Description
Written for researchers focusing on general relativity, supergravity, and cosmology, this is a self-contained exposition of the structure of the cosmological singularity in generic solutions of the Einstein equations, and an up-to-date mathematical derivation of the theory underlying the Belinski–Khalatnikov–Lifshitz (BKL) conjecture on this field. Part I provides a comprehensive review of the theory underlying the BKL conjecture. The generic asymptotic behavior near the cosmological singularity of the gravitational field, and fields describing other kinds of matter, is explained in detail. Part II focuses on the billiard reformulation of the BKL behavior. Taking a general approach, this section does not assume any simplifying symmetry conditions and applies to theories involving a range of matter fields and space-time dimensions, including supergravities. Overall, this book will equip theoretical and mathematical physicists with the theoretical fundamentals of the Big Bang, Big Crunch, Black Hole singularities, the billiard description, and emergent mathematical structures.
Chapter
1.5 Kasner-Like Singularities of Power Law Asymptotics
1.6 Instability of Kasner Dynamics
1.7 Transition to the New Regime
1.8 Oscillatory Nature of the Generic Singularity
1.9 Rotation of Kasner Axes
2 Homogeneous Cosmological Models
2.1 Homogeneous Models of Bianchi Types IX and VIII
2.2 Equations of Motion for Homogeneous Models
2.3 Models of Types IX and VIII with Fixed Kasner Axes
2.4 Models of Types IX and VIII with Rotating Axes
2.5 On the Extension to the Inhomogeneous Case
3 On the Cosmological Chaos
3.1 Stochasticity of the Oscillatory Regime
3.3 Gravitational Turbulence
4 On the Influence of Matter and Space-Time Dimension
4.3 Perfect Fluid of Stiff Matter Equation of State
4.4 Yang–Mills and Electromagnetic Fields
4.6 Pure Gravity in Higher Dimensions
4.7 Generalized Kasner Solutions: Rigorous Results
4.8 On the Influence of Viscous Matter
Part II Cosmological Billiards
5 The Billiard of Four-Dimensional Vacuum Gravity
5.1 Hamiltonian Form of the Action
5.3 More on Hyperbolic Space H[sub(2)]
5.4 Kasner Solution Revisited
5.5 Hamiltonian in Pseudo-Gaussian Gauge
5.6 BKL Limit and Emergence of Billiard Description
5.9 Chaos and Volume of the Billiard Table
5.10 Coxeter Group for Pure Gravity in Four Dimensions
6 General Cosmological Billiards
6.1 Models – Hamiltonian Form of the Action
6.2 Geometry of the Space of Scale Factors
6.3 Hyperbolic Space in M Dimensions
6.4 Hamiltonian in Iwasawa Variables and BKL Limit
7 Hyperbolic Coxeter Groups
7.2 Convex Polyhedra in Hyperbolic Space
7.3 Coxeter Groups: General Considerations
7.4 Coxeter Groups: Examples
7.5 Coxeter Groups and Weyl Groups
7.6 Coxeter Groups Associated with Gravitational Theories
7.7 The Kac–Moody Symmetry Conjecture
A Various Technical Derivations
A.1 Perturbations to Kasner-Like Asymptotics
A.2 Frame Components of the Ricci Tensor
A.3 Exact Solution for Transition Between Two Kasner Epochs
A.4 The Derivation of the Rotation Effect of the Kasner Axes
B Homogeneous Spaces and Bianchi Classification
B.1 Homogeneous Three-Dimensional Spaces
B.2 Bianchi Classification
B.4 On the Freezing Effect in Bianchi IX Model
C.1 Equations of the Gravitational and Spinor Fields
C.2 An Exact Homogeneous Solution for the Massless Case
C.3 The General Solution in the Vicinity of the Singularity
D Lorentzian Kac–Moody Algebras
D.3 The Chevalley Involution
D.6 The Invariant Bilinear Form
D.8 Hyperbolic Kac–Moody Algebras
D.9 Overextensions of Finite-Dimensional Lie Algebras
The Cosmological Singularity
( Cambridge Monographs on Mathematical Physics )
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.