« We find them smaller and fainter, in constantly increasing numbers, and we know that we are reaching into space, farther and farther, until, with the faintest nebulae that can be detected with the greatest telescopes, we arrive at the frontier of the known universe. »
— Edwinn Hubble. Quoted in E Maor, To infinity and beyond (Princeton 1991)
Image credit: NASA, The Hubble Heritage Team (STScI)
Historically, the first models of our universe in the framework of General Relativity, which originated the Theory of Big-Bang, are Friedman-Lemaître space-times (also called Robertson-Walker space-times). Despite of their relative simplicity, these models reveal nontrivial behavior: the spatial slices can be shrunk in the past or in the future. This complicated feature of space-time is even more apparent in the context of Bianchi Cosmological Models, to which an important literature in physics is devoted (see for example [Dynamical systems in cosmology, Cambridge University Press, 1997]). These models provide examples of chaotic behaviors in which the phases of expansion and contraction of the spatial slices alternate in an unpredictable way.
In our study space-time means a Lorentzian manifold M which decomposes (non uniquely!) as an orthogonal product S x R. Let M(S) be the space of Riemannian metrics on S: the restriction to every level S x {t} defines a path in M(S). The requirement that M obeys the Einstein equation admits a dynamical formulation: the path in M(S) is the projection of a trajectory of some flow (the Einstein flow) defined on a subset of the cotangent bundle T*M(S). Of course, this point of view is quite limited due to the infinite dimension of T*M(S). But Friedman-Lemaître or Bianchi examples make clear that this Einstein flow admits very interesting finite-dimensional subsystems (set of trajectories), and a crucial observation for this project is that the family of space-times with constant curvature (abbr. CC-space-times) is also a natural finite-dimensional subsystem.
The constant curvature hypothesis, if mathematically very natural and appealing, has long been considered as physically unrelevant. However, this reluctance decreased since the 90's, mainly due to the development by 't Hooft and E. Witten of classical and quantum gravity in dimension 2+1, in which solutions to the Einstein equations automatically have constant curvature (see e.g. [Carlip, Quantum gravity in 2+1 dimensions, Cambridge University Press, 1998]). At this period, V. Moncrief addressed some questions concerning what we call here the Moncrief flow which is a projection of the Einstein flow in dimension 2+1 into the cotangent bundle of the Teichmüller space of S.
In the early 90's, G. Mess revealed deep connections between gravity in dimension 2+1 and notions more familiar to geometers such as measured geodesic laminations and earthquakes. If we except Scannell PhD thesis, it was almost 10 years before these ideas being pushed further in mathematics. Members of our team had a crucial role in this renewal, but let's also mention here among others the contributions of R. Benedetti, K. Krasnov, K. Scannell, J.M. Schlenker... All this work opened the the way to the study of the Einstein flow reduced to CC-spacetimes, which is the main goal of our project.
The choice of the time function t, which is not unique, is a crucial matter. There are some natural choices, to which all other choices of time has to be compared with: the cosmological time, the CMC time (also called ``York time'' in physics), and the K time. In summary, here are some problems we intend to solve:
Describe the peculiar geometry of level sets of cosmological time, CMC time and K time in a given CC space-time and determine to what extent any other slicing of space-time can be compared to these special ones
Study the dynamical properties of the Moncrief flow,
Determine for t going to 0 (near the ``Big-Bang'') the (Hausdorff - Gromov) metric limit of the metric on S x {t} , and also the limit of the ``Moncrief lines'' in the various compactifications of Teichmüller space.
Another goal of our project, wich has also a dynamico-geometrical flavour, is to prove the Cosmic Censorship conjecture in the special context of CC space-times, in the following formulation: generically, a maximal globally hyperbolic spatially compact CC space-time admits no non-trivial isometric embedding into any bigger space-time. We strongly believe that this result can be achieved through a dynamical and geometrical analysis of the action on the horizon and the initial singularity of the holonomy group.
All the questions above, formulated in the context of CC space-times, admit natural generalizations to singular CC space-times (i.e. with particles) or to conformally flat space-times: these extensions will also be considered in the development of the project. Finally, the last objective of our project is to write a handbook of Bianchi cosmology accessible to geometers and topologists, and to analyze the associated dynamical systems.