Overview
This lecture explores black holes: how they can be fully described by just three numbers, their thermodynamic properties, the information paradox, and the emergence of the holographic principle in theoretical physics.
Describing Celestial Objects
- The Moon requires thousands or millions of numbers to describe features like craters, coordinates, and composition.
- Black holes, regardless of size or history, are fully described by only three numbers: mass, electric charge, and angular momentum.
Early History of Black Holes
- John Michell first theorized gravitation could trap light, defining a "dark star" (later known as a black hole).
- Escape velocity calculations show that if an object's density is high enough, not even light can escape.
Einstein and Schwarzschild
- Einstein’s general relativity describes gravity as the deformation of spacetime by mass.
- Karl Schwarzschild found the first exact black hole solution, introducing the concept of the event horizon (point of no return).
Black Hole Uniqueness and No-Hair Theorem
- Only three parameters (mass, charge, spin) are needed to describe a black hole—proven by the black hole uniqueness theorem.
- No matter how a black hole forms, all other information is lost ("black holes have no hair").
Thermodynamics and Entropy
- Entropy measures the number of internal microstates corresponding to observable parameters.
- Jacob Bekenstein and Stephen Hawking showed black holes possess entropy proportional to the area of their event horizon, not volume.
- The second law of thermodynamics applies: black hole entropy never decreases.
Hawking Radiation and Black Hole Temperature
- Hawking predicted black holes emit radiation ("Hawking radiation") due to quantum effects near the event horizon.
- Black hole temperature is inversely proportional to its mass: smaller black holes are hotter.
- Black holes can eventually evaporate due to Hawking radiation.
Information Paradox and Holographic Principle
- The apparent loss of information in black hole evaporation leads to the information paradox.
- The holographic principle states that all information about a volume can be encoded on its boundary (event horizon).
- This suggests black hole evaporation preserves information in subtle quantum correlations in the radiation.
Open Questions and String Theory Insights
- The microscopic structure of black holes is not fully understood for real astrophysical cases.
- String theory in higher dimensions allows calculation of black hole microstates, matching entropy predictions in some cases.
- The true internal composition of real black holes remains unknown, but the holographic principle offers hope for future understanding.
Key Terms & Definitions
- Event Horizon — the boundary around a black hole beyond which nothing can escape.
- Escape Velocity — the speed required to break free from a body's gravitational pull.
- Entropy — a measure of the number of internal microstates corresponding to observable properties.
- Hawking Radiation — theoretical radiation emitted by black holes due to quantum effects.
- No-Hair Theorem — the statement that black holes are fully described by mass, charge, and angular momentum.
- Holographic Principle — the idea that all information in a volume can be represented on its boundary.
Action Items / Next Steps
- Review the definitions and implications of black hole entropy and Hawking radiation.
- Study the basics of the holographic principle and its connection to black holes.
- Prepare questions on the information paradox for further discussion.