Lectures on General Relativity, Cosmology and Quantum Black Holes (Second Edition)
Description
The primary goal of this book is to allow students to understand in a critical way two pillars of modern theoretical physics: inflationary theory, and quantum black holes and the information-loss problem. Accordingly, the book extensively introduces black holes and cosmology before proceeding to the important issues found in inflation and the information-loss problem. Other important topics discussed in detail include the cosmological constant and its relation to dark energy and an introduction to quantum field theory on curved backgrounds.
In this updated and extended edition in addition to including new exercises and developments in the field, the author extends the coverage by including new chapters on loop quantum gravity, the impact of anti-de Sitter/conformal field theory correspondence and quantum entanglement on the origins of space time and the black-hole information problem, and quantum gravity in the context of matrix quantum mechanics. Including numerous exercises, the material provides a single rigorous text for advanced students in theoretical physics and mathematics requiring an introduction to the implications and interpretation of general relativity in areas of cosmology. Readers of this text will be well prepared to follow the theoretical developments in the field and undertake research projects as part of an MSc or PhD programme.
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Badis Ydri is a professor of theoretical physics at the Institute of Physics, Annaba University, Algeria. He earned his PhD from Syracuse University, New York, USA, in 2001. He is also a research associate at the Dublin Institute for Advanced Studies, Ireland. His postdoctoral work includes a Marie Curie fellowship at Humboldt University in Berlin, Germany, and a Hamilton fellowship at the Dublin Institute for Advanced Studies, Ireland. His general areas of expertise encompasses quantum field theory, general relativity, string theory, and philosophy of physics. His ongoing research explores: 1) matrix quantum mechanics approaches to quantum black holes and quantum gravity, 2) gauge/gravity duality and M-theory, 3) noncommutative geometry and matrix models, 4) renormalization group equation and Monte Carlo methods, 5) artificial intelligence in computational physics, 6) quantum philosophy, and 7) hard physical philosophy of consciousness and existence. He is the author of seven books in theoretical physics.
Content
- Outline placeholder
- Introduction
- Organization
- Proposal
- Author biography
- Badis Ydri
- Chapter General relativity essentials
- 1.1 Equivalence principle
- 1.2 Relativistic mechanics
- 1.3 Differential geometry primer
- 1.3.1 Metric manifolds and vectors
- 1.3.2 Geodesics
- 1.3.3 Tensors
- 1.4 Curvature tensor
- 1.4.1 Covariant derivative
- 1.4.2 Parallel transport
- 1.4.3 The Riemann curvature tensor
- 1.5 The stress-energy-momentum tensor
- 1.5.1 The stress-energy-momentum tensor
- 1.5.2 Perfect fluid
- 1.5.3 Conservation law
- 1.5.4 Minimal coupling
- 1.6 Einstein's equation
- 1.6.1 Tidal gravitational forces
- 1.6.2 Geodesic deviation equation
- 1.6.3 Einstein's equation
- 1.6.4 Newtonian limit
- 1.7 Killing vectors and maximally symmetric spaces
- 1.8 The Hilbert-Einstein action
- 1.9 Exercises
- References
- Chapter Black holes
- 2.1 Spherical star
- 2.1.1 The Schwarzschild metric
- 2.1.2 Particle motion in Schwarzschild spacetime
- 2.1.3 Precession of perihelia and gravitational redshift
- 2.1.4 Free fall
- 2.2 Schwarzschild black hole
- 2.3 The Kruskal-Szekres diagram: maximally extended Schwarzschild solution
- 2.4 Various theorems and results
- 2.5 Reissner-Nordström (charged) black hole
- 2.5.1 Maxwell's equations and charges in GR
- 2.5.2 Reissner-Nordström solution
- 2.5.3 Extremal Reissner-Nordström black hole
- 2.6 Kerr spacetime
- 2.6.1 Kerr (rotating) and Kerr-Newman (rotating and charged) black holes
- 2.6.2 Killing horizons
- 2.6.3 Surface gravity
- 2.6.4 Event horizons, ergosphere and singularity
- 2.6.5 Penrose process
- 2.7 Black hole thermodynamics
- 2.8 Exercises
- References
- Chapter Cosmology I: the expanding Universe
- 3.1 Homogeneity and isotropy
- 3.2 Expansion and distances
- 3.2.1 Hubble law
- 3.2.2 Cosmic distances from standard candles
- 3.3 Matter, radiation, and vacuum
- 3.4 Flat universe
- 3.5 Closed and open universes
- 3.6 Aspects of the early Universe
- 3.7 Concordance model
- 3.8 Friedmann-Lemaître-Robertson-Walker metric
- 3.9 Friedmann equations
- 3.9.1 The first Friedmann equation
- 3.9.2 Cosmological parameters
- 3.9.3 Energy conservation
- 3.10 Examples of scale factors
- 3.10.1 Matter and radiation-dominated universes
- 3.10.2 Vacuum-dominated universes
- 3.10.3 Milne universe
- 3.10.4 The static universe
- 3.10.5 Expansion versus recollapse
- 3.11 Redshift, distances and age
- 3.11.1 Redshift in a flat universe
- 3.11.2 Cosmological redshift
- 3.11.3 Comoving and instantaneous physical distances
- 3.11.4 Luminosity distance
- 3.11.5 Other distances
- 3.11.6 Age of the Universe
- 3.12 Exercises
- References
- Chapter Cosmology II: the inflationary Universe
- 4.1 Cosmological puzzles
- 4.1.1 Homogeneity/horizon problem
- 4.1.2 Flatness problem
- 4.2 Elements of inflation
- 4.2.1 Solving the flatness and horizon problems
- 4.2.2 Inflation
- 4.2.3 Amount of inflation
- 4.2.4 End of inflation: reheating and scalar-matter-dominated epoch
- 4.3 Perfect fluid revisited
- 4.4 Cosmological perturbations
- 4.4.1 Metric perturbations
- 4.4.2 Gauge transformations
- 4.4.3 Linearized Einstein equations
- 4.4.4 Explicit calculation of dG^?µ
- 4.4.5 Matter perturbations
- 4.5 Matter-radiation equality
- 4.6 Hydrodynamical adiabatic scalar perturbations
- 4.7 Quantum cosmological scalar perturbations
- 4.7.1 Slow-roll revisited
- 4.7.2 Mukhanov action
- 4.7.3 Quantization and inflationary spectrum
- 4.8 Rederivation of the Mukhanov action
- 4.8.1 Mukhanov action from ADM
- 4.8.2 Power spectra and tensor perturbations
- 4.8.3 CMB temperature anisotropies
- 4.9 Exercises
- References
- Chapter Quantum field theory on curved backgrounds, vacuum energy and quantum gravity
- 5.1 Dark energy
- 5.2 The cosmological constant
- 5.3 Elements of quantum field theory in curved spacetime
- 5.4 Calculation of vacuum energy in curved backgrounds
- 5.4.1 Quantization in Friedmann-Lemaître-Robertson-Walker (FLRW) universes
- 5.4.2 Instantaneous vacuum
- 5.4.3 Quantization in de Sitter spacetime and Bunch-Davies vacuum
- 5.4.4 QFT on curved background with a cutoff
- 5.4.5 The conformal limit ??1/6
- 5.5 Is vacuum energy real?
- 5.5.1 The Casimir force
- 5.5.2 The Dirichlet propagator
- 5.5.3 Another derivation using the energy-momentum tensor
- 5.5.4 From renormalizable field theory
- 5.5.5 Is vacuum energy really real?
- 5.6 The ADM formulation
- 5.7 A brief introduction of Horava-Lifshitz quantum gravity
- 5.7.1 Lifshitz scalar field theory
- 5.7.2 Foliation-preserving diffeomorphisms and kinetic action
- 5.7.3 Potential action and detailed balance
- 5.8 Exercises
- References
- Chapter Hawking radiation, the information paradox and black hole thermodynamics
- 6.1 Introduction and summary
- 6.2 Rindler spacetime and general relativity
- 6.2.1 Rindler spacetime
- 6.2.2 Review of general relativity
- 6.3 Schwarzschild black holes
- 6.3.1 Schwarzschild black holes
- 6.3.2 Near horizon coordinates
- 6.4 Kruskal-Szekres diagram
- 6.4.1 Kruskal-Szekres extension and Einstein-Rosen bridge
- 6.4.2 Euclidean black hole and thermal field theory
- 6.5 Density matrix and entanglement
- 6.5.1 Density matrix: pure and mixed states
- 6.5.2 Entanglement, decoherence and von Neumann entropy
- 6.6 Rindler decomposition and Unruh effect
- 6.6.1 Rindler decomposition
- 6.6.2 Unruh temperature
- 6.7 Quantum field theory in curved spacetime
- 6.8 Hawking radiation
- 6.8.1 The Unruh effect revisited
- 6.8.2 From quantum scalar field theory in rindler background
- 6.8.3 Summary
- 6.9 Hawking radiation from quantum field theory in Schwarzschild background
- 6.9.1 Kruskal and Schwarzschild (Boulware) observers and field expansions
- 6.9.2 Bogolubov coefficients
- 6.9.3 Hawking radiation and Hawking temperature
- 6.10 The Unruh versuss Boulware vacua: pure to mixed
- 6.10.1 The adiabatic principle and trans-Planckian reservoir
- 6.10.2 The Unruh method revisited and greybody factor
- 6.10.3 Unruh vacuum state │U>
- 6.11 The information problem in black hole Hawking radiation
- 6.11.1 Information loss, remnants and unitarity
- 6.11.2 Information conservation principle
- 6.11.3 Page curve and Page theorem
- 6.12 Black hole thermodynamics
- 6.12.1 Penrose diagrams
- 6.12.2 Bekenstein-Hawking entropy formula
- 6.12.3 Brick wall and stretched horizon
- 6.12.4 Conclusion
- 6.13 Exercises
- References
- Chapter Quantum black holes and gauge/gravity duality
- 7.1 Outline
- 7.2 The JT-CFT2 system: a first look
- 7.2.1 Eternal AdS2 black hole
- 7.2.2 AdS2 black hole formed from gravitational collapse
- 7.2.3 Quantum correction and coupling to a heat bath
- 7.2.4 More on the boundary theory and the Schwarzian
- 7.3 More on JT gravity coupled to conformal matter
- 7.3.1 The eternal AdS2 black hole
- 7.3.2 The coupling to conformal matter
- 7.3.3 The Hartle-Hawking state
- 7.3.4 An evaporating AdS2 black hole
- 7.4 Entropy and quantum extremal surface in JT gravity
- 7.4.1 Entropies, Ryu-Takayanagi formula and the island conjecture
- 7.4.2 Entanglement entropy of an interval in CFT2 and AdS2
- 7.4.3 The late time quantum extremal surfaces
- 7.4.4 The early time quantum extremal surfaces
- 7.5 The 'island' conjecture
- 7.5.1 The Randall-Sundrum model: a digression
- 7.5.2 Holographic conformal matter
- 7.5.3 The island and entanglement wedges at late times
- 7.5.4 The entanglement wedges at early times and Page curve
- 7.5.5 The AdS2 eternal black hole: the two-intervals entropy of radiation
- 7.5.6 The replica wormhole picture
- 7.5.7 Replica wormhole versus Hawking saddle
- 7.6 Summary
- References
- Chapter Loop quantum gravity and Banks-Fischler-Shenker-Susskind approaches to quantum gravity
- 8.1 Loop quantum gravity
- 8.1.1 Outline
- 8.1.2 Review of general relativity
- 8.1.3 The Hilbert-Einstein action
- 8.1.4 The vielbein formalism
- 8.1.5 ADM formulation and geometrodynamics
- 8.1.6 The Palatini action and Ashtekar variables
- 8.1.7 General relativity as an SU(2) gauge theory of self-dual spin connections
- 8.1.8 The real SU(2) gauge theory
- 8.1.9 Loop representation and spin networks
- 8.2 BFSS matrix model and gauge/gravity duality
- 8.2.1 The action and symmetries
- 8.2.2 The non-perturbative dynamics
- 8.2.3 Dp-branes or how matrices arise
- 8.2.4 D0-branes or how gravitons arise
- 8.2.5 The DLCQ quantization
- 8.2.6 DLCQ quantization of M-theory and BFSS model
- 8.2.7 Supermembranes and non-commutative geometry
- 8.2.8 Conclusion
- 8.3 Causal dynamical triangulation and multitrace matrix models
- References
- Chapter
- A.1 Manifolds
- A.1.1 Maps, open set and charts
- A.1.2 Manifold: definition and examples
- A.1.3 Vectors and directional derivative
- A.1.4 Dual vectors and tensors
- A.1.5 Metric tensor
- A.2 Curvature
- A.2.1 Covariant derivative
- A.2.2 Parallel transport
- A.2.3 The Riemann curvature
- A.2.4 Geodesics
