Mathematical Relativity

NOTA: (Questa è la prima delle nove lezioni (per un totale di 20 ore) del corso di dottorato "Mathematical Relativity"). Il calendario delle lezioni successive è disponibile qui: https://phd.unibo.it/matematica/en/teaching/2023-2024 ABSTRACT: This course covers the basic principles of differential geometry and its application to special and general relativity, some important exact solutions of the Einstein field equations, the singularity theorems, the Cauchy problem in general relativity, and some generalities about (classical) black holes. In detail: - Short overview of Special Relativity; - Non-inertial motion and the notion of observer; - The Equivalence Principle: Newton’s gravity vs. Special Relativity, weak and Einstein’s equivalence principles, physical implications; - Short review of Differential Geometry, if necessary; - Postulates of General Relativity; - Einstein’s field equations: variational derivation; - Exact solutions & Carter-Penrose diagrams: Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr; - Uniqueness (no-hair) theorems; - Causality theory on Lorentzian manifolds; - Singularity theorems: Hawking and Penrose; - Cauchy problem in General Relativity: Cauchy horizons and loss of predictability.