Müller I. A history of thermodynamics. The doctrine of energy and entropy (Müller I. A history of thermodynamics. The doctrine of energy and entropy.pdf), страница 11

The problem is somewhatsimplified – but not oversimplified – by the assumption that the frames haveparallel axes and that their relative motion, with speed V, is along the x-axis.For that case Einstein obtainsx1 Vtx1c2x2c,1 Vc 2x1x1c Vt c21 Vc 2,x2 ,x2x 2c ,x3cx3 , t cx3x3c , ttVc2x12, or inversely1 Vc 2tc Vc2x1c2.1 Vc 2This is the Lorentz transformation, so called, because Lorentz69 derived itfrom the requirement that the Maxwell equations of electro-magnetismA.A.

Michelson, E.W. Morley: “Influence of motion of the medium on the velocity oflight” American Journal of Science 31 (1886), p. 377.66 H.A. Lorentz: “Versuch einer Theorie der elektrischen und optischen Erscheinungen inbewegten Körpern” [Attempt of a theory of electrical and optical phenomena in movingbodies.] Leiden 1895 §§89–92. Translation of 1923 in: “The principle of relativity, ...”under the title “Michelson’s interference experiment” Dover Publications.

loc.cit.Lorentz acknowledges FitzGerald’s priority grudgingly by saying: As FitzGerald kindlytells me, he has for a long time dealt with his hypothesis in his lectures. The thenhypothetical phenomenon became known as the FitzGerald contraction, but is more oftencalled the Lorentz contraction.67 I believe that Lorentz fools himself here. Indeed in Michelson’s experiments the rodcarrying the light source and the mirror were of brass and stone in different experiments;it seems quite inconceivable that the ether would have affected both materials in the samemanner.68 A. Einstein: “Zur Elektrodynamik...” loc.cit.69 H.A.

Lorentz: “Electro-magnetic phenomena in a system moving with any velocity lessthan light.” English version of Proceedings of the Academy of Sciences of Amsterdam, 6(1904). Reprinted in: “The principle of relativity, …” Dover Publications. loc. cit.Lorentz Transformation39should have the same form in all uniformly translated frames. Einstein doesnot mention Lorentz except in a later reprinting of his papers, where he saysin a footnote: The memoir by Lorentz was not at this time known to theauthor [i.e.

Einstein]. 70For Vc  1 the Lorentz transformation becomes the Galilei transformation of classical mechanics. But generally, for higher velocities, it differsfrom the Galilei transformation subtly, and in a manner difficult to grasp intuitively. Let us consider this:A sphere of radius R at rest in frame Kƍ with the centre in the origin has the surfacexƍ12 + xƍ22 + xƍ32 = R2. According to the Lorentz transformation that sphere, seenfrom the frame K, has the surface of an ellipsoid with a contracted axis in thedirection of the motion, viz.2x122 x32 x21 Vc 2R2.Conversely a sphere at rest in K is given by x12 + x22 + x32 = R2, but viewed fromframe Kƍ it appears as the ellipsoid2x 1c22 x 2c x 3cV21 c2R2.Thus, according to Einstein, no frame of absolute rest exists; it is therelative motion of the frames that is responsible for the contractions.

Noether is mentioned and no suggestive explanation is offered. This is cold comfortfor people who understand and argue intuitively. Einstein presents reason, pure andundiluted, a mathematical deduction from a convincing observation, that is all, – nospeculation.What happens with time intervals is even more counter-intuitive: Let there be twoevents at some fixed point with xƍ1 which are apart in time by ǻtƍ in frame Kƍ. By1the Lorentz transformation the interval is equal to 't't c ! 't c .21 V 2cThus the observer in K will see the time interval lengthened, a phenomenon that isknown as time dilatation. The phenomenon is often discussed in scientificfeuilletons as giving rise to the twin paradox: Twin 1 remains at home – at a fixedplace xƍ1 – while twin 2 goes on a long trip with high speed along the x1-axis andthen returns, again with high speed.

His heart beat is lengthened by the timedilatation and therefore his metabolism is slowed down, so that after hisreturn he is still a young man, while his brother, twin 1, has aged. That observation is amazing, and strange, but not paradoxical yet. The paradox appears when70I have never been able to see Einstein’s papers in the Annalen der Physik, becausewhenever I looked for them – in the libraries of several countries – they were stolen; cutout, or torn out, the ultimate accolade! But then, the papers have been reprinted manytimes and some re-printings carry footnotes by Einstein, so also the Dover publicationcited above. That is a good thing, because some of the footnotes are quite illuminating.402 Energywe realize that both twins are in relative motion.

Thus twin 2 remains firmly atsome point x1 and considers his brother as travelling – relative to him. The intervalbetween heart beats of twin 2 is therefore't21 Vc 2 't c  't c in his frame K,so that he has aged, while twin 1 is still young after the return. That is a genuineparadox, if there ever was one.71Eleven years later, in 1916, Einstein would declare himself not entirelysatisfied with the arid reasoning exhibited in his work on special relativity.At the beginning of his memoir on general relativity he says:72 In classicalmechanics, and no less in special relativity, there is an inherent epistemological defect which was, perhaps for the first time, clearly pointed outby Ernst Mach.

It is not enough to state that uniformly moving frames –inertial frames – are special; we should like to know what makes them so,irrespective of whether they are related by Galilei- or Lorentz-transformations. Einstein explains that he sees distant masses and the motion offrames with respect to those as the seat of the causes for the phenomenaoccurring in frames. Thus non-inertial frames feel gravitational forces fromthe distant masses, while inertial frames feel no effect at all, – and thatdefines them.E = m c2Maxwell’s ether relations are invariant under Lorentz transformations,73while the general set of Maxwell equations in Fig.

2.8 is generally invariant,against all analytic transformations, see above. Einstein felt that there was aproblem, because Newton’s equation – the basis of mechanics – are Galileiinvariant. He says somewhat awkwardly: 74 … the laws of electrodynamics… should be valid for all frames of reference for which the equations ofmechanics hold good. We will raise this conjecture the purport of whichwill hereafter be called “Principle of Relativity” to the status of a postulate.Since electrodynamics was trustworthy – not least because of Michelson’s71I have been told that the twins will turn out to be equally old after their reunion when theinevitable periods of acceleration at the beginning, middle and end of the trip are takeninto account.

And, of course, that acceleration is only suffered by the twin who reallytravels. Accelerations are the subject of the general theory of relativity, and we shall notgo into this any further.72 A. Einstein: “Die Grundlage der allgemeinen Relativitätstheorie” Annalen der Physik 49(1916). English translation: “The foundation of the general theory of relativity” in: “Theprinciple of relativity, …” Dover Publications. loc. cit.73 The invariance of the speed of light in Lorentz frames is, of course, a corollary of theinvariance of the ether relations.74 A.

Einstein: “Zur Elektrodynamik ...” loc.cit.E = m c241experiment – mechanics had to be modified so as to become Lorentzinvariant. The question was: How?Mechanics and electrodynamics are largely separate, of course, but theydo have points of contact, like when a moving charge e is accelerated by anelectro-magnetic force Fi in an electric field Ei and a magnetic flux densityBi. This force is called the Lorentz force and we havedx j Ødx „j ØÈÈe É Ei ε ijkBk Ù or Fi „ e É Ei„ ε ijkBk„dtdt „ ÚÙÊÚÊFiin frame K and Kƍ respectively. Thus Newton’s equations in K and Kƍshould readmd 2 x1dt 2Fi or m „d 2 x1„dt „ 2Fi „ ,and one should follow from the other one by a Lorentz transformation.

Itturned out that this requirement could not be satisfied, not even withdifferent masses m and mƍ as indicated in the equations. If, for simplicity,dxthe charge is at rest in Kƍ – so that its velocity in K equals ( dt1 ,0,0) – it ispossible to show, cf. Insert 2.4, that the Lorentz transformation from Kƍ toK givesmc11c2(dx1 2dt)3d 2 x1dt 2F1andmcd 2 x2,31 c12 ( dxdt1 ) 2 dt 2F2,3 .That result led Einstein to postulate a longitudinal mass for the x1direction and a transverse mass for the other two directions.75The distinction between two masses – a transversal and a longitudinalone – can be avoided.