
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 Quasilinear Systems 

2.1 
Existence and uniqueness of
smooth solutions 

2.2 
First order systems 

2.3 
Generalization to variable coefficient
and nonlinear systems 

2.4 
Hyperbolicity and numerical simulations 
3 
The
Problem of hyperbolicity in general relativity 

3.1 
The standard approach,
or the 4D covariant approach 

3.2 
The modification of the field equations
outside the constraint submanifold, 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 initialboundary value problem 

References 

Footnotes 