General Relativity – Benjamin Crowell

  • A geometrical theory of spacetime
    • Time and causality
    • Experimental tests of the nature of time
      • The Hafele-Keating experiment
      • Muons
      • Gravitational red-shifts
    • Non-simultaneity and the maximum speed of cause and effect
    • Ordered geometry
    • The equivalence principle
      • Proportionality of inertial and gravitational mass
      • Geometrical treatment of gravity
      • Eotvos experiments
      • The equivalence principle
      • Gravitational red-shifts
      • The Pound-Rebka experiment
  • Geometry of flat spacetime
    • Affine properties of Lorentz geometry
      • Parallelism and measurement
    • Relativistic properties of Lorentz geometry
      • Geodesics and stationary action
    • The light cone
      • Velocity addition
      • Logic
    • Experimental tests of Lorentz geometry
      • Dispersion of the vacuum
      • Observer-independence of c
      • Lorentz violation by gravitational forces
    • Three spatial dimensions
      • Lorentz boosts in three dimensions
      • Gyroscopes and the equivalence principle
      • Boosts causing rotations
      • An experimental test: Thomas precession in hydrogen
  • Differential geometry
    • Tangent vectors
    • Affine notions and parallel transport
      • The affine parameter in curved spacetime: a rough sketch
      • The affine parameter in more detail
      • Parallel transport
    • Models
    • Intrinsic quantities
      • Coordinate independence
    • The metric
      • The Euclidean metric
      • The Lorentz metric
      • Isometry, inner products, and the Erlangen program
      • Einstein’s carousel
    • The metric in general relativity
      • The hole argument
      • A Machian paradox
    • Interpretation of coordinate independence
      • Is coordinate independence obvious?
      • Is coordinate independence trivial?
      • Coordinate independence as a choice of gauge
  • Tensors
    • Lorentz scalars
    • Four-vectors
      • The velocity and acceleration four-vectors
      • The momentum four-vector
      • The frequency vector and the relativistic Doppler shift
      • A non-example: electric and magnetic fields
      • The electromagnetic potential four-vector
    • The tensor transformation laws
    • Experimental tests
      • Universality of tensor behavior
      • Speed of light differing from c
      • Degenerate matter
    • Conservation laws
      • No general conservation laws
      • Conservation of angular momentum and frame dragging
    • Things that aren’t quite tensors
      • Area, volume, and tensor densities
      • The Levi-Civita symbol
      • Spacetime volume
      • Angular momentum**
  • Curvature
    • Tidal curvature versus curvature caused by local sources
    • The stress-energy tensor
    • Curvature in two spacelike dimensions
    • Curvature tensors
    • Some order-of-magnitude estimates
      • The geodetic effect
      • Deflection of light rays
    • The covariant derivative
      • The covariant derivative in electromagnetism
      • The covariant derivative in general relativity
    • The geodesic equation
      • Characterization of the geodesic
      • Covariant derivative with respect to a parameter
      • The geodesic equation
      • Uniqueness
    • Torsion
      • Are scalars path-dependent?
      • The torsion tensor
      • Experimental searches for torsion
    • From metric to curvature
      • Finding the Christoffel symbol from the metric
      • Numerical solution of the geodesic equation
      • The Riemann tensor in terms of the Christoffel symbols
    • Manifolds
      • Why we need manifolds
      • Topological definition of a manifold
      • Hausdorff property
      • Local-coordinate definition of a manifold
      • Differentiable manifolds
      • The tangent space
  • Vacuum solutions
    • Event horizons
      • The event horizon of an accelerated observer
      • Information paradox
      • Radiation from event horizons
    • The Schwarzschild metric
      • The zero-mass case
      • Geometrized units
      • A large-r limit
      • The complete solution
      • Geodetic effect
      • Orbits**
      • Deflection of light
    • Black holes
      • Singularities
      • Event horizon
      • Infalling matter
      • Expected formation
      • Observational evidence
      • Singularities and cosmic censorship
      • Hawking radiation
      • Black holes in d dimensions
    • Degenerate solutions
  • Symmetries
    • Killing vectors
      • Conservation laws
    • Spherical symmetry
    • Penrose diagrams and causality
      • Flat spacetime
      • Schwarzschild spacetime
      • Astrophysical black hole
      • Penrose diagrams in general
      • Global hyperbolicity
    • Static and stationary spacetimes
      • Stationary spacetimes
      • Isolated systems
      • A stationary field with no other symmetries
      • A stationary field with additional symmetries
      • Static spacetimes
      • Birkhoff’s theorem
      • No-hair theorems
      • The gravitational potential
    • The uniform gravitational field revisited
      • Closed timelike curves
  • Sources
  • Gravitational waves
    • The speed of gravity
    • Gravitational radiation