The Formation of Black Holes in General Relativity (Ems Monographs in Mathematics)

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. 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. A major challenge since that time 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 the present 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. Contents......Page 8 Prologue......Page 12 1 The Optical Structure Equations......Page 40 2 The Characteristic Initial Data......Page 74 3 L^\infty Estimates for the Connection Coefficients......Page 102 4 L^4(S) Estimates for the 1st Derivatives of the Connection Coefficients......Page 126 5 The Uniformization Theorem......Page 158 6 L^4(S) Estimates for the 2nd Derivatives of the Connection Coefficients......Page 192 7 L^2 Estimates for the 3rd Derivatives of the Connection Coefficients......Page 226 8 The Multiplier Fields and the Commutation Fields......Page 286 9 Estimates for the Derivatives of the Deformation Tensors of the Commutation Fields......Page 310 10 The Sobolev Inequalities on the C_u and the C_u......Page 336 11 The S-tangential Derivatives and the Rotational Lie Derivatives......Page 354 12 Weyl Fields and Currents. The Existence Theorem......Page 376 13 The Multiplier Error Estimates......Page 434 14 The 1st-Order Weyl Current Error Estimates......Page 452 15 The 2nd-Order Weyl Current Error Estimates......Page 492 16 The Energy-Flux Estimates. Completion of the Continuity Argument......Page 528 17 Trapped Surface Formation......Page 584 Bibliography......Page 594 Index......Page 598