|
Abstract |
1 |
Introduction and Outline |
|
1.1 |
Outline, basic material |
|
1.2 |
Outline, the
asymptotics in the direction towards the singularity |
|
1.3 |
Outline, the expanding
direction |
|
1.4 |
Strong cosmic censorship |
2 |
Gowdy Spacetimes |
|
2.1 |
Symmetry in
cosmology |
|
2.2 |
Definition of the Gowdy class |
|
2.3 |
Coordinate systems |
|
2.4 |
The
polarized subcase |
3 |
Foliations |
|
3.1 |
CMC foliations |
|
3.2 |
Areal foliation |
|
3.3 |
Existence
of foliations, related symmetry classes |
|
3.4 |
Prescribed mean curvature |
4 |
Strong
Cosmic Censorship |
|
4.1 |
The initial value problem |
|
4.2 |
Strong cosmic
censorship |
|
4.3 |
Curvature blow up |
|
4.4 |
Pathological examples in the case of
Gowdy |
5 |
BKL, Fuchsian Methods and Asymptotic Expansions |
|
5.1 |
The BKL
picture |
|
5.2 |
Asymptotic expansions, Fuchsian methods |
6 |
Equations |
|
6.1 |
Expanding
direction |
|
6.2 |
The direction towards the singularity |
|
6.3 |
Wave-map
structure |
|
6.4 |
Conserved quantities, kinetic energy density |
7 |
Singularity,
Polarized Case |
|
7.1 |
Equations, polarized T3-Gowdy |
|
7.2 |
Associated Velocity
Term Dominated system |
|
7.3 |
Asymptotics of the solution to the polarized
T3-Gowdy equations |
|
7.4 |
Curvature blow up, polarized T3-case |
|
7.5 |
Asymptotic
velocity, polarized T3-case |
|
7.6 |
S2 × S1 and S3 cases |
8 |
Asymptotic Expansions
Using Fuchsian Methods, General T3-Case |
|
8.1 |
Geometric interpretation of
va |
|
8.2 |
Restriction on the velocity |
|
8.3 |
Geodesic loop |
|
8.4 |
Existence of expansions
using Fuchsian methods, T3-case |
|
8.5 |
Existence of expansions using Fuchsian
methods, S2 × S1 and S3 cases |
9 |
Spikes |
|
9.1 |
Inversion |
|
9.2 |
Gowdy to Ernst
transformation |
|
9.3 |
False spikes |
|
9.4 |
High velocity spikes |
10 |
Asymptotic Velocity,
General T3-Gowdy |
|
10.1 |
Existence of an asymptotic velocity |
|
10.2 |
Relevance of
the asymptotic velocity to the issue of curvature blow up |
|
10.3 |
Interpretation
of the asymptotic velocity as a rate of convergence to the boundary
in hyperbolic space |
|
10.4 |
Two dimensional version of the asymptotic
velocity |
|
10.5 |
Dominant contribution to the asymptotic velocity |
|
10.6 |
Value
of the asymptotic velocity as a criterion for the existence of
expansions |
11 |
The Generic Set, General T3-Gowdy |
|
11.1 |
Nondegenerate
true spikes |
|
11.2 |
Nondegenerate false spikes |
|
11.3 |
The generic set,
definition |
|
11.4 |
Verification of genericity, openness |
|
11.5 |
Verification of
genericity, density |
12 |
Expanding Direction, Polarized Gowdy |
|
12.1 |
Asymptotic
behavior |
|
12.2 |
Comparison with spatially homogeneous solutions |
13 |
Expanding
Direction, The General Case |
|
13.1 |
Energy decay |
|
13.2 |
Proof of decay of the
energy |
|
13.3 |
Asymptotic ODE behavior |
|
13.4 |
Geometric interpretation of the
asymptotics |
|
13.5 |
Concluding remarks |
|
13.6 |
Geodesic completeness |
14 |
Strong
Cosmic Censorship in Gowdy Spacetimes |
|
14.1 |
The polarized case |
|
14.2 |
General
T3-case |
15 |
Acknowledgments |
 |
References |
|
Figures |