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 10

In a hundred years you can tax them saidFaraday.The scientific impact of Maxwell’s equations was equally great, althoughalso delayed. When the equations were closely studied – by H.A. Lorentzand A. Einstein – it turned out that the main set, shown in Fig. 2.8, isinvariant under any space-time transformation whatsoever, while the etherrelations are invariant only under Lorentz transformations, see below.The true nature of the Maxwell equations as conservation laws of chargeand magnetic flux was identified even later by Gustav Adolf FeodorWilhelm Mie (1868–1957).59 Mie put Lorentz’s and Einstein’s transformation rules into an elegant four-dimensional form.

This crowningachievement in electro-magnetism is reviewed by Claus Hugo Hermann57J.C. Maxwell: “On Faraday’s lines of force.” Transactions of the Cambridge PhilosophicalSociety, X (1856).J.C. Maxwell: “On physical lines of force” Parts I and II, Philosophical Magazine XXI(1861), parts III and IV, Philosophical Magazine (1862).J.C.

Maxwell: “A dynamical theory of the electro-magnetic field” Royal SocietyTransactions CLV (1864).58 J.C. Maxwell: “Treatise on electricity and magnetism” (1873).J.C. Maxwell: “An elementary treatise on electricity” William Garnett (ed.) (1881).59 G. Mie: “Grundlagen einer Theorie der Materie” [Foundations of a theory of matter]Annalen der Physik 37, pp.

511-534; 39, pp. 1-40; 40, pp. 1–66 (1912).342 EnergyWeyl (1885–1955)60 and I shall give the briefest possible summary, cf.Insert 2.3. This will help us to appreciate the eventual recognition of energyas mass, or of mass as energy.Transformation properties of electro-magnetic fieldsThe most appropriate formulation of electro-magnetism is four-dimensional so thatAx (A = 0,1,2,3) equals (t,x1,x2,x3) where t is time and xi are Cartesian spatialcoordinates of an event. If we introduce the electro-magnetic field tensor ij and thecharge density vector ı asM ABª 0«E« 1«E2«¬E 3 E1 E20B3 B3B20 B1 E3º B2 »»B1 »»0 ¼VandA(q, J 1, J 2 , J 3 )the local form of the conservation laws of magnetic flux and of charge readw M CDwV AH ABCD0and0.BAwxwxThe latter is formally solved by settingABVAwKwx B0,whereK ABª 0« D1«« D2«¬ D3D10D2H3 H3H20H 1D3 º H2»»H1 »»0 ¼is called charge-current potential.

For that reason D and H are also known ascharge potential and current potential, respectively, as well as by the earlierconventional names dielectric displacement and magnetic field.Upon inspection the underlined equations are the general Maxwell equations ofFig. 2.8 which are thus recognized as conservation laws of magnetic flux andABcharge61 respectively. If ijAB are covariant components and Ș contravariant ones, asindicated by the customary position of the indices, we have for any arbitrary spaceAABtime transformation xƍ = xƍ (x )cM CDwx A wx BM ABwx cC wx c DandK c CDw x c C w x c D AB ,Kwx A wx Band therefore the general Maxwell equations retain their forms in all frames.60H. Weyl: “Raum-Zeit-Materie” [Space-time-matter] Springer, Heidelberg (1921) Englishtranslation: Dover Publications, New York (1950).61 For the integral form of these equations of balance the reader might consult I.

Müller:“Thermodynamics” Pitman, Boston, London (1985) Chap. 9. Another instructive accountof Mie’s and Weyl’s treatment of electrodynamics and relativity may be found in thememoir by C.A. Truesdell and R. Toupin: “The classical field theories” Handbuch derPhysik III/1 Springer.

Heidelberg (1960). pp. 660–700 and 736–744.Albert Einstein (1879–1955)35In particular the transformation rules of E and B readE„i˜x A ˜x Bϕand B „i˜t „ ˜x „ ABi1˜x A ˜x Bεϕ2 ijk ˜x „ ˜x „ ABjkThis defines the components Ei ƍ and Bi ƍ in all frames. Similarly Di ƍ and Hi ƍ can becalculated from Di and Hi.Once the transformation laws of E, B, D, H are known, we may ask for thetransformations that leave the ether relations D = İ0 E and H = µ0 B invariant.

Itturns out that these are Lorentz transformations, see below.Insert 2.3Albert Einstein (1879–1955)Mayer’s haphazard collection of forces – fall force, motion, tensile force,heat, magnetism, electricity, and force of chemical separation, three of themimponderables, cf. Fig.

2.9 – were now confirmed, actually within Mayer’slifetime as different types of energy: potential, kinetic, elastic, internal,electro-magnetic, and chemical respectively. And energy as a whole wasrecognized as being conserved, when one type changed into another one.This was a great step of unification, and to a new generation of physicistsenergy became a familiar concept, like mass, or momentum, which werealready well-established conserved quantities of old. In some way all typesof energy had to be considered imponderable, because a compressed spring(say) did not seem to weigh more than a relaxed one.But then it turned out – through the work of Einstein – that energy E andmass m were the same; or rather they were two quantities strictly related toeach other by the equationE = m c2 ,where c is the speed of light. Thus, if energy is mass, and since mass hasweight, now it turned out that all energies were ponderable.Indeed, if a body has potential energy or kinetic energy, it is only becauseits mass is bigger at a height, or when it moves.

A compressed springweighs more than a relaxed one. And, if a body is hot, it is also heavier thanif it were cold, because its particles have a bigger speed in the mean. If twoatoms are bound together chemically – so that their potential energy issmaller than when they are apart – they have a smaller mass.362 EnergyFig. 2.9. Mayer’s collection of forcesTo be sure, the factor of proportionality c2 between E and m is so big, andthe energy differences are so small, that the mass- and weight-changes in allmentioned cases are too small to be detected.

However, this is not so whennuclear forces are involved. Thus the nuclear force between protons andneutrons in a He4 nucleus – an Į-particle – is so strong, and the bindingenergy is so large, that there is an appreciable mass defect: Namely, themasses of two protons and two neutrons are 2·1.67239·10–27g and2·1.67470·10–27 g respectively and the mass of the Į-particle which theyform is 6.64373·10–27 g; consequently there is a mass defect of 0.76% andthat is quite noticeable.The introduction of a “luminiferous ether” willprove to be superfluous inasmuch as the viewhere to be developed will not require an“absolute stationary space” provided with specialproperties, nor assign a velocity-vector to a pointof the empty space in which electromagneticprocesses take place.62Fig.

2.10. Albert Einstein. Dismissal of the ether62A. Einstein: “Zur Elektrodynamik bewegter Körper” [On the electrodynamics of movingbodies] Annalen der Physik 17 (1905) Translation of 1923 in: “The principle of relativity,a collection of original memoirs of the special and general relativity” W. Perrett, G.B.Jeffrey (eds.) Dover Publications. Introductory remarks.Lorentz Transformation37It is true though, that this phenomenon had not yet been noticed in theyear 1905, when Einstein presented his paper on what we now call SpecialRelativity.63 That paper must concern us at this point, because it establishesthe relation between energy and mass, which was later used to explain themass defect, after that phenomenon had been detected. The formula E = mc2came up in the paper at the very end, almost as an afterthought, andcertainly not at all with the fanfare which it deserves for being the mostimportant equation of physics, as which we now recognize it.

Actually, themain issue of Einstein’s paper was not mass or energy at all, but ether andabsolute space. We have to digress in order to explain.Lorentz TransformationAt that time, the beginning of the 20th century, the universe was supposedto be filled with ether – the luminiferous ether – through which lighttravelled with the speed c. The ether was supposed to be at rest in absolutespace, and all bodies moved through the ether without disturbing its state ofrest; so also the earth and the sun.

The question arose whether the speed ofthe earth through the ether – the absolute speed, as it were – could bemeasured, and that was the question asked by Albert Abraham Michelson(1852–1931), first alone and then in collaboration with Edward WilliamsMorley (1838–1923). They sent out a light ray to a mirror at the distance Land measured the time interval before it returned. If the earth, and the lightsource, and the mirror moved with speed V through the ether, the biggesttime interval should have been64'tLLc V c V2L 12 .c 1 Vc2In the experiment, however, the interval was found to be 2cL , irrespectiveof the direction of the ray, just as if the earth were at rest with the etherwhich, of course, was unlikely to such a degree that the possibility was notseriously considered.65 So, the experiment showed that the speed of light isindependent of the motion of the source.63A.

Einstein: “Zur Elektrodynamik …” loc. cit.The time interval should have depended on the angle between the light ray and thevelocity of the earth. The biggest interval would occur, if that angle were zero.65 The actual details of Michelson’s measurement are ingenious and cumbersome, because itis not easy to measure ǻt. For details the reader may consult Michelson’s papers which,incidentally, earned him the Nobel prize of physics in 1907.A.A. Michelson: “The relative motion of the earth and the luminiferous ether.” AmericanJournal of Science 22 (1881), p. 122.64382 EnergyAmong the attempts of an explanation there was one that turned out to beheuristically important: George Francis FitzGerald (1851–1901) suggested,in 1890, or so, that distances in the direction of the ether wind – theonrushing ether – should be shortened so as to offset the discrepancybetween the expected and the measured results of Michelson’s experiments.Hendrik Anton Lorentz (1853–1928) made the same assumption in 1895.66Lorentz expounds on it by speculating about the influence of the ether onthe action between two molecules or atoms [so that] there cannot fail to bea change of dimension as well.67Einstein does not mention Michelson, but he accepts his experimentalresult when he speaks about the unsuccessful attempts to discover anymotion of the earth relatively to the “light medium”.68 And he does notattempt to explain Michelson’s failure by speculating about the ether; hesimply proceeds to identify the transformation of spatial and timecoordinates, that is required for two frames K and Kƍ in uniform relativetranslation, if the speed of light equals c in both.