Wladimir-Georges Boskoff - A Mathematical Journey to Relativity: Deriving Special and General Relativity with Basic Mathematics

UNITEXT for Physics series, formerly UNITEXT Collana di Fisica e Astronomia, publishes textbooks and monographs in Physics and Astronomy, mainly in English language, characterized of a didactic style and comprehensiveness. The books published in UNITEXT for Physics series are addressed to upper undergraduate and graduate students, but also to scientists and researchers as important resources for their education, knowledge and teaching.

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This book is an approach to Special and General Relativity from a full mathematical point of view. When Physics is studied, there is the need for understanding its language, that is, Mathematics. Diracs words describe very well what we want to do: God used beautiful Mathematics in creating the world; therefore, we present a part of this divine plan, the beautiful Mathematics of Special and General Relativity. We wrote a textbook which, we believe, can be easily used by students in Mathematics, Physics and Engineering studies; by teachers or by some other people who are interested in this subject. If someone already knows Mathematics, that is, both basic Geometry and Differential Geometry, this person can neglect the first six chapters. She/he can start from Gravity in Newtonian Mechanics. People who study Physics should start from the very beginning in order to understand the development of Geometry. The improvement of mathematical language, in more than two thousand and five hundred years, allowed to produce a common language for both Calculus and Linear Algebra; this approach ends up to a dialect, the Differential Geometry, which constitutes the basic tool of Relativity. Without the effort to understand the nature of the Non-Euclidean Geometry, the Differential Geometry could not occur. Without Differential Geometry, General Relativity could not exist. The first six chapters represent the adventure of Geometry from axioms until the Non-Euclidean Geometries through Differential Geometry. A lot of examples and solved exercises help the reader to understand the theory. Actually, the entire book, which is written in a unitary way, offers clear statements and proofs. About the proofs: it offers complete proofs; all computations are presented. In our opinion, this is the only way to understand the complicated computations which depend on the Differential Geometry language. Reading line by line, the reader can understand every single proof. The references which inspired us are mentioned not only at the beginning of each chapter but also in the text. Some proofs and some approaches of the theory are completely original. If somebody is reading from the beginning to the end of this book, it becomes understandable why each subject presented is important for the topic. We hope that our humble efforts are useful, first of all, for learning people to whom this book is mainly dedicated. We thank our colleagues; our teachers; our friends and, first of all, our students whose questions, discussions and remarks allowed us to enter the perfect world of Geometry towards its amazing realization which is General Relativity. We also want to thank Dr. Marina Forlizzi and the Springer staff for invaluable support in publishing this book.

As a final remark, we want to say that this book was conceived about two years ago during pleasant discussions on Mathematics and General Relativity in scientific congresses and meetings between the authors and was concluded during the severe period of the global Coronavirus disease. We hope that Science and its high values can be comforting even in difficult situations like the present one, as happened so many times in history.

What does a mathematical journey towards the general theory of Relativity look like? The authors propose an original itinerary moving from Euclidean and non-Euclidean Geometry created from axioms to models of geometric Euclidean and non-Euclidean worlds. Differential Geometry of surfaces and then abstract Differential Geometry are special stops for two reasons: