Chapter 24: Singularities
Singularities are points where spacetime curvature becomes infinite and classical general relativity breaks down. The Penrose-Hawking singularity theorems prove they're inevitable under reasonable physical conditions.
Types of Singularities
Coordinate Singularities
Apparent singularities due to bad coordinate choice (e.g., r = 2M in Schwarzschild). Can be removed by coordinate transformation. Not physical.
Curvature Singularities
True physical singularities where RμνρσRμνρσ → ∞. Cannot be removed. Example: r = 0 in Schwarzschild.
Naked vs Clothed
Clothed: hidden behind event horizon (Schwarzschild r=0).
Naked: visible to external observers (forbidden by cosmic censorship?).
Penrose-Hawking Theorems
Conditions for Singularities
- Strong energy condition: Rμνuμuν ≥ 0 for all timelike u
- Global hyperbolicity (no closed timelike curves)
- Existence of a trapped surface or sufficient matter concentration
⟹ Geodesic incompleteness (singularity) is inevitable!
These theorems don't tell us what happens at the singularity — only that classical GR predicts incomplete geodesics. Quantum gravity is needed to understand the singularity itself.