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The Formation of Black Holes in General Relativity

Demetrios Christodoulou

TL;DR

Christodoulou investigates black-hole formation in pure general relativity by focusing ingoing gravitational waves within a vacuum spacetime. He develops a short-pulse method within a two-parameter double-null (optical) foliation, deriving sharp $L^\infty$, $L^4$, and $L^2$ estimates for connection coefficients and Weyl curvature to prove a local existence theorem for large data and a separate trapped-surface formation result. The work unifies an existing theory of null geometry with a novel analytical hierarchy that scales with a small pulse duration, leading to a dynamically formed trapped surface and a first rigorous demonstration of black-hole formation without symmetry. The results illuminate long-time dynamics of GR and contribute to cosmic censorship by showing that sufficiently focused incoming waves can generate trapped surfaces and eventually singular boundaries.

Abstract

The subject of this work is the formation of black holes in pure general relativity, by the focusing of incoming gravitational waves. The theorems established 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 suitably small neighborhood of Minkowskian data. The theorems are general, no symmetry conditions on the initial data being imposed.

The Formation of Black Holes in General Relativity

TL;DR

Christodoulou investigates black-hole formation in pure general relativity by focusing ingoing gravitational waves within a vacuum spacetime. He develops a short-pulse method within a two-parameter double-null (optical) foliation, deriving sharp , , and estimates for connection coefficients and Weyl curvature to prove a local existence theorem for large data and a separate trapped-surface formation result. The work unifies an existing theory of null geometry with a novel analytical hierarchy that scales with a small pulse duration, leading to a dynamically formed trapped surface and a first rigorous demonstration of black-hole formation without symmetry. The results illuminate long-time dynamics of GR and contribute to cosmic censorship by showing that sufficiently focused incoming waves can generate trapped surfaces and eventually singular boundaries.

Abstract

The subject of this work is the formation of black holes in pure general relativity, by the focusing of incoming gravitational waves. The theorems established 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 suitably small neighborhood of Minkowskian data. The theorems are general, no symmetry conditions on the initial data being imposed.

Paper Structure

This paper contains 88 sections, 4141 equations, 5 figures.

Table of Contents

  1. The Optical Structure Equations
  2. The Basic Geometric Setup
  3. The Optical Structure Equations
  4. The Bianchi Equations
  5. Canonical coordinate systems
  6. The Characteristic Initial Data
  7. The characteristic initial data
  8. Construction of the solution in an initial domain
  9. $L^\infty$ Estimates for the Connection Coefficients
  10. Introduction
  11. $L^\infty$ Estimates for $\chi^\prime$
  12. $L^\infty$ Estimates for $\underline{\chi}^\prime$
  13. $L^\infty$ Estimates for $\eta,\underline{\eta}$
  14. $L^\infty$ Estimates for $\omega, \underline{\omega}$
  15. The smallness requirement on $\delta$
  16. ...and 73 more sections

Figures (5)

  • Figure 1: Case $c^*\geq u_0+\delta$
  • Figure 2: Case $c^*< u_0+\delta$
  • Figure 3: The domain considered in regard to the first inequality
  • Figure 4: The domain considered in regard to the second inequality, case $t< u_0+\delta$
  • Figure 5: The domain considered in regard to the second inequality, case $t\geq u_0+\delta$