Jonathan Luk- Singularities in General Relativity

Explore the fascinating world of singularities in general relativity through this comprehensive lecture by Jonathan Luk. Delve into Lorentzian geometry and Einstein equations, examining the fundamental theorem of general relativity and key concepts like Minkowski and Schwarzschild spacetimes. Investigate Penrose's theorem, asymptotically flat initial data, and the Kerr spacetime, while considering the strong cosmic censorship conjecture and stability of Kerr exterior. Analyze blue shift instability, spacelike singularity conjecture, and the Reissner-Nordström spacetime. Discover solutions arising from spherically symmetric data, weak null singularities, and the formation of trapped surfaces in vacuum. Conclude by examining black hole spacetimes arising from gravitational collapse and the potential breakdown of weak null singularities.

Intro
General relativity: Lorentzian geometry
General relativity: Einstein equation
Fundamental theorem of general relativity
Minkowski spacetime
The Schwarzschild spacetime
How general are singularities?
Construction of Schwarzschild-like singularities
Penrose's theorem
Example: geodesically incomplete but non-singular
Asymptotically flat initial data
The Kerr spacetime
Strong cosmic censorship conjecture
Stability of Kerr exterior
Blue shift instability
Spacelike singularity conjecture
Singularity structure for perturbations of Kerr?
Stability of the Kerr Cauchy horizon
The Reissner-Nordström spacetime
Solutions arising from spherically symmetric data
Weak null singularities and an instability conjecture
Formation of trapped surface in vacuum
Black hole spacetimes arising from gravitational collapse
Breakdown of weak null singularities?