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James Isenberg

Publications and source records attributed to James Isenberg.

86 records · Page 5Linked to original sources
arXiv2002

On the topology of vacuum spacetimes

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. For any compact manifold which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [IMP] (gr-qc/0109045), which is restricted to constant mean curvature data.

gr-qc↗
arXiv2002

Gluing and wormholes for the Einstein constraint equations

We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with suitable corresponding conditions on the extrinsic curvature. In the compact setting a mild nondegeneracy condition is required. In the final section of the paper, we list a number ways this construction may be used to produce new types of vacuum spacetimes.

gr-qc↗
arXiv2002

Constructing Solutions of the Einstein Constraint Equations

The first step in the building of a spacetime solution of Einstein's gravitational field equations via the initial value formulation is finding a solution of the Einstein constraint equations. We recall the conformal method for constructing solutions of the constraints and we recall what it tells us about the parameterization of the space of such solutions. One would like to know how to construct solutions which model particular physical phenomena. One useful step towards this goal is learning how to glue together known solutions of the constraint equations. We discuss recent results concerning such gluing.

gr-qc↗
arXiv2002

Asymptotic Behavior of Polarized and Half-Polarized U(1) symmetric Vacuum Spacetimes

We use the Fuchsian algorithm to study the behavior near the singularity of certain families of U(1) Symmetric solutions of the vacuum Einstein equations (with the U(1) isometry group acting spatially). We consider an analytic family of polarized solutions with the maximum number of arbitrary functions consistent with the polarization condition (one of the ``gravitational degrees of freedom'' is turned off) and show that all members of this family are asymptotically velocity term dominated (AVTD) as one approaches the singularity. We show that the same AVTD behavior holds for a family of ``half polarized'' solutions, which is defined by adding one extra arbitrary function to those characterizing the polarized solutions. (The full set of non-polarized solutions involves two extra arbitrary functions). We begin to address the issue of whether AVTD behavior is independent of the choice of time foliation by showing that indeed AVTD behavior is seen for a wide class of choices of harmonic time in the polarized and half-polarized (U(1) Symmetric vacuum) solutions discussed here. \

gr-qc↗
arXiv2001

Singularity Formation in 2+1 Wave Maps

We present numerical evidence that singularities form in finite time during the evolution of 2+1 wave maps from spherically equivariant initial data of sufficient energy.

math-ph↗
arXiv2000

Global existence for wave maps with torsion

Wave maps (i.e. nonlinear sigma models) with torsion are considered in 2+1 dimensions. Global existence of smooth solutions to the Cauchy problem is proven for certain reductions under a translation group action: invariant wave maps into general targets, and equivariant wave maps into Lie group targets. In the case of Lie group targets (i.e. chiral models), a geometrical characterization of invariant and equivariant wave maps is given in terms of a formulation using frames.

math-ph↗
arXiv2000

Critical Phenomena in Nonlinear Sigma Models

We consider solutions to the nonlinear sigma model (wave maps) with target space S^3 and base space 3+1 Minkowski space, and we find critical behavior separating singular solutions from nonsingular solutions. For families of solutions with localized spatial support a self-similar solution is found at the boundary. For other families, we find that a static solution appears to sit at the boundary. This behavior is compared to the black hole critical phenomena found by Choptuik.

math-ph↗
arXiv1999

Einstein Constraints on Asymptotically Euclidean Manifolds

We consider the Einstein constraints on asymptotically euclidean manifolds $M$ of dimension $n \geq 3$ with sources of both scaled and unscaled types. We extend to asymptotically euclidean manifolds the constructive method of proof of existence. We also treat discontinuous scaled sources. In the last section we obtain new results in the case of non-constant mean curvature.

gr-qc↗
arXiv1998

The Singularity in Generic Gravitational Collapse Is Spacelike, Local, and Oscillatory

A longstanding conjecture by Belinskii, Khalatnikov, and Lifshitz that the singularity in generic gravitational collapse is spacelike, local, and oscillatory is explored analytically and numerically in spatially inhomogeneous cosmological spacetimes. With a convenient choice of variables, it can be seen analytically how nonlinear terms in Einstein's equations control the approach to the singularity and cause oscillatory behavior. The analytic picture requires the drastic assumption that each spatial point evolves toward the singularity as an independent spatially homogeneous universe. In every case, detailed numerical simulations of the full Einstein evolution equations support this assumption.

gr-qc↗
arXiv1998

Mixmaster Behavior in Inhomogeneous Cosmological Spacetimes

Numerical investigation of a class of inhomogeneous cosmological spacetimes shows evidence that at a generic point in space the evolution toward the initial singularity is asymptotically that of a spatially homogeneous spacetime with Mixmaster behavior. This supports a long-standing conjecture due to Belinskii et al. on the nature of the generic singularity in Einstein's equations.

gr-qc↗
arXiv1997

Global Foliations of Vacuum Spacetimes with $T^2$ Isometry

We prove a global existence theorem (with respect to a geometrically- defined time) for globally hyperbolic solutions of the vacuum Einstein equations which admit a $T^2$ isometry group with two-dimensional spacelike orbits, acting on $T^3$ spacelike surfaces.

gr-qc↗