Spacetime and Geometry:
Contents

Contents

• Preface

• Special Relativity and Flat Spacetime
  • Prelude
  • Space and time, separately and together
  • Lorentz transformations
  • Vectors
  • Dual vectors (one-forms)
  • Tensors
  • Manipulating tensors
  • Maxwell's equations
  • Energy and momentum
  • Classical field theory
  • Exercises
  
  
• Manifolds
  • Gravity as geometry
  • What is a manifold?
  • Vectors again
  • Tensors again
  • The metric
  • An expanding universe
  • Causality
  • Tensor densities
  • Differential forms
  • Integration
  • Exercises
  
  
• Curvature
  • Overview
  • Covariant derivatives
  • Parallel transport and geodesics
  • Properties of geodesics
  • The expanding universe revisited
  • The Riemann curvature tensor
  • Properties of the Riemann tensor
  • Symmetries and Killing vectors
  • Maximally symmetric spaces
  • Geodesic deviation
  • Exercises
  
  
• Gravitation
  • Physics in curved spacetime
  • Einstein's equation
  • Lagrangian formulation
  • Properties of Einstein's equation
  • The cosmological constant
  • Energy conditions
  • The Equivalence Principle revisited
  • Alternative theories
  • Exercises
  
  
• The Schwarzschild Solution
  • The Schwarzschild metric
  • Birkhoff's theorem
  • Singularities
  • Geodesics of Schwarzschild
  • Experimental tests
  • Schwarzschild black holes
  • The maximally extended Schwarzschild solution
  • Stars and black holes
  • Exercises
  
  
• More General Black Holes
  • The black hole zoo
  • Event Horizons
  • Killing Horizons
  • Mass, charge, and spin
  • Charged (Reissner-Nordström) black holes
  • Rotating (Kerr) black holes
  • The Penrose process and black-hole thermodynamics
  • Exercises
  
  
• Perturbation Theory and Gravitational Radiation
  • Linearized theory and gauge transformations
  • Degrees of freedom
  • Newtonian fields and photon trajectories
  • Gravitational wave solutions
  • Production of gravitational waves
  • Energy loss due to gravitational radiation
  • Detection of gravitational waves
  • Exercises
  
  
• Cosmology
  • Maximally symmetric universes
  • Robertson-Walker metrics
  • The Friedmann equation
  • Evolution of the scale factor
  • Redshifts and distances
  • Gravitational lensing
  • Our universe
  • Inflation
  • Exercises
  
  
• Quantum Field Theory in Curved Spacetime
  • Introduction
  • Quantum mechanics
  • Quantum field theory in flat spacetime
  • Quantum field theory in curved spacetime
  • The Unruh effect
  • The Hawking effect and black hole evaporation
  
  
• Appendices
  • Maps between manifolds
  • Diffeomorphisms and Lie derivatives
  • Submanifolds
  • Hypersurfaces
  • Stokes's theorem
  • Geodesic congruences
  • Conformal transformations
  • Conformal diagrams
  • The parallel propagator
  • Non-coordinate bases