The Formation of Black Holes in General RelativityEuropean Mathematical Society, 2009 - 589 σελίδες In 1965 Penrose introduced the fundamental concept of a trapped surface, on the basis of which he proved a theorem which asserts that a spacetime containing such a surface must come to an end. The presence of a trapped surface implies, moreover, that there is a region of spacetime, the black hole, which is inaccessible to observation from infinity. Since that time a major challenge has been to find out how trapped surfaces actually form, by analyzing the dynamics of gravitational collapse. The present monograph achieves this aim by establishing the formation of trapped surfaces in pure general relativity through the focusing of gravitational waves. The theorems proved in this monograph constitute the first foray into the long-time dynamics of general relativity in the large, that is, when the initial data are no longer confined to a suitable neighborhood of trivial data. The main new method, the short pulse method, applies to general systems of Euler-Lagrange equations of hyperbolic type and provides the means to tackle problems which have hitherto seemed unapproachable. This monograph will be of interest to people working in general relativity, geometric analysis, and partial differential equations. |
Περιεχόμενα
Prologue | 1 |
The Characteristic Initial Data | 69 |
L | 91 |
estimates for ŋ | 102 |
L4S Estimates for the 1st Derivatives of the Connection Coefficients | 115 |
The Uniformization Theorem | 147 |
3 | 158 |
L4 S Estimates for the 2nd Derivatives of the Connection Coefficients | 181 |
3 | 281 |
Estimates for the Derivatives of | 299 |
Estimates for the 2nd derivatives of the deformation | 313 |
The Sobolev Inequalities on the Cu and the | 326 |
The Stangential Derivatives and the Rotational Lie Derivatives | 343 |
Weyl Fields and Currents The Existence Theorem | 365 |
The Multiplier Error Estimates | 423 |
The 1stOrder Weyl Current Error Estimates | 441 |
29 | 185 |
888 | 192 |
128 | 198 |
L² Estimates for the 3rd Derivatives of the Connection Coefficients | 215 |
The Multiplier Fields and the Commutation Fields | 241 |
221 | 247 |
Introduction | 275 |
The 2ndOrder Weyl Current Error Estimates | 475 |
as the 1storder estimates | 481 |
The EnergyFlux Estimates Completion of the Continuity Argument | 517 |
Trapped Surface Formation | 573 |
Bibliography | 583 |
Συχνά εμφανιζόμενοι όροι και φράσεις
2-covariant according apply apply Lemma arbitrary assumptions Bianchi Bianchi identities boundary bounded components conclude condition connection coefficients Consider constant continuous contribution coordinates corresponding covariant curvature deduce defined definition denote derivatives domain error integral estimates expression fact factor field Finally function future given hence holds hypersurface identical implies inequality initial data integral J¹(R L² Su,u L²(Cu Lemma mapping metric Moreover non-negative norms Note null obtain pair Proof propagation equation Proposition quantities respect results of Chapter role satisfies side So,uo solution spacetime Strx Su,u Su.u Substituting suitably small depending surface symmetric tables taking tensor tensorfield term theorem third u₁ vanishes vectorfield Weyl current Weyl field yields ΜΟ