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.
ADDENDUM TO "LORENZ TRANSFORMATION"
June 5, 2010
How is it possible that two of the world’s greatest physicists would come up with a concept that makes no sense physically? It is almost beyond belief and needs to be explained — as simply as possible. Lorentz preceded Einstein, so his error has historical priority. Here is what happened:
Lorentz came, somehow, using difficult math, to an expression,
c²/(c²-v²)
that, because of having so many squared terms, he called, simply, beta squared, and then without further deliberation or question he took the square-root to obtain beta, or as it eventually became known — the ‘gamma’ factor — a factor that makes no sense physically. But how did he get this fraction in the first place?
If Lorentz had been more of a physicist and less of a mathematician he might have argued thus: the earth travels through the aether with about the velocity v, so if light travels with velocity c with respect to the aether, then with respect to the earth it will have velocity c-v when the beam of light is moving in the direction of the earth’s motion, but on the return path it is moving, relative to the earth, with velocity c+v. The time of travel is given through the formula,
velocity = distance/time, or time = distance/velocity
So for the first leg we get T1=d/(c-v) and for the second leg we get T2=d/(c+v), and the sum can be written with a common denominator (c+v)(c-v) = c²–v² and a numerator that needs to be reduced, essentially, to c². This is the expression Lorentz called beta squared. But the numerator, according to this intuitive approach, is actually 2cd and reduces to c² only if the distance d happens to have the magnitude c/2. For this value of d, and only this value, beta squared represents, numerically but not dimensionally, the time of the round trip. Since this does not get us to ‘gamma’, as yet, we need to search on.
There is another, more general meaning for beta squared. If we take the ratio of the round trip time with the assumption of a stationary aether, to the two way time, 2T, in the case where there is no aether ,we get for the denominator the trip time 2T = 2d/c. In other words we get the ratio
(T1 + T2)/2T = [2dc/(c² – v²)] / [2d/c] = c²/(c² – v²)
which, as we see, reduces to beta squared. Since v and c are positive and c is larger than v, what beta squared actually tells us is that the round trip time in the case there is an aether is always greater than the two way time without the aether. But just for the record, M&M found this ratio to be exactly one, that is, v=0. But in any case the square root of this ratio has no physical meaning. The square root of time is physically senseless, as is the square root of the ratio of two durations.
Lorentz would have been the first to admit that the square root of time (or the square root of the ratio of two durations) has no physical meaning – but in doing the math he lost sight of the physics. He never felt the need to ask about the meaning of beta. Einstein simply followed in his footsteps. He thought he had avoided the question of the existence of the aether, but his math implicitly evoked this same asymmetry in the velocity of light.
The Michelson Morley experiment points decisively to the absence of the aether. Lorentz’s attempt to nevertheless maintain the aether as a physical fact, and his use of deformation of matter to rationalize the negative results of M&M is what inspired Einstein. Had Einstein accepted the empirical evidence of M&M, instead of the obscure teachings of the celebrated Nobel Prize recipient Lorentz, his relativity theory would never have been born.
The moral is: physicists should do physics — not just mathematics; and we should not expect mathematicians to do our physics for us.
I want to express my sincere thanks to my good friend Dr Peter Marquardt for his aid and assistance in this long wild quest for clarity and truth in the treacherous world of rtheoretical physics.
E = mc²
Added August 18, 2010
Einstein derives his most important formula, E = mc², in a short sequel to his June 1905 paper on special relativity, by applying the gamma factor to his relativity principle.
The math is trivial once we accept these, but the relativity principle deserves as much scrutiny as the source and meaning of the gamma factor. His failure to define the concept ‘energy’ and relate it without contradiction to the concept ‘light’ constitutes the root problem in this work.
He begins by stating his relativity principle: the laws regarding the changes in the conditions (zustaende) of a physical system are independent of which of two coordinate systems, that are in relative linear motion, we use to describe these facts, or conditions.
In trying to understand his relativity principle consider this: Suppose we have a gun at rest on the earth that we want to fire at a target ten feet away. But now we want to relate this situation to a train moving at twenty feet per second from left to right. Are the gun and the target to be now considered to be on the train, or is the train, with the observer in it, just going by and we are observing the situation while seated on the train? Which situation represents his relativity principle.
He continues by referring to a body at rest in one of these systems and asserting it has energy Eo, and then he wants to find its energy, in the other system. The other system is moving with velocity v with respect to the first system.
Energy, in so far as it is classical kinetic energy, relates to mass and motion. In the case of a gun at rest on the ground we would have to know the velocity of the bullet with respect to the target and the mass of the bullet, and then we could compute mv². But we still would need to know which scenario to use in the case that we are on a moving train. Nothing in his math that follows his remarks clarifies the scenario.
The scenario is important. If the observer and the gun are both on the train instead of on the ground, then Einstein’s relativity principle is just a restatement of the principle that all motion is relative – and that has nothing to do with Special Relativity. If, on the other hand, the gun and target are on the ground and the observer is on the train then this relativity principle is invalid since the motion of the bullet as perceived on the train will not be the same as when perceived and measured by an observer on the ground. If, for example, the train is moving at the speed of the bullet, the bullet is perceived to be stationary with respect to the train. So we have a choice: Einstein’s relativity principle is either trivial or wrong.
If a body is at rest, and we want to associate with it some quantity or type of energy, independent of its movement or relationship to another body, it is not its mass but its atomic or nuclear character that can yield such energy or energies.
To the extent that a substance is destroyed and its mass, m, is carried off by radiation and nuclear fragments moving at close to the speed of light, to that extent we can hope to generate classical kinetic energy approaching mc², but that will not happen to a mass of clay or iron, so it is the nature and character of the mass that is relevant. We are outside the domain of classical physics – and outside the domain of special relativity.
He speaks of a body at rest in the first system as emitting light energy. What is the meaning of the phrase ‘to emit light energy’? In classical physics we transfer energy from one body to another by means of matter – for example from the bullet to the target. So what is the light energy that is ‘emitted’? (On page 640 he talks about light waves emitted from a resting body that have energy L/2.) According to Einstein there is energy residing in the light — but also, according to Einstein, light has no mass. But, as we know, energy requires, and is coupled to mass. There is an internal contradiction in his calculations and concepts, and it appears his idea of energy is clouded or confused. But out of nowhere, by using an approximation derived from the gamma factor, he comes up with the assertion that this loss of energy represents a loss of mass, and that gets him
E = mc².
Einstein was right in his intuition that there can be an energy associated with an object at rest. That recognition is the fact that pulled physics into the atomic age. But this is a new kind of energy, or better, a new family of energies and forces that were not suspected before the discovery of radioactivity.
If mass is ‘lost’ in a nuclear reaction or transformation we should not seek to explain it by means of a deformation of space or time — it is not relativity theory that we need but a detailed look at the nuclear fragments and radiation that takes the place of the ‘missing’ mass. If we allow radiation to have mass along with the rest of the nuclear debris than nothing is missing, and then light can carry energy as well.
The bottom line is E = mc² raises more questions than answers. It is derived from the dubious gamma factor. It does not clarify the relation between matter, energy and light. It does not prove the validity of special relativity, but is suggestive of the immense power of atomic energy.
Note: That mass might change due to speed was under intense discussion in 1904. The Kaufmann experiments indicated that as the speed of electrons increases e/m decreases. Einstein’s results favored an increase in mass instead of a decrease in the effect of charge. That became the prevailing view. Whether Einstein was aware of these experiments is not clear, but highly probable.
No amount of math can hide the absurdity, or contradiction — on the one hand, mass retains its quantity and its identity in the classical concept of kinetic energy, even as the velocity, v, of its impact increases and gets close to the velocity of light; but on the other hand the mass increases and becomes infinite according to special relativity (not to mention one interpretation of the Kaufmann experiments), but on the other hand (assuming a third hand), on reaching the velocity of light, it vanishes, and turns into energy (of some sort or other) — the meaning of the term ‘energy’ is left to the imagination of the reader.
An example from a different context may help in understanding the concept ‘energy’: If I am making pancakes I take water from one container and flour from another and combine them in a third container that then contains a mixture from which I can make pancakes. But in the case of ‘energy’ we don’t have a container of mass and a container of energy that we can combine to make a third and separate entity called ‘energy’. Kinetic energy IS that combination of mass and velocity — it is not just ‘equal’ to that combination. That is a distinction that the word ‘equals’ does not contain — and it can be the source of confusion when we use the concept ‘energy’. This is not physics, it is metaphysics or linguistics or common sense. But without this distinction much of physics can be very confusing indeed. What the word ‘energy’ can mean apart from ‘mass times the square of the velocity’ has not been clearly defined by Einstein or other physicists of that time.
“Energy cannot exist except in connection with matter.”
(James Clerk Maxwell, “Matter and Motion”, 1877, Chapter 6, Paragraph 108, "Test of a Material Substance")
