Foliated Rigidity and Dynamics of Space-Time

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Proposal: DMS-0072165 PI Scot Adams The study of Lorentz manifolds may be motivated by physical considerations, since the Einstein field equations determine Lorentz metrics on space-time. Following F.~Klein's Erlanger program, one method of approaching a geometric object is to study its group of isometries, and it has long been known that the isometry group of a Lorentz manifold is a Lie group. Moreover, any Lie group acts on itself (by left translation) preserving any left-invariant Lorentz metric. However, as originally noticed by R.~Zimmer, M.~Gromov, and N.~Kowalsky, if one requires even a small amount of complication to the dynamics, then the list of Lie groups admitting a Lorentz action becomes quite restricted. My work over the last few years has focused on quantifying exactly which connected Lie groups admit a complicated action by isometries of a connected Lorentz manifold. As the definition of ``complicated'' varies, the answer varies, and I intend to look at a few more of these variations over the coming years. In particular, I would like to determine the collection of connected, noncompact, simple Lie groups admitting a locally faithful, nontame, isometric action on a connected Lorentz manifold. I believe it should be true that any such group either has infinite center or is locally isomorphic to the Lie group of 2 by 2 real unimodular matrices. A Lorentz manifold, or ``space-time'' is one of the basic object of general relativity. Some of these objects have many symmetries, while most have none at all. Fix a space-time having a large collection of symmetries. For any point in this space-time, if we move it around by all the symmetries under consideration, the collection of image points so obtained is called the ``orbit'' of the point. A number of researchers have noticed that it is difficult to construct a space-time with chaotic orbits, by which we mean, broadly speaking, that some orbits repeatedly return (or nearly return) to the same place. There are, in fact, many ways to define precisely the word chaotic, and this means that a number of different theorems have been obtained. I intend to continue developing results along these lines.