Part V: Advanced Topics
Beyond the classic solutions lie deep theoretical questions and cutting-edge applications: singularity theorems, black hole thermodynamics, numerical relativity, gravitational lensing, and the interface with quantum mechanics.
Part Overview
This part explores the frontiers of general relativity. Penrose and Hawking proved that singularities (points of infinite curvature) are generic features of GR, not just artifacts of symmetry. Black hole thermodynamics reveals deep connections between gravity, quantum mechanics, and thermodynamics. The ADM formalism casts GR as a Hamiltonian system, enabling numerical simulations of black hole mergers. Gravitational lensing uses spacetime curvature as a cosmic telescope. Finally, we glimpse the challenges of quantum gravity.
Key Topics
- • Singularity theorems: Penrose, Hawking, and the inevitability of singularities
- • Black hole thermodynamics: Bekenstein-Hawking entropy, Hawking radiation
- • ADM formalism: 3+1 split of spacetime, Hamiltonian GR
- • Numerical relativity: simulating black hole mergers and gravitational waves
- • Gravitational lensing: weak lensing, strong lensing, microlensing
- • Path to quantum gravity: the problem of quantizing GR
6 chapters | Frontiers of GR | From black hole entropy to quantum gravity
Chapters
Chapter 1: Singularity Theorems
Penrose's 1965 theorem: gravitational collapse leads to singularities. Hawking's extension to cosmology: the Big Bang is a singularity. Conditions: energy conditions, trapped surfaces, causality. Singularities are not coordinate artifacts—they're genuine spacetime pathologies. Cosmic censorship conjecture: singularities are hidden inside black holes.
Chapter 2: Black Hole Thermodynamics
The four laws of black hole mechanics parallel thermodynamics. Bekenstein-Hawking entropy: where is the horizon area. Black holes have temperature: Hawking radiation makes them radiate with . Information paradox: does Hawking radiation preserve information? Holographic principle.
Chapter 3: ADM Formalism (3+1 Split)
Arnowitt-Deser-Misner formulation: splitting spacetime into space + time. Foliation by spacelike hypersurfaces. Lapse function and shift vector. ADM Hamiltonian and constraints. Initial value problem in GR: specifying data on a Cauchy surface. The Hamiltonian formulation is essential for canonical quantization and numerical relativity.
Chapter 4: Numerical Relativity
Solving Einstein's equations on computers. Finite differencing, spectral methods. Stability and convergence. Simulating binary black hole mergers: inspiral, merger, ringdown. Extracting gravitational waveforms for LIGO. The 2005 breakthroughs that enabled long-term stable simulations. Applications: testing LIGO signals, understanding astrophysical systems.
Chapter 5: Gravitational Lensing
Light bends around massive objects. Weak lensing: statistical distortions of galaxy shapes, mapping dark matter. Strong lensing: multiple images, Einstein rings, time delays. Microlensing: detecting dark objects via brightness fluctuations. Lensing equation. Magnification and shear. Applications: measuring the Hubble constant, finding exoplanets, probing dark matter.
Chapter 6: Path to Quantum Gravity
GR is a classical theory; quantum mechanics governs the microscopic world. What happens when both matter? The Planck scale: m. Problems with naively quantizing GR: non-renormalizability. Approaches: string theory, loop quantum gravity, asymptotic safety. Black hole entropy hints at a quantum theory of gravity. The path forward.
Course Navigation
Prerequisites:
- • All previous GR parts (Parts I-IV)
- • Quantum Mechanics
- • Thermodynamics and statistical mechanics