RELATIVITY: THE MADNESS OF 20th CENTURY SCIENCE

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RELATIVITY: THE MADNESS OF 20^th CENTURY PHYSICS

Pentcho Valev

In his book "The Tragicomical History of Thermodynamics 1822-1854" C. Truesdell characterises this science as "a prime example to show that physicists are not exempt from the madness of crowds". Since thermodynamics can be defined as a 19^th century science, Truesdell could have called it, accordingly, "the 19^th century prime example". However the logical paradigm set out by 19^th century thermodynamicists has developed so that the 20^th century "prime example" is undoubtedly relativity. The present paper contains a malicious scenario presenting the main protagonist, Albert Einstein, as a plagiarist and juggler. Needless to say, the story is biased. Yet it is realistic. This means that the events described could be interpreted in terms of plagiarism or/and juggling.

Initially, young and ambitious Einstein bumps into a couple of equations which are known today as Lorentz transforms:

ξ = γ(x – vt)

τ = γ(t – vx/c²)

where γ = (1 – v²/c²)^-1/2. It should be noted that, in this form of the equations, x, t, ξ and τ refer to separations between two events and the first event is the origin in both inertial frames.

It is easy to see that, according to Lorentz transforms, if x=ct, then ξ=cτ and vice versa. This means that Lorentz transforms imply that the speed of light is constant (independent of the speed of the source or observer). Einstein notices that and decides to reverse the logic: the constancy of the speed of light becomes a "postulate" which, together with another postulate (the principle of relativity), should produce equations identical to Lorentz transforms. In the case of success the fame is guaranteed: the deduction of the equations from two simple postulates is so valuable that any other approach or authorship would be regarded as insignificant.

Yet there is a snag: Lorentz transforms cannot be deduced from the two postulates simply because the latter do not contain enough information. Two additional postulates are needed:

Postulate 3: The variables x, t, ξ and τ obey the equations

ξ = Ax + Bt

τ = Cx + Dt

Postulate 4: The coefficients A, B, C and D are independent of x, t, ξ and τ.

There is a significant difference between "announcing the two original postulates and then deducing the transforms from them" and "announcing the four postulates and then deducing the transforms from them". Both cases amount to plagiarism but in the former the deduction can, under favourable circumstances, be presented as a work of genius whereas in the latter the plagiarism is obvious, under any circumstances. The problem is extremely serious. In 1916 Einstein will know how to overcome it but in 1905 he has no idea. So he decides to apply what might be called the principle of maximum absurdity: the more absurdity is introduced, the more religious awe and less criticism is evoked. This maximum absurdity is concentrated in the quantity x’ in the equation

(1/2)[τ(0,0,0,t) + τ(0,0,0, t + x’/(c–v) + x’/(c+v))] = τ(x’,0,0, t + x’/(c-v))

in Einstein’s 1905 paper. It is easy to see that x’ undergoes at least four metamorphoses:

1. x’ is the length of the (unidirectional) path of the ray as measured in the stationary system K. In this capacity x’ can only be finite and constant.

2. x’ is a variable since τ is defined as a function of x’, y, z and t.

3. x’ is a function of x and t: x’ = x – vt

4. x’ is chosen infinitesimally small.

The metamorphoses of x’ allow Einstein to "deduce" almost anything, Lorentz transforms included. For the last 100 years this particular application of the principle of maximum absurdity has evoked a lot of admiration and no criticism at all (rather, the criticism has been quite enough but since it has come from dissident sources the relativity cult have just ignored it).

In 1916 Einstein is much more confident (see Appendix 1 in his "Relativity"). The process of deification is advancing, mathematicians willingly help him and he does not need to use the outrageous principle of maximum absurdity in the deduction of Lorentz transforms (although he fiercely continues to use this principle in other cases). Rather, in order to obtain Postulates 3 and 4 as corollaries of the original two postulates (otherwise he would be forced to tell the truth – that Postulates 3 and 4 are just axioms - and everything would crumble), he applies a much milder trick known in logic as "the fallacy of affirming the consequent". If a proposition (consequent) follows from another proposition (antecedent), we can affirm the antecedent (as an axiom) and deduce the consequent. But this does not allow us to affirm the consequent and deduce the antecedent. In short, the antecedent does not necessarily follow from the consequent. Deducing the antecedent from the consequent is fallacious.

In Appendix 1 the antecedent and the consequent are as follows:

Antecedent: ξ – cτ = λ(x – ct), where λ is constant.

Consequent: If x – ct = 0, then ξ –cτ = 0, and vice versa.

Clearly, the consequent does follow from the antecedent, but the antecedent does not follow from the consequent. Rather, the equation ξ–cτ = λ(x–ct), where λ is constant, belongs to an infinite set of various equations all of which satisfy the condition expressed by the consequent. However only this particular equation (the antecedent) can provide Postulates 3 and 4 in the form of corollaries and Einstein resorts to the fallacy of affirming the consequent in order to "deduce" it. Otherwise he would have to start the deduction of Lorentz transforms from four explicit postulates and the sadness in the relativity cult might increase dramatically.

No more problems with the "deduction" of Lorentz transforms from two simple postulates – it remains one of Einstein’s greatest achievements forever. But are Lorentz transforms true? More precisely, are the corollaries of these transforms true? Consider a rest frame in which, as a stick passes by at constant speed, two knives simultaneously (in the rest frame) leave marks on the stick. If, in the rest frame, the distance between the knives is L, the marks on the stick will be a distance γL apart when viewed in the stick’s frame. This distance (γL) is length-contracted down to the distance L in the rest frame, in accordance with Lorentz transforms. The example is trivial and can be found in textbooks.

What if the knives are somewhat more decisive and, instead of just leaving marks on the stick, cut it up so that a part of the stick remains trapped between the knives? How long will be this part? Its length cannot be γL since γ > 1 and the knives are only a distance L apart. But the length of the trapped part cannot be L either since, once measured to be γL in the stick’s frame, it must remain so in that frame despite the fact that the relative speed of the two frames has become zero (the proper length does not depend on the relative speed). In a sane science this would be called reductio ad absurdum and the premise producing the contradiction (Lorentz transforms) would be rejected. Luckily (for relativists), there is no such nuisance as reductio ad absurdum in the relativity cult.

True or false, Lorentz transforms predict symmetrical time dilation. This means that, as two inertial frames pass one another, the observer in either of them sees clocks in the other run slow relative to his own clocks. Yet that is not the only metamorphosis of time in Einstein’s 1905 paper. Apart from symmetrical time dilation, Einstein introduces asymmetrical time contraction. If one of the frames is at rest whereas the other moves with constant velocity in a closed curve, the observer in the moving frame (as well as the observer in the frame at rest) sees clocks in the frame at rest running fast (relative to clocks in the moving frame) precisely by a factor of γ. Why? This question may seem difficult but Einstein has offered a simple answer (see the end of Section 23 in his “Relativity”) that any relativist should find absolutely satisfactory: because the moving frame is not inertial. If Einstein was somewhat fairer, since neither the existence nor the magnitude of the time contraction factor γ follow from “because the moving frame is not inertial", his 1905 paper would contain the following explicit postulate:

Postulate 5: When one of the frames is not inertial and its relative speed is constant as it moves in a closed curve (possibly with the exception of very short accelerating intervals where the speed is not constant), the original time dilation characteristic of clocks in an inertial frame turns into a time contraction. Clocks in the inertial frame run fast precisely by a factor of γ with respect to clocks in the non-inertial frame, and γ is absolutely independent of the type and magnitude of the accelerations (gravitational fields) experienced in the non-inertial frame.

In "Time and the Space-Traveller" L. Marder describes mechanisms that make the running of clocks in the non-inertial frame dependent on the acceleration. Yet, in accordance with Einstein’s 1905 arbitrariness (embodied in Postulate 5), even if the running of clocks in the non-inertial frame depends on the acceleration, clocks in the inertial frame still run fast precisely by a factor of γ.

Let us now test the postulate of constancy of the speed of light. More precisely, we are interested in the value an observer (receiver) on the ground obtains as he (indirectly) measures the speed of light emitted by the top of a tower with a height h. If the equivalence principle is correct, the same value must be obtained by a receiver at the back end of a rocket with length h as he measures the speed of light emitted by the front end (it is assumed that at the moment of the emission the rocket is at rest and then accelerates with constant acceleration g). The experiment at the tower has been performed (in 1960) and the frequency measured by the receiver has been found to obey the formula

f_r = (1 + gh/c²)f_s (1)

where f_s and f_r are the frequencies of light at the source (the top of the tower or the front end of the rocket) and the receiver respectively. Given the experimentally confirmed formula (1), the receiver must now calculate the speed of the received light.

The receiver is going to use the formula f_r = V/λ, where V is the relative speed of the light and the receiver (V is expected to be either c or different from c) and λ is the wavelength. He also knows that, at the moment the light reaches the back end of the rocket, this end (together with the receiver at it) will have a speed v=gh/c relative to the original frame at rest. Substituting both formulas, as well as the obvious f_s = c/λ, in (1) yields

V = c + v (2)

That is, the receiver measures the speed of light to be variable rather than constant (it depends on the speed of the receiver). Einstein’s postulate of constancy of the speed of light is false.

Of course, we have implicitly assumed that the wavelength, λ, does not change as the light travels between the source and the receiver. This should be obvious since, especially in the rocket, there is no physical factor at all that could be responsible for such a change. Of course, relativists are not very interested in the physical meaning of concepts. Still, now that the postulate of constancy of the speed of light is at stake, they should immediately introduce:

Postulate 6: As the light travels in a vacuum its wavelength constantly decreases so that, in the particular case where the light covers the distance between accelerating source and receiver, against the direction of the acceleration, its wavelength changes in accordance with λ_r=λ_s/(1+gh/c²).

This postulate is just as important as the postulate of constancy of the speed of light. It can even be regarded as an integral part of the postulate of constancy of the speed of light so far as the latter automatically becomes false if the wavelength does not change in the way prescribed by Postulate 6. However the great importance of Postulate 6 does not interfere with the obvious fact that this postulate is absurd. If the light was travelling in the direction of the acceleration, another postulate would require the wavelength to increase.

In Appendix 3 in his "Relativity" Einstein deduces the redshift factor essentially from the concept of time contraction, that is, from Postulate 5. If our deduction of the redshift factor from the hypothesis that the speed of light is variable is valid, Postulate 5 turns out to be inconsistent with the postulate of constancy of the speed of light (rather, Postulate 5 looks consistent with the negation of the postulate of constancy of the speed of light affirming that the speed of light is variable). This could have been suspected even earlier: the principle of constancy of the speed of light is consistent with the concept of symmetrical time dilation whereas Postulate 5 introduces, in accordance with the respective text in Einstein’s 1905 paper, asymmetrical time contraction. It makes sense to check this inconsistency once more in so far as Postulate 5 plays a crucial role in the development of general relativity.

The concept of time dilation has its classical example of light travelling vertically on a train. A light source on the floor of the train emits a flash of light. The light travels up to a mirror on the ceiling and then back to the source, thus forming a "tick" of a light clock. From the ground frame, it takes longer for the light to make the roundtrip because the path of the light, as judged from the ground frame, is longer (whereas the speed of light is postulated to be the same in both frames). Therefore, as the light clock situated on the train passes two light clocks situated on the ground, the following scenario is plausible. When the train clock passes the first ground clock, they are both set to zero. Then the second ground clock is synchronised with the first. Finally, as the train clock passes the second ground clock, it reads less than the second ground clock.

Will this scenario remain relevant if the clocks in the ground frame are replaced by clocks situated on the periphery of a rotating disk and their readings are compared with the reading of a standing-by clock (corresponding to the clock on the train)? If it will, we have a contradiction and Postulate 5 will prove inconsistent with the postulate of constancy of the speed of light. The reason is obvious: the light-travelling-vertically-on-a-train scenario predicts symmetrical time dilation whereas Postulate 5 postulates asymmetrical time contraction (the standing-by clock must run faster than clocks on the periphery of the rotating disk).

At first sight, the light-travelling-vertically-on-a-train scenario cannot remain relevant for the rotating disk – the periphery of the disk is not an inertial frame and the synchronisation of clocks situated on it is problematic. Yet there is a procedure allowing the rotating periphery to approach "inertiality" as closely as desirable. As we increase the diameter of the disk while keeping the linear velocity of the periphery constant, any arc with a fixed length approaches a straight line with zero gravitational field – that is, the arc becomes an inertial frame. Accordingly, clocks situated at the ends of the arc can be synchronised and, compared with them, the standing-by clock will undergo time dilation, in accordance with the light-travelling-vertically-on-a-train scenario and the postulate of constancy of the speed of light. On the other hand, the development of general relativity (in particular, the deduction of the redshift factor in Appendix 3 in Einstein’s "Relativity") strictly requires the standing-by clock to undergo time contraction, in accordance with Postulate 5.

There are theories for which the coexistence of inconsistent concepts is essential. If, in an attempt to restore consistency, one of the inconsistent concepts is abandoned, the theory will simply die.

A concluding remark devoted to the ethics of relativists. The concept of time contraction, just as the postulate of constancy of the speed of light, is sacred in relativity – nothing can force relativists to doubt that the standing-by clock will run fast by a factor of γ relative to clocks fixed on the periphery of the rotating disk. Yet, apart from this particular “absolute truth”, any application of the theory of relativity to rotating systems leads to contradictory and confusing conclusions (see http://edu.supereva.it/solciclos/gron_d.pdf or http://arxiv.org/PS_cache/physics/pdf/0404/0404027.pdf ). Initiated relativists even go as far as to say that, with respect to such applications, rotating systems are "inherently ambiguous". How long is the periphery of the rotating disk as judged from the non-rotating frame? Ehrenfest says the periphery is shorter than 2πR, Einstein says it is longer than 2πR, others demonstrate their moderation by saying that the length of the periphery is equal to 2πR. A fourth party build careers by writing long papers where the numerous confusions and contradictions are “disentangled". And no relativist would ever find it suitable to ask the simple question: If any application of relativity to rotating systems is contradictory and confusing, who gives us the right to regard the claim that a non-rotating clock runs fast by a factor of γ as nothing short of an absolute truth?