| Preface |
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vii | |
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1 | (5) |
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1 | (1) |
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Some global results, recent developments |
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2 | (1) |
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3 | (3) |
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6 | (13) |
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6 | (6) |
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The Fourier transform and Sobolev spaces |
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8 | (1) |
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Symmetric hyperbolic systems |
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9 | (1) |
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Linear and non-linear wave equations |
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10 | (2) |
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Geometry, global hyperbolicity and uniqueness |
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12 | (2) |
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Geometry and global hyperbolicity |
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12 | (1) |
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13 | (1) |
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14 | (3) |
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Constraint equations and local existence |
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15 | (2) |
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17 | (1) |
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Existence of a maximal globally hyperbolic development |
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17 | (1) |
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Pathologies, strong cosmic censorship |
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17 | (2) |
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Part I Background from the theory of partial differential equations |
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19 | (74) |
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21 | (5) |
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21 | (1) |
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22 | (4) |
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26 | (7) |
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Schwartz functions, the Fourier transform |
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27 | (2) |
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The Fourier inversion formula |
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29 | (4) |
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33 | (12) |
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34 | (1) |
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Weak differentiability, Wk, p spaces |
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35 | (3) |
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Temperate distributions, Hs spaces |
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38 | (5) |
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43 | (2) |
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45 | (12) |
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47 | (2) |
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49 | (1) |
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Gagliardo-Nirenberg inequalities |
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50 | (7) |
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Symmetric hyperbolic systems |
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57 | (12) |
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57 | (1) |
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The basic energy inequality |
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58 | (5) |
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63 | (1) |
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64 | (5) |
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69 | (7) |
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71 | (2) |
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Existence of solutions to linear wave equations |
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73 | (3) |
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Local existence, non-linear wave equations |
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76 | (17) |
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76 | (3) |
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79 | (4) |
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83 | (4) |
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Continuation criterion, smooth solutions |
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87 | (2) |
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89 | (4) |
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Part II Background in geometry, global hyperbolicity and uniqueness |
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93 | (52) |
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95 | (16) |
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95 | (5) |
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100 | (11) |
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100 | (1) |
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Covariant differentiation |
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101 | (4) |
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Coordinate expressions for curvature |
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105 | (2) |
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107 | (1) |
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108 | (1) |
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108 | (1) |
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109 | (1) |
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109 | (2) |
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Characterizations of global hyperbolicity |
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111 | (20) |
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Existence of a Cauchy hypersurface |
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111 | (6) |
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117 | (3) |
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120 | (4) |
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Smooth temporal functions adapted to Cauchy hypersurfaces |
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124 | (4) |
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128 | (3) |
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Uniqueness of solutions to linear wave equations |
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131 | (14) |
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Preliminary technical observations |
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131 | (4) |
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Uniqueness of solutions to tensor wave equations |
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135 | (6) |
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141 | (4) |
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Part III General relativity |
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145 | (40) |
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147 | (5) |
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147 | (2) |
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149 | (2) |
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Constraint equations, non-linear scalar field case |
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151 | (1) |
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152 | (12) |
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152 | (2) |
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154 | (2) |
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Existence of a globally hyperbolic development |
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156 | (2) |
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Two developments are extensions of a common development |
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158 | (6) |
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164 | (12) |
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Sobolev spaces on manifolds |
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164 | (2) |
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166 | (1) |
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Cauchy stability in general relativity |
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167 | (9) |
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Existence of a maximal globally hyperbolic development |
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176 | (9) |
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Background from set theory |
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176 | (1) |
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Existence of a maximal globally hyperbolic development |
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177 | (8) |
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Part IV Pathologies, strong cosmic censorship |
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185 | (80) |
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187 | (9) |
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187 | (1) |
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188 | (4) |
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The asymptotically flat case |
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189 | (1) |
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189 | (2) |
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191 | (1) |
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Construction of extensions in the unimodular case |
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192 | (1) |
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Sketch of the proof of existence of inequivalent extensions |
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193 | (1) |
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194 | (2) |
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196 | (10) |
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196 | (4) |
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Constant mean curvature hypersurfaces |
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200 | (3) |
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Conditions ensuring maximality |
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203 | (1) |
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Conditions ensuring inextendibility |
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204 | (2) |
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206 | (7) |
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206 | (3) |
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209 | (1) |
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210 | (3) |
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Einstein's vacuum equations |
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213 | (12) |
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213 | (1) |
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214 | (4) |
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Elementary properties of developments |
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218 | (3) |
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Causal geodesic completeness and incompleteness |
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221 | (4) |
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Closed universe recollapse |
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225 | (7) |
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Recollapse for an open set of initial data |
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230 | (2) |
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232 | (11) |
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The Wainwright-Hsu variables |
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233 | (1) |
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Relation between the time coordinates |
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234 | (1) |
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Terminology, asymptotic behaviour |
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235 | (1) |
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Criteria ensuring curvature blow up |
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236 | (1) |
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Limit characterization of the Taub solutions |
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236 | (2) |
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Asymptotic behaviour, Bianchi type I and II |
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238 | (1) |
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239 | (1) |
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240 | (1) |
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241 | (1) |
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242 | (1) |
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LRS Bianchi class A solutions |
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243 | (9) |
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243 | (2) |
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245 | (1) |
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246 | (1) |
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Bianchi type II, VIII and IX |
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247 | (5) |
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252 | (8) |
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Construction of an embedding |
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252 | (2) |
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Basic properties of the extensions |
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254 | (4) |
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SCC, unimodular vacuum case |
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258 | (2) |
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Existence of inequivalent extensions |
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260 | (5) |
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265 | (20) |
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267 | (10) |
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267 | (1) |
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Different notions of measurability |
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268 | (2) |
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270 | (1) |
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271 | (1) |
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272 | (1) |
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Smooth functions with compact support |
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273 | (2) |
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Differentiability in the infinite dimensional case |
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275 | (2) |
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277 | (8) |
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Identities concerning permutation symbols |
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277 | (2) |
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279 | (1) |
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279 | (1) |
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280 | (1) |
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280 | (1) |
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281 | (1) |
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282 | (3) |
| Bibliography |
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285 | (6) |
| Index |
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291 | |
