Conformal notions provide valuable tools for the analysis of global properties of spacetimes. In Part IV of this book it has been shown how a conformal point of view leads to proofs of the global existence and non-linear stability of de Sitterlike spacetimes, of the semiglobal existence and non-linear stability of Minkowskilike spacetimes, and how they provide a systematic procedure for the construction of anti-de Sitter-like spacetimes. Moreover, conformal methods provide a robust framework for the analysis of the gravitational field of isolated systems in a neighbourhood of both null and spatial infinity.

The application of conformal methods in general relativity is a mature area of research with a considerable number of open problems. Several of these have been discussed in various places of this book. Unavoidably, there are other problems and aspects of the subject which, for reasons of space, could not be covered in the main text. This last chapter presents a list of ideas and problems which, in the opinion of the author, may play a role in the future development of the subject.

Stability of cosmological models

The global non-linear stability of the de Sitter spacetime was discussed in Chapter 15. This exact solution can be regarded as a basic cosmological model. The analysis of Chapter 15 can be extended to include a non-vacuum matter content with good conformal properties: for example, the Maxwell, Yang-Mills and conformally coupled scalar field; see Friedrich (1991) and Lübbe and Valiente Kroon (2012). More recently, the ideas behind these proofs have been adapted in Lübbe and Valiente Kroon (2013b) to provide an analysis of the future stability of Friedman-Robertson-Walker cosmological models with a perfect fluid with the equation of state of radiation; see Section 9.4. A natural question is whether conformal methods could be adapted to more general matter models, that is, matter models with a non-vanishing trace.