Motion of Isolated bodies

It is shown that sufficiently smooth initial data for the Einstein-dust or the Einstein-Maxwell-dust equations with non-negative density of compact support develop into solutions representing isolated bodies in the sense that the matter field has spatially compact support and is embedded in an exterior vacuum solution

Geometrical Hyperbolic Systems for General Relativity and Gauge Theories

The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom [the spatial shift vector $\beta^{i}(t,x^{j})$ and the spatial scalar potential $\phi(t,x^{j})$, respectively] are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of $(t,x^{j})$ from subsidiary equations that are not part of the evolution equations. Propagation of certain (``radiative'') dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by $(1)$ taking a further time derivative of the equation of motion of the canonical momentum, and $(2)$ adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier $\beta^{i}$) or of the Gauss's law constraint of electromagnetism (Lagrange multiplier $\phi$). General relativity also requires a harmonic time slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first order symmetric form. The dynamically propagating gravity fields straightforwardly determine the ``electric'' or ``tidal'' parts of the Riemann tensor.Comment: 24 pages, latex, no figure

Future complete spacetimes with spacelike isometry group and field sources

We extend to the Einstein Maxwell Higgs system results first obtained previously in collaboration with V. Moncrief for Einstein equations in vacuum.Comment: to appear in proceedings of the greek relativity meeting 200

Proof of the Thin Sandwich Conjecture

We prove that the Thin Sandwich Conjecture in general relativity is valid, provided that the data $(g_{ab},\dot g_{ab})$ satisfy certain geometric conditions. These conditions define an open set in the class of possible data, but are not generically satisfied. The implications for the ``superspace'' picture of the Einstein evolution equations are discussed.Comment: 8 page

Cosmological spacetimes not covered by a constant mean curvature slicing

We show that there exist maximal globally hyperbolic solutions of the Einstein-dust equations which admit a constant mean curvature Cauchy surface, but are not covered by a constant mean curvature foliation.Comment: 11 page

Hamiltonian Time Evolution for General Relativity

Hamiltonian time evolution in terms of an explicit parameter time is derived for general relativity, even when the constraints are not satisfied, from the Arnowitt-Deser-Misner-Teitelboim-Ashtekar action in which the slicing density $\alpha(x,t)$ is freely specified while the lapse $N=\alpha g^{1/2}$ is not. The constraint ``algebra'' becomes a well-posed evolution system for the constraints; this system is the twice-contracted Bianchi identity when $R_{ij}=0$. The Hamiltonian constraint is an initial value constraint which determines $g^{1/2}$ and hence $N$, given $\alpha$.Comment: 4 pages, revtex, to appear in Phys. Rev. Let

The constraint equations for the Einstein-scalar field system on compact manifolds

We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.Comment: Minor changes, final version. To appear: Classical and Quantum Gravit

Covariant Poisson equation with compact Lie algebras

The covariant Poisson equation for Lie algebra-valued mappings defined in 3-dimensional Euclidean space is studied using functional analytic methods. Weighted covariant Sobolev spaces are defined and used to derive sufficient conditions for the existence and smoothness of solutions to the covariant Poisson equation. These conditions require, apart from suitable continuity, appropriate local integrability of the gauge potentials and global weighted integrability of the curvature form and the source. The possibility of nontrivial asymptotic behaviour of a solution is also considered. As a by-product, weighted covariant generalisations of Sobolev embeddings are established.Comment: 31 pages, LaTeX2

On completeness of orbits of Killing vector fields

A Theorem is proved which reduces the problem of completeness of orbits of Killing vector fields in maximal globally hyperbolic, say vacuum, space--times to some properties of the orbits near the Cauchy surface. In particular it is shown that all Killing orbits are complete in maximal developements of asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact manifold. This result gives a significant strengthening of the uniqueness theorems for black holes.Comment: 16 pages, Latex, preprint NSF-ITP-93-4

Global solutions of the Einstein-Maxwell equations in higher dimensions

We consider the Einstein-Maxwell equations in space-dimension $n$. We point out that the Lindblad-Rodnianski stability proof applies to those equations whatever the space-dimension $n\ge 3$. In even space-time dimension $n+1\ge 6$ we use the standard conformal method on a Minkowski background to give a simple proof that the maximal globally hyperbolic development of initial data sets which are sufficiently close to the data for Minkowski space-time and which are Schwarzschildian outside of a compact set lead to geodesically complete space-times, with a complete Scri, with smooth conformal structure, and with the gravitational field approaching the Minkowski metric along null directions at least as fast as $r^{-(n-1)/2}$