On a previous page, The Minkowski Interval predicts the Michelson-Morley experiment, I explained why the geometric mean was the suitable average of reflected light beam journeys with respect to earth motion, longitudinally and transversely.
On this page, 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.
In other words, the previous page gave proof positive that the geometric mean is the suitable calculation in the M-M experiment. This page gives 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. Actually 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, as their own calculation, using the arithmetic mean for the longitudinal journey, did not.
The Lorentz transformation of times can be used to relate times taken. There is the time taken by a reflected light beam unaffected by any supposed wave medium drag expressed in terms of its possible relations to the earth's motion. Of all the possible orientations, the longitudinal and transverse motions are the simplest to measure and the most distinct, geometricly 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 distances are most simply treated as one unit of distance.
Time is measured as distance divided by velocity. The average transverse or "cross-stream" time, t, taken is given by simple geometry as distance one 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 = 1/(c² - v²)^1/2.
The longitudinal time, t', is given over the same distance one, covered by the part of the light beam, not split-off "across-stream", but reflected up and down "stream" against and with the earth's motion. This gives the beam's 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 one by the light speed minus the earth's velocity: 1/(c - v). The tail-wind time is given by: 1/(c + v).
Michelson and Morley used the arithmetic mean, adding the back and forth times and dividing by two, for the (wrong) average time, t', for the head and tail wind journey. This gives: t' = {2c/(c² - v²)}/2 = c/(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).
Substituing 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:
{c/(c² - v²)}²(c² - v²) = {1/(c² - v²)^1/2}²(c² - v²).
Therefore:
{c/(c² - v²)}²(c² - v²) = {1/(c² - v²)^1/2}²(c² - v²).
So:
c²/(c² - v²) = {1/(c² - v²)^1/2}²(c² - v²).
And:
c² = (c² - v²).
Therefore v = 0 for the transverse beam's journey. This is inconsistent with the beam's journey being transverse to earth velocity, v. Some will say, as I have said myself before, that the transverse beam has zero relative motion to the earth. But that is not what the Michelson-Morley experiment is supposed to be measuring.
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. Why didnt Hendrik Lorentz see this?
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 Michelson and Morley's 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 Michelson and Morley's 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.
But the point is that 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', may seem simplistic. However, there is a further way of showing that the arithmetic mean calculation does not make sense. This is the main topic of this web page.
A further time, to be taken into consideration, is the unaffected light beam journey time, T. This is simply one divided by the light speed, c, both ways, which gives the same value for the average time. That is: T = 1/c.
The Lorentz transformation of times can be applied to relate the two times, earth-transverse time, t and earth-unaffected time, T, to find the relative motion, v, between a supposedly dragged and undragged light beam.
Hence:
t = T(1 + cv/c²)/(1 - v²/c²)^1/2.
This is the additive Lorentz transformation of times, T being faster than t. Substituting in T = 1/c and t = 1/(c² - v²)^1/2, then:
1/(c² - v²)^1/2 = (1/c)(1 + cv/c²)/(1 - v²/c²)^1/2.
Therefore:
1 = (c/c)(1 + v/c) = 1 + v/c.
Therefore: v = 0.
This means that there is no relative motion or ether drag between the supposedly dragged and undragged light beams. This is in accord with Einstein's interpretation of his Special Theory of Relativity. Albert Einstein drops the notion of a universal ether that has a universal velocity. The notion of a universal velocity follows from the assumption of universal space and universal time. Einstein replaced this assumption with local observers having local measures of space and time, which could be related to each other thru the Lorentz transformations.
I read somewhere that Lorentz himself didnt accept relativity theory. Never, in his long and famous career did Einstein ever receive a Nobel prize for his relativity theory, special or general.
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.
Using the geometric mean for the longitudinal M-M time, t', gives the same time as transverse time,t, so there is no further problem of relating t' to T, in that case. But in the case of the M-M calculation of an arithmetic mean longitudinal time, t' differs from t, and so another Lorentz transformation is required to relate t' to T:
c/(c²-v²) = (1/c)(1 + cv/c²)/(1 - v²/c²)^1/2.
Therefore:
c² = (c + v)(c² - v²)^1/2.
Therefore:
c^4 = (c + v)²(c² - v²) = (c² + v² + 2cv)(c² - v²)
= c^4 + c²v² + 2vc^3 - v²c² - v^4 - 2cv^3.
Therefore:
0 = 2vc^3 - v^4 - 2cv^3.
Therefore:
0 = 2c^3 - v^3 - 2cv².
This is a cubic equation, which can be solved by a reduction method to a cubic term and a first order term without a second order term:
0 = 2c^3/v^3 - 1 - 2c/v.
Re-arranging:
2c^3/v^3 - 2c/v = 1.
Let c/v = r. cos Q.
Then:
2(r.cos Q)^3 - 2r.cos Q = 1.
From a trigonometric identity:
2^3.cos^3 Q = 2(1.cos 3Q + 3.cos Q).
Canceling 2 and re-arranging:
4.cos^3 Q - 3.cos Q = cos 3Q.
Solving the cubic equation in terms of the trigonometric identity:
(2r^3)/4 = 2r/3 = 1/cos 3Q.
Therefore r² = 4/3. And r = ±2/3^1/2 = ±1.15 to two decimal places.
And:
cos 3Q = 3/2r = ±1.73 to two dec. places.
Another solution is r = 0 but this would also make cos 3Q greater than unity.
As cosines only tabulate between the magnitudes of one and zero, all three solutions of the cubic equation are meaningless for the M-M arithmetic mean calculation of earth-longitudinally reflected light beam journey time, t', related to an unaffected light beam time, T, by the Lorentz transformation. QED.
I should point out that only, after I worked out this cubic equation's incompatibility of the arithmetic mean Michelson-Morley calculation with the Lorentz time transformation between t' and T, did I realise the simplistic incompatibilities between t' and t, explained previously. It took seeing thru the complicated equation and its meaning, before I could see thru the simple equations and their similar meaning.
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 longitude calculation with the equations of special relativity.
Of course, one always risks making a fool of oneself in saying someone
else is wrong, particularly if those someones are the practical geniuses,
Michelson and Morley and the ever-so-distinguished physics profession that
followed in their giant foot-steps without noticing their simple but natural
failure to use the less obvious but suitable average.
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 too excited to heed such a small omission.
Nevertheless, without the experimenters, there is no science. And this page - whatever its truth - is meant as a tribute to them.
This extra section is a reply to a member of the Theoretical Physics e-mail group on this web page's subject.
----- Original Message ----- From: "Jose Molina"
<donmagufo@gmail.com>
To: <theoretical_physics@yahoogroups.com>Sent: Monday, July 27, 2009
5:36 PM
Subject: RE: [Theoretical_Physics] Lorentz transfm. non-solns. of M-M
calcn.
Do you think the LISA experiment (similar to Michelson Morley but in the space) will give the same results as the Michelson Morley?
Global Mechanics states that it will be the opposite.
Jose Molina
http://www.molwick.com/es/experimentos/125-experimentos-luz.html
My reply was as follows:
Mr Jose Molina, your question takes some answering! Right or wrong, I have
tried to answer it satisfactorily.
The English link to your cited page doesnt work, which is a pity because it
looks a nice simple explanation. Anyway, I looked at the Wikipedia article. I
must admit I hadnt thought of the LISA experiment of a 3 satellite M-M
set-up. But its main concern doesnt seem to be to replicate the M-M expt but
to detect gravitational waves thru-out space.
So, to answer your question, I'm going to imagine a 3 satellite experiment
(call it M-M 2) that does, if practical, repeat the conditions of the
original M-M expt. For instance, the earth's velocity round the sun might be
replaced by the sun's velocity round the center of the galaxy, the Milky Way.
Then the split laser beam from one satellite reflects longitudinally, with
respect to the sun's velocity, to a second satellite, and transversely to the
third satellite.
(LISA will form an equilateral triangle, which modifies the most simple and
distinct perpendicular calculation.)
According to my claim, the longitudinal reflection, with respect to some massive velocity, creates a change in velocity or acceleration. This is why the suitable average is the geometric mean, which gives the correctprediction that the split light beam takes the same time both ways. As I mentioned and speculated some (on a web page previous to the two cited) this acceleration implies in principle Einstein's equivalence principle.
From his famous accelerating lift thought-experiment, such an acceleration
implies a bending of light from the inside-lift reference frame. But it is
still the same beam as the straight light line seen exterior to the lift.
However, gravity, if strong enough, can slow light. I have no idea whether a
gravitational bending effect could be produced on a sun-longitudinal
satellite laser reflection.
(Previously I guessed there might be some sortof decoherence effect on the
reflected beam but I really dont know what I'm talking about!)
Anyway, according to my reasoning, the correct mathematics, using the geometric mean, should hold on any scale: if M-M 2 replicates the original experiment, the light beams should still take the same time.
Suppose the split beam times were shown to be not the same. I would be most surprised if they conformed to the Michelson-Morley use of the arithmetic mean for the longitudinal reflection. This produces a longer time but it should still be incorrect, because an arithmetic mean can only measure constant change or velocity, not acceleration under gravity.
Suppose, for argument's sake, I am right. That would mean that the
Minkowski Interval has the mathematical form of a geometric mean that
correctly predicts an acceleration effect even on a solar system scale of the
M-M expt that might be measurably gravitational - however!
If so, this would seem a bit puzzling. After all, the Minkowski Interval is
the province of Special Relativity and Euclid-like flat space-time and is not
supposed to measure gravitational acceleration requiring a geometry of
curvature, the province of General Relativity.
Perhaps, I should explain that the Interval is only a geometric mean form in a qualified way. It is more like a combination of arithmetic mean and geometric mean form. I have suggested that geometries might be classified on a scale, like the binomial series of terms, from pure arithmetic mean form or straight line geometry, to pure geometric mean forms of presumably extreme curvature.
I admit that it should be possible to make or observe two light beams,
going the same distance, take different times under different gravitational
conditions. It just seems to me that the mathematics of the peculiar
Michelson-Morley experimental conditions dont work out that way.
That's my guess but I am not a physicist, as you no doubt have correctly
guessed.
Sincerely,
Richard Lung.
Richard Lung
26 july 2009;
8 august 2009.