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**The Cauchy problem in general relativity.**
*(English)*
Zbl 1169.83003

ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-053-1/pbk). xiii, 294 p. (2009).

This monograph deals with almost all aspects of the Cauchy problem in general relativity, starting with a short historical overview, and inserting almost all necessary ingredients: from the theory of partial differential equations, differential geometry and relativity theory. Appendices list several notions and conventions used in the book, bibliography and index are quite useful, too. Unfortunately, only very few figures are given even for such parts of the material, where they would be quite helpful, and are usually given in other literature on the topic.

Part I gives the background from the theory of partial differential equations, like Fourier transforms, Sobolov space, and wave equations. Part II presents the Lorentz geometry background, especially the notion of global hyperbolicity. Part III covers the constraint equation, local existence, Cauchy stability and the existence of a maximal globally hyperbolic development for the Einstein equation.

Part IV titles “Pathologies, strong cosmic censorship”, and is an example of the proverb “A good theory lives from its counterexamples.”, in this case it means a lot of examples which contradict every-day experience, like the non-uniqueness of the possible maximal extensions of given initial data. Another example is the closed universe recollapse conjecture.

From the Publisher’s description: “This book first provides a definition of the concept of initial data and a proof of the correspondence between initial data and development. It turns out that some initial data allow non-isometric maximal developments, complicating the uniqueness issue. The second half of the book is concerned with this and related problems, such as strong cosmic censorship. The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those wishing to enter the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included.”

Part I gives the background from the theory of partial differential equations, like Fourier transforms, Sobolov space, and wave equations. Part II presents the Lorentz geometry background, especially the notion of global hyperbolicity. Part III covers the constraint equation, local existence, Cauchy stability and the existence of a maximal globally hyperbolic development for the Einstein equation.

Part IV titles “Pathologies, strong cosmic censorship”, and is an example of the proverb “A good theory lives from its counterexamples.”, in this case it means a lot of examples which contradict every-day experience, like the non-uniqueness of the possible maximal extensions of given initial data. Another example is the closed universe recollapse conjecture.

From the Publisher’s description: “This book first provides a definition of the concept of initial data and a proof of the correspondence between initial data and development. It turns out that some initial data allow non-isometric maximal developments, complicating the uniqueness issue. The second half of the book is concerned with this and related problems, such as strong cosmic censorship. The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those wishing to enter the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included.”

Reviewer: Hans-Jürgen Schmidt (Potsdam)

### MSC:

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

35Q75 | PDEs in connection with relativity and gravitational theory |

83A05 | Special relativity |

83C25 | Approximation procedures, weak fields in general relativity and gravitational theory |

83C75 | Space-time singularities, cosmic censorship, etc. |

83C15 | Exact solutions to problems in general relativity and gravitational theory |

83F05 | Relativistic cosmology |