# Book Review: On the topology and future stability of the universe, by Hans Ringström

Mihalis Dafermos is Professor of Mathematics at Princeton University

This impressive new book is first and foremost an original and thought-provoking contribution to the study of cosmology in research monograph form, in the best tradition of the kind of deep mathematical work which has played a crucial role in the development of the subject. At the same time, the book doubles as a dependable introduction and  reference for several foundational results in the analysis of the Einstein equations and relativistic kinetic theory which are hard to find elsewhere but which form the basis of so much current (and hopefully, future!) work.

Both these roles are most welcome.

Let me first discuss what this book offers as an original contribution to a broader perspective on our current understanding of cosmological models of the universe.

The author considers the fully coupled Einstein-Vlasov scalar field system with no exact symmetries assumed on the solutions. This system, coupling gravity, freely streaming particles and a nonlinear scalar field, is sufficiently versatile so as to effectively approximate perfect fluids  with various equations of state and contains both the case of a positive cosmological constant $\Lambda>0$ as well as more exotic dark energy models. The main new result proven in the book is a statement of future stability of ‘standard cosmological models’ (i.e.~spatially-flat homogeneous spacetimes with accelerated expansion) thought of as solutions of this complicated  system of nonlinear partial differential equations:

All spacetimes (with no underlying symmetries assumed!)~solving the above equations and initially sufficiently close to such a reference model (in a suitable atlas of coordinate patches) will again be future geodesically complete, exhibit accelarated expansion, and share the model’s most basic qualitative causal features.

It is precisely due to the accelerated expansion that the proof of the above result is quite tractable, once the problem has been correctly set up, and in fact eminently suitable to be read by beginners in the field. For the expansion effectively localises the problem to the domain of dependence of one coordinate patch at a time and moreover allows the otherwise formidable non-linearities of the Einstein equations to be understood as time-integrable error terms.

This phenomenon already ocurred in the proof of the non-linear stability of pure de Sitter space by H.~Friedrich from the 1980’s,  generalised more recently in previous work by the author of the book under review  and subsequently in work of Rodnianski and Speck which considered the more difficult case of perfect fluids. This analytically favourable situation of accelerated expansion should be contrasted with the much more difficult proof of the non-linear stability of Minkowski space, the monumental work of Christodoulou and Klainerman which appeared in 1993. There, the rate of decay of waves towards null infinity is exactly borderline to controlling the non-linearities by simply exploiting dispersion, and the precise miraculous structure of the Einstein equations thus  becomes paramount.

Returning to the present book, the author puts an attractive additional twist on his results by giving an elegant argument, using little more than the domain of dependence property and the fact that all Riemannian manifolds are approximately flat at a sufficiently small scale, to construct spacetimes satisfying his closeness assumption with arbitrary global spatial topology.

This highlights an important point for the proper formulation of the ‘Copernican principle’: In the case of approximately spatially flat universes, the statement that one is initially locally close to homogeneity in a suitable atlas of coordinate patches turns out to be a much weaker assumption than the statement that one is initially globally close, for the latter would include the requirement that the global spatial topology admits a flat metric.

I suspect that even an epistemologically conservative reader, who may otherwise reject  altogher study of the question of the global spatial topology of the universe at a scale inaccessible to observation, will welcome the precise and elegant statement the author proves, together with his intellectually sophisticated discussion of it, already in Chapter 1.1.

Turning now to the book’s second role as an introduction or reference work on the initial value problem in general relativity, let me say at the outset that in this context this is a book that I am particularly happy to have on my shelf.

The foundational results in the subject are written carefully in exemplary form, further improving on the author’s more elementary textbook ‘The Cauchy problem in General Relativity’, EMS publications, 2009 which I also strongly recommend! In particular, the proofs of local existence, local uniqueness (interpreted first in coordinates and then geometrically), and  finally existence of a unique maximal Cauchy development are done in all detail. Cauchy stability is given its due–and its ubiquitous applications in this book  are a very nice illustration  of why it is important. The book’s elegant treatments of foundations are welcome  even when specialised to the more basic vacuum case, but the author’s inclusion of Vlasov matter, in particular his original and attractive approach for dealing with some of the technicalities arising, could be particularly important for future work. Moreover, the book contains some very nice general discussions of the relation of kinetic theory to fluids, as well as a summary of observational cosmology easily digestible by mathematical readers (Chapter 5).

Let me add a very concrete  testament to one immediate tangible result connected to this book’s dependable exposition:

As is well known, the original proof of the existence of the above mentioned ‘maximal Cauchy development’ in the vacuum case was given by Choquet-Bruhat and Geroch, in a pioneering 1969 paper in CMP. Their proof relied,  however, on the axiom of choice (in the form of Zorn’s lemma), making it  appear fundamentally non-constructive. This situation has given rise to various quasi-philosophical speculations: Is dynamics in general relativity really something that exists but cannot be constructed? While the book under review did not resolve this issue, its carefully written account (see Chapter 23) of the Choquet-Bruhat-Geroch proof, including many details omitted from the original paper, served as a starting point for a bright young student in Cambridge to do so. See J. Sbierski, On the existence of a maximal Cauchy development for the Einstein equations-a dezornification to appear Ann. Henri Poincaré, for the elusive constructive proof. I cannot think of a better vindication of this book’s  dedication to completeness than the quick appearance of Sbierski’s new proof, and I am sure that the latter will make its way into a possible later second edition as the final definitive approach to this foundational theorem.

I look forward to many other advances – going well beyond the mere foundations – by students of the future who learn from this book.

One final comment: The double role that this book quite successfully serves may at the same time make it seem, at slightly over seven hundred pages, a bit too long for some readers’ taste. This length however is in many ways a mirage. The book’s very conversational style  and its various leisurely discussions, aimed alternatingly at researchers with a more mathematical or physical background, make it not nearly as heavy as might first seem.

Whether one wishes to delve right into the author’s unique perspective on standard cosmological models  and the details of his non-linear stability proof, or one simply seeks a dependable introduction and reference on the foundations to the Cauchy problem for the Einstein equations coupled to kinetic theory, there  is much to learn from and enjoy in this very nice accessible book.

On the Topology and Future Stability of the Universe
by Hans Ringström
Oxford Mathematical Monographs
736 pages | Hardback
978-0-19-968029-0
Published: May 2013
Also available as: eBook

Price: £82.00