SPRINGER BRIEFS IN PHYSICS Pierre Fleury Gravitation From Newton to Einstein SpringerBriefs in Physics Series Editors BalasubramanianAnanthanarayan,CentreforHighEnergyPhysics,IndianInstitute of Science, Bangalore, India EgorBabaev,PhysicsDepartment,UniversityofMassachusettsAmherst,Amherst, MA, USA MalcolmBremer,HHWillsPhysicsLaboratory,UniversityofBristol,Bristol,UK Xavier Calmet, Department of Physics and Astronomy, University of Sussex, Brighton, UK FrancescaDiLodovico,DepartmentofPhysics,QueenMaryUniversityofLondon, London, UK Pablo D. Esquinazi, Institute for Experimental Physics II, University of Leipzig, Leipzig, Germany Maarten Hoogerland, University of Auckland, Auckland, New Zealand Eric Le Ru, School of Chemical and Physical Sciences, Victoria University of Wellington, Kelburn, Wellington, New Zealand Hans-Joachim Lewerenz, California Institute of Technology, Pasadena, CA, USA Dario Narducci, University of Milano-Bicocca, Milan, Italy James Overduin, Towson University, Towson, MD, USA Vesselin Petkov, Montreal, QC, Canada Stefan Theisen, Max-Planck-Institut für Gravitationsphysik, Golm, Germany Charles H.-T. Wang, Department of Physics, The University of Aberdeen, Aberdeen, UK James D. Wells, Physics Department, University of Michigan, Ann Arbor, MI, USA Andrew Whitaker, Department of Physics and Astronomy, Queen’s University Belfast, Belfast, UK SpringerBriefs in Physics are a series of slim high-quality publications encom- passing the entire spectrum of physics. 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More information about this series at http://www.springer.com/series/8902 Pierre Fleury Gravitation From Newton to Einstein 123 Pierre Fleury Department ofTheoretical Physics University of Geneva Geneva, Switzerland Instituto deFísica Teórica Madrid,Spain ISSN 2191-5423 ISSN 2191-5431 (electronic) SpringerBriefs inPhysics ISBN978-3-030-32000-3 ISBN978-3-030-32001-0 (eBook) https://doi.org/10.1007/978-3-030-32001-0 ©TheAuthor(s),underexclusivelicencetoSpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To the students of Africa Preface TheAfricanInstituteforMathematicalSciences(AIMS)isapan-Africannon-profit educational organisation founded by the South African cosmologist Neil Turok, with the purpose of promoting mathematical sciences in Africa. It proposes an intensive 1-year master-level programme for excellent and highly motivated African students, with courses ranging from fundamental to applied mathematics, theoretical physics and languages. The AIMS network consists of six centres in Cameroon, Ghana, Rwanda, Senegal, South Africa and Tanzania. Each of them trains a cohort of about 50 students per year. This document gathers the lecture notes of a course entitled Gravitation: from Newton to Einstein, which I gave in January 2018 and January 2019 at AIMS-Cameroon. The course was initially designed to fit in 30 h, each section correspondingtoatwo-hourlecture.Mymaingoal,inthiscourse,wastoproposea big picture of gravitation, where Einstein’s theory of relativity arises as a natural increment to Newton’s theory. The students are expected to be familiar with the fundamentals of Newton’s mechanics and gravitation, for the first chapter to be a merereformulationofknownconcepts.Thesecondchapterthenintroducesspecial and general relativity at the same time, while the third chapter explores concrete manifestations of relativistic gravitation, notably gravitational waves and black holes.Thenumerousexercisesmustbeconsideredpartofthecourseitself;theyare intended to stimulate active reading. Geneva, Switzerland Pierre Fleury vii Acknowledgements Iwouldnothavehadtheopportunitytodeliverthiscoursewithoutmymentorand friend Jean-Philippe Uzan, who both introduced me to the AIMS network and helped me designing the structure of the course itself. I also thank the academic director of AIMS-Cameroon, Marco Garuti, for his warm welcome and for having trusted me to take care of his students 2 years in a row. Many thanks to the tutors Peguy Kameni Ntseutse, Hans Fotsing and Pelerine Nyawo, for their daily assis- tance, and to my fellow lecturers, notably Patrice Takam, Charis Chanialidis, Jane Hutton and Julia Mortera. Finally, I would like to express my sincere congratula- tionstotheAIMSstudentsfortheirremarkableattitude,dedicationandhardwork. Influential References Theorganisationandcontentofthiscourse,especiallyinthefirstchapter,arepartly inspired from Relativity in Modern Physics [1] by Nathalie Deruelle and Jean-Philippe Uzan.They also reflect mypersonal approach torelativityand grav- itation, which has been influenced by Special Relativity in General Frames [2] by Eric Gourgoulhon, A Relativist’s Toolkit [3] by Eric Poisson, and a remarkable doctoralcourseongeneralrelativitythatGilles Esposito-FarèsegaveattheInstitut d’AstrophysiquedeParisin2013.Ialsousedbitsandpiecesofacoursegivenbymy esteemed colleague Martin Kunz at the University of Geneva in 2017 and 2018, itselfbasedontheverycomprehensiveGeneralRelativity[4]byNorbertStraumann. ix Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Newton’s Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Time and Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Scalar Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.4 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.5 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Newton’s Three Laws of Dynamics . . . . . . . . . . . . . . . . . 13 2.2.2 Conserved Quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.3 Non-inertial Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Euler–Lagrange Equation. . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.2 Variational Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.3 Hamilton’s Least Action Principle . . . . . . . . . . . . . . . . . . 21 2.4 Gravitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Universal Gravity Law. . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Gravitational Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.3 Lagrangian Formulation of Newton’s Gravity . . . . . . . . . . 28 2.5 Application to the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.1 Orbits of Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.2 Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Einstein’s Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1 Separation of Two Events . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 Minkowski Metric and Four-Vectors. . . . . . . . . . . . . . . . . 39 3.1.3 Relativity of Time and Space . . . . . . . . . . . . . . . . . . . . . . 42 xi xii Contents 3.2 Physics in Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.1 Motion and Frames in Relativity . . . . . . . . . . . . . . . . . . . 46 3.2.2 Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.3 Nordström’s Theory of Gravity . . . . . . . . . . . . . . . . . . . . 53 3.3 Differential Geometry Tool Kit . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.1 Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.2 Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.3 Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 Space-Time Tells Matter How to Fall . . . . . . . . . . . . . . . . . . . . . 65 3.4.1 Equivalence Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.2 Geodesic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.3 Physics in Curved Space-Time . . . . . . . . . . . . . . . . . . . . . 71 3.5 Matter Tells Space-Time How to Curve. . . . . . . . . . . . . . . . . . . . 74 3.5.1 Energy–Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . 74 3.5.2 Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5.3 Action Principle for Gravitation . . . . . . . . . . . . . . . . . . . . 82 4 The General-Relativistic World . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.1 Weak Gravitational Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.1.1 Linearised Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . 87 4.1.2 Newtonian Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.1.3 Gravitational Dilation of Time . . . . . . . . . . . . . . . . . . . . . 92 4.2 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.1 Transverse Trace-Less Gauge. . . . . . . . . . . . . . . . . . . . . . 95 4.2.2 Effect on Matter and Detection. . . . . . . . . . . . . . . . . . . . . 97 4.2.3 Production of Gravitational Waves . . . . . . . . . . . . . . . . . . 102 4.3 The Schwarzschild Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.1 The Schwarzschild Solution . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3.3 Event Horizon and Black Hole. . . . . . . . . . . . . . . . . . . . . 109 4.3.4 Black Holes in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 References.... .... .... .... ..... .... .... .... .... .... ..... .... 119