# 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

- Spacetime
- Manifolds
- Vectors and vector fields
- Lorentz vector spaces
- Fiber bundles
- The metric
- Derivatives and connections
- Spacetime submanifolds
- Singularities
- Distributions on manifolds
- The Weyl transform and conformal spacetimes
- Spacetime orientability, spin structures and spinors
- Killing vectors and symmetries
- Petrov and Segre classification
- Causal relations
- Causal sets
- Causal properties
- The causal hierarchy
- Abstract causal spaces
- Spacetime boundaries and asymptotic behaviours
- Horizons and trapped surfaces
- Topology change and Lorentz cobordisms
- Continuous metrics
- Exotic spacetimes
- Non-Hausdorff manifolds

- General relativity
- The Einstein field equations
- The Lagrangian formalism
- Alternative formulations of general relativity
- The stress-energy tensor
- Geodesic deviation and congruence
- Energy conditions
- Hamiltonian formulation
- Mass in general relativity
- Causality, topology and general relativity
- General relativity in $n$ dimensions
- Conceptual notions
- Linearized gravity
- Distributional general relativity
- Classical mechanics
- Thermodynamic
- Classical field theory

- The Cauchy problem
- The Cauchy problem for classical fields and particles
- The Cauchy problem in general relativity

- Alternative theories
- Theories of a more general connection
- Scalar and vector theories
- $f(R)$ gravity
- Brans-Dicke gravity
- Pauli-Fierz gravity
- Massive gravity
- Kaluza-Klein
- Supergravity
- Regge calculus
- Theories of a different signature
- Misc. theories

- Quantum theory
- Quantum theory in general relativity
- Quantum field theory in curved spacetime
- Conformal field theory
- The AdS/CFT correspondance

- Quantum gravity
- Generalities on quantum gravity
- Semiclassical gravity
- Stochastic gravity
- Covariant quantum gravity
- Canonical quantum gravity
- Loop quantum gravity
- Lorentzian and Euclidian gravity
- String theory
- Causal dynamical triangulation
- Non-commutative geometry
- Causal sets
- Entropic gravity

- Specific spacetimes
- Minkowski space
- de Sitter space
- Anti-de Sitter space
- Friedmann-Lemaître-Robertson-Walker metric
- Schwarzschild spacetime
- Reissner-Nordström metric
- Kerr and associated spacetimes
- Wave solutions
- The Gödel spacetime
- Wormholes
- Faster-than-light spacetimes
- Causality violating and time machine solutions

- The real world
- Experimental tests of general relativity
- Astrophysics
- Cosmology
- Post-Newtonian parametrization
- Tests of quantum gravity
- History