Hence:
t = T(1 + 0.v/c²)/(1 - v²/c²)^1/2.
Where, again, the earth-unaffected light beam has earth-relative velocity
zero.
Like the Interval, this requires the earth-relative time to be reduced by the
contraction factor to equal the earth-unaffected time.
Explanations of relativity theory have sometimes said that it abolished the ether. It might be truer to say that it abolished the ether as having a property of universal or absolute velocity. Nowadays, a universal "sea" of virtual energy, that exists even in the remotest vacuum of space, is a candidate for a modern notion of the ether. But it is not supposed to have an absolute velocity.
Using the Interval to relate T to t', with the M-M calculation of an arithmetic mean longitudinal time, t', produces the incorrect result that T has the same value as t.
Using the Lorentz transformation to relate M-M calculation of t' to T = d/c gives the result that the contraction factor is equal to unity, which is only true when relative velocity, v, is insignificant compared to light speed. (That is the classical, not the relativistic, case.)
The Michelson-Morley experiment is sometimes called a null result. This is with reference to the defeated expectation that the longitudinal light journey would take longer than the transverse journey. The irony is that the real null result is in the inconsistency of the M-M arithmetic mean calculation with the equations of special relativity.
My web pages, which first set out my workings, until carrying out renovations, have been riddled with errors (as I once admitted, more truly than I knew, to an unresponsive physics forum).
The minor calculation short-coming of the Michelson-Morley experiment, the
technical triumph of its day, is rather as if the NASA astronauts got to the
Moon, in 1969, without remembering to bring back some rock samples, and
everyone was too excited by the magnitude of the achievement, to heed the
over-sight. (I learnt as much from Albert Michelson as I did from Albert
Einstein.)
Without the experimenters, there is no science.
Richard Lung
26 july; august 2009;
re-written with corrections, july 2015.