Part IV: Classic Solutions
The Einstein field equations are notoriously difficult to solve. However, by exploiting symmetries, we can find exact solutions that describe black holes, rotating black holes, charged black holes, expanding universes, and gravitational waves.
Part Overview
This part covers the most important exact solutions to Einstein's field equations. The Schwarzschild solution describes non-rotating black holes and is the first exact solution discovered (1916). The Kerr solution generalizes to rotating black holes. The FLRW metric describes homogeneous, isotropic cosmology. Gravitational waves are ripples in spacetime that travel at the speed of light, recently detected by LIGO.
Key Topics
- • Schwarzschild solution: spherically symmetric vacuum spacetime
- • Event horizons, singularities, and black holes
- • Kerr solution: rotating black holes and the ergosphere
- • Reissner-Nordström solution: electrically charged black holes
- • FLRW cosmology: expanding universe, Hubble law, cosmological constant
- • Gravitational waves: linearized theory, polarizations, LIGO detections
6 chapters | Exact solutions | Black holes to the cosmos
Chapters
Chapter 1: Schwarzschild Solution
The Schwarzschild metric: . Spherical symmetry and Birkhoff's theorem. The Schwarzschild radius . Event horizon, photon sphere, geodesics. Gravitational redshift. Radial infall and coordinate singularities.
Chapter 2: Kerr Solution (Rotating BH)
The Kerr metric describes rotating black holes. Boyer-Lindquist coordinates. Angular momentum parameter . The ergosphere: region where nothing can remain stationary. Frame dragging. The Penrose process: extracting rotational energy. Extremal Kerr: . Most astrophysical black holes rotate.
Chapter 3: Reissner-Nordström Solution
Electrically charged black holes. The metric: . Inner and outer horizons. Extremal case: . Naked singularities and cosmic censorship. Why astrophysical black holes are nearly neutral.
Chapter 4: FLRW Cosmology
The Friedmann-Lemaître-Robertson-Walker metric: . Homogeneity and isotropy. Scale factor . Curvature parameter . Friedmann equations: . Hubble's law, cosmological redshift, the Big Bang.
Chapter 5: Gravitational Waves
Ripples in spacetime that travel at c. Linearized gravity: perturbations on flat space. Transverse-traceless gauge. Two polarizations: plus (+) and cross (×). Quadrupole formula for radiation. Energy carried by gravitational waves. LIGO detections: binary black hole mergers, binary neutron star mergers. GW150914 and the new era of gravitational wave astronomy.
Chapter 6: Plane Wave Solutions
Exact plane-parallel (pp) wave solutions. The pp-wave metric: . Null coordinates. Brinkmann coordinates. These spacetimes have vanishing Ricci tensor but nonzero Weyl tensor. Applications to gravitational wave propagation and quantum field theory in curved spacetime.