Modeling Black Hole Evaporation

The following sections are included:

• Preface

• Contents

Black holes are among the most fascinating predictions of Einstein's theory of general relativity. They are an exotic, but natural outcome of the very basic feature of the gravitational interaction. The universality of gravity, nicely expressed by the equivalence principle, allows to produce an accumulative effect that can result in a very strong gravitational field. Indeed, it can be so strong that it cannot be counterbalanced by other forces and continues to grow up until extreme situations. Since gravity affects the spacetime geometry, the gravitational field produced by matter could become so strong as to substantially modify the ordinary causal structure of spacetime, and produce a region where even light is trapped and no particle can escape to infinity…

The following sections are included:

• Modeling the Gravitational Collapse
  • The Oppenheimer-Snyder model
  • Non-spherical collapse

• The Schwarzschild Black Hole
  • Eddington-Finkelstein coordinates
  • Kruskal coordinates

• Causal Structure and Penrose Diagrams
  • Minkowski space
  • Schwarzschild spacetime

• Kruskal and Locally Inertial Coordinates at the Horizons
  • Redshift factors and surface gravity

• Charged Black Holes

• The Extremal Reissner–Nordström Black Hole

• Rotating Black Holes
  • Energy extraction from rotating black holes: the Penrose process

• Trapped Surfaces, Apparent and Event Horizons
  • The area law theorem

• The Laws of Black Hole Mechanics

• Stringy Black Holes

The following sections are included:

• Canonical Quantization in Minkowski Space

• Quantization in Curved Spacetimes
  • Bogolubov transformations and particle production

• Hawking Radiation in Vaidya Spacetime
  • Ingoing and outgoing modes
  • Wave packets
  • Wick rotation
  • Planck spectrum
  • Uncorrelated thermal radiation
  • Where are the correlations?
  • Thermal density matrix

• Including the Backscattering
  • Waves in the Schwarzschild geometry
  • Late-time basis to accommodate backscattering
  • Thermal radiation and grey-body factors
  • Estimations for the luminosity
    • Emission of massless and massive particles

• Importance of the Backreaction

• Late-Time Independence on the Details of the Collapse
  • The example of two shock waves
  • Geometric optics approximation
  • General collapse
  • Adding angular momentum and charge

• Black Hole Thermodynamics

• Physical Implications of Black Hole Radiance and the Information Loss Paradox
  • Black hole evaporation
  • Breakdown of quantum predictability
  • Alternatives to restore quantum predictability

The following sections are included:

• Rindler Space

• Conformal Symmetry, Stress Tensor and Particle Number
  • The normal ordered stress “tensor” operator
  • The SO(d,2) conformal group and Möbius transformations
    • Infinitesimal transformations
  • The particle number operator

• Radiation in Rindler Space: Hawking and Unruh Effects
  • The Hawking effect
    • Correlation functions and thermal radiation
  • Radiation through the horizon
  • The Unruh effect
    • Unruh modes
    • Möbius invariance of the vacuum state
  • Three different vacuum states

• Anti-de Sitter Space as a Near-Horizon Geometry
  • Extremal black holes
  • Near-extremal black holes

• Radiation in Anti-de Sitter Space: the Hawking Effect
  • Three vacuum states

• The Moving-Mirror Analogy for the Hawking Effect
  • Exponential trajectory: thermal radiation
  • Radiationless trajectories
    • Hyperbolic trajectories
  • Asymptotically inertial trajectories and unitarity

The following sections are included:

• Relating the Virasoro and Trace Anomalies via Locally Inertial Coordinates

• The Polyakov Effective Action
  • The role of the Weyl-invariant effective action

• Choice of the Quantum State
  • Boulware state
  • Hartle–Hawking state
  • The “in” and Unruh vacuum states: the Hawking flux
    • “in” vacuum state
    • Unruh state

• Including the Backscattering in the Stress Tensor: the s-wave
  • Two-dimensional symmetries
  • The normal ordered stress tensor
    • Transformation laws
  • The covariant quantum stress tensor
  • Effective action
    • Anomaly induced effective action
    • The Weyl-invariant effective action
  • Quantum states and Hawking flux
    • Boulware state
    • Hartle-Hawking state
    • The “in” and Unruh vacuum states: Hawking flux with backscattering

• Beyond the s-wave Approximation

• The Problem of Backreaction
  • Backreaction equations from spherical reduction
  • The problem of determining the state-dependent functions
  • Returning to the near-horizon approximation

The following sections are included:

• The Near Horizon Approximation
  • Schwarzschild
  • Near-extremal Reissner–Nordström black holes: the JT model
  • Near-extremal dilaton black holes
    • Einstein frame
    • String frame: the CGHS model

• The CGHS Model
  • The CGHS black hole
    • Free field
    • Linear dilaton vacuum
    • Black hole solutions
    • CGHS and four-dimensional dilaton black holes
    • CGHS and Schwarzschild black holes
  • Including matter fields
    • Dynamical black hole solutions
    • Singularities, event and apparent horizons
  • Bogolubov coefficients and Hawking radiation
  • Quantum states for the CGHS black hole
    • Boulware state
    • Hartle-Hawking state
    • Unruh state
    • “in” vacuum state

• The Problem of Backreaction in the CGHS Model
  • State-dependent functions for evaporating black holes
  • The RST model
    • Liouville fields
    • General solution
    • Semiclassical static solutions
    • Backreaction in the Unruh state
  • Black hole evaporation in the RST model
    • Information loss in the RST model
    • Is the picture of RST black hole evaporation generic?

• The Semiclassical JT Model
  • Extremal solution
  • Semiclassical static solutions
  • General solutions
  • Black hole evaporation in the JT model
    • First approximation for the state-dependent functions
    • Exact treatment for the state-dependent functions
  • Beyond the near-horizon approximation and Hawking radiation
  • Information loss in the JT model
  • Unitarity in the semiclassical approximation?

The following sections are included:

• Bibliography

• Index