Part III: BH Thermodynamics & Quantum
Black holes are thermodynamic objects with temperature and entropy. They radiate via Hawking radiation, slowly evaporating. These quantum properties reveal deep connections between gravity, quantum mechanics, and information theory.
Part Overview
In the 1970s, Jacob Bekenstein and Stephen Hawking discovered that black holes have entropy proportional to their horizon area and radiate thermally with a temperature inversely proportional to their mass. This marriage of general relativity, quantum field theory, and thermodynamics reveals profound connections and deep paradoxes, including the black hole information paradox and the holographic principle.
Key Topics
- • Four laws of black hole mechanics and thermodynamic analogy
- • Bekenstein-Hawking entropy: $S_{BH} = \frac{k_Bc^3A}{4G\hbar}$
- • Hawking radiation and black hole evaporation
- • Black hole information paradox and the fate of information
- • Holographic principle: 3D information on 2D surfaces
- • AdS/CFT correspondence and black hole microstates
6 chapters | Quantum black holes | Entropy to information
Chapters
Chapter 1: Laws of Black Hole Mechanics
The four laws parallel thermodynamics: (0) Surface gravity is constant on the horizon (zeroth law). (1) First law relates mass changes to area, angular momentum, and charge. (2) Horizon area never decreases (second law). (3) Cannot reduce surface gravity to zero in finite steps (third law). Bardeen, Carter, Hawking 1973.
Chapter 2: Bekenstein Entropy
Bekenstein (1973) argued black holes must have entropy to preserve the second law of thermodynamics. The Bekenstein-Hawking formula $S_{BH} = \frac{k_Bc^3A}{4G\hbar} = \frac{k_BA}{4\ell_P^2}$ relates entropy to horizon area, where $\ell_P = \sqrt{G\hbar/c^3}$ is the Planck length. For a solar mass black hole,$S_{BH} \sim 10^{54}k_B$—enormous! This is the maximum entropy any object of that size can have. The area law suggests a holographic nature of gravity: 3D information encoded on 2D surfaces.
Chapter 3: Hawking Radiation
Black holes emit thermal radiation at temperature $T_H = \frac{\hbar c^3}{8\pi Gk_BM}$. For solar mass,$T_H \sim 60$ nanokelvin—far below the CMB at 2.7 K. Smaller black holes are hotter: $T_H \propto M^{-1}$. Derivation via quantum field theory in curved spacetime: vacuum fluctuations near the horizon create particle-antiparticle pairs. Black holes evaporate, losing mass: evaporation time $t_{evap} \propto M^3$. For $M_\odot$, $t_{evap} \sim 10^{67}$ years. Final stages: explosive?
Chapter 4: Information Paradox
Pure quantum state collapses to black hole, which then evaporates via thermal Hawking radiation. Thermal radiation carries no information about the initial state. Where did the information go? This violates unitarity (information conservation) in quantum mechanics. Proposed resolutions: information in correlations, remnants, soft hair, holography. Recent progress: island formula, Page curve. Still unsolved!
Chapter 5: Holographic Principle
Bekenstein bound: maximum entropy in a region is proportional to its surface area, not volume. 't Hooft and Susskind: the degrees of freedom of a spatial region are encoded on its boundary—a hologram. All information about a 3D volume is contained on its 2D surface. This suggests spacetime itself is emergent. Deep implications for quantum gravity.
Chapter 6: AdS/CFT and Black Holes
The Anti-de Sitter/Conformal Field Theory correspondence (Maldacena 1997): a quantum gravity theory in AdS space is dual to a CFT on its boundary. Black holes in the bulk correspond to thermal states in the CFT. This provides a microscopic description of black hole entropy by counting CFT states. Strominger-Vafa counted microstates for extremal black holes in string theory, reproducing Bekenstein-Hawking entropy exactly. A triumph for string theory.
Course Navigation
Prerequisites:
- • Part I: Black Hole Basics
- • Part II: Rotating & Charged BHs
- • Quantum Mechanics
- • Thermodynamics and statistical mechanics