Living Reviews in Relativity

"Hyperbolic Methods for Einstein’s Equations"
Oscar A. Reula  

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1 Introduction
1.1 Background and history
1.2 Main subject and plan of the review
2 The Theory of Linear Constant Coefficients Evolution Equations and Generalizations to Quasi-linear Systems
2.1 Existence and uniqueness of smooth solutions
2.2 First order systems
2.3 Generalization to variable coefficient and non-linear systems
2.4 Hyperbolicity and numerical simulations
3 The Problem of hyperbolicity in general relativity
3.1 The standard approach, or the 4-D covariant approach
3.2 The modification of the field equations outside the constraint sub-manifold, or the 3+1 decomposition point of view
4 Recent Approaches to the Problem
4.1 The ADM representation
4.2 The frame representation
4.3 Ashtekar’s representation
5 Beyond the prescribed gauge
5.1 Trial and error method
5.2 Hyperbolic extensions
5.3 Elliptic extensions
6 The Role of the Constraints
6.1 The constraints in the harmonic gauge
6.2 The constraints in the new systems: Theoretical considerations
6.3 The constraints in the new systems: Numerical considerations
6.4 The constraints in the initial-boundary value problem
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