Yvonne Choquet-Bruhat

This reference book, which has found wide use as a text, provides an answer to the needs of graduate physical mathematics students and their teachers. The present edition is a thorough revision of the first, including a new chapter entitled ``Connections on Principle Fibre Bundles'' which includes sections on holonomy, characteristic classes, invariant curvature integrals and problems on the geometry of gauge fields, monopoles, instantons, spin structure and spin connections. Many paragraphs have been rewritten, and examples and exercises added to ease the study of several chapters. The index includes over 130 entries.

General Relativity has passed all experimental and observational tests to model the motion of isolated bodies with strong gravitational fields, though the mathematical and numerical study of these motions is still in its infancy. It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. In this monograph, aimed at researchers in mathematics and physics, the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.

Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in volume 1 of the 2-Volume set Analysis, Manifolds and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number of parenthesis following the title.

This is an open access title. It is made available under a Creative Commons Attribution-Non Commercial-No Derivatives 4.0 International licence. It is available to read and download as a PDF version on the Oxford Academic platform.General Relativity is a beautiful geometric theory, simple in its mathematical formulation but leading to numerous consequences with striking physical gravitational waves, black holes, cosmological models, and so on.This introductory textbook is written for mathematics students interested in physics and physics students interested in exact mathematical formulations (or for anyone with a scientific mind who is curious to know more of the world we live in), recent remarkable experimental and observational results which confirm the theory are clearly described and no specialised physics knowledge is required. The mathematical level of Part A is aimed at undergraduate students and could be the basis for acourse on General Relativity. Part B is more advanced, but still does not require sophisticated mathematics.Based on Yvonne Choquet-Bruhat's more advanced text, General Relativity and the Einstein Equations, the aim of this book is to give with precision, but as simply as possible, the foundations and main consequences of General Relativity. The first five chapters from General Relativity and the Einstein Equations have been updated with new sections and chapters on black holes, gravitational waves, singularities, and the Reissner-Nordström and interior Schwarzchild solutions.The rigour behind this book will provide readers with the perfect preparation to follow the great mathematical progress in the actual development, as well as the ability to model, the latest astrophysical and cosmological observations. The book presents basic General Relativity and provides a basis for understanding and using the fundamental theory.

"Ds que ma raison s'est veille, j'ai dsir comprendre quelque chose de cet univers trange o nous vivons, et de ce qu'y font les tres humains, moi en particulier. J'ai flirt avec des domaines varis de la connaissance : sciences naturelles, physique, histoire, philosophie. Je suis finalement devenue une mathmaticienne travaillant sur des problmes poss par la physique. Comme l'ont expriment tous les chercheurs, une dcouverte, mme minime, est toujours une grande joie, comme celle d'une terre inconnue." Y. C.-B.

Ce volume contient des contributions nouvelles à la solution des problèmes posés par la relativité générale classique et aussi de nombreux exposés concernant les problèmes nouveaux de géométrie différentielle, d'analyse et de topologie soulevés par les théories physiques les plus actuelles.

« Dès que ma raison s’est éveillée, j’ai désiré comprendre quelque chose de cet univers étrange où nous vivons, et de ce qu’y font les êtres humains, moi en particulier. J’ai flirté avec des domaines variés de la connaissance : sciences naturelles, physique, histoire, philosophie. Je suis finalement devenue une mathématicienne travaillant sur des problèmes posés par la physique. Comme l’ont expérimenté tous les chercheurs, une découverte, même minime, est toujours une grande joie, comme celle d’une terre inconnue. » Y. C.-B. Yvonne Choquet-Bruhat est la première femme élue à l’Académie des sciences. C’est l’une des plus grandes mathématiciennes et physiciennes de notre temps. Elle nous livre ici un récit intime où s’entremêlent vie personnelle, découvertes scientifiques et grande histoire. Ses recherches couvrent un domaine très large de la connaissance, allant de la première preuve mathématique de l’existence de solutions de la théorie relativiste de la gravitation d’Einstein à l’étude de la conversion d’ondes électromagnétiques en ondes gravitationnelles (ou l’inverse) au voisinage d’un trou noir.