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The Big Book of Spacetime

The Big Book of Spacetime is a project for a compilation of various theorems, properties, metrics and experimental facts related to general relativity that may not have a book of their own, be only present in obscure sources or with vague proofs. It aims to be a step by step construction of the mathematical frameworks of general relativity along with applications for it.

Table of content

  1. Spacetime
    1. Manifolds
    2. Vectors and vector fields
    3. Lorentz vector spaces
    4. Fiber bundles
    5. The metric
    6. Derivatives and connections
    7. Spacetime submanifolds
    8. Singularities
    9. Distributions on manifolds
    10. The Weyl transform and conformal spacetimes
    11. Spacetime orientability, spin structures and spinors
    12. Killing vectors and symmetries
    13. Petrov and Segre classification
    14. Causal relations
    15. Causal sets
    16. Causal properties
    17. The causal hierarchy
    18. Abstract causal spaces
    19. Spacetime boundaries and asymptotic behaviours
    20. Horizons and trapped surfaces
    21. Topology change and Lorentz cobordisms
    22. Continuous metrics
    23. Exotic spacetimes
    24. Non-Hausdorff manifolds
  2. General relativity
    1. The Einstein field equations
    2. The Lagrangian formalism
    3. Alternative formulations of general relativity
    4. The stress-energy tensor
    5. Geodesic deviation and congruence
    6. Energy conditions
    7. Hamiltonian formulation
    8. Mass in general relativity
    9. Causality, topology and general relativity
    10. General relativity in $n$ dimensions
    11. Conceptual notions
    12. Linearized gravity
    13. Distributional general relativity
    14. Classical mechanics
    15. Thermodynamic
    16. Classical field theory
  3. The Cauchy problem
    1. The Cauchy problem for classical fields and particles
    2. The Cauchy problem in general relativity
  4. Alternative theories
    1. Theories of a more general connection
    2. Scalar and vector theories
    3. $f(R)$ gravity
    4. Brans-Dicke gravity
    5. Pauli-Fierz gravity
    6. Massive gravity
    7. Kaluza-Klein
    8. Supergravity
    9. Regge calculus
    10. Theories of a different signature
    11. Misc. theories
  5. Quantum theory
    1. Quantum theory in general relativity
    2. Quantum field theory in curved spacetime
    3. Conformal field theory
    4. The AdS/CFT correspondance
  6. Quantum gravity
    1. Generalities on quantum gravity
    2. Semiclassical gravity
    3. Stochastic gravity
    4. Covariant quantum gravity
    5. Canonical quantum gravity
    6. Loop quantum gravity
    7. Lorentzian and Euclidian gravity
    8. String theory
    9. Causal dynamical triangulation
    10. Non-commutative geometry
    11. Causal sets
    12. Entropic gravity
  7. Specific spacetimes
    1. Minkowski space
    2. de Sitter space
    3. Anti-de Sitter space
    4. Friedmann-Lemaître-Robertson-Walker metric
    5. Schwarzschild spacetime
    6. Reissner-Nordström metric
    7. Kerr and associated spacetimes
    8. Wave solutions
    9. The Gödel spacetime
    10. Wormholes
    11. Faster-than-light spacetimes
    12. Causality violating and time machine solutions
  8. The real world
    1. Experimental tests of general relativity
    2. Astrophysics
    3. Cosmology
    4. Post-Newtonian parametrization
    5. Tests of quantum gravity
    6. History