Statistics – Applications
Scientific paper
Feb 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23.1023c&link_type=abstract
Classical and Quantum Gravity, Volume 23, Issue 3, pp. 1023-1024 (2006).
Statistics
Applications
Scientific paper
In 1952, Mme Yvonne Choquet-Bruhat published a major paper, Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires (Acta Math. 88 141-225), which laid the foundation for modern studies of the Cauchy problem in general relativity. The fiftieth anniversary of this event was celebrated with an eponymous Cargèse Summer School in 2002. The proceedings of that summer school are summarized electronically (as audio, video, transparencies and lecture notes, where available) on a DVD archive included with this volume, and are also available on the internet.
However the organizers decided that a separate volume describing the 'state of the art in mathematical general relativity' would be useful, and this book is the result. It includes some material not covered in the school and excludes some school material which has been covered adequately elsewhere. Unfortunately, I was unable to find, electronically, a table of contents, which every prospective purchaser would wish to see, and so this review does in fact list all the articles, ordered, roughly, by length.
About one fifth of the book is devoted to a survey of Smoothness at Null Infinity and the Structure of Initial Data by Helmut Friedrich. This is a modern study of gravitational radiation, and the analysis of Einstein's equations. It is extremely helpful to survey all of this material, including some of the latest developments, using a consistent notation. This article is strongly recommended to anyone hoping to gain a foothold in this area. Note also that 47 pages of transparencies have become 84 book pages.
Lars Andersson has surveyed, in The Global Existence Problem in General Relativity, some results and conjectures about the global properties of 3+1-dimensional spacetimes with a compact Cauchy surface. Again it is very useful to have essentially all of the known results presented in a consistent notation. This material is not on the DVD.
Yvonne Choquet-Bruhat has contributed a long research paper, Future Complete U(1) Symmetric Einsteinian Spacetimes, the Unpolarized Case. There is a non-linear stability theorem due to her and Vincent Moncrief in which spacetime is of the form M × R where M is a circle bundle over a compact orientable surface of genus >1 and the 4-metric admits a Killing symmetry along the spacelike circular fibres. The new result removes the polarization condition, i.e., the orthogonality of the fibres to quotient 3-manifolds. This is a classic example of how to derive results in this field. It is also available, in full, on the DVD.
The article Group Actions on Lorentz Spaces, Mathematical Aspects: A Survey, contributed by Thierry Barbot and Abdelghani Zeghib, extends the second author's school presentation on the classification of large isometry groups of Lorentz manifolds, which brings together comprehensively material of surprising diversity, well worth a perusal.
Robert Bartnik and Jim Isenberg have contributed a brilliant review of The Constraint Equations, greatly extending the first author's one hour school presentation. This is a worthy successor to the landmark survey of James W York, 26 years earlier.
There is a survey of systems of partial differential equations which admit an initial value formulation up to gauge diffeomorphisms by Robert Geroch entitled Gauge, Diffeomorphisms, Initial-Value Formulation, Etc, which appears to be a reworking and update of an earlier review by the author. The school presentation is very nearly the same as the book version.
Hubert Bray presented a trio of lectures on global inequalities at the summer school. These have been expanded into a review The Penrose Inequality, with a second author Piotr Chrusciel, which deals with its generalization as the Riemannian Penrose inequality. It is very satisfying to see an information flow from general relativity to Riemannian geometry!
In Future Complete Vacuum Spacetimes, Lars Andersson and Vincent Moncrief contribute a global existence theorem for small perturbations of K = -1 vacuum Friedmann-Robertson-Walker spacetimes. The novelty here is the use of the Bel-Robinson energy and its generalizations.
At a more specialized level, Michael Anderson has contributed a review of Cheeger-Gromov Theory and Applications to General Relativity, which is an update of the lectures he gave at the summer school. This reviewer is unfamiliar with the material presented here, but it looks to be potentially important.
Luis Lehner and Oscar Reula have presented a rather useful but concise review of numerical relativity for mathematical relativists, entitled Status Quo and Open Problems in the Numerical Construction of Spacetimes, which is very different to the first author's school presentation. This should perhaps be worked up to a more encyclopaedic review.
Gregory Galloway has contributed Null Geometry and the Einstein Equations, an extended version of his summer school lectures, which surveyed how techniques from global Lorentzian geometry and causality theory can be used to obtain results about the global behaviour of solutions of the Einstein equations, thus generalizing and extending the classic Hawking-Penrose results.
The reader should recall that the articles have been ordered by objective length rather than subjective quality. Alan Rendall's is a masterly if terse review of The Einstein-Vlasov System, a laudable attempt to bring the mathematical status of results on the non-vacuum field equations up to that of their vacuum counterparts. This duplicates precisely his school presentation.
Last but not least, John Friedman has given an excellent succinct review of what can go wrong if one abandons the requirement of time-orientability, The Cauchy problem on Spacetimes That Are Not Globally Hyperbolic, which seems to be identical to his school presentation. As always, this experienced author's presentation has exemplary clarity.
This is an impressive volume presenting clearly the current state of the art. No relativity group should be without a copy.
No associations
LandOfFree
Book Review: does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Book Review:, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Book Review: will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-1488776