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p"> The Big Book of Spacetime
A bit of everything

# 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
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