# Vincent Moncrief

The Gravitation Physics group at Yale currently consists of one graduate student and two postdoctoral fellows. The interests of this group cover a broad range of topics ranging from the perturbation theory of cosmological models with non-trivial topologies to the study of the Cosmic Censorship and Hoop conjectures for Einstein’s theory of General Relativity to the treatment of sufficiently small but fully non-linear perturbations of certain background cosmological spacetimes.

Postdoctoral fellow Masayuki Tanimoto and graduate student Mary Vasu are currently investigating different aspects of the perturbation theory of certain cosmological solutions of Einstein’s equations having non-trivial spatial topology. This research involves the expansion of the perturbations in suitably defined tensor harmonics, the derivation of the associated perturbation equations and the reformulation of these equations in terms of naturally defined gauge invariant quantities. The aim of these projects is to study the asymptotic properties, for expanding universes, of the spacetime metric and matter perturbations and to determine how these properties are related to the underlying spatial topology. The understanding of linearized perturbations is a necessary first step towards the ultimate goal of treating sufficiently small but fully non-linear perturbations. Some examples of this generalization to the treatment of non-linear perturbations are currently under study by Vincent Moncrief and his collaborators.

Postdoctoral fellow Sergio Goncalves is studying aspects of the Cosmic Censorship and Hoop conjectures by analyzing the behavior of solutions of Einstein’s equations which have either cylindrical or translational symmetry. An important question in this regard is whether such spacetimes can form trapped surfaces in the course of their evolution and, if not, how the evolution conspires (if it does) to avoid the formation of naked singularities. An important tool in these studies has been the Hamiltonian reduction of Einstein’s equations and the identification of the unconstrained degrees of freedom for the gravitational field and the derivation of the field equations which these variables satisfy.

Vincent Moncrief’s own research is mainly concerned with the global existence and asymptotic properties of cosmological solutions of Einstein’s equations and especially the question of how these properties depend upon the topology of spacetime. He is also interested in how a study of the “Einstein flow” on various manifolds might shed light on open questions in 3-mainfold topology itself. Most of this research involves the treatment of sufficiently small but nevertheless fully non-linear perturbations of certain special backgrounds and includes an analysis of higher as well as lower dimensional spacetimes in addition to physical 3 + 1 dimensional spacetime. A key result (obtained jointly with Arthur Fischer of the University of California at Santa Cruz) was to relate the reduced Hamiltonian for Einstein’s equations to a topological invariant known as the Yamabe invariant (or sigma constant) for the spatial manifold and to show that the reduced Hamiltonian is monotonically decreasing along all solutions of the field equations (in the direction of cosmological expansion) and therefore evidently seeking to attain its infimum which in turn is expressible in terms of the sigma constant. A discussion of this and related work (with Lars Andersson of the University of Miami and Yvonne Choquet-Bruhat of the Universite’ Paris VI) may be found in Moncrief’s and Choquet-Bruhat’s lectures at the Cargese summer school on *50 years of the Cauchy Problem in General Relativity:* http://fanfreluche.math.univ-tours.fr

**References**

- Arthur E. Fischer and Vincent Moncrief, “Conformal Volume Collapse of 3- Manifolds and the Reduced Einstein Flow” in
*Geometry, Mechanics and Dynamics*, Springer 2002 - Yvonne Choquet-Bruhat and Vincent Moncrief, “Future Global in Time Einsteinien Spacetimes with U(1) Isometry Group”, Ann. Henri Poincare’ 2 (2001 )1007-1064.
- Masayuki Tanimoto, Vincent Moncrief and Katsuhito Yasuno, “Perturbations of Spatially Closed Bianchi III Spacetimes”, Yale preprint, 2002, submitted to Classical and Quantum Gravity.
- Sergio Goncalves, “No Horizon Theorem for Vacuum Gravity with Spacelike G1 Isometry Groups” Yale preprint, 2002.