Hyperbolic Methods for Einstein’s Equations

"Hyperbolic Methods for Einstein’s Equations"

Oscar A. Reula

	Abstract
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
	References
	Footnotes