Collisionless matter in general relativity

General relativity is a theory of gravity first proposed by Einstein in 1915. Despite being over 100 years old, many great mathematical challenges remain in understanding the theory and its governing equations -- the Einstein equations -- which, appropriately viewed, are a nonlinear system of hyperbolic partial differential equations. One particularly striking aspect of the theory, which is in sharp contrast to the preceding Newtonian theory of gravity, is that it is nontrivial in the vacuum, i.e. in the absence of any matter. Tremendous progress has been made on the Einstein equations in vacuum recent years. Progress in the presence of matter, however, has been considerably slower. The proposed research attempts to bridge this gap.

Much of the mathematical study of general relativity is driven by the so called cosmic censorship conjectures, first formulated by Penrose, which posit that any singularities arising in the theory must have certain desirable properties. In the language of partial differential equations the conjectures can be viewed as statements of global existence and global uniqueness. A fruitful direction of study, which indeed constitutes much of the progress to date on the conjectures, involves exhibiting regimes, close to certain explicit stationary solutions, in which the conjectures hold.

The slow progress in the presence of matter is due to the fact that one has to not only understand difficulties arising in the vacuum and difficulties arising from the matter equations alone, but also the new difficulties which arise due to the coupling of these difficulties. This proposal concerns a particularly simple type of matter, which nonetheless is regarded as physical and exhibits a wide range of new phenomena in general relativity, described by the Vlasov equation.