CODE 98825 ACADEMIC YEAR 2024/2025 CREDITS 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA SCIENTIFIC DISCIPLINARY SECTOR MAT/07 LANGUAGE Italian TEACHING LOCATION GENOVA SEMESTER 2° Semester TEACHING MATERIALS AULAWEB OVERVIEW This course offers an advanced study of the curved space-time of general relativity. We will see how, under very general condition, such a space-time necessarily has singularities. This means that black holes and the Big-Bang are not pathologies of general relativity, but are intrinsic to the theory. AIMS AND CONTENT LEARNING OUTCOMES The purpose of this course is to show Hawking's and Penrose's singularity theorems in general relativity (for which Penrose won the Nobel Prize in Physics in 2020). To get there, we will first study the notion of completeness and extensibility for pseudo-Riemannian varieties, then the cuasale struttsara of this variety. A key notion will be that of globally hyperbolic space, the starting point of numerous advanced arguments in general relativity. AIMS AND LEARNING OUTCOMES Ability to confront well known mathematical concepts (completeness, metric space) in a new context (lorentzian geometry). Knowledge of the mathematical tools required to study the causal structure of spacetime (global iperbolicity). Developing a multidisciplinary scientific culture, at the cutting edge of recent advances in the field (study of black holes is currenbtly under a revolution, due to the discovery of black holes and the photographies of Event Horizon Telesecope). PREREQUISITES Previous course of differential geometry and/or general relativity. TEACHING METHODS traditional SYLLABUS/CONTENT • Completeness and extendibilty manifold vs metric space geodetic completeness completeness of pseudo-riemannian geometry Singular spaces cartesian and deformed product Rindler space and constant acceleration Kruskal extension and white/black holes FRLW space and the Big-Bang Singularity theorems causal structure in lorentzian geometry variation and geodesic congruence singularity theorem RECOMMENDED READING/BIBLIOGRAPHY O'Neill "Semi-riemannian geometry" Hawking & Ellis "The large scale structure of spacetime" Wald "General relativity" Lectures notes TEACHERS AND EXAM BOARD PIERRE OLIVIER MARTINETTI Ricevimento: On appointment Exam Board PIERRE OLIVIER MARTINETTI (President) NICOLA PINAMONTI SIMONE MURRO (President Substitute) MARCO BENINI (Substitute) NICOLO' DRAGO (Substitute) LESSONS LESSONS START According to the academic calendar Class schedule The timetable for this course is available here: Portale EasyAcademy EXAMS Exam schedule Data appello Orario Luogo Degree type Note 19/09/2025 09:00 GENOVA Esame su appuntamento