General Relativity

General Relativity in 28 chapters — from “Newton’s gravity and why it cannot be the final word” to “Frontiers: black-hole thermodynamics, information, quantum gravity”.

Chapters

  1. Newton’s gravity and why it cannot be the final word
    • What Newton got right
    • Conceptual cracks
    • The Keplerian content of Newton’s law
    • Empirical cracks
    • What a successor theory must do
    • Summary and forward view
    • Problems
  2. Special relativity, geometrically
    • Events, worldlines, and spacetime
    • The invariant interval and its three types
    • Lorentz transformations from interval invariance
    • Four-vectors, proper time, and four-velocity
    • Momentum, energy, and E=mc2
    • The relativistic Doppler effect
    • From Minkowski to a general metric
    • Summary and forward view
    • Problems
  3. The equivalence principle
    • The equality of inertial and gravitational mass
    • From a coincidence to a principle
    • Three principles, and the differences between them
    • How local is “local”? The tidal residue
    • Clocks and light in a lift: gravitational redshift
    • The bending of light, and the missing half
    • Why the principle makes gravity geometry
    • Summary and forward view
    • Problems
  4. A first look at curved spacetime
    • Geometry intrinsic to a surface
    • From surface to spacetime
    • Curvature equals tidal force
    • Summary and forward view
    • Problems
  5. The Schwarzschild metric
    • The line element
    • What the metric describes
    • The horizon: coordinate singularity, not physical
    • Proper time and proper distance
    • Christoffel symbols and the connection
    • Derivation of the Schwarzschild metric
    • Summary and forward view
    • Problems
  6. Orbits, perihelion precession, and Mercury
    • The Mercury anomaly
    • Conservation laws and the radial equation
    • Reading the effective potential
    • The perihelion shift
    • The Newtonian limit and the rosette
    • Modern tests
    • Summary and forward view
    • Problems
  7. Light in the Schwarzschild geometry
    • The eclipse that made Einstein famous
    • What is a light ray in GR?
    • Conservation laws for null geodesics
    • The photon sphere
    • Light deflection
    • Gravitational lensing
    • Shapiro time delay
    • Modern tests
    • Summary and forward view
    • Problems
  8. Manifolds, tangent spaces, tensors
    • Manifolds
    • Tangent spaces
    • Tensors
    • The metric: the central tensor
    • Summary and forward view
    • Problems
  9. The metric, proper time, and volume
    • The metric as an inner product
    • Proper time and proper distance
    • Volume and the volume element
    • Putting the metric to work
    • Problems
  10. Covariant derivative and parallel transport
    • The problem with the partial derivative
    • The Christoffel symbols
    • Parallel transport
    • The Lie derivative
    • Covariant derivative as physics
    • Problems
  11. The geodesic equation, revisited
    • Variational principle
    • The geodesic equation and its Newtonian limit
    • Worked example: geodesics on the 2-sphere
    • Schwarzschild geodesics: the formal reduction
    • Back to Mercury: consistency with Chapter 6
    • Problems
  12. Curvature: Riemann, Ricci, Einstein tensor
    • Curvature as failure of derivatives to commute
    • Contractions: Ricci and the scalar curvature
    • Worked example: the 2-sphere
    • Geodesic deviation: the physics of the Riemann tensor
    • Bianchi identities and the path to field equations
    • The Weyl tensor: trace-free curvature
    • Problems
  13. Stress-energy: the sources of gravity
    • The stress-energy tensor
    • Examples
    • Energy conditions
    • Conservation: ∇μ​Tμν=0
    • Putting it all together
    • Problems
  14. The Einstein field equations
    • The equations
    • The Intuition Lab
    • Math Decoder
    • The constraint count and physical degrees of freedom
    • Uniqueness: the Lovelock argument
    • The Einstein-Hilbert action
    • The Newtonian limit, in detail
    • Worked example: the metric outside a stationary mass distribution
    • The cosmological constant: dimensional analysis and modern interpretation
    • Phenomenology Bridge
    • Material card: Einstein’s path to the equations
    • Summary and forward view
    • Problems
  15. Linearised gravity and the Newtonian limit
    • The weak-field expansion
    • Gravitoelectromagnetism
    • Frame dragging and the Lense-Thirring effect
    • Preview of gravitational waves
    • Summary and forward view
    • Problems
  16. Event horizons, Kruskal extension, Penrose diagrams
    • Crossing the horizon: Eddington-Finkelstein coordinates
    • The horizon as a one-way membrane
    • The Kruskal-Szekeres extension
    • Penrose diagrams: conformal compactification
    • The no-hair theorem
    • Summary and forward view
    • Problems
  17. Charged black holes: Reissner-Nordström
    • The Reissner-Nordström metric
    • Deriving the RN solution from Einstein-Maxwell
    • Geodesics and the effective potential
    • The extremal limit and the inner horizon
    • Reissner-Nordström as a step toward Kerr
    • Problems
  18. Rotating black holes: Kerr
    • The Kerr metric
    • Where does the Kerr metric come from?
    • The Intuition Lab
    • Math Decoder
    • The static limit and the ergosphere
    • The ZAMO frame
    • Geodesics, conserved quantities, and the Carter constant
    • ISCO and energy extraction
    • Astrophysical Kerr
    • Phenomenology Bridge
    • Material card: Kerr’s discovery and astrophysical relevance
    • Summary and forward view
    • Problems
  19. Gravitational collapse and the singularity theorems
    • The collapse picture
    • Oppenheimer-Snyder collapse in detail
    • Trapped surfaces
    • Penrose’s singularity theorem
    • The BKL conjecture
    • Cosmic censorship
    • What the theorems leave open
    • Problems
  20. Generation of gravitational waves: the quadrupole formula
    • The wave equation, sourced
    • Retarded Green’s function: derivation
    • Multipole expansion
    • The quadrupole formula
    • The two polarisations
    • Emission pattern from binaries
    • Energy carried by gravitational waves
    • Problems
  21. Binary inspirals: chirp, merger, ringdown
    • Post-Newtonian inspiral
    • The Hulse-Taylor binary pulsar
    • The IMR waveform
    • Phenomenology Bridge
    • Problems
  22. Detection and gravitational-wave astronomy
    • The interferometer
    • Matched filtering and signal extraction
    • GW150914
    • Standard sirens and the Hubble constant
    • GW170817 and multimessenger astronomy
    • The detector network
    • Future detectors
    • Problems
  23. The FRW geometry of a homogeneous universe
    • The cosmological principle and FRW metric
    • Deriving the FRW form from the cosmological principle
    • The Intuition Lab
    • The scale factor and Hubble’s law
    • Cosmological redshift from null geodesics
    • Distance measures in cosmology
    • Math Decoder
    • The Hubble tension
    • Phenomenology Bridge
    • Material card
    • Summary and forward view
    • Problems
  24. The Friedmann equations and the standard model
    • Deriving the Friedmann equations
    • The conservation law and equations of state
    • The cosmic eras
    • Single-component scale factors
    • The age of the universe
    • Math Decoder
    • Phenomenology Bridge
    • Material card: ΛCDM at a glance
    • Problems
  25. Inflation
    • The puzzles
    • Inflation: the mechanism
    • The slow-roll formalism
    • Reheating in detail
    • Solving the puzzles
    • The primordial power spectrum
    • Predictions and tests
    • Problems
  26. Dark energy and the accelerating universe
    • The 1998 discovery
    • Inferring w and ΩΛ​ from data
    • The cosmological constant problem
    • Alternatives to a cosmological constant
    • The Hubble tension
    • The history of Λ
    • Summary and forward view
    • Problems
  27. The ADM formalism and numerical relativity
    • The ADM 3+1 split
    • Gauss-Codazzi and the Einstein equations
    • Why plain ADM fails, and what BSSN changes
    • Gauge: choosing the lapse and the shift
    • Initial data: solving the constraints
    • Finding the horizons
    • Extracting the waveform
    • Summary and forward view
    • Problems
  28. Frontiers: black-hole thermodynamics, information, quantum gravity
    • Black-hole thermodynamics
    • The information paradox
    • Why general relativity must fail, and where
    • Holography
    • Approaches to quantum gravity
    • Summary and forward view
    • Problems