General Relativity and the Hawking-Penrose Singularity Theorems
This course provides a rigorous analysis of the 1970 landmark paper by Stephen Hawking and Roger Penrose, 'The Singularities of Gravitational Collapse and Cosmology.' Students will explore the mathematical foundations of space-time singularities, focusing on how the authors generalized previous results to prove that singularities are an inevitable feature of general relativity under physically reasonable conditions. The curriculum covers the transition from symmetrical models to global differential geometry, the role of trapped surfaces, and the implications for both black hole physics and Big Bang cosmology. Participants will examine the four critical physical assumptions—Einstein’s field equations, the strong energy condition, causality (no closed timelike curves), and the generality of curvature—while investigating the Raychaudhuri effect and the nature of geodesic incompleteness. By the end of the course, students will be able to articulate why singularities are not merely artifacts of high symmetry but fundamental predictions of gravitational theory in both collapsing stars and expanding universes.
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