The Minkowski Interval predicts the Michelson-Morley experiment.

Relative acceleration.

In special relativity, the role of acceleration was over-looked for removing the paradox from the twin paradox. (Paul Davies explained it well in Space and Time in the Modern Universe.) How to resolve which twin does not age relative to the other, when they are both separated by speeds significantly approaching light and then brought back together? The twins are moving relatively to each other, so the supposed paradox goes, so why should one be any more liable to age than the other?

The key is that the twin sent off in a rocket, from earth, has to return, and that means a change in velocity of the rocketeer, relative to the earth. A change in velocity is an acceleration of the rocketeer. So, the rocketeer is the one to under-go the relativistic effects of slowed time, at significant approaches to light speed.

Even if the rocketeer did not do an abrupt about-turn but circled back, circular motion still has a component of acceleration. This has been tested on the atomic scale, where a particle orbiting its twin is measured to have a lengthened life.
By the way, the accelerated twin in the rocket should not experience having lived any longer than normal, tho he might return to an earth which was much older than when he left it, and his earth-bound twin long since dead.

The geometric mean is an average, which resembles the form known to special relativity as the Fitzgerald-Lorentz contraction factor (sometimes just called the gamma factor and symbolised by the Greek letter gamma).

The Michelson-Morley calculation should have used a geometric mean to average the reflected light beam's time with and against the earth's motion. The reflection involves a relative acceleration, if the light beam reflection is from going with, to going against, the earth's motion. (It would be a relative deceleration if the reflection were against, followed by with, earth motion. But it amounts to the same thing.)

An acceleration involves a geometric series, which is averaged by a geometric mean. In graphical terms, the slower velocity of the light (having to catch up) with earth motion, would describe a less steep slope than the faster velocity of light (arriving sooner when) against earth velocity. That is, respectively, like the difference between head-wind and tail-wind speeds.

Michelson and Morley averaged the "head" and "tail" wind journeys using the familiar average known as the arithmetic mean. But an arithmetic mean applies, in graphical terms, only to a constant slope, that represents a constant velocity or zero acceleration.

A change in velocity is exemplified in a change from head-wind to tail-wind velocity (or vice versa) that defines acceleration (or deceleration). In that case, the two different slopes, of the two different velocities, with head and tail winds, make-up a "curve". It is not accurate enough to represent them by some compromise straight line, which is all an arithmetic mean form of average could do.

Hence, the Michelson-Morley calculation gave the wrong prediction that light would take longer, to go with and against the earth's motion, than crossways or transversely to it.
One could use the arithmetic mean to average the time for that part of the light beam that was split off and reflected "cross-wind" or "cross-stream" (rather than "head" and "tail" wind or "up" and "down" stream) of the earth's motion. That is because, cross-stream, the velocity of the light beam remains constant with respect to the earth's motion. The two transverse trips produce the same measures, so the geometric mean averages them to the same effect as the arithmetic mean.

Using the geometric mean to average the up and down stream (or head and tail wind) motion of the light beam, with respect to earth motion, gives the same time taken as the average time taken by the cross-stream beam.
This corrected Michelson-Morley calculation gives the correct prediction for the Michelson-Morley experiment which shows that there is no difference in the times taken by the split light beam to reflect transversely or longitudinally to earth motion.

(As discussed on another page, a test in principle is offered of relative acceleration, in the longitudinal direction. Einstein's equivalence principle of acceleration to gravity would apply to the Michelson-Morley experiment. A gravitational bending and decoherence of a reflected longitudinal laser would be too small to measure. Until technology and ingenuity allow, Nature would have to be kind enough to set-up such an improbable experiment, on an astronomical scale.)

The Fitzgerald-Lorentz contraction factor was actually an ad hoc hypothesis or a patch to make calculation conform with experiment. It innocently covered-up a mistake, thus blocking the way to understanding.

How the Interval demonstrates the Michelson-Morley experiment.

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The equation for Minkowski's Interval is (1):

I = (ct) - vt = (ct') - v't'.

There I stands for Interval, which is a common space-time measure shared by observers of a given event at speeds approaching light speed, c, which is constant. Observer O has a frame of reference which measures time, t, and velocity, v, and distance x = vt. Observer O' in a different frame of reference has the different, hence indexed, measures of time, t', and velocity, v', and space, x'.
The distances can be considered in one dimension or as a combination or vector of two or three spatial dimensions.

The Michelson-Morley experiment showed that light takes the same time to travel over the same distance, either at right angles with, or aligned with the earth's motion.

The Interval shows this too, in terms of two observations that take the same time. That is when t = t'. Moreover, the Interval, I, can be considered as the common distance that both the split beams have to travel.

Also the velocity, v, has a counterpart velocity, at right angles, because the M-M experiment has split light beams, one aligned and the other at right angles to earth velocity, v.

The right-angled velocity is given by an "imaginary" velocity, iv. The misleading term, imaginary, simply means that the velocity is multiplied by i, the square root of minus one. Using more jargon, i is an "operator," which places the velocity, v, at right angles to light velocity, c.

Therefore from equation (1), v' = iv. Let iv = u.
(If vt = x and as t = t', then x' = ix.)

Firstly, considering the Interval in terms of longitudinal motion or the light beam aligned with and against earth's motion. This can be expressed in terms of equation (2):

I = (ct) - vt = t(c + v)(c - v).

The factorised values, on the right side of the equation, represent a light beam, c, moving with or against the earth's velocity, v. This is the scenario for the part of the light beam in the Michelson-Morley experiment that is aligned with or against the earth's motion. The beam is reflected, so an out-going journey, with the earth's motion, means the return journey must be against it. And vice versa.
The part of the beam that is split off to travel cross-ways to the earth's motion will be considered afterwards.

Re equation (2), if earth's velocity and the beam's velocity are moving in the same direction, the light has to catch up with its destination (which is the reflecting mirror, on the journey's first leg) that the earth is carrying away from it. And the light's velocity is that much reduced by the earth's velocity.
That is: (c - v).

If the beam is moving in the opposite direction to the earth, the beam arrives that much sooner at its goal (which is back to the source, on the journey's second leg) and earth velocity must be added, as speeding the arrival. That is (c + v).

Thus, the Interval as equation (2) may represent the longitudinal journey of the light beam in the Michelson-Morley experiment.
No, this isnt the way Michelson and Morley calculated the time taken by the light beam. It was some years before Minkowski formulated the Interval.

Reformulate equation (2) to equation (3):

t = I/(c + v)(c - v) = {I/(c + v)}{I/(c - v)}.

The right side of the equation is the multiple of two ratios. A constant distance, I, is divided either by the faster velocity (c + v) or the slower velocity, (c - v), representing relative motion of the light beam, respectively, against and with the earth's motion.
The arithmetic says these ratios measure respectively faster and slower times.

Using the geometric mean to average these times means multiplying the ratios, just as on the right side of equation (3). Then taking the square root, for the geometric mean time, t, as in equation (4):

t = ({I/(c + v)}{I/(c - v)})^1/2.

Anyone of the thousands of books on relativity (or indeed the relevant page on my own web site) will show that this result for the average time taken by the reflected light beam in alignment with earth's motion is NOT the result calculated by Michelson and Morley.
Later, the Fitzgerald-Lorentz contraction was cooked-up to make their calculation agree with the result of their experiment that light takes the same time to return when reflected cross-ways and in alignment to earth's motion.

But equation (4) for the earth-aligned light beam DOES agree with the Michelson-Morley calculation for the cross-ways light beam to earth motion. And this agreement matches the result of the M-M experiment. It makes no difference which average, arithmetic mean or geometric mean, is used for the cross-ways journey's time. But to get agreement with the aligned journey's time, the geometric mean must be used, as explained above.

Equations (2), (3) and (4) show two things. Firstly that the Interval calculates the true time for the aligned journey in the Michelson-Morley experiment. And secondly that the Interval is a geometric mean. That is special relativity is a theory of statistical averages.

As is well known, the irony is that Albert Einstein, who used statistics in so much of his research, baulked at the idea of special relativity being a statistical theory. I dont know whether he would have regarded this perspective as an infringement of his ideal of relativity as a progression from the classical physics of Newton.

Now, consider Michelson-Morley's cross-ways relative motion of light beam to earth motion. Velocity v stands for earth motion that is aligned to the light beam's direction of motion. As explained, a motion at right angles to aligned velocity, v, is given by iv = u. Substituting in the Interval as equation (5):

I = (ct) - vt = t{c + (iv)} = t(c + u).

Notice about equation (5) that the Interval squared has lost its negative sign with respect to transverse velocity, u. In other words, the conventional form of the Interval is with respect to aligned motion. The Interval for transverse motion more nearly resembles Euclid's geometry as given by Pythagoras' theorem. This theorem says the sum of the squares of the sides of a right-angled triangle equal the square of the hypotenuse. This theorem extends to the sum of the squares of three sides of a cube equaling the hypotenuse in three-dimensions.

Historicly, the Interval is considered as a break, from extending Pythagoras' rule, in that the fourth dimension is a subtraction of its square from up to three spatial dimensions.
(The present treatment simply considers the velocity in one spatial dimension. It could be considered as a distance vector, vt = x, of up to three dimensions, with ct as the fourth dimension and of opposite sign. The intrusion of the negative value, in the Pythagorean formula, represents aligned motion, back and forth, with respect to the other one, two or three spatial dimensions.)
But for transverse motion, as given by velocity, u = iv, the summing of a fourth square gives the Interval as no more than the fourth-dimensional hypotenuse of Pythagoras' theorem for four dimensions.

Equation (5) transforms to equation (6):

I = (ct) - vt = t{c + (iv)} = t(c + u) = t(c + iu)(c - iu).

In equation (6) the Interval has been transformed in terms of the two factors (c + iu) and (c - iu). Their opposite signs, plus and minus, tell of two opposite motions, by way of the light beam being reflected. But in this case, the light velocity, c, and the earth velocity, u, are at right angles, as given by each operator sign, i.

The fact that equation (2) is a transformation to equation (6) meets the conditions of the Michelson-Morley experiment. The Interval, I, has the dimensions of distance. It represents the same distance that the split light beams have to travel whether in earth-aligned or earth-transverse motion. Moreover, the condition of the aligned and the transverse journeys taking the same time was met by making t = t'.

To calculate the average time taken by the reflected transverse light beam, requires the geometric mean used on equation (6). On the right side, the two factors, (c + iu) and (c - iu) are considered in terms of the limits of a range. To get the average of a range using the geometric mean, multiply the two limits and take their square root.

Hence, equation (7):

t = I/(c + iu)(c - iu).

Therefore (7):

t = I/{(c + iu)(c - iu)}^1/2.

This equation (7) for the average time taken by the earth-transverse beam, with its velocity, iu, agrees with equation (4), for the average time taken by the earth-aligned beam, with its velocity v, as v = iu.

This agreement of times, for the orthogonally split beams, shows how Minkowski's Interval gives the correct prediction for the Michelson-Morley experiment.

It may need to be explained why the geometric mean had to be used to average the transverse motion, as well as to average the aligned motion. The transverse motion was put in the form of complex variables.

A "complex variable" is, say, z = c + iu, where light speed, c, is the so-called "real" part and earth velocity, iu, the "imaginary" part. The point about these forms is that they involve rotation, as implied in the use of the operator, i, as meaning: turn thru ninety degrees. Acceleration, that is non-uniform motion, is a component of rotation, and this requires the use of the geometric mean for averaging. (My web page on Mach's principle etc exemplifies this.) The arithmetic mean can be used only to average constant change, such as uniform motion.

Richard Lung.
20 september 2008.

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