The Michelson-Morley (M&M) experiment is the key to understanding the development of physics early in the 20 the century. It inspired Lorentz to conjecture, in 1895, that matter deforms when in motion. It resulted in his mathematical derivation of the so-called Lorentz transformation, and the associated gamma factor, in 1904. The Lorentz Transformation, in turn, was used by Einstein to claim that space and time are deformable due to motion, as formulated in his Special Relativity Theory (SRT) in 1905.

The Lorentz Transformation, LT for short, is based on a mathematical error that has not been detected for one hundred years. The associated gamma factor is an untenable fiction.

MICHELSON-MORLEY AND LORENTZ

The prevalent belief in the nineteenth century was that light is a wave, carried by a subtle medium, the aether, which is at rest in the universe. The sun is at rest, in the center of the universe, and the earth moves through the aether and around the sun at about 30 km/sec. The Michelson-Morley (M&M) experiment was designed to verify this belief. If light is sent back and forth on earth, in the direction of the earth’s movement, then the round trip should take longer than it would if there were no aether. It should also take longer than it would for light moving back and forth in the direction perpendicular to the earth’s movement. The M&M experiment used two identical rods perpendicular to each other along which the light moved back and forth. This experiment, first performed in 1881, repeated in 1887, and often thereafter, could find no indication of a difference. But the belief could not be shaken, and Hendrik Lorentz, in 1895 and again in 1904, thought the explanation could possibly be that the rod placed in the direction of the earth’s movement might contract due to the earth’s movement, just enough to make the round trip time equal to the case when there is no movement, or when the movement is in the perpendicular direction.

To understand the challenge, and the significance, of the Michelson-Morley (M&M) experiment we can look at an analogous situation. We know that the velocity of a plane is measured with respect to the air stream through which it moves. If we fly from, say, San Francisco to New York and back, and the air is still, it will take six hours each way, at 500 miles per hour to complete the 6000 miles roundtrip. If the air is moving at 100 miles an hour, from west to east, and the plane flies at 500 miles per hour, the distance going is covered in five hours (at 500+100 miles per hour), and the return trip takes 7.5 hours (at 500-100 miles per hour). So the total time is not 12 hours but 12.5 hours. The gain and loss, due to the movement of the air, do not quite cancel. Analogously, that would be the situation if the velocity of light is measured with respect to the aether, and the earth moves through the aether. But, of course, the M&M experiment found no such thing. The total time for a light beam to go back and forth between two fixed points was the same as if the earth were standing still (with respect to the aether).

In the above analogy, if we call s the velocity of the plane, and v the velocity of the jet stream, we can calculate the increase as a function of s and v. The formula is [1 – v^2/s^2] (^ indicates an exponent), as the fraction by which we would have to shrink the total length to eliminate the increase in time. If, in the above experiment, the jet stream reached 250 miles per hour (a ratio v/s = 0.5) we would have to shorten the distance to 2250 miles, 3000x(1-.25). The reader can easily verify that we then need 3 hours for the flight from SF to NY and 9 hours for the return flight – a total of 12 hours

A quick review of how we get this formula should be helpful. We can let L be the distance from SF to NY, and s the speed of the plane (to distinguish it from the speed of light c). We can then write: L = sT, L = (s-v)T’, and L = (s+v)T’’, where T is the time when he air is still, T’ the time flying against the jet stream, and T” the time flying with the jet stream. The fractional increase in time due to the jet stream is then given by the ratio (T”+ T’)/2T which is easily seen to be [s/(s-v) + s/(s+v)]/2. It takes a little algebra to arrive at the formula:

T”+T’ = 2T/[1 – v^2/s^2].

This formula holds true for light by substituting c for s. Since the denominator is always less than one, it shows that the round trip time is always increased due to the result of a jet stream (or the aether).

Both Lorentz and Einstein then proceed to split this expression into two equal parts that multiply together, and obtain for each leg an average increase(and therefore a needed contraction factor) given by:

sqrt[(T"+T')/2T] = 1/sqrt(1-v^2/c^2).

This is the so called gamma factor which plays a crucial role in 20th century physics.

But there is a subtle point that must be noted. If we want to deal with a single direct path, that is in some sense the ‘average’ of coming and going, we can, mathematically, take the square root of the total path reduction that is needed. The problem with this step lies in the relationship between mathematics and physics: We must remember the important distinction between an arithmetic and a geometric mean. We use a geometric mean when two factors multiply, and an arithmetic mean when two quantities add. In the case of two-way Doppler for light we get a multiplication of the Doppler factors; but in the case of the M&M experiment we have an addition of the transit times of the light, back and forth..

The shorter time on one leg of a journey is not further modified on the return leg. We simply get a longer time of transit - the two times add, they do not multiply. In the above example, using the case v = c/2, we find that the total distance must be reduced by the factor (1 - .25) = .75, and this is also the reduction for each leg. In the case of the plane flying between SF and NY, we need a distance of 2250 miles on each leg (3000x.75) to bring the total time back to12 hours. If we tried to reduce each path by the square root of .75 we would only reduce the distance to 2598 miles - not enough to reduce the round trip to 12 hours. But that, in essence, is what Lorentz did!

Incidentally, Lorentz made this mistake because he took an expression made up of squared quantities, c^2/(c^2 – v^2), and introduced for this expression a new variable, that he called beta-squared. That made it natural and tempting to take the square root – without bothering to ask about the sense or legitimacy of this operation. This is equation 3 in section 4 of his 1904 paper.

Einstein maintained that he developed the Lorentz transformation independently. He does not cite Lorentz in his 1905 paper. It is most curious, therefore, that he made the same mathematical mistake of taking the square root at the same point in the development of the transform as his predecessor. Those physicists who derived the LT in the 20th century followed suit.

FEYNMAN AND RINDLER

Richard Feynman in his famous book “Lectures on Physics“, vol 1, 1963, presents an analysis of the M&M experiment. Although he gets to the same formula it does not have the same meaning as for Lorentz and Einstein. He does not take the square root. Instead, he argues that, in the perpendicular direction, in the M&M experiment, the time is also increased. The M&M experiment does not correspond to the requirement that the path in the perpendicular direction cancels out the movement of the earth. The perpendicular path of the light to the mirror, from the source, is increased due to the movement of the apparatus while the light is in motion – and similarly for the return path. Feynman introduces another variable T’’’ to denote this increased time in the perpendicular direction. Because it is the hypotenuse of a right triangle, he finds the increae T’’’/T is given by sqrt[1/(1 – v^2/c^2)], (the gamma factor). He then compares T’ + T’’ to 2T’’’ and since

(T’ + T’’)/2T’’’ = (T’ + T’’)/2T x T/T’’’

he gets

1/sqrt(1 – v^2/c^2) = 1/(1 – v^2/c^2) x sqrt(1 – v^2/c^2).

His argument can be found on pp. 15-3 to 15-5.

Feynman correctly answers the question: Given the set up of the M&M experiment how much greater is the time of the round trip parallel to the motion of the earth in comparison with the round trip perpendicular to that motion. He does not answer the question: How great is the effect of the motion of the earth compared to no motion. These are two different questions – they cannot have the same answer.

It is important to realize that the ratio of (T’ + T’’)/2 to T’’’ that Feynman obtains cannot be the same as the ratio of (T’ + T’’)/2 to T which Lorentz and Einstein seek – so either Lorentz or Feynman gets the wrong answer. Rindler, also analyzes the M&M experiment. He takes note of the increase in time in the perpendicular direction, and comes up with the same formula as Feynman, [see Wolfgang Rindler, „Introduction to Special Relativity“, 1991, p.3].

So Lorentz and Einstein came up with the wrong answer to the right question, while Feynman and Rindler came up with the right answer to the wrong question. And all get the same formula! Notice that Feynman and Rindler deal with an artifact of the experiment rather than the essence of the theory.

Conclusion: The Lorentz Transformation as derived by Lorentz and Einstein is not physically meaningful or justified. It is mathematically incorrect. The same formula, derived by Feynman and Rindler, does not have the same meaning as the formula derived by Lorentz and Einstein. It should not be called, or taken to be, the Lorentz Transformation.

Table of Contents

PUBLISHED BOOK 2005

HOME PAGE
INTRODUCTION
THE PRINCIPLE OF RELATIVITY
SPECIAL RELATIVITY - 1905
SPECIAL RELATIVITY - 1917

THE MICHELSON-MORLEY EXPERIMENT
THE LORENTZ TRANSFORMATION
MICHELSON-MORLEY PLUS LORENTZ, CONDENSED

THE DOPPLER EFFECT
A COUNTER EXAMPLE TO SRT
A COUNTER EXAMPLE TO GRT
THE AGE OF THE UNIVERSE

CONCLUSIONS
APPENDIX I: TYPE 1A SUPERNOVAE
APPENDIX II: A EUCLIDEAN MODEL OF THE UNIVERSE
APPENDIX III: MASS AND ENERGY
All contents copyright 1997, 2005
Revised 6/30/2005