This book opens with an axiomatic description of Euclidean and non-Euclidean geometries. Euclidean geometry is the starting point to understand all other geometries and it is the cornerstone for our basic intuition of vector spaces. The generalization to non-Euclidean geometry is the following step to develop the language of Special and General Relativity. These theories are discussed starting from a full geometric point of view. Differential geometry is presented in the simplest way and it is applied to describe the physical world. The final result of this construction is deriving the Einstein field equations for gravitation and spacetime dynamics. Possible solutions, and their physical implications are also discussed: the Schwarzschild metric, the relativistic trajectory of planets, the deflection of light, the black holes, the cosmological solutions like de Sitter, Friedmann-Lemaître-Robertson-Walker, and Gödel ones. Some current problems like dark energy are also scketched. The book is self-contained and includes details of all proofs. It provides solutions or tips to solve problems and exercises. It is designed for undergraduate students and for all readers who want a first geometric approach to Special and General Relativity.

Autorentext

Wladimir-Georges Boskoff graduated from the Faculty of Mathematics, University of Bucharest in 1982 and completed his Ph.D. at the same university in 1994. Since 1990 he has been a member of the Department of Mathematics at the Ovidius university, teaching courses on various subjects, including Euclidean and non-Euclidean geometry, differential geometry, calculus on manifolds, mechanics and relativity, astronomy, and other subjects. His scientific papers and books relate to the foundations of geometry, Euclidean and hyperbolic geometry, metric geometry, differential geometry, modified theories of gravity, general relativity, and the history of mathematics. He has been an invited speaker at conferences in France, Japan, the USA, Greece, Italy, and Chile. He is a recipient of the Academy of Sciences of Romania's G. Tzitzeica Prize for contributions to geometry (1996), and the Romanian Mathematical Society Medal for contributions at mathematical education (2010). Salvatore Capozziello is a Full Professor of Astronomy and Astrophysics at the Department of Physics at the Universita' di Napoli "Federico II", Visiting Professor at Gran Sasso Science Institute (L'Aquila), and Honorary Professor at Tomsk State Pedagogical University (Russia). He also holds research appointments at Istituto Nazionale di Fisica Nucleare (INFN), Istituto Nazionale di Astrofisica (INAF), and Gruppo Nazionale di Fisica Matematica (GNFM-INDAM). From 2012 to 2018, he was President of the Italian Society for General Relativity and Gravitation (SIGRAV). Professor Capozziello has spent periods of his scientific career in Germany, Poland, the UK, Mexico, the USA, South Africa, Canada, France, and Japan. His research focuses on general relativity, cosmology, relativistic astrophysics, and physics of gravitation in their theoretical and phenomenological aspects, in particular extended theories of gravity and their cosmological and astrophysical applications. His main scientific achievement has been in 2002 when he introduced the concept of gravitational curvature quintessence, f(R) gravity, to explain the cosmological dark energy. He is the author of almost 550 refereed papers and six books and listed as one of the Top Italian Scientists.

Inhalt

1. Euclidean and Non-­Euclidean Geometries: How they appear.- 2. Basic Facts in Euclidean and Minkowski Plane Geometry.- 3. Geometric Inversion, Cross Ratio, Projective Geometry and Poincaré Disk Model.- 4. Surfaces in 3D-Spaces.- 5. Basic Differential Geometry.- 6. Non-Euclidean Geometries and their Physical Interpretation.- 7. Gravity in Newtonian Mechanics.- 8. Special Relativity.- 9. General Relativity and Relativistic Cosmology.- 10. A Geometric Realization of Relativity: The Affine Universe and de Sitter Spacetime.