The Global Nonlinear Stability of the Minkowski Space (PMS-41)

The Global Nonlinear Stability of the Minkowski Space (PMS-41)

Demetrios Christodoulou
Sergiu Klainerman
Copyright Date: 1993
Pages: 432
https://www.jstor.org/stable/j.ctt7zthns
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  • Book Info
    The Global Nonlinear Stability of the Minkowski Space (PMS-41)
    Book Description:

    The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to the Einstein Vacuum Equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. The work contains a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions. It thus provides the mathematical framework in which we can give a rigorous derivation of the laws of gravitation proposed by Bondi. Moreover, it establishes other important conclusions concerning the nonlinear character of gravitational radiation. The authors obtain their solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the flat initial data set corresponding to the Minkowski space-time. They thus establish the global dynamic stability of the latter.

    Originally published in 1994.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-6317-4
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. I-IV)
  2. Table of Contents
    (pp. V-VIII)
  3. Acknowledgments
    (pp. IX-X)
  4. CHAPTER 1 Introduction
    (pp. 1-28)

    The aim of this book is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, our work accomplishes the following goals:

    1. It provides a constructive proof of global, smooth, nontrivial solutions to the Einstein-Vacuum equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities.

    2. It provides a detailed description of the sense in which these solutions are close to the Minkowski space-time in all directions and gives a rigorous derivation of the laws of gravitational radiation proposed by Bondi. It also describes our...

  5. I. Preliminary Results in 2- and 3-Dimensional Riemannian Geometry
    • CHAPTER 2 Generalized Hodge Systems in 2-D
      (pp. 31-52)

      Throughout this chapter we assume that (S,γ) is a compact, 2-dimensional Riemannian manifold. We denote by ∇̸ the covariant differentiation onSand byKits Gauss curvature. RecallGauss-Bonnet Theorem\[\int_{S}{Kd{{\mu }_{\gamma }}=2\pi \chi (S)}\]whereγis the volume element of (S, γ) andχ(S) is the Euler characteristic ofS. We also recall the following:

      Uniformization Theorem

      There exists a conformal transformation of the metric γ̊ = Ω2γ such that the Gauss curvature$\overset{\circ}{\mathop{K}}$of the new metrics is\[\overset{\circ}{\mathop{K}}=\left\{ \begin{matrix} 1\text{ if }\chi (S)=2 \\ 0\text{ if }\chi (S)=0 \\ -1\text{ if }\chi (S)\le -2. \\ \end{matrix} \right.\]

      Though most of the results of this chapter hold for general compact surfaces, the case of interest to us is that...

    • CHAPTER 3 General Results in 3-D Geometry
      (pp. 53-77)

      Throughout this chapter we assume (Σ,g) to be a 3-dimensional Riemannian manifold diffeomorphic toR3on which there exists a generalizedradialfunction. By this we understand a differentiate, real, functionudefined on all points of Σ, outside a center pointO, which takes values onto the interval left(u0, ∞) and verifies the following assumptions:

      1.uhas no critical points.

      2. The level surfaces ofu, to be denoted bySu, are diffeomorphic to the 2-dimensional spheresS2.

      Also, denoting by IntSuthe component of Σ \Suthat containsO, and by ExtSuthe other one, we require...

    • CHAPTER 4 The Poisson Equation in 3-D
      (pp. 78-109)

      Throughout this chapter we assume (Σ,g) to be a 3-dimensional Riemannian manifold diffeomorphic toR3on which there exists a generalized radial functionuwith second fundamental formθand Gaussian curvatureK. We require that the radial functionuisquasiconvex, by which we mean that

      trθ> 0,K> 0

      This is in fact implied by the stronger assumptions we make, namely, that the fundamental constantskm,kM,aM,am,hm,hM,ς (see 3.1.3) verify that

      k−1m,kM,a−1m,aM,h-1m,hM,ς are finite. (4.1.1a)

      Also, we make the assumption that\[{{r}_{0}}=r(0)\ge 1 \caption{(4.1.1\text {b})}\]

      whereris the function ofudefined by 3.1.1h. Other assumptions that will...

    • CHAPTER 5 Curvature of an Initial Data Set
      (pp. 110-120)

      In this chapter we use the results of the previous chapter to analyze the global smallness assumption of our main theorem (see 1.0.15).

      In view of the remark following the statement of the second version of the main theorem in the introduction we can assume that, given an initial data set Σ,g,kverifying the constraint equations and the global smallness assumption, there exists a pointx0∈ Σ such that\[Q({{x}_{(0)}},1)\le \varepsilon \caption{(5.0.1\text {a})}\]where\[Q({{x}_{(0)}},b)=\underset{\Sigma }{\mathop{\sup }}\,\{{{(d_{0}^{2}+1)}^{3}}{{\left| Ric \right|}^{2}}\}+\left\{ \int_{\Sigma }{\sum\limits_{l=0}^{3}{{{(d_{0}^{2}+1)}^{l+1}}{{\left| {{\nabla }^{l}}k \right|}^{2}}+\int_{\Sigma }{\sum\limits_{l=0}^{1}{{{(d_{0}^{2}+1)}^{l+3}}{{\left| {{\nabla }^{l}}B \right|}^{2}}}}}} \right\}\]wheredis the distance function fromx0on Σ andBis the Bach tensor;\[{{B}_{ij}}={{(\text{curl}\,\hat{R})}_{ij}}\]as defined in the introduction.

      The traceless symmetric...

    • CHAPTER 6 Deformation of 2-Surfaces in 3-D
      (pp. 121-132)

      The aim of this chapter is to present a method of producing foliations of a 3-dimensional Riemannian manifold.

      Definition 6.0.1 A surface S in Σ, diffeomorphic toS2, is an equivalent class of embeddingsi:S2→ Σ. Two embeddingsi1,i2are said to be equivalent if there exists a diffeomorphismh:S2S2s.t.i2=i1h.

      Definition 6.0.2 A homotopy of embeddings ofS2into Σ is a differentiable mapf: [0, 1] ×S2→ Σ such that, for eacht∈ [0, 1], the mapft:S2→...

  6. II. Bianchi Equations in Space-Time
    • CHAPTER 7 The Comparison Theorem
      (pp. 135-204)

      This part provides the main ideas of our treatment of the Bianchi identities of an Einstein-Vacuum space-time, in order to control its curvature tensor. These ideas are at the heart of this book.

      Given an Einstein space-time (M, g), we consider Weyl tensorsW, which are four tensors verifying all the symmetry properties of the Riemann curvature tensor, that is,\[{W}_{\alpha \beta \gamma \delta }}=-{{W}_{\beta \alpha \gamma \delta }}=-{{W}_{\alpha \beta \delta \gamma }} \caption{(7.1.1\text {a})\]\[{{W}_{\alpha \beta \gamma \delta }}+{{W}_{\alpha \gamma \delta \beta }}+{{W}_{\alpha \delta \beta \gamma }}=0 \caption{(7.1.1\text {b})}\]\[{{W}_{\alpha \beta \gamma \delta }}={{W}_{\gamma \delta \alpha \beta }} \caption{(7.1.1\text {c})}\](7.1.1d) plus the trace condition\[{{W}_{\beta \delta }}=W_{\beta \alpha \delta }^{\alpha }=0. \caption{(7.1.1\text {e})}\]

      We recall the well-known fact that 7.1.1c is in fact a consequence of 7.1.1a and 7.1.1b. Thus,$W_{\alpha \beta \gamma \delta }^{'}=W_{\gamma \delta \alpha \beta }^{'}$.

      The left and right duals ofWhave been defined by\[^{\star}{{W}_{\alpha \beta \gamma \delta }}=\frac{1}{2}{{\in }_{\alpha \beta \mu \nu }}{{W}^{\mu \nu }}_{\gamma \delta } \caption {(7.1.1\text {f})}\]...

    • CHAPTER 8 The Error Estimates
      (pp. 205-258)

      In this chapter we provide the error estimates generated in the process of estimating the normsQ1andQ2(see the definition in 7.6.1g and 7.6.1h). We prove here the main result of Part II, which we call theboundedness theorem.

      We start with the following:

      Definition 8.1.1 Given an arbitrary vectorfieldXwith deformation tensor(X)π, we introduce the following notation:\[{{(X)}_{p\gamma }}={{({{\mathbf {Div}}^{(X)}}\hat{\pi })}_{\gamma }}={{\mathbf D}^{\alpha (X)}}{{\hat{\pi }}_{\alpha \gamma }} \caption{(8.1.1\text {a})}\]\[^{(X)}{{q}_{\alpha }}_{\beta \gamma }={\mathbf D}_{\beta }^{(X)}{{\hat{\pi }}_{\gamma \alpha }}-{\mathbf D}_{\gamma }^{(X)}{{\hat{\pi }}_{\beta \alpha }}-\frac{1}{3}{{(}^{(X)}}{{p}_{\gamma }}{{\mathbf g}_{\alpha \beta }}{{-}^{(X)}}{{p}_{\beta }}{{\mathbf g}_{\alpha \gamma }})\caption{(8.1.1\text {b})}\]where$^{(X)}{\hat{\pi }}$is the traceless part of$^{(X)}{\pi }$.

      LetWbe a Weyl tensor satisfying the homogeneous Bianchi equations 7.1.2, and consider${{\hat{\mathcal L}}_{X}}W$. Then, recalling Proposition 7.1.2, we write\[{{D}^{\alpha }}{{({{\hat{\mathcal L}}_{X}}W)}_{\alpha \beta \gamma \delta }}=J{{(X;W)}_{\beta \gamma \delta }}\caption {(8.1.2\text {a})}\]where\[J(X;W)=\frac{1}{2}({{J}^{1}}(X;W)+{{J}^{2}}(X;W)+{{J}^{3}}(X;W))\caption {(8.1.2\text {b})}\]\[\begin{array}{*{35}{l}} {{J}^{1}}{{(X;W)}_{\beta \gamma \delta }}{{=}^{(X)}}{{{\hat{\pi }}}^{\mu \nu }}{{D}_{\nu }}{{W}_{\mu \beta \gamma \delta }} \\ {{J}^{2}}{{(X;W)}_{\beta \gamma \delta }}{{=}^{(X)}}{{p}_{\lambda }}{{W}^{\lambda }}_{\beta \gamma \delta } \\ {{J}^{3}}{{(X;W)}_{\beta \gamma \delta }}{{=}^{(X)}}{{q}_{\alpha \beta \lambda }}{{W}^{\alpha \lambda }}_{\gamma \delta }{{+}^{(X)}}{{q}_{\alpha \gamma \lambda }}{{W}^{\alpha }}{{_{\beta }}^{\lambda }}_{\delta }{{+}^{(X)}}{{q}_{\alpha \delta \lambda }}{{W}^{\alpha }}{{_{\beta \gamma }}^{\lambda }}. \\ \end{array}\]Taking...

  7. III. Construction of Global Space-Times.: Proof of the Main Theorem
    • CHAPTER 9 Construction of the Optical Function
      (pp. 261-283)

      In this chapter we shall assume that a space-time slab$\bigcup\nolimits_{t\in [0,{{t}_{\star}}]}{{{\Sigma }_{t}}}$has been constructed, each Σtbeing a level set of the unique time function defined in the introduction, Σ0corresponding to the initial hypersurface, and Σt*to the final hypersurface. Our main objective here is to construct an optical functionuand use it in order to define the vectorfieldsK, Sand the rotation vector fields Ω. The construction of this global optical function is obtained by matching an exterior optical function to an interior one. The exterior optical function is by far the more important one for...

    • CHAPTER 10 Third Version of the Main Theorem
      (pp. 284-310)

      In this section we assume a space-time slab$\bigcup\nolimits_{[0,{{t}_{\star}}]}{{{\Sigma }_{t}}}$, foliated by a maximal time foliationtand an optical functionu, and we introduce our basic norms involving the curvature tensor R, second fundamental formk, and lapse functionϕ, as well as the optical quantitiesχ, ζ, ω.

      GivenS-tangent tensorfieldVwe first define\[{{\left| V \right|}_{p,S}}(t,u)={{\left( \int_{{{S}_{t,u}}}{{{\left| V \right|}^{p}}d{{\mu }_{\gamma }}} \right)}^{1/p}}\quad \text{if}\ 1\le p<\infty \caption {(10.1.1\text {a})}\]\[= \underset{{{S}_{t,u}}}{\mathop{\sup }}\,\left| V \right|\quad \text{if}\ p=\infty .\]

      We also introduce the following norms defined in the interior and exterior regions$\Sigma _{t}^{i},\,\Sigma _{t}^{e}$of each slice:\[\begin{matrix} {{\left\| V \right\|}_{p,i}}={{\left( \int_{\Sigma _{t}^{i}}{{{\left| V \right|}^{p}}} \right)}^{1/p}}\ \text{if}\ 1\le p<\infty \\ =\underset{\Sigma _{t}^{i}}{\mathop{\sup }}\,\left| V \right|\quad \text{if}\ p=\infty \\ {{\left\| V \right\|}_{p,e}}(t)={{\left( \int_{\Sigma _{t}^{e}}{{{\left| V \right|}^{p}}} \right)}^{1/p}}\quad \text{if}\ 1\le p<\infty \\ =\underset{\Sigma _{t}^{e}}{\mathop{\sup }}\,\left| V \right|\quad \text{if}\ p=\infty \\ {{\left\| \left| V \right| \right\|}_{p,e}}(t)=\underset{u\ge {{u}_{0}}(t)}{\mathop{\sup }}\,{{\left| V \right|}_{p,S}}(t,u)\quad \text{if}\ 1\le p<\infty \\ =\underset{\Sigma _{t}^{e}}{\mathop{\sup }}\,\left| V \right|\quad \text{if}\ p=\infty , \\ \end{matrix}\caption {(10.1.1\text {b})}\]where we recall that$\Sigma _{t}^{i}=I$consists of points for which$r\le \frac{{{r}_{0}}(t)}{2}$while$\Sigma _{t}^{e}=E$consists of those for which$r\ge \frac{{{r}_{0}}(t)}{2}$, withr0(t) the...

    • CHAPTER 11 Second Fundamental Form
      (pp. 311-340)

      The aim of this section is to derive the estimates for the second fundamental form of the time foliationkneeded in Step 3b, of the proof of the main theorem. We recall (see 1.0.14b, 1.0.14c) thatksatisfies the elliptic system, on each slice Σt\[\begin{matrix} \text{tr}k = 0 \\ \text{curl}k = H \\ \text{div}k = 0. \\ \end{matrix}\caption {(11.1.1\text {a})}\]

      Also\[{{R}_{ij}}={{k}_{ia}}k_{j}^{a}+{{E}_{ij}}\caption {(11.1.1\text {b})}\]whereE, Hare the electric and magnetic parts of the space-time curvature relative to the time foliation. The Bianchi identities of the space-time imply that, in particular,EandHverify, on Σt, the following divergence equations:¹\[{\text {div}}E=k\wedge H\caption {(11.1.1\text {c})}\]\[{\text {div}}H=-k\wedge E.\caption {(11.1.1\text {d})}\]

      We will show that equations 11.1.1a and 11.1.1b completely determine the...

    • CHAPTER 12 The Lapse Function
      (pp. 341-350)

      In this chapter we make use of the estimates for the Poisson equation of Chapter 4 to estimate the lapse functionϕ. We recall (see 1.0.13) thatϕverifies the lapse equation\[\Delta \phi ={{\left| k \right|}^{2}}\phi\caption {(12.0.1\text {a})}\]andϕ→ 1 at infinity on each Σt. In view of the maximum principle and Harnack inequality we infer that 0 <ϕ≤ 1. Throughout this chapter we assume that the bootstrap BA1and BA2hold and that the estimates for the second fundamental formkhave already been derived. In other words, we have, with the notation of the previous chapter,\[\overline{{{\mathcal K}_{[2]}}}(t)\le c{{\mathcal R}_{[1]}}(t)\le c{{\epsilon }_{0}}\caption {(12.0.1\text {b})}\]\[\overline{{{\mathcal K}_{[3]}}}(t)\le c{{\mathcal R}_{[2]}}(t)\le c{{\epsilon }_{0}}\caption {(12.0.1\text {c})}\]

      The estimates...

    • CHAPTER 13 Derivatives of the Optical Function
      (pp. 351-410)

      The aim of this section is to derive estimates in the exterior for the higher derivatives of the optical functionu.

      Throughout this section we use as null pair thel-null pair of (t,u), that is,\[l=-{{\mathbf g}^{\mu \nu }}{{\mathbf D}_{\nu }}u={{a}^{-1}}(T+N)\quad \quad \underline l=a(T-N)\]whereais the lapse function of the foliation on${{\sum }_{t}}$given by the level hypersurfaces ofu,\[{{a}^{-1}}=\left| \nabla u \right|={{\mathbf D}_{T}}u.\]Relative to thel-pair, the Ricci coefficients defined by formulas 7.3.1b in Chapter 7, take the form\[\begin{array}{*{35}{l}} H=\chi & \underline H=\underline \chi \\ Z=\zeta & \underline Z=\underline \zeta \\ Y=0 & \underline Y=\underline \xi \\ \Omega =0 & \underline \Omega =-\omega \\ V=\zeta . & {} \\ \end{array}\caption {(13.1.1\text {a})}\]

      Thus, for a corresponding null frame${{e}_{4}}=l,{{e}_{3}}=\underline l,{{({{e}_{A}})}_{A=1,2}}$we have\[\begin{array}{*{35}{l}} {{\mathbf D}_{A}}{{e}_{3}}={\underline {\text{ }\!\!\chi\!\!\text{ }}_{AB}}{{e}_{B}}+{{\text{ }\!\!\zeta\!\!\text{ }}_{A}}{{e}_{3}} & {{\mathbf D}_{A}}{{e}_{4}}={{\text{ }\!\!\chi\!\!\text{ }}_{AB}}{{e}_{B}}-{{\text{ }\!\!\zeta\!\!\text{ }}_{A}}{{e}_{4}} \\ {{\mathbf D}_{3}}{{e}_{3}}=2{\underline {\text{ }\!\!\xi\!\!\text{ }}_{A}}{{e}_{A}}+2\omega {{e}_{3}} & {{\mathbf D}_{3}}{{e}_{4}}=2{{\text{ }\!\!\zeta\!\!\text{ }}_{A}}{{e}_{A}}-2\omega {{e}_{4}} \\ {{\mathbf D}_{4}}{{e}_{3}}=2{\underline {\text{ }\!\!\zeta\!\!\text{ }}_{A}}{{e}_{A}} & {{\mathbf D}_{4}}{{e}_{4}}=0. \\ \end{array}\caption {(13.1.1\text {b})}\]

      Also,\[\begin{array}{*{35}{l}} {{\mathbf D}_{B}}{{e}_{A}}={{\not{\nabla}}_{B}}{{e}_{A}}+\frac{1}{2}{{\text{ }\!\!\chi\!\!\text{ }}_{AB}}{{e}_{3}}+\frac{1}{2}{\underline{\text{ }\!\!\chi\!\!\text{ }}_{AB}}{{e}_{4}} \\ {{\mathbf D}_{3}}{{e}_{A}}={{\not{\mathbf D}}_{3}}{{e}_{A}}+{{\text{ }\!\!\zeta\!\!\text{ }}_{A}}{{e}_{3}}+{\underline{\text{ }\!\!\xi\!\!\text{ }}_{A}}{{e}_{4}} \\ {{\mathbf D}_{4}}{{e}_{A}}={{\not{\mathbf D}}_{4}}{{a}_{A}}+{\underline{\text{ }\!\!\zeta\!\!\text{ }}_{A}}{{e}_{4}}. \\ \end{array}\caption {(13.1.1\text {c})}\]

      The Ricci coefficients$\chi,\underline \chi, \zeta, \underline \zeta, \underline \xi, \omega$of thel-frame are connected to the Ricci coefficients...

    • CHAPTER 14 The Last Slice
      (pp. 411-442)

      In this chapter we construct the functionuon the last slice${{\sum }_{{t}_{\star}}}$. Given the surface${{S}_{{{t}_{\star}},0}}$, the intersection of the standard coneC0with${{\sum }_{{t}_{\star}}}$, we defineu*to be the solution of the following inverse lapse problem:\[{{\left| \nabla {{u}_{\star}} \right|}^{-1}}=a,\quad \quad {{u}_{\star}}\left| {{S}_{{{t}_{\star}},0}} \right.=0\caption {(14.0.1\text {a})}\]whereasatisfies on each level surface${{S}_{{{t}_{\star}},{{u}_{\star}}}}$ofu*the equation\[{\not \Delta} \log a=f-\bar{f}-{\not{\text{div}}}\varepsilon \text{,}\quad \overline{\log a}=0\caption {(14.0.1\text {b})}\]with\[f=K-\frac{1}{4}{{(tr\chi )}^{2}}.\caption {(14.0.1\text {c})}\]

      In this chapter the optical quantities and the curvature components are expressed relative to the normalized null normals\[e_{3}^{'}=T-N={{a}^{-1}}{{e}_{3}},\quad e_{4}^{'}=T+N=a{{e}_{4}},\]Nbeing the outward unit normal to${{S}_{{{t}_{\star}},{{u}_{\star}}}}$relative to${{\sum }_{{t}_{\star}}}$. However, for convenience of notation we have dropped the primes. We...

    • CHAPTER 15 The Matching
      (pp. 443-465)

      In this chapter we shall match the exterior optical functionuEwith the interior optical functionuIto obtain a globally defined functionu. Leth1be aCfunction on ℜ such that\[{{h}_{1}}=\left\{ \begin{array}{*{35}{l}} =1;\quad t\le \frac{7}{20} \\ =0;\quad t\ge \frac{8}{20}. \\ \end{array} \right.\caption {(15.0.1)}\]

      Let\[\vartheta =\frac{{{u}_{E}}+1+t}{1+t},\quad f=h\circ \vartheta\caption {(15.0.2\text {a})}\]and let\[u=(1-{{f}_{1}}){{u}_{E}}+{{f}_{1}}{{u}_{I}}.\caption {(15.0.2\text {b})}\]

      As the support of 1 −f1is included in the domain of definition ofuEand the support off1is included in the domain of definition ofuI, the functionuis defined globally. To estimate the Hessian ofu, we have to compareuEanduIin the matching region\[\]

      We do this by comparing each...

    • CHAPTER 16 The Rotation Vectorfields
      (pp. 466-490)

      In this chapter we construct the rotation vectorfields(a)Ω and estimate their deformation tensors. We assume that the estimates derived in the previous chapters for the exterior optical functionuE, interior optical functionuI, and time functionthold true. We recall that the global optical functionuwas defined by matchinguEwithuIaccording to the following formula:\[u=(1-{{f}_{1}}){{u}_{E}}+{{f}_{1}}{{u}_{I}}\]where${{f}_{1}}={{h}_{1}}\circ \vartheta$and\[\vartheta =\frac{{{u}_{E}}+(1+t)}{1+t}\]andh1(t) is a smooth function oftdefined such that,\[{{h}_{1}}=\left\{ \begin{array}{*{35}{l}} =1;\quad t\le \frac{7}{20} \\ =0;\quad t\ge \frac{8}{20}. \\ \end{array} \right.\caption {16.0.1\text {a}}\]

      In the matching regionM1, defined as the set of points where$\mathbf D{{f}_{1}}$is supported, the properties of the two optical functions are...

  8. CHAPTER 17 Conclusions
    (pp. 491-512)

    As we have shown in the proof of the main theorem given in Chapter 10, the exterior optical function$^{({{t}_{\star}})}u$defined on the slab${{\bigcup }_{t\in [0,{{t}_{\star}}]}}{{\sum }_{t}}$converges as${{t}_{\star}}\to \infty $to a global exterior optical functionu. For each${{t}_{\star}}\ge 0$, the 0-level set of$^{({{t}_{\star}})}u$is the part ofC0, the 0-level set ofu, contained in the slab${{\bigcup }_{t\in [0,{{t}_{\star}}]}}{{\sum }_{t}}$. We define for eacht≥ 0 a diffeomorphism${{\phi }_{t,0}}$ofS2ontoSt,0such that${{\phi }_{{{t}_{2}},0}}\circ \phi _{{{t}_{1}},0}^{-1}$is the diffeomorphism of${{S}_{{{t}_{1}},0}}$onto${{S}_{{{t}_{2}},0}}$given by the generators ofC0. Given any${{t}_{\star}}\ge 0$, we consider the exterior optical function...

  9. Bibliography
    (pp. 513-514)