General Relativity - Affine Spaces, Einstein's Equations, and Spacetime Models

Explore the mathematical foundations and physical principles of Einstein's theory of general relativity through this comprehensive lecture series delivered in Torino during April-May 2018. Begin with fundamental concepts including affine spaces as manifolds and pre-Einstein models of spacetime, then progress through essential mathematical tools such as tensors, pseudotensors, densities, and tensor fields on manifolds. Master the theory of connections, curvature, and torsion before examining Newton's spacetime model and its limitations. Delve into Einstein's revolutionary spacetime concepts and learn to linearize Einstein's field equations for practical applications. Study gravitational waves within the framework of linearized Einstein gravity and explore spacetime symmetries through the de Sitter model. Analyze the famous Schwarzschild solution and related spherically symmetric spacetimes, understanding their physical significance through Kruskal coordinates. Calculate geodesics in Schwarzschild spacetime and derive the deflection of light by massive objects. Apply general relativity to astrophysical contexts by modeling spherical stars and conclude with an examination of cosmological models that describe the large-scale structure and evolution of the universe.