THE LORENTZ TRANSFORMATION – COMPARING LIGHT AND SOUND
Added 7-12-08
The comparison of light with sound has, historically considered, done physics more harm than good. Sound requires a carrier, primarily air, which not only carries sound but also determines its speed. The more dense the air the faster, and the more effective the propagation of sound. There is no use in trying to analyze air to see where the sound is hidden in it. In a sense the medium is the sound - it does not contain it.
In the Nineteenth Century, with the discovery that light, i.e. electromagnetic radiation, has wavelike properties, the search was on for its carrier, the ‘aether’. But neither the attempts of Michelson and Morley, nor those of many others, could detect such a carrier. Lorentz was willing to have matter, even time, deform rather than give up the idea of a carrier of light. That his mathematics contained a fatal flaw was even more unfortunate than his inability to give up the idea of a carrier of light.
Einstein was courageous enough to abandon the idea of the aether, but in so doing, he invested photons with the ability to determine their own speed, (he even made c into a universal constant) and in addition postulated that the speed was independent of the movement of the source, his second principle of Special Relativity.
Einstein did not see the connection of this principle with the idea of the aether. But this principle invests photons with a capacity to generate motion, much as the air has that capacity for sound. The second principle emancipated the movement of light from the movement of the source. This is comparable to making the movement of a bullet independent of the movement of the gun from which it is fired. The problem this brings with it is the need for reconciling this view with the first principle, that all movement is relative to the platform or system from which it is observed, or in which it originates, and is defined. Without some trick that removes the contradiction these two principles are logically contradictory. Einstein’s reconciliation of this inconsistency requires a distortion of time and space and we are back to the Lorentz Transformation. Einstein adopted the LT together with the fatally flawed mathematical derivation of the ‘Gamma Factor’.
There is no other occurrence in the history of science where a logical inconsistency was ‘removed’ by mathematical devices and a redefinition of physical reality – and was adopted and celebrated by the scientific community as inspired work.
We are better off abandoning the aether along with the notion that photons are masters of their own fate – Einstein’s Second Principle - but this may require a new generation of physicists who have not been indoctrinated in 20th Century Physics.
DERIVING THE LORENTZ TRANSFORMATION
All derivations of the Lorentz Transformation (LT) are based, like the Michelson-Morley experiment, on two-way travel. An additional assumption is that the same length contraction applies to both paths, coming and going.
Lorentz thought that perhaps the molecules of a solid might be compressed in the direction of motion, and this could explain why the time of travel is the same in both directions. His transformation attempted to quantify this contraction of matter.
Lorentz conjectured as early as 1895 that perhaps the rod parallel to the earth’s movement might shrink because ‘electrons would lose their spherical shape due to the earth’s motion’. ‘Electron’ at that time was a word denoting some elementary particle that is the ultimate constituent of matter.
Lorentz quantified his conjecture in a paper in 1904. He showed how much shorter the round trip, in the direction parallel to the earth’s motion, needed to be so that the effect of the earth’s motion, in relation to the stationary aether, would be counteracted. He then tried to find how much shorter the rod would have to be on each leg of the roundtrip in the direction parallel to the earth’s motion. Here is where he got into trouble that no one, to date, has appreciated. The formula he ended up with, called the gamma factor, is one of the most widely used expressions in astronomy and particle physics. It is also the basis of Einstein’s SRT – and it is wrong.
A quick review of how Lorentz gets to his result, is the following: We can let L be the distance that light moves in one direction, and c the speed of light. We can then write: L = cT, L = (c-v)T’, and L = (c+v)T’’, where T is the time if there is no motion of the earth relative to the aether, T’ the time required when the light is moving in the opposite direction to the earth’s motion, and T’’ the time in the reverse direction. The fractional increase in time due to the movement of the earth is then given by the ratio (T’’+ T’)/2T which is easily seen to be [c/(c-v) + c/(c+v)]/2. It takes a little algebra to arrive at the formula:
T’’+T’ = 2T/[1 – v2/c2].
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-v2/c2).
This is the so-called gamma factor that plays a crucial role in 20th century physics.
That there is no need to take the square root becomes obvious if, using the first equation, we write (T’ + T’’)/2, the average increase due to motion, on the left hand side, and divide this by T, the time it takes for each leg of the journey in the absence of motion. The right side is then simply 1/ [1 – v2/c2], which is the fractional increase, on the average, for each leg of the journey - no square root is involved - this is the square of gamma.
Incidentally, Lorentz made this mistake because he took an expression made up of squared quantities, c2/(c2 – v2), and introduced for this expression a new variable, 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. The reader may recognize beta-squared as the fraction by which the total time must be shortened, and half of the shortening should pertain to each leg of the round trip – not the square root. The Lorentz transformation is a science fiction fantasy, but it has captured 20th century physics and cosmology.
EINSTEIN AND THE LT
Einstein never mentioned Lorentz in his 1905 paper. Either he followed in his footsteps, or he had independently the same bad idea. Einstein’s second principle asserts that the speed of light is independent of the movement of the source.
In other words an observer stationary in one system that is in motion relative to the system in which the source is stationary will measure the speed of light to be the same as if he were stationary in the system in which the source is stationary.
So, to satisfy the second principle, we must assume that in the M&M experiment Einstein would be implicitly concerned with an observer on the sun rather than on earth, (the source in the M&M experiment is attached to the earth, which is in motion with respect to the sun). His actual thinking is too vague and unclear as to how the coordinate system of the observer and the earth’s coordinate system (the system of the source) are related. Working out the consequences leads him to the conclusions that time must also contract and mass must increase – it must become infinite at the speed of light. So light, i.e.radiation, cannot have mass.
FEYNMAN AND THE LT
Richard Feynman in his famous book “Lectures on Physics“, vol. 1, 1963, presents an analysis of the M&M experiment. He arrives at the same formula as Lorentz and Einstein, but it does not have the same physical meaning. It is clear that he is uncomfortable with Lorentz’s derivation.
Feynman does not take the square root. Instead, he argues that, for light traveling in two perpendicular directions, as was the case in the M&M experiment, the total time for the two legs is also increased (but not by the same amount as for a light beam traveling back and forth in one direction!).
The M&M experiment does not correspond to the requirement that the path in the perpendicular direction cancels out the movement of the earth. Feynman introduces another variable, T’’’, to denote this increased time, for a one-way path, in the perpendicular direction. Because it is the hypotenuse of a right triangle, he finds the relative increase in the perpendicular direction T’’’/T is given by sqrt[1/(1 – v2/c2)]. He then compares the round trip time in the direction of the earth’s movement to the round trip time in the perpendicular direction, i.e he compares T’ + T’’ to 2T’’’, (rather than to 2T), and since
(T’ + T’’)/2T’’’ = (T’ + T’’)/2T x T/T’’’
is a truism, he gets
1/sqrt(1 – v2/c2) = 1/(1 – v2/c2) x sqrt(1 – v2/c2).
Feynman gets the gamma factor as the result of his derivation. (His argument can be found on pp. 15-3 to 15-5.) But although it is the same formula, Feynman’s formula does not have the same meaning as the formula that Lorentz develops.
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 motion of the earth compared to a situation with no motion”. These are two different questions – they cannot have the same answer. It is important to realize that the ratio (T’ + T’’)/2 to T’’’ which Feynman obtains cannot be the same as the ratio (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].
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!
What is true is that the M&M experiment should have come up with a result consistent with the gamma factor, in the result of Feynman – if the hypothesis, as to the propagation of light, of Lorentz were realized. Of course the M&M experiment, and its null effect, is inconsistent with this theory, - which is why a shrinkage of space (and time) is required to account for the negative result, if one insists, in spite of the evidence, on maintaining the validity of the assumptions. But the correction needs to be applied with respect to the viewpoint of Lorentz and Einstein – not Feynman and Rindler, since the last two deal with an artifact of the experiment rather than the essence of the theory.
The essential point is this. The one directional Lorentz transform is a geometric mean of a sum of two quantities that represent time of travel. That result produces nonsense – nonsense physically if not mathematically.
EINSTEIN’S MATH
Einstein’s mathematical derivations are not always clear, but that makes it even more imperative to see whether the math has, in fact, a sound basis in physics. The math, as such, is no more than high school algebra.
SRT does not even rise to the level of calculus. What is obscure is the meaning of the individual equations in a physics sense, and the physical rational for Einstein taking the mathematical steps in arriving, finally, somehow, at the Lorentz Transformation.
Reviewing Einstein’s proof of the Lorentz Transformation, it appears that he confuses mathematical and physical simultaneity. Mathematically, two linear equations that are both true can be added or subtracted; but physically light cannot simultaneously move in two opposite directions – especially at different rates. Consequently to combine two equations that represent such alternative possibilities is not physically meaningful. Lorentz adds the back and forth movements of light. The two movements are presumed not to be identical because the earth moves through the aether – Einstein combines them, without use of the aether, and by thinking of them as ‘simultaneous’ (but, somehow, not identical). His equations 3 and 4, in his appendix 1, suggest that light simultaneously has different speeds in opposite directions. To make this point clear: The equation x' - ct' = 0 (equation 2) can be written as x' = ct' and says simply that a light pulse starting at the origin will arrive at the horizontal coordinate x' after traveling at the speed c for a time t' (in the coordinate system K). The next equation, (3), is (x' - ct') = lambda(x - ct). It says that you can arrive at the point x' in the coordinate system he calls K' (he mistakenly writes k’), by multiplying x and t in the coordinate system K by the constant lambda.
So much so good, but then he proceeds to let the light propagate, as well, in the opposite horizontal direction (the negative direction). Here he uses a different constant, mu instead of lamda, to show that you reach the point -x' (which is as far from the origin as x’) in the same manner as you reach x' in the positive direction.
That is the fundamental mistake! Without the aether to carry it, light propagates at the same rate in all directions, in any coordinate system. So to reach -x', in the coordinate system K', the light pulse, starting at the origin, should make use of the same constant lambda as for the positive direction. This, however gets Einstein nowhere - no LT, no shrinkage of space, no dilation of time, no SRT.
