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Progress in Mathematical Physics Volume 47 Editors-in-Chief Anne Boutet de Monvel, Université Paris VII Denis Diderot, France Gerald Kaiser, Center for Signals and Waves, Austin, TX, USA Editorial Board

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Progress in Mathematical Physics Volume 47 Editors-in-Chief Anne Boutet de Monvel, Université Paris VII Denis Diderot, France Gerald Kaiser, Center for Signals and Waves, Austin, TX, USA Editorial Board Sir M. Berry, University of Bristol, UK C. Berenstein, University of Maryland, College Park, USA P. Blanchard, University of Bielefeld, Germany A.S. Fokas, University of Cambridge, UK D. Sternheimer, Université de Bourgogne, Dijon, France C. Tracy, University of California, Davis, USA Einstein, Poincaré Seminar 2005 Thibault Damour Olivier Darrigol Bertrand Duplantier Vincent Rivasseau Editors Birkhäuser Verlag Basel Boston Berlin Editors: Thibault Damour IHÉS 35, Route de Chartres, Bures-sur-Yvette France Olivier Darrigol Université Denis Diderot (Paris 7) 2, place Jussieu Paris Cedex 05 France Bertrand Duplantier Service de Physique Théorique Orme des Merisiers CEA - Saclay Gif-sur-Yvette Cedex France Vincent Rivasseau Laboratoire de Physique Théorique Université Paris XI Orsay Cedex France Mathematics Subject Classification 83-03, 83A05, 83B05, 83Cxx, 60Hxx A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de . ISBN Birkhäuser Verlag, Basel Boston Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF Printed in Germany ISBN-10: ISBN-13: Contents Foreword ix Olivier Darrigol The Genesis of the Theory of Relativity 1 1 Maxwell s theory as it was Flashback: The optics of moving bodies Lorentz s theory Poincaré s criticism The Lorentz invariance Einstein s theory The inertia of energy Conclusions Short bibliography Clifford M. Will Special Relativity: A Centenary Perspective 33 1 Introduction Fundamentals of special relativity Einstein s postulates and insights Time out of joint Spacetime and Lorentz invariance Special relativistic dynamics Classic tests of special relativity The Michelson-Morley experiment Invariance of c Time dilation Lorentz invariance and quantum mechanics Consistency tests of special relativity Special relativity and curved spacetime Einstein s equivalence principle Metric theories of gravity Effective violations of local Lorentz invariance Is gravity Lorentz invariant? vi Contents 6 Tests of local Lorentz invariance at the centenary Frameworks for Lorentz symmetry violations Modern searches for Lorentz symmetry violation Concluding remarks References Jacques Bros and Ugo Moschella The Geometry of Relativistic Spacetime 59 Jacques Bros From Euclid s Geometry to Minkowski s Spacetime 60 Introduction and general survey On the use of geometry in mathematical physics and the concept of spacetime Geometry of description and geometry of representation The use of geometry in more than three dimensions Galilean spacetime as a geometry of representation of motion phenomenons Postulates and construction of Minkowski s spacetime The postulates and the light-cone structure of spacetime Simultaneousness revisited Space-ships flight: the anniversary curve Minkowskian (pseudo-)distance and the inverse triangular inequality: the twin paradox Spatial equidistance and the Lorentz contraction of lengths Lorentz transformations in the Minkowskian plane and two-dimensional Lorentz frames The four-dimensional Minkowski s spacetime; tetrads, Lorentz group and Poincaré group Accelerated motions and curved world-lines Curvilinear distances and the slowing down of clocks Minkowski s description of accelerations A comfortable trip for the Langevin traveler On the visual appearance of rapidly moving objects: Lorentz contraction revisited The Minkowskian energy-momentum space: E = mc 2 and particle physics Toward simple geometries of curved spacetimes References Ugo Moschella The de Sitter and anti-de Sitter Sightseeing Tour 120 Introduction Contents vii 1 An analogy: non-euclidean spaces of constant curvature The de Sitter universe Anti-de Sitter Epilogue References Philippe Grangier Experiments with Single Photons Back to the beginning: Einstein s 1905 and 1909 articles Quantum optics and the photon Using single photons: Quantum Key Distribution Single photon sources Coalescing photons En guise de conclusion : towards entangled photons on demand. 147 References Thibault Damour Einstein : His Approach to Physics On Einstein s Epistemology Einstein and Philosophy Hume, Kant, Mach and Poincaré Scientific Philosophy and Einstein s Conceptual Innovation Einstein and the Theories of Relativity Einstein and the Kantian Quantum A Crucial Conversation Waves Over Here, Quanta Over There! Einstein s Ghost Field, Born s Probability Amplitude, and Heisenberg s Uncertainty Relations A Watershed Moment Adventurers in Entangled Reality The Mouse and the Universe The Multiple World The Kantian Quantum References Albert Einstein On Boltzmann s Principle and Some Immediate Consequences Thereof Translation by Bertrand Duplantier and Emily Parks from the original German text into French and English 183 Bertrand Duplantier Commentary 194 General potential Moments of any order viii Contents Bertrand Duplantier Brownian Motion, Diverse and Undulating A brief history of Brownian motion Robert Brown and his precursors The period before Einstein William Sutherland, Albert Einstein, Marian von Smoluchowski Louis Bachelier Paul Langevin Jean Perrin s experiments Measurements by Brownian fluctuations Micromanipulation of DNA molecules Measurement of force by Brownian fluctuations Theory Potential theory and Brownian motion Introduction Newtonian potential Harmonic functions and the Theorem of the Mean The Dirichlet problem Relation between potential theory and Brownian motion Recurrence properties of Brownian motion The fine geometry of the planar Brownian curve The Brownian boundary Potential theory in a neighborhood of a Brownian curve Multifractality Generalized multifractality Foreword This book is the fourth in a series of lectures of the Séminaire Poincaré, whichis directed towards a large audience of physicists and of mathematicians. The goal of this seminar is to provide up-to-date information about general topics of great interest in physics. Both the theoretical and experimental aspects are covered, with some historical background. Inspired by the Bourbaki seminar in mathematics in its organization, hence nicknamed Bourbaphi, the Poincaré Seminar is held twice a year at the Institut Henri Poincaré in Paris, with contributions prepared in advance. Particular care is devoted to the pedagogical nature of the presentations so as to fulfill the goal of being readable by a large audience of scientists. This volume contains the seventh such Seminar, held in It is devoted to Einstein s 1905 papers and their legacy. After a presentation of Einstein s epistemological approach to physics, and the genesis of special relativity, a centenary perspective is offered. The geometry of relativistic spacetime is explained in detail. Single photon experiments are presented, as a spectacular realization of Einstein s light quanta hypothesis. A previously unpublished lecture by Einstein, which presents an illuminating point of view on statistical physics in 1910, at the dawn of quantum mechanics, is reproduced. The volume ends with an essay on the historical, physical and mathematical aspects of Brownian motion. We hope that the publication of this series will serve the community of physicists and mathematicians at the graduate student or professional level. We thank the Commissariat àl Énergie Atomique (Division des Sciences de la Matière), the Centre National de la Recherche Scientifique (Sciences Physique et Mathématiques), and the Daniel Iagolnitzer Foundation for sponsoring the Seminar. Special thanks are due to Chantal Delongeas for the preparation of the manuscript. Thibault Damour Olivier Darrigol Bertrand Duplantier Vincent Rivasseau Einstein,1 31 c Birkhäuser Verlag, Basel, 2005 Poincaré Seminar 2005 The Genesis of the Theory of Relativity Olivier Darrigol The most famous of Albert Einstein s papers of 1905 is undoubtedly the one concerning the theory of relativity. Any modern physicist knows that this theory imposes a strict and general constraint on the laws of nature. Any curious layman wonders at the daring reform of our ancestral concepts of space and time. As often happens for great conceptual breakthroughs, the theory of relativity gave rise to founding myths whose charm the historian must resist. The first of this myth is that Einstein discovered the theory of relativity in a single stroke of genius that defies any rational analysis. Some of Einstein s reminiscences favor this thesis, for instance his allusion to a conversation with Michele Besso in which he would have suddenly realized that a reform of the concept of time solved long standing paradoxes of electrodynamics. One could also argue that the historical explanation of a deep innovation is by definition impossible, since a radically new idea cannot be derived from received ideas. In the case of Einstein s relativity the rarity of pre-1905 sources further discourages historical reconstruction, and invites us to leave this momentous discovery in its shroud of mystery. This romantic attitude does not appeal to teachers of physics. In order to convey some sort of logical necessity to relativity theory, they have constructed another myth following which a few experiments drove the conceptual revolution. In this empiricist view, the failure of ether-drift experiments led to the relativity principle; and the Michelson-Morley experiment led to the constancy of the velocity of light; Einstein only had to combine these two principles to derive relativity theory. As a counterpoise to this myth, there is a third, idealist account in which Einstein is supposed to have reached his theory by a philosophical criticism of fundamental concepts in the spirit of David Hume and Ernst Mach, without even knowing about the Michelson-Morley experiment, and without worrying much about the technicalities of contemporary physics in general. A conscientious historian cannot trust such myths, even though they may contain a grain of truth. He must reach his conclusions by reestablishing the contexts in which Einstein conducted his reflections, by taking into account his education and formation, by introducing the several actors who shared his interests, by identifying the difficulties they encountered and the steps they took to solve them. In this process, he must avoid the speculative filling of gaps in documentary sources. Instead of rigidifying any ill-founded interpretation, he should offer an 2 O. Darrigol open spectrum of interpretive possibilities. As I hope to show in this paper, this sober method allows a fair intelligence of the origins of relativity. A first indication of the primary context of the early theory of relativity is found in the very title of Einstein s founding paper: On the electrodynamics of moving bodies. This title choice may seem bizarre to the modern reader, who defines relativity theory as a theory of space and time. In conformity with the latter view, the first section of Einstein s paper deals with a new kinematics meant to apply to any kind of physical phenomenon. Much of the paper nonetheless deals with the application of this kinematics to the electrodynamics and optics of moving bodies. Clearly, Einstein wanted to solve difficulties he had encountered in this domain of physics. A survey of physics literature in the years shows that the electrodynamics of moving bodies then was a widely discussed topic. Little before the publication of Einstein s paper, several studies with similar titles appeared in German journals. Much experimental and theoretical work was being done in this context. The greatest physicists of the time were involved. They found contradictions between theory and experience or within theory, offered mutually incompatible solutions, and sometimes diagnosed a serious crisis in this domain of physics. Since Heinrich Hertz s experiments of on the electric production of electromagnetic waves, Maxwell s field theory was the natural frame for discussing both the electrodynamics and the optics of moving bodies. In order to understand the evolution of this subject, one must first realize that the theory that Maxwell offered in his treatise of 1873 widely differed from what is now meant by Maxwell s theory. 1 Maxwell s theory as it was Like most of his contemporaries, Maxwell regarded the existence of the ether as a fundamental and undeniable fact of physics. He held this medium responsible for the propagation of electromagnetic actions, which included optical phenomena in his view. His theory was a phenomenological theory concerned with the macroscopic states of a continuous medium, the ether, which could combine with matter and share its velocity v. These states were defined by four vectors E, D, H, B that obeyed a few general partial differential equations as well as some relations depending on the intrinsic properties of the medium. In the most complete and concise form later given by Oliver Heaviside and Heinrich Hertz, the fundamental equations read E = DB/Dt, H = j + DD/Dt D = ρ, B =0, (1) where j is the conduction current and D/Dt is the convective derivative defined by D/Dt = / t (v )+v( ). (2) The Genesis of the Theory of Relativity 3 In a linear medium, the forces E and H were related to the polarizations D and B by the relations D = ɛe and B = µh, and the energy density (1/2)(ɛE 2 +µh 2 ) of the medium had the form of an elastic energy. For Maxwell and his followers, the charge density and the conduction current j were not primitive concepts: the former corresponded to the longitudinal gradient of the polarization or displacement D, and the latter to the dissipative relaxation of this polarization in a conducting medium. The variation DD/Dt of the displacement constituted another form of current. Following Michael Faraday, Maxwell and his disciples regarded the electric fluids of earlier theories as a naïvely substantialist notion. 1 The appearance of the convective derivative D/Dt in Maxwell s theory derives from his understanding of the polarizations D and B as states of a single medium made of ether and matter and moving with a well-defined velocity v (that may vary from place to place): the time derivatives in the fundamental equations must be taken along the trajectory of a given particle of the moving medium. The resulting law of electromagnetic induction, E = DB/Dt = B/ t + (v B), (3) contains the (v B) contribution to the electric field in moving matter. By integration around a circuit and through the Kelvin-Stokes theorem, it leads to the expression E dl = d B ds (4) dt of Faraday s law of induction, wherein the integration surface moves together with the bordering circuit. When the magnetic field is caused by a magnet, the magnetic flux only depends on the relative position of the magnet and the circuit so that the induced current only depends on their relative motion. In sum, the conceptual basis of Maxwell s original theory widely differed from what today s physicists would expect. Electricity and magnetism were field-derived concept, whereas modern electromagnetism treats them as separate entities. A quasi-material ether was assumed. The fundamental equations (1) only correspond to our Maxwell equations in the case of bodies at rest, for which the velocity v is zero and the convectivederivative D/Dt reduces to the partial derivative / t.one thing has not changed, however: the theory s ability to unify electromagnetism and optics. In a homogenous insulator at rest, Maxwell s equations imply the existence of transverse waves propagating at the velocity c =1/ ɛµ. Having found this electromagnetic constant to be very close to the velocity of light, Maxwell identified these waves with light waves. The resulting theory automatically excludes the longitudinal vibrations that haunted the earlier, elastic-solid theories of optics. Within a few years after Maxwell s death (in 1879), a growing number of British physicists saluted this achievement and came to regard Maxwell s theory as 1 J.C. Maxwell, A treatise on electricity and magnetism, 2 vols. (Oxford, 1973); H. Hertz, Über die Grundgleichungen der Elektrodynamik für bewegte Körper, Annalen der Physik, 41 (1890), 4 O. Darrigol philosophically and practically superior to earlier theories. The Germans had their own theories of electricity and magnetism, based on electric and magnetic fluids (or Amperean currents) directly acting at a distance. They mostly ignored Maxwell s theory until in 1888 Heinrich Hertz demonstrated the emission of electromagnetic waves by a high-frequency electric oscillator. After this spectacular discovery was confirmed, a growing number of physicists adopted Maxwell s theory in a more or less modified form. Yet this theory was not without difficulties. Maxwell had himself noted that his phenomenological approach led to wrong predictions when applied to optical dispersion, to magneto-optics, and to the optics of moving bodies. In these cases he suspected that the molecular structure of matter had to be taken into account. 2 Flashback: The optics of moving bodies Maxwell s idea of a single medium made of ether and matter implied that the ether was fully dragged by moving matter, even for dilute matter. Whereas this conception worked very well when applied to moving circuits and magnets, it was problematic in the realm of optics. The first difficulty concerned the aberration of stars, discovered by the British astronomer James Bradley in 1728: the direction of observation of a fixed star appears to vary periodically in the course of a year, by an amount of the same order as the ratio (10 4 ) of the orbital velocity of the earth to the velocity of light. 2 The old corpuscular theory of light simply explained this effect by the fact that the apparent velocity of a light particle is the vector sum of its true velocity and the velocity of the earth (see Fig. 1). In the early nineteenth century, the founders of the wave theory of light Thomas Young and Augustin Fresnel saved this explanation by assuming that the ether was completely undisturbed by the motion of the earth through it. Indeed, rectilinear propagation at constant velocity is all that is needed for the proof. 3 Fresnel s assumption implied an ether wind of the order of 30km/s on the earth s surface, from which a minute modification of the laws of optical refraction ought to follow. As Fresnel knew, an earlier experiment of his friend François Arago had shown that refraction by a prism was in fact unaffected by the earth s annual motion. Whether or not Arago had reached the necessary precision of 10 4,Fresnel took this result seriously and accounted for it by means of a partial dragging of the ether within matter. His theory can be explained as follows. According to an extension of Fermat s principle, the trajectory that light takes to travel between two fixed points (with respect to the earth) is that for 2 J. Bradley, A new apparent motion discovered in the fixed stars; its cause assigned; the velocity and equable motion of light deduced, Royal Society of London, Proceedings, 35 (1728), A. Fresnel, Lettre d Augustin Fresnel àfrançois Arago s

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