Part I: Black Hole Basics
Black holes are the simplest macroscopic objects in the universe—completely characterized by mass, charge, and angular momentum. This part explores the Schwarzschild solution, the first exact black hole solution to Einstein's equations, discovered in 1916.
Part Overview
The Schwarzschild solution describes a non-rotating, electrically neutral black hole. It's the exterior geometry of any spherically symmetric mass distribution. We'll study the event horizon—a one-way membrane from which even light cannot escape—the central singularity, geodesics of particles and photons, the photon sphere, and coordinate systems that reveal the true structure of black hole spacetime.
Key Topics
- • Schwarzschild metric and Birkhoff's theorem
- • Event horizon at $r_s = 2GM/c^2$: the point of no return
- • Singularities: coordinate vs. curvature singularities
- • Timelike and null geodesics near black holes
- • Photon sphere at $r = 3GM/c^2$: unstable circular light orbits
- • Eddington-Finkelstein coordinates: crossing the horizon
6 chapters | Foundation of black hole physics | From Schwarzschild to the horizon
Chapters
Chapter 1: Schwarzschild Solution
The Schwarzschild metric: $ds^2 = -\left(1 - \frac{r_s}{r}\right)c^2dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1}dr^2 + r^2d\Omega^2$, where $r_s = 2GM/c^2$ is the Schwarzschild radius. Derivation from spherical symmetry. Birkhoff's theorem: this is the unique spherically symmetric vacuum solution. Comparison to Newtonian gravity.
Chapter 2: Event Horizon
At $r = r_s$, the metric appears to diverge—but this is a coordinate singularity, not physical. The event horizon is a null surface: light cones tip over so that all future-directed paths point inward. Crossing the horizon is a one-way trip. The horizon as a causal boundary. Properties: area $A = 4\pi r_s^2$, surface gravity $\kappa = c^4/(4GM)$.
Chapter 3: Singularities
At $r = 0$, the curvature invariant $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ diverges to infinity—a true singularity. Coordinate singularities (like $r = r_s$) can be removed by coordinate transformations; curvature singularities cannot. Penrose-Hawking singularity theorems: singularities are generic in gravitational collapse. Cosmic censorship: singularities are hidden inside horizons.
Chapter 4: Geodesics Near Black Holes
Solving the geodesic equation in Schwarzschild spacetime. Conserved quantities from Killing vectors. Effective potential for radial motion. Circular orbits stable outside the ISCO (innermost stable circular orbit). Radial infall dynamics. Gravitational redshift and time dilation effects near the horizon.
Chapter 5: Photon Sphere and Light Rings
Photons can orbit at $r = 3GM/c^2 = 1.5r_s$, the photon sphere. These orbits are unstable: slight perturbations send photons either to infinity or into the black hole. The photon sphere determines the size of the black hole shadow seen by distant observers. EHT image of M87*: the dark region is bounded by light from the photon sphere. Light bending and gravitational lensing near black holes.
Chapter 6: Eddington-Finkelstein Coordinates
Schwarzschild coordinates break down at the horizon. Eddington-Finkelstein coordinates are regular there: ingoing E-F uses $v = t + r^* = t + r + r_s\ln|r/r_s - 1|$, outgoing uses $u = t - r^*$. In these coordinates, infalling observers smoothly cross the horizon in finite proper time. Kruskal-Szekeres coordinates: maximal analytic extension revealing the full causal structure, including the white hole region.
Course Navigation
Prerequisites:
- • General Relativity (Parts I-III)
- • Special relativity and geodesic equations