Black holes: from horizon to singularity

Black holes represent one of the most extreme predictions of general relativity, yet they are now routinely observed through gravitational waves and direct imaging. Understanding them is essential for grasping how gravity shapes the universe, from stellar death to galaxy formation. This course provides a clear, rigorous foundation in black hole physics, connecting theoretical concepts to observational evidence without oversimplification. The programme begins with stellar evolution and the conditions that lead to gravitational collapse. You will examine the Chandrasekhar limit and the Oppenheimer-Volkoff equation to understand when a neutron star must become a black hole. The core of the course covers the Schwarzschild and Kerr solutions, including the event horizon, the ergosphere, and the Penrose process. Detailed attention is given to the singularity theorems of Penrose and Hawking, as well as the information paradox and Hawking radiation. Observational topics include the detection of gravitational waves from binary black hole mergers (LIGO/Virgo) and the Event Horizon Telescope images of M87* and Sagittarius A*, including how interferometry reconstructs the shadow of a black hole. The methodology combines formal derivations with conceptual explanations. You will work through key equations—such as the Schwarzschild metric and the geodesic equation—to understand light bending, time dilation, and orbital precession near a black hole. Common pitfalls, such as misinterpreting the event horizon as a physical surface or conflating coordinate singularities with real ones, are addressed explicitly. The course also clarifies the distinction between stellar-mass and supermassive black holes, and why Hawking radiation is negligible for the latter. Applied exercises involve interpreting spacetime diagrams and calculating basic quantities like the Schwarzschild radius. This course is designed for advanced undergraduate students in physics or astronomy, graduate students transitioning into general relativity or astrophysics, researchers in adjacent fields (e.g., high-energy physics, cosmology) who need a compact yet thorough treatment of black holes, and science educators who teach relativity or modern astrophysics and want to deepen their own understanding. By the end, you will be able to derive the Schwarzschild solution from the Einstein field equations, explain the causal structure of a black hole spacetime using Penrose diagrams, interpret the physical meaning of the event horizon and singularity, and critically discuss the observational evidence for black holes. You will have a working vocabulary of key terms (e.g., ISCO, photon sphere, Killing horizon) and be prepared to read current research literature or pursue further study in quantum gravity.