Lecture Notes: Black Holes, Holography, and Quantum Information

Introduction

• Course Title: Black Holes, Holography, and Quantum Information
• Instructor: Chetan Krishnan, CHP (Center for High Energy Physics)
• Contact: Email is preferred for communication.
• Objective: Understanding quantum aspects of gravity through the lens of black holes.

Course Overview

• Black holes are referred to as the hydrogen atom of quantum gravity.
  • Helps understand fundamental aspects of quantum gravity despite our incomplete knowledge.
• Current understanding is limited; we have theories but lack a complete framework.
• Focus on questions surrounding black holes:
  • What happens at the horizon of a black hole?
  • What occurs at the singularity?
• Key Theoretical Framework: Area CFT correspondence (a way to quantize gravity).

Structure of the Course

• Today's lecture will introduce terminologies and concepts.
• Next lectures will be more systematic and detailed.
• Emphasis on theoretical understanding over astrophysical aspects.

Black Holes in Context

• Black holes are formed by gravitational collapse in astrophysics.
• Interest lies in understanding them as solutions to general relativity to inform theories of quantum gravity.
• Connection with recent advancements in astrophysics:
  • Detection of gravitational waves, imaging black hole horizons, etc.

Prerequisites for the Course

• Essential Background:
  • General Relativity: Familiarity with Einstein's equations and the Schwarzschild solution is necessary.
  • Quantum Field Theory: Basic understanding, particularly of free scalar field theory (first chapter of standard texts).

Recommended Reading

• General Relativity: "Spacetime and Geometry" by Sean Carroll.
• Quantum Field Theory: "Quantum Field Theory in a Nutshell" by Gerald S. Brown.

Course Evaluation

• Grading based on attendance, participation, and a presentation at the end of the course (50/50 split).

Key Concepts in Black Hole Physics

Black Hole Entropy

• Entropy of black holes is proportional to the area of their event horizon: [ S = \frac{A}{4G} ]
  • Beckenstein-Hawking Entropy: Introduced by Jacob Beckenstein, later confirmed by Stephen Hawking's calculations.
  • Surprising aspect: Black holes retain thermodynamic properties despite having no parameters (no hair theorem).

Information Paradox

• Major issue in understanding black holes: How does information behave as a black hole evaporates?
• Hawking's original conclusion: Information is lost during black hole evaporation, contradicting quantum mechanics' unitarity.
• Unit of information is related to the coherence of quantum states.
  • Pure states should evolve into pure states, but black hole evaporation leads to a mixed thermal state.

Smoothness of Black Hole Horizons

• Key Assumption: Smoothness of the event horizon is crucial for Hawking's calculations.
  • A freely falling observer at the horizon experiences local physics akin to Minkowski space.
• If smoothness breaks down, it can lead to contradictions in understanding black hole evaporation and information retention.

Conclusion and Future Topics

• The course will delve deeper into the theoretical underpinnings of black holes.
• Next lectures will cover:
  • Singularities and their implications.
  • The causal structure of black holes.

Final Notes

• Questions are encouraged at any time during the course for clarification.
• Reminder that the implications of findings in black hole physics are still under active research and discussion.