Arthur E. Fischer

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I

A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented. A number of sharp regularity and smoothness properties of the solutions are obtained. The present paper is devoted to the case ofRn with suitable asymptotic conditions imposed. As an example, we apply this theory to give new proofs of the existence and uniqueness theorems for the Einstein equations in general relativity, due to Choquet-Bruhat and Lichnerowicz. These new proofs usingfirst order techniques are considerably simplier than the classical proofs based onsecond order techniques. Our existence results are as sharp as had been previously known, and our uniqueness results improve by one degree of differentiability those previously existing in the literature.

Linearization stability of the Einstein equations

1. Introduction. An important problem in general relativity is the question of whether or not a solution of the linearized Einstein field equations (relative to a given background solution) actually approximates to first order a curve of exact solutions to the nonlinear equations. Here we announce that under certain geometrical conditions on the background solution the problem can be answered affirmatively; however, in certain exceptional cases the answer may be negative. In the affirmitive case we shall say that the background metric is linearization-stable. Let (4) # be a Lorentz metric (signature —, +, +, +) on a 4-manifold V. The empty space Einstein field equations of general relativity are that the Ricci tensor of (4) # vanish :

Resolving the singularities in the space of Riemannian geometries

A method is described for unfolding the singularities of superspace, G=M/D, the space of Riemannian geometries of a manifold M. This extended, or unfolded superspace, is described by the projection GF(M)=(M×F(M))/D→M/D=G, where F(M) is the frame bundle of M. The unfolded space GF(M) is an infinite‐dimensional manifold without singularities. Moreover, as expected, the unfolding of GF(M) at each geometry [ g0]∈G is parametrized by the isometry group Ig0(M) of g0. The construction is completely natural, gives complete control and knowledge of the unfolding at each geometry necessary to make GF(M) a manifold, and is generally covariant with respect to all coordinate transformations. A similar program is outlined, based on the methods of this paper, of desingularizing the moduli space of connections on a principal fiber bundle.

The reduced Einstein equations and the conformal volume collapse of 3-manifolds

We consider the problem of the Hamiltonian reduction of Einsteins equations on a (3 + 1)-vacuum spacetime that admits a foliation by constant mean curvature compact spacelike hypersurfaces M of Yamabe type -1. After a conformal reduction process, we find that the reduced Einstein flow is described by a time-dependent non-local dimensionless reduced Hamiltonian Hreduced which is strictly monotonically decreasing along any non-constant integral curve of the reduced Einstein system. We discuss relationships between Hreduced, the σ-constant of M, the Gromov norm |M|| and the hyperbolic σ-conjecture. As examples, we consider Bianchi models that spatially compactify to manifolds of Yamabe type -1. For these models we show that under the reduced Einstein flow, Hreduced asymptotically approaches either the σ-constant or in the hyperbolizable case, the conjectured σ-constant, as suggested by our general theory. In the non-hyperbolizable cases, the conformal metric of the reduced Einstein flow volume-collapses M along either circular fibres, embedded tori, or collapses the entire manifold to a point, and in each case, the collapse occurs with bounded curvature. We consider applications of these results to future all-time small-data existence theorems for spatially compact spacetimes.

The Reduced Hamiltonian of General Relativity and the σ-Constant of Conformal Geometry

For the problem of the Hamiltonian reduction of Einstein’s equations on a 3+1 vacuum spacetime that admits a foliation by constant mean curvature (CMC) compact spacelike hypersurfaces M that satisfy certain topological restrictions, we introduce a dimensionless non-local time-dependent reduced Hamiltonian system

The internal symmetry group of a connection on a principal fiber bundle with applications to gauge field theories

H_{reduced} :R^ - \times P_{reduced} \to R

Hamiltonian reduction and perturbations of continuously self-similar (n + 1)-dimensional Einstein vacuum spacetimes

where the reduced Hamiltonian is given by