The Formation of Black Holes in General Relativity

Εξώφυλλο
European 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.
 

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Περιεχόμενα

Prologue
1
The 5tangential Derivatives and the Rotational Lie Derivatives
11
The Optical Structure Equations
29
The Characteristic Initial Data
63
L4 S Estimates for the 1st Derivatives of the Connection Coefficients
115
The Uniformization Theorem
147
L4S Estimates for the 2nd Derivatives of the Connection Coefficients
181
L2 Estimates for the 3rd Derivatives of the Connection Coefficients
215
The Sobolev Inequalities on the C and the
325
Weyl Fields and Currents The Existence Theorem
365
The Multiplier Error Estimates
423
The IstOrder Weyl Current Error Estimates
441
The 2ndOrder Weyl Current Error Estimates
481
The EnergyFlux Estimates Completion of the Continuity Argument
517
Trapped Surface Formation
573
Bibliography
583

The Multiplier Fields and the Commutation Fields
275
Estimates for the Derivatives of
299

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