Alma Mater Studiorum Universita` di · Bologna DOTTORATO DI RICERCA IN Matematica Ciclo XXV Settore concorsuale di afferenza: 01/A4 Settore Scientifico disciplinare: MAT/07 Max Abraham’s and Tullio Levi-Civita’s approach to Einstein Theory of Relativity Presentata da: Michele Mattia Valentini Coordinatore Dottorato: Relatore: Chiar.ma Prof.ssa Chiar.mo Prof. Giovanna Citti Sandro Graffi Esame finale anno 2014 Introduction This thesis deals with the theory of relativity and its diffusion in Italy in the first decades of the XX century. Albert Einstein’s theory of Special and General relativity is deeply linked with Italy and Italian scientists. Not many scientists really involved themselves in that theory understanding, but two of them, Max Abraham and Tullio Levi-Civita left a deep mark in the theory development. Max Abraham engaged a real battle against Ein- stein between 1912 and 1914 about electromagnetic theories and gravitation theories, while Levi-Civita played a fundamental role in giving Einstein the correct mathematical instruments for the general relativity formulation since 1915. Many studies have already been done to explain their role in the develop- ment of Einstein theory from both a historical and a scientific point of view. This work, which doesn’t have the aim of a mere historical chronicle of the events, wants to highlight two particular perspectives. 1. Abraham’s objections against Einstein focused onthree important con- ceptual kernels of theory of relativity: the constancy of light speed, the relativity principle and the equivalence hypothesis. Einstein was forced by Abraham to explain scientific and epistemological reasons of the formulation of his theory. In that occasion Einstein gave also rig- orous logical structure to the reasoning sequence appeared in different instants in the papers published in those years. The possibility of tak- ing la careful look at the basics of the theory explained by the author i ii himself makes the contents of this contrast an exceptional resource for understanding Einstein’s thinking. 2. Tullio Levi-Civita, after some years of waiting and seeing, accepted to be involved by Abraham himself in the discussion about Einstein the- ory of relativity since 1915. As known, Levi-Civita’s involvement gave Einstein the possibility to correct some errors and come to a definite version of his theory using the absolute differential calculus by Ricci and Levi-Civita himself. Levi-Civita’s contribution to the theory of Relativity has already been analyzed by many authors from both a scientific and a historical point of view. The aim of this work is to underline that Levi-Civita’s partic- ular approach to relativity was a significant interpretation of the same theory, a little different from the original one in some aspects. Levi-CivitabegantoinvolvehimselfinRelativity, whenEinsteinhadal- ready published the first Entwurf of General Relativity. In that period the research of an invariant form for the gravitational field equations led physicists and mathematicians to propose a modification of Hamil- ton variational principle in order to make it invariant with respect to every coordinates transformation. We are going to show that this kind of approach, shared by many authors in facing General Relativity, is historically proposed by Levi-Civita as key even to Special Relativity. The first chapter presents Relativity spread in Italy just from a historical point of view, highlighting its main characters and events.These historical remarks will allow us to have a correct background in order to understand the meaning of the scientific themes discussed in the following chapters. The second chapter presents the analysis of the three conceptual kernels of Max Abraham’s objections against Einstein’s theory of Relativity men- tioned before. The scientific nodes of the debate are analyzed from 1905 to 1914, through an exam of the original articles. The aim of this part of the work is mainly Introduction iii to compare authors ideas as they were originally explained. The third chapter focuses the attention on Levi-Civita’s interpretation of Special Relativity. Levi-Civita played a fundamental role in giving a correct form to General Relativity. This special kind of approach made him formu- late Special Relativity in a very original way, as it arises from his lecture entitled How can a conservative get to the threshold of new mechanics. In this work part a deep analysis of the contents of this lecture is made together with the attempt to reconstruct the mathematical steps that were neglected. Intheconclusionweoutlinesomegeneralreflectionswhichcouldbefuture research perspectives. Contents Introduction iii 1 Historical Remarks 1 2 Max Abraham’s objections against relativity 7 2.1 Scientific preliminary remarks . . . . . . . . . . . . . . . . . . 8 2.2 The constancy of the velocity of light . . . . . . . . . . . . . . 11 2.3 The principle of relativity and the aether . . . . . . . . . . . . 15 2.4 The equivalence hypothesis and the gravitation . . . . . . . . 24 2.5 Conclusive remarks . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Relativity in Levi-Civita’s thinking 30 3.1 The contents of the lecture “How could a conservative get to the threshold of new mechanics” . . . . . . . . . . . . . . . . . 33 3.1.1 Hamilton principle . . . . . . . . . . . . . . . . . . . . 33 3.1.2 Hamilton principle invariance with respect to general transformation of coordinates . . . . . . . . . . . . . . 35 3.1.3 Introducing time change into Hamilton principle . . . . 37 3.1.4 Loss of Hamilton principle invariance with respect to a general transformation of time-dependent coordinates 42 3.1.5 Hamilton principle modification in order to obtain in- variance with respect to a general transformation of coordinates involving time . . . . . . . . . . . . . . . . 44 iv Contents v 3.1.6 Lorentz transformations as a particular application of the previous method . . . . . . . . . . . . . . . . . . . 47 3.1.7 The optics problem . . . . . . . . . . . . . . . . . . . . 49 3.1.8 Confluence of optics and mechanics . . . . . . . . . . . 51 3.1.9 Light rays curving under the action of masses . . . . . 52 3.1.10 Equations of mathematical physics corrected by Ein- stein i.e. Restricted relativity . . . . . . . . . . . . . . 53 3.1.11 Influence of allthephysics phenomena onthemeasures of space and time i.e General Relativity . . . . . . . . . 55 3.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2 3.2.1 On the meaning of ds invariance . . . . . . . . . . . . 58 3.2.2 The hidden postulates of the Einstein relativity . . . . 58 3.2.3 Differences from Einstein theory of relativity . . . . . . 59 3.2.4 Uniqueness of Levi-Civita’s reasoning . . . . . . . . . . 60 Conclusion 61 Bibliography 64 Chapter 1 Historical Remarks It’s useful to give some historical background of the Italian Relativity reception before entering into details of Max Abraham and Levi-Civita role 1 in it. One of the first traces of Italian diffusion of Relativity is 1906 Roberto Marcolongo’s paper about Lorentz transformations [? ]. It witnesses that the mathematicians and not the physicists were the first to give attention to the theory of Relativity. In 1907 Physics Review entitled “Le recenti teorie elettromagnetiche e il moto assoluto” by Orso Mario Corbino [5], Einstein theory is not even mentioned among the contributions about moving elec- trons behavior. According to Corbino, the experiments which had revealed the existence of the corpuscles called electrons, led, as consequence, to suppose electrons hav- ing “una massa apparente di natura elettromagnetica, variabile col cambiare 2 della velocit`a”. Lorentz theory of deformable electron and Abraham theory of rigid electron were the two theories which contended these phenomena explanation. 1For a more complete chronicle of the Relativity events in Italy see [1], [2] and [3]. About the same arguments see also [4] 2“ apparent electromagnetic mass, changing with speed” 1 Historical remarks 2 In 1911, Tullio Levi-Civita, describing Italian mathematical physics evo- lution [6] between 1860 and 1910, touched on mechanics basics critique by Einstein, writing: c’`e anche in meccanica la tendenza rivoluzionaria. Tale apparisce agli ortodossi seguaci di Newton e Lagrange quella che, in nome del principio di relativit`a di Lorentz-Einstein, `e condotta a fondere i con- cetti dispazio editempoe anegarel’invariabilita` dellamassa. Siren- derebbeinconseguenzanecessariaunaricostruzioneab imis dituttala filosofianaturale. Attendiamopergiudicare. Bastiintantoriconoscere l’importanza dell’attuale movimento relativista e l’influsso innovatore 3 che esso va suscitando Levi-Civita is taking time, but recognizes Einstein relativity principle to beinherentlyuntouchable, evenifitbringsusualwaysofthinkinginquestion. He stays in such a waiting attitude until Max Abraham, become Rational Mechanics professor at the Milan Politecnico in 1909, started a real contro- versy against Albert Einstein about Special Relativity contents. Even if with criticism, Max Abraham’s objections against Einstein theory contribute to spread Special Relativity contents on Italian journals. How- ever the number of Italian scientists really interested in debate’s contents is very small. The debate mentioned, developed between 1912and 1914, will be analyzed in the next chapter with big attention, because all the conceptual kernels of Einstein theory are discussed, clarified and developed in it. During those years, Einstein has published, among the other important 4 papers , thefamousEntwurf[12], writtenincollaborationwithMarcelGross- 3There’s a revolutionary tendency in mechanics too. That is the one which leads to merge space and time concepts together and to refuse mass invariance, in Lorentz- Einstein relativity principle’s name. This idea appears to be revolutionary to Newton andLagrange’sdisciples.Asconsequenceallthenaturalphilosophyhasto bebuiltab imis again. Let’s wait and see. It suffices, for now, to acknowledge the relativistic movement importance and the innovator influence it’s raising. 4See [7], [8], [9], [10], [11] Historical remarks 3 man. That is the nearest model to the General Einstein theory, in which he defined the mathematical structure and the physical basics almost com- pletely. In this paper Einstein and Grossman considered the possibility of using the Ricci curvature scalar as the Hamilton function inside a suitable variational principle and obtaining field equations generally covariant. Even in the ex- 5 positions and explanations published in 1914 Einstein kept this variational approach, extending the invariance group of gravitational equations to the maximum. The choice of the correct Hamilton function to be put as the basis of the variational principle was the reason of the last Abraham attack against Einstein. In 1915, in fact, Abraham was sent off from Italy because of his German nationality. Levi-Civita was involved by Abraham in this controversy. So he decided to devote in person in the theory and started a correspondence with Einstein which became of fundamental historical importance. OnlythankstoLevi-Civita’shelp, Einsteinmanagedtosolvetheproblems of his General theory. Einstein and Levi-Civita’s collaboration continued for some years, even with moments of controversy and independent work. Many studies have already been done about this collaboration and its importance, so it won’t be examined in depth here. It’s important to underline, instead, that, since 1917, while Levi-Civita tookpart as a protagonist at the most important international debates about Einstein theories, in Italy, with the war ongoing, every German expression is totally refused. That is the case of theory of Relativity. The mathematical physicists, Roberto Marcolongo, Gian Antonio Maggi e Tommaso Boggio were the only ones interested in it, even if with very differ- 5See [? ] and [? ] Historical remarks 4 ent attitudes and approaches. All of them had Levi-Civita’s works as point of reference and approached the theory of Relativity directly as a general gravitation theory. Despite of hate for Germans, the theory of Relativity spread among people scientifically closest to Levi-Civita. Attilio Palatini was one of the most bril- liant of them. In 1919 Palatini, influenced by Levi-Civita’s work, published a paper[13] about the deduction of the field equations from an invariant variational principle. He criticized Einstein methods, judging them as un- satisfactory because they weren’t invariant at every step of reasoning and proposed his own method. In the same year (1919) a series of treatments was organized in Rome, at the Mathematical Seminar of the Science Faculty of Rome University, di- rected by Vito Volterra, in order to explain the basics of Relativity and of the mathematical instrument it used, that is the absolute differential calculus by Ricci and Levi-Civita. The series of treatments was mainly organized by Castelnuovo, who invited Levi-Civita and Marcolongo to speak. Levi-Civita was ordered to do the introductory lecture in order to attract even the new mechanics opponents’ attention, while the two following lectures, which presented respectively the contents of Special and General Relativity were committed to Marcolongo. The particular aim of Levi-Civita’s lecture makes it extremely interesting from both a scientific and historical point of view. It’s an introduction to the theory of Relativity in which the author tried to omit the revolutionary nature of Einstein physics postulates, deciding to obtain Relativity results through quantitative modifications of classic Hamilton principle. This lecture will be analyzed in details in the third chapter in order to highlight its innovative and, in some aspects, unique nature. This series of treatments is very important even because it’s the act which