The Special Theory of Relativity

Part One: The Michelson-Morley Experiment and the Lorentz transformations.

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Introduction.

The Michelson-Morley Experiment.

The Michelson-Morley calculation (using inappropriate average).

The Lorentz transformations.


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John Gribbin, in one of his many books on popular science, once derided special relativity as "ancient history." So great was the growth of natural science, in the twentieth century compared to the rest of history, a theory from the beginning of that century must seem out-moded. But he still had to deal with it, under whatever caption. It is still a starting point of modern physics.

This treatment of special relativity is not by a professional scientist. It is just an explanation by a reader of books on popular science. These, quite rightly, go into the wierd and wonderful consequences of the theory, to inspire the imaginations of new-comers. This account is more concerned with what are the (quite simple) formulas behind these consequences. So, no claim is made, here, to be either inspirational or authoritative. Nor did I pick up any natural science as a social science student, tho I was particularly interested in scientific method.

Thousands of books have been written about relativity. Often, a good idea is given of the mathematics of special relativity. This can be done without going much, if at all, beyond high school standard. The following account does not attempt to go beyond fairly basic algebra. (As if I could.)

The math of general relativity is much too advanced for me. And I dont know of any good popular account of it. Roger Penrose, in The Emperor's New Mind, gives a thumb-nail sketch of the math involved. He also prefaces Six Not So Easy Pieces, by Richard Feynman, which gives the best conceptual explanation of General Relativity, that Ive come across. The "pieces" are, in fact, taken from his under-graduate text-book on physics. Don't let that put you off. Feynman approach gives much insight.
If you are a beginner in the subject, it would be better to read first several introductory works, to fully appreciate Feynmans account of both special and general relativity. I am glad I didnt come across Six Not So Easy Pieces early in my reading. And, in fact, my following explanation of special relativity is not influenced by that book.


Introduction

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The three related topics to be treated here are the Michelson-Morley experiment, the Lorentz transformations, and Minkowski Interval. These topics give precise information, about special relativity, in terms of simple algebra and geometry.
When I say simple, I mean simple by professional standards. I don't mean that the math was simple for me to teach myself -- far from it. But I hope my efforts, such as they are, will give others an easier time of it than I had.

The name "the theory of special relativity" (as well as of "general relativity") comes from Albert Einstein. As far as special relativity was concerned, he gave the physical meaning of what was going on, in such as the Lorentz transformations.

Minkowski was Einsteins former teacher, who supplied a neater mathematical form to the equations. At first, Einstein didnt see the point of the new formalism. But it was to become a point of departure for his general theory.

As for the Michelson-Morley experiment, Einstein said he had heard of it, when he wrote his famous 1905 paper on special relativity. It was reputed that he hadnt. Anyway, this experiment is a land-mark to the origin of modern physics.


The Michelson-Morley experiment (1887)

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Clerk-Maxwell equations showed that electro-magnetic waves moved at the speed of light, suggesting that light is an electro-magnetic wave. A wave normally waves some material medium it moves thru. To talk of a water wave without water wouldnt seem to make much sense. So, a light wave was held to manifest the matter of which the universe is made, the so-called universal "ether." This ether was a supposed "ocean" of material reality thru which light waves traveled.

(I dont know whether light itself couldnt be considered the medium of light waves, as water is of water waves. In Feynman quantum theory, light is all over the place or "oceanic," on the sub-atomic scale. But this is not to the point of the history of the "ether" as a hypothesis in physics.)

Michelson and Morley set out to detect the ether with an experiment. Given that the ether was the all-pervading medium of the universe, its characteristics, such as velocity, could not be related to any thing else. The ether would have its own definitive or absolute velocity, like the velocity of a stream relative to ones position on the bank. (Absolute velocity seems a self-contradiction. It can only be conceived in relative terms and therefore is not absolute. This is perhaps a cue for Einstein principle of relativity.)

In the Michelson-Morley experiment, ones position on the "bank" is akin to ones position on Earth relative to a universal ether "stream."
The experiment can be explained by continuing the analogy with the stream. One could take two return journeys of equal distance on the stream. Taking into account the velocity of the stream, simple geometry shows which journey would take the longest and which would take the shortest time.

The return journey, which would be most slowed by stream velocity, is the one going directly up and down stream. Going down stream (like having a tail-wind) would be the fastest way for a boat to travel. But having to make a return trip (like facing a head-wind) would so out-weigh that advantage, that, on average, the combined up-stream and down-stream trip would be the slowest.

The quickest return journey, over an equal distance, is the one taken across-stream and back. Actually, this wouldn't follow a path at right angles to the banks, because the velocity of the stream is pulling the boat some way down-stream, while the crossing is being made. And the boat goes adrift by an equal amount, on the way back to the bank that one embarked from. So, one would land back some way down the bank, from ones embarkation point.

In the Michelson-Morley experiment, a beam of light stands in for the boat. This beam is split to do the two different "boat" journeys described. Mirrors effect the split light beams return journeys.
One light beam was expected to return slightly more slowly than the other, as a result of a difference in "ether drag" upon the two beams. This is analgous to the boat going up and down stream taking longer than the boat going across stream and back.

Of course, no physicist knew the direction of the supposed ether "wind" (to change the analogy from water to air). But this experiment, with the light beam split at right angles to itself and reflected, was repeated in all directions, and at all times in the Earths annual orbit. So, the ether wind direction and velocity could presumably be infered from the experiment, in that series, which showed the maximum and minimum delays of a split beam subject to ether drag, analgous to wind drag.

In fact, the Michelson-Morley experiment showed always the same result: a null result. The split beams of light always returned with their waves still in step. An interferometer would have measured interference effects, if they hadnt. Therefore, the split light waves took the same time to make their two return journeys, traveling the same distance, without their different directions causing a more powerful ether drag on one beam than the other. The speed of light remained constant.

(Eventually, Einstein special theory of relativity would assume there is no ether and that the speed of light is constant. By this, physicists mean constant in a vacuum, disregarding that light is slightly slowed in transparent media, like air and water.)


The Michelson-Morley calculation (using inappropriate average).

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This section gives the historic and currently accepted calculation. I showed elsewhere that if you substitute the geometric mean for the arithmetic mean, to average the mirror reflected light beam journeys, the correct prediction is achieved for the result of the famous Michelson-Morley experiement.

The Michelson and Morley calculation, for the experimental result they expected, is essentially nothing more than simple arithmetic, involving the fact that light velocity, c, equals distance, d, traveled, divided by the time, t, taken.

Two different times were predicted, the shortest time, t, for a beam of light, split to pass athwart the ether stream, and the longest time, which can be designated t', for the rest of a beam, aligned to the ether stream.

But experiment showed that the times, t and t', were the same. Therefore, to make the calculation agree with experiment, a factor had to be written into the calculation, to show t and t' as equal. This factor, would, in effect, contract the longest time, t', so it equalled the shortest time, t. Hence, its name, the Fitzgerald-Lorentz contraction factor, after the two physicists, who independently suggested it.

Eventually, Einstein would dispense with the assumption of ether absolute velocity and an absolute space and absolute time, presupposed since Newton. Einstein would explain this contraction factor, in terms of observers, in different situations or frames of reference, measuring distinct times, distances and velocities. Different observers could relate their spatio-temporal measurements, basically thru the so-called contraction factor. But no one observers frame of reference could be considered absolute, or prior, to any-one elses. Observations were all relative to each other and no one was absolute. Hence, the theory of relativity.

How, at first, did Michelson and Morley work out the times, t and t'? The longest time, t', is the return journey, up and down the ether stream. In other words, the Earth, at a certain point in its orbit, is supposed to be traveling in line with the stream of universal ethereal matter. Thereby, the experimenters light beam is also in line with the ether stream (besides being split off at right angles to measure the shortest delayed time, t, of light travel across the ether stream).

Going with the ether stream, the velocity of light, usually given the letter c, for ("celerity" but I prefer) "constant" velocity, should have a tail-wind or tail-stream of ether velocity added to it. The ether velocity is given by earth velocity, just as the velocity of a stream can be measured by the speed one is running beside its bank to make the stream appear to be standing still. This ether velocity may be given the letter, u.

The velocity of light with an ether tail-stream was calculated as its own velocity, c, combined with the ether velocity, u. That is (c+u). Therefore, the time light takes to travel a distance, d, down-stream was given as the length divided by the combined velocity, or, d/(c+u).

When the light is reflected, thought of as a boat now going up-stream, it would be slowed down by the speed of an ether "head-wind," to be subtracted from the light speed: the light beam speed up-stream was calculated at (c-u). The up-stream time taken is the slower one of d/(c-u).

Therefore, the combined time, t', for light to travel up and down an ether "stream" would be

d/(c+u) plus d/(c-u). This works out as 2dc/(c-u)(c+u), or,
2dc/(c-u).

For working out the cross-ways return journey, over a comparable distance, d, of the light beam with respect to the ether stream, please refer to diagram, below. The light source, S, is the point where the "boat," as light-beam, sets off across the stream, reflected back by a mirror, M, analgous to a far "bank."

Diagram of Michelson-Morley calculation.

Diagram of the Michelson-Morley calculation.

The arrow shows the earths direction, here also defining an "ether stream," reckoned to cause light the greatest delay, in traveling up and down stream. The least delay is caused that part of the light beam that splits across stream. But it is still subject to ether drift downstream, both to and from M'.

But, by the time the light, from source, S, has reached mirror, M, the mirror has moved to position, M', in the diagram. This is like down-stream drift on a boat crossing. The same is true for the journey back. So, when the light-beam reaches M', it has completed half the journey in half the time, t/2. This is because crossing the ether stream affects journeys, both ways, equally. There is no element of a head wind one way or a tail wind, the other way.

By the time, the light beam has reached the mirror, at M', the light source, S, has also traveled half way, S', to its rendezvous, S", with the reflected light-beam it emitted.

On the diagram, the light beam travels at speed, c, from S to M', given as distance, z. Meanwhile, the mirror has traveled from M to M', given as distance, y, down the ether stream, at a speed equated (as explained above) to earth velocity, u. The two times of travel are equal, actually time t/2, as mentioned already. Therefore, t/2 equals both distance, z, divided by velocity, c, and distance, y, divided by velocity, u.

Or, t/2 = z/c = y/u. The distance, z, can be found in terms of Pythagoras theorem. Namely, z = y + d. Therefore, z = (zu/c) + d.

And: z{1 - (u/c)} = d.

So: z = d.c/square root(c-u).

Therefore: z = d{c/square root(c-u)}^1/2

The journey from S to M' is similar to the return from M' to S", also of distance, z. Therefore, the total distance traveled by a crossing light-speed "boat" is twice z, or,

2d{c/square root(c-u)}^1/2square root

Dividing this distance, both ways across, by the light-beam cross-stream velocity, c, gives its total crossing time, t, as 2d/(c-u)^1/2.

This was how Michelson and Morley calculated the two times, t' and t, as a predicted out-come of maximal and minimal ether drag, respectively.
The fame of their experiment rested on the fact that it showed the two times to be equal. The two times could only be made equal, in theory as well as practise, by introducing a factor, F (the so-called Fitzgerald-Lorentz contraction factor) so that the shorter time, t, equals the longer time, t', multiplied by the contraction factor.

The contraction factor, F, equals t/t' = {2d/square root(c-u)^1/2}{(c-u)/2dc}

= square root{(c-u)^1/2}/c = square root{1-(u/c)}^1/2.


The Lorentz transformations ( 1895 ).

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Prof. H A Lorentz showed that Maxwell electrodynamic equations still held, whether or not subject to a prevailing ether "stream" or "ether wind." But the co-ordinate systems or frames of reference, of two observers would differ according as to the extent their measurements were affected by the ether wind.

Supposing, for the sake of argument, the measurements taken in a laboratory on earth were not subject to the ether wind. The measurements there are given by time, t, distance, x, (as well as two other dimensions of space that dont essentially affect the argument). Lorentz showed that a laboratory elsewhere, say on a rocket, could still discover Maxwells equations. But the measurements in the rocket would be in a different frame of reference, with different time, say, t', different distance, x' (and its two other spatial dimensions). Consequently, velocities would measure differently: x' = u't', instead of x = ut.

However, these different co-ordinates could be related by a set of equations, to become known as the Lorentz transformations. These involve the Fitzgerald-Lorentz contraction factor, F, which we've already met.

The Lorentz transformation for the times is:

t' = t(1-uv/c)/square root(1-v/c)^1/2.

The velocity, v, stands for the relative velocity of two observers frames of reference. The velocities, u or u', are the velocities of an object, seen by observer 0 or observer O', respectively.
We have to bear in mind that the speed of light is so great in comparison with our normal earth-bound movements, that it is in effect infinite. That being the case, the Lorentz transformation, for two local times, reduces to just the one ("absolute") time, we are familiar with.

A Lorentz transformation applies to mass, also previously thought an absolute. As a body approaches the speed of light, its mass increases. As this speed is outside our normal experience, no such mass increases are observed. But this phenomenum, as well as time dilation effects to particles normal life-times, is regularly measured in laboratory experiments with atomic and sub-atomic particles, moving very close to the speed of light.

In theory, a body could never reach the speed of light, because that would involve acquiring infinite mass. Hence, the speed of light is a limiting maximum on all objects with mass.

It turns out that momentum, or mass times velocity, has a similar form of Lorentz transformation to that for distance measurements. Energy has a similar transformation to that for the different times observers measure, in high energy physics. (Given as a post-script to his 1905 paper on special relativity, Einstein's E = mc is the most famous consequence of these transformations. Henri Poincar also derived this equation in his close anticipation of Einstein theory.)

The velocity, v, in the above Lorentz transformation of times, relates to the difference between the observers two measured velocities, u and u'. But, as things work out, it is generally not a simple subtraction between the two, to allow for the fact that one laboratory is moving faster, relative to another.

But, if one of the frames of reference is considered at rest, so the velocity is zero, or u = 0, the velocity, measured in the other frame, u' = v. The Lorentz transformation of times then reduces to the Michelson-Morley calculation, modified by the contraction factor, F. For,

t' = t(1 + 0.v/c)/F = t/F.

Two observers, with differences in relative motion, that are significant compared to the speed of light, would observe different speeds in their clocks. Moreover, these would be real effects, resulting in twins, say, ageing at different rates.

The so-called "twin paradox" raises the question of why one twin, rather than the other, since they are both in relative motion, should be the one to age more slowly than normal. The retarded ager went off in the rocket, at a velocity, that was a large fraction of the speed of light, and returned to earth to find his twin long dead. The twins were just in relative motion -- most of the time.

The catch, that resolves the "paradox," is to do with the fact that the rocket man would have to turn the rocket to return to earth, under-going the kind of accelerative forces, not experienced on earth.
This is properly explained in popular physics books, such as by Paul Davies.


Richard Lung.
Early 2001; minor correction, sept. 2003.
Corrections, 16 june 2015.



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