The 2nd edition of this textbook features more than 100 pages of new material, including four new chapters, as well as an improved discussion of differential geometry concepts and their applications. The textbook aims to provide a comprehensive geometric description of Special and General Relativity, starting from basic Euclidean geometry to more advanced non-Euclidean geometry and differential geometry. Readers will learn about the Schwarzschild metric, the relativistic trajectory of planets, the deflection of light, the black holes, and the cosmological solutions like de Sitter, Friedman-Lemaître-Robertson-Walker, and Gödel ones, as well as the implications of each of them for the observed physical world. In addition, the book provides step-by-step solutions to problems and exercises, making it an ideal introduction for undergraduate students and readers looking to gain a better understanding of Special and General Relativity. In this new edition, a wide discussion on metric-affine theories of gravity and equivalent formulations of General Relativity is reported. The aim is presenting also topics which could be useful for PhD students and researchers studying General Relativity from an advanced point of view.

Updates and new material are offered Explains how special and general relativity are derived from basic mathematics Presents Einstein's field equations and their physical implications

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 Full Professor of Astronomy and Astrophysics at the Department of Physics at the Università di Napoli "Federico II" where he teaches General Relativity and Cosmology. He is visiting Professor at the Gran Sasso Science Institute (L'Aquila, Italy), Honorary Professor at the Tomsk State Pedagogical University (Tomsk, Russia), Member of the Lepage Research Institute (Presov, Slovakia) and Foreign Member of the Serbian Academy of Non-linear Sciences (Belgrade, Serbia). He is the Coordinator of the PhD in Cosmology, Space Science & Space Technology (SPACE) at the Scuola Superiore Meridionale, Napoli. He is the Managing Editor of the International Journal of Geometric Methods in Modern Physics (World Scientific). 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). He spent periods of his scientific career in Germany, Poland, UK, Mexico, 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 achievements have been the introduction of the concept of the curvature quintessence to explain the cosmological dark energy and the use of Noether symmetries as selection rules for observable universes. In November 2023, he received the Yang-Hui Award by the ICMAACS, University of Shanghai, China. He is the author of more than 650 refereed papers and 9 books. He is listed as one of the Top Italian Scientists.

Inhalt

Euclidean and Non-Euclidean Geometries: How they appear.- Basic Facts in Euclidean and Minkowski Plane Geometry.- From Projective Geometry to Poincaré Disk. How to carry out a Non-Euclidean Geometry Model.- Revisiting the Differential Geometry of Surfaces in 3D-Spaces.- Basic Differential Geometry Concepts and their Applications.- Differential Geometry at Work: Two Ways of Thinking the Gravity. The Einstein Field Equations from a Geometric Point of View.- Differential Geometry at Work: Euclidean, Non-Euclidean and Elliptic Geometric Models from Geometry and Physics.- Gravity in Newtonian Mechanics.- Special Relativity.- General Relativity and Relativistic Cosmology.- A Geometric Realization of Relativity: the de Sitter Spacetime.- Another Geometric Realization of Relativity: the Anti-de Sitter Spacetime.- **** More than Metric: Geometric Objects for Alternative Pictures of Gravity.- Metric-Affine Theories of Gravity.- Conclusions