THEORYOFRELATIVITY W ~PAULI Translated fromtheGermanby G.FIELD WithSupplementary Notes by theAuthor PERGAMONPRESS LONDON.NEWYORK~PARISLOSANGELES 1958 PERGAMONPRESSLTD. 4 & 5 Fitzroy Square,LondonW.l. PERGAMONPRESSINC. 122 East 55th Street,NewYork 22,N. Y. P.O. Box 47715, Los Angeles,California. PERGAMONPRESSS.A.R.L. 24 Rue des Beoles, ParisV" Translatedfromthearticle" Relativiiatstheorie" in Encyklopiidie dermathematjschenWissenscha/ten,Vol.V19,(B.G.Teubner, Leipzig1921).Sole authorized English translation of the original GermaIl' editionbypermissionof theB.G.Teubner Publishing House,Stuttgart. Copyright 1958 PERGAMONPRESSLTD. Library of CongressCard Number 58-7692 PRINTEDIN(.JREATBRITAINBYJ.W.ARROWSMITHLTD.,BRISTOL PREFACE THIRTY-FIVEyearsagothisarticleonthetheoryof relativity,written by me at the rather young age of 21years forthe MathematicalEncyclo-pedia, was first published as a separate monograph together with a preface by Sommerfeld, who as the editor of this volume of the Encyclopedia was responsibleformyauthorship.It wastheaimof thearticletogivea completereviewof the wholeliteratureonrelativitytheoryexistingat thattime(1921).Meanwhile,theproductionof textbooks,reportsand papers on the theory of relativity has grown into a flood,which rose anew at the50thAnniversaryof the firstpapersof Einstein onrelativity,in the same year 1955.in which all physicists were mourning hisdeath. In this situation any idea of completenessregardingthe now existing literature in a revised new edition of the book had to be given up from the beginning.Idecidedtherefore,in ordertopreservethecharacter of the book as an historical document, to reprint the .old text in its original form, buttoaddanumber ofnotesattheendofthebook,whichreferto certain passages of the These notes give to the reader selected linfor-mationaboutthelaterdevelopmentsconnectedwithrelativitythe.ory and also my personal views upon some controversial/questions. Especially in the' last of thesenot.eson unifiedfield theories,Ido not conceal to the reader my scepticismconcerning all attempts of t4is kind which have been madeuntil now,and alsoaboutthe futurechancesof success of theories with such aims.These' questions are closely connected with the problem of the range validity of the classical fieldconcept in its application to the atomic featuresof Nature.Thecritical view,whichI uttered in the last section of the original text with respect to any solution on these classical lines., has since been very much deepened by the epistemo-logicalanalysisof quantummechanics,orwavemechanics,whichwas formulated in 1927.On the other hand Einstein maintained the hope for total solution onthelines of aclassicalfieldtheory until the end of his life. These differences of opinion are merging into the great open problem of therelationof relativity theory toquantumtheory, which willpre-sumably occupy physicists for a long while to come. In particular, aclear connection between the general theory of relativity and quantum mec}l-anics is not yetsight. JustbecauseIemphasizeinthelastof thenotesacertain between the viewson problems beyond the original frameof special and general relativity held by Einstein himself on the one hand, and by most of the physicists,includingmyself,on the other,Iwishto concludethis prefacewithsomeconciliatoryremarksonthepositionofrelativity theoryin thedevelopmentof physics. Thereisapointof viewaccordingtowhichrelativityisthe end'-pointof"classical whichmeansphysicsinthestyleof Newton-Faraday-Maxwell,governedbythe"deterministic"formof v vi Preface causality in space and time, while afterwards the new quantum-mechanical style of the laws of Nature came into play. This point of view seems to me only partly true, and does not sufficiently do justice to the great influence of Einstein, the creator of the theory of relativity,on the general way of thinking of the physicists of today. By its epistemological analysis of the consequences of the finitenessof the velocity of light(and with it,of all signal-velocities),the theory of special wasthe firststepaway fromnaivevisualization.Theconceptofthestateofmotion ,ofthe "luminiferousaether",asthehypotheticalmediumwas:calledearlier, hadtobegivenup, notonly becauseit turned to be unobservable, butbecauseitbecamesuperfluousasanelementofa,mathematical formalism,thegroup-theoreticalpropertiesof which wouldonlybe dis-turbed by it.. By the widening of the transformation group in general relativity the idea of distinguished inertial coordinate systems could alsobe eliminated byEinsteinasinconsistent with the group-theoretical properties of the theory.Withoutthisgeneralcriticalattitude,whichabandonednaive visualizationsinfavourof aconceptualanalysisof thebetween observational data and the mathematical quantities in atheore-tical formalism,the establishment of the modern form of quantum theory would not have been possible.In the "complementary" quantum theory, the epistemological analysis of the finiteness of the quantum of action led to further steps away fromnaive visualizations.In thiscase it wasboth the classical field concept, and the concept of orbits of particles (electrons) in space and time, which had to be given up in favour of rational general-izations. Again,these concepts were rejected, not only because the orbits are unobservable,but alsobecausethey becamesuperfluousand would disturb the symmetry inherent in the general transformation group under-lying the mathematical formalismof the theory. Iconsiderthetheoryof relativitytobeanexampleshowinghowa fundamental scientific discovery, sometimes even against the resistance of its creator,givesbirth to further fruitful developments, following its own autonomous course. IamgratefultotheInstituteforAdvancedStudyinPrincetonforaffordingmethe opportunityof writing the Supplementary Notes, pp.207-232, during my stay there early in1956.AndIshouldliketothankmycolleaguesat PrincetonwithwhomIdiscussed many of the problems in these notes. If gratefullyacknowledgetheexcellenthelpof the translator,Dr.GerardField,of the Departmentof MathematicalPhysics,Universityof Birmingham. Zurich,18 November1956W.P. ACKNOWLEDGMENTSINTHEORIGINALARTICLE Iwishto express my warm gratitudeto Geheimrat Klein, forthe great interest he has shown inthisarticle,forhis activehelpin and for his valuable advice on many occasions.My thanks are also due to Herr Bessel Hagen, for his careful proof.reading of part of this article. CONTENTS PREFACEBYW.PAULI PREFACEBYA.SOMMERFELD BIBLIOGRAPHY PartI.Thefoundationsofthespecialtheoryofrelativity p. xi xiii 1.Historical background (Lorentz,Poincarb, Einstein)1 2.The postulate of relativity4 3.Thepostulateoftheconstancyofthevelocityoflight.Ritz'sandrelated theories5 4.Therelativityofsimultaneity.DerivationoftheLorentztransformation fromthetwopostulates.AxiomaticnatureoftheLorentztransformation9 5.Lorentz contraction and time dilatation11 6.Einstein'sadditiontheoremforvelocitiesanditsapplicationtoaberration and the drag coefficient.The Doppler effect15 PartII.Mathematicaltools 17.The four-dimensional space.time world(Minkowski)21 8.More general transformation groups22 9.Tensor calculus for affine tr8llSformations24 10.GeomE-tricalmeaningofthecontravariantandcovariantcomponentsofa vector27 ll."Surface'; and "volume" tensors.Four-dimensional volumes30 12.Dual tensors33 13.Transition to Riemannian geometry34 14.Parallel displacemebt of avector37 15.Geodesiclines39 16.Space curvature41 17.Riemannian coordinates and their applications44 18.The special cases of Euclidean geometry and of constant curvature48 19.The integral theorems of Gauss and Stokes in afourdimensionalRiemannian manifold52 20.Derivationof invariantdifferentialoperations,usinggeodesiccomponents56 21.Affine tensors and freevectors60 22.Reality relations62 23.Infinitesimal coordinate transformations and variational theorems64 PartIII.Specialtheoryofrelativity.Furtherelaborations (a)Kinematics 24.Four-dimensioIUJl representation of the Lorentz transformation 25.The addition theorem for velocities 26.Transformation law for acceleration.Hyperbolic motion (b)Electrodynamics 71 73 74 27.Conservation of charge.Four-current density76 28.Covariance of the basic equations of electron theory78 29.Ponderomotive forces.Dynamics of the electron81 30.Momentum and energy of the electromagnetic field.Differentialandintegral forms of the conservation laws_85 31.The invariant action principle of electrodynamics88 vii viiiContents 32.Applications to specia.l cases, (e