Analytical and Numerical Approaches to Mathematical Relativity

The 319th Wilhelm-and-Else-Heraeus Seminar 'Mathematical Relativity: New Ideas and Developments' took place in March 2004. Twelve of the invited speakers have expanded their one hour talks into the papers appearing in this volume, preceded by a foreword by Roger Penrose. The first group consists of four papers on 'differential geometry and differential topology'. Paul Ehrlich opens with a very witty review of global Lorentzian geometry, which caused this reviewer to think more carefully about how he uses the adjective 'generic'. Robert Low addresses the issue of causality with a description of the 'space of null geodesics' and a tentative proposal for a new definition of causal boundary. The underlying review of global Lorentzian geometry is continued by Antonio Masiello, looking at variational approaches (actually valid for more general semi-Riemannian manifolds). This group concludes with a very clear review of pp-wave spacetimes from Jose Flores and Miguel Sanchez. (This reviewer was delighted to see a reproduction of Roger Penrose's seminal (1965) picture of null geodesics in plane wave spacetimes which attracted him into the subject.) Robert Beig opens the second group 'analytic methods and differential equations' with a brief but careful discussion of symmetric (regular) hyperbolicity for first (second) order systems, respectively, of partial differential equations. His description is peppered with examples, many specific to relativstic continuum mechanics. There follows a succinct review of linear elliptic boundary value problems with applications to general relativity from Sergio Dain. The numerous examples he provides are thought-provoking. The 'standard cosmological model' has been well understood for three quarters of a century. However recent observations suggest that the expansion in our Universe may be accelerating. Alan Rendall provides a careful discussion of the changes, both mathematical and physical, to the standard model which might be needed. This reviewer found the exposition much clearer than much of the phenomenological literature. Finally Laszlo Szabados gives a very systematic spacetime discussion of the group theoretical analysis of general relativity by Beig and O Murchadha, addressing the Poincare structure and the centre-of-mass of asymptotically flat spacetimes. The third and final group is entitled 'numerical methods'. Beverly Berger summarizes her 'Living Review' on numerical approaches to spacetime singularities and includes more recent analytical results emphasizing the synergy between mathematical and numerical approaches. For numerical evolutions on a domain of compact spatial support boundary conditions will be needed and, as pointed out by this reviewer, for unconstrained evolutions it is essential that the boundary conditions ensure constraint conservation. This aspect is discussed in a clear elementary way by Simonetta Frittelli and Roberto Gomez. Dave Neilsen, Luis Lehner, Olivier Sarbach and Manuel Tiglio review algorithms adopted recently by the Louisiana group, particularly summation by parts and constraint monitoring with applications to bubble and black hole spacetimes. Finally Maria Babiuc, Bela Szilagyi and Jeffrey Winicour discuss the Pittsburgh group's approach with particular reference to harmonic gauge conditions. It is perhaps unfortunate for the editors of this work that published proceedings of more recent numerical relativity meetings exist already. However, as this reviewer has tried to indicate, this slim (but not inexpensive) volume contains a wealth of diverse, fascinating material which needs to be perused by research students and others new to this field. Many will wish to buy it, but even if you do not, make sure your institution's library purchases a copy! (book review)