Description
This primer proposes a journey from Newton’s dynamics to Einstein’s relativity. It constitutes a pedagogical, rigorous, and self-contained introduction to the concepts and mathematical formulation of gravitational physics. In particular, much attention is devoted to exploring and applying the basic tools of differential geometry, that is the language of general relativity. Real-world manifestations of relativity, such as time dilation, gravitational waves, and black holes, are also discussed in detail. This book is designed for third-year bachelor or first-year master students in theoretical physics, who are already familiar with Newton’s physics, possibly had an introductory course on special relativity, and who are seeking to learn general relativity on a firm basis. Pierre Fleury is currently a post-doctoral fellow at the Instituto de Fsica Terica of Madrid (Spain), specialized in relativistic cosmology and gravitational lensing. He concluded his PhD in 2015 at the Institut d’Astrophysique de Paris, and was successively a post-doctoral researcher at the University of Cape Town and at the University of Geneva. His main research interest consists in using cosmological observations to learn about the fundamental laws of nature. He is also actively involved in the diffusion of knowledge, through both academic teaching and public outreach. He likes acting, cooking, playing music, and watching French movies. I Newton’s physics I.A Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I.B Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 I.C Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 I.D Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 I.E Application to the Solar System . . . . . . . . . . . . . . . . . . . . . . . . 23 Epilogue: when Newtonian physics fails . . . . . . . . . . . . . . . . . . . . . . . 28 II Einstein’s theory of relativity 29 II.A Space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 II.B Physics in four dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 II.C Differential geometry tool kit . . . . . . . . . . . . . . . . . . . . . . . . . 45 II.D Space-time tells matter how to fall . . . . . . . . . . . . . . . . . . . . . . 52 II.E Matter tells space-time how to curve . . . . . . . . . . . . . . . . . . . . . 59 Newton versus Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 III The general-relativistic world 69 III.AWeak gravitational fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 III.B Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 III.C The Schwarzschild black hole . . . . . . . . . . . . . . . . . . . . . . . . . 83 References
