The evolution problem in general relativity. (English) Zbl 1010.83004

Progress in Mathematical Physics. 25. Boston, MA: Birkhäuser. xii, 385 p. (2003).
The authors start their preface by writing: “The main goal of this work is to revisit the proof of the global stability of the Minkowski space by C. Christodoulou and S. Klainerman”.
In fact, the present book could also seen as a revised and enlarged version of the book: C. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton (1993); see the detailed book reviews by V. Perlick in Zbl 0827.53055 and in General Relativity Gravitation 32, 761-763 (2000).
The first chapter gives an introduction about Lorentz manifolds. This chapter can be used independently of the remaining text simply as a short textbook on relativity-related differential geometry. Unfortunately, the authors refer to different sources for their sign conventions, so it remains open at different places, what conventions they are using, e.g. do they assume for the volume form \[ \varepsilon_{0123} = 1 \quad \text{ or} \quad \varepsilon^{0123} = 1? \] which produces an additional factor \((-1)\) in contrast to the Euclidean signature, where both equations simultaneously hold.
Chapters 2 until 7 are dealing with the statement and the proof of the fact, that, in a well-defined sense, the Einstein gravitational field equation has such a structure, that empty space is stable against those perturbations, which are decaying quickly enough at large spatial distances. The proof applies Cauchy’s development of suitably chosen initial conditions in a, again suitably chosen, \(3+1\)-foliation of space-time.
The final chapter 8 deals with other applications, e.g., the Bondi mass and ADM mass of an asymptotically flat space-time, by use of a double null foliation.


83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C40 Gravitational energy and conservation laws; groups of motions


Zbl 0827.53055