Vincent Moncrief

Symmetry and bifurcations of momentum mappings

The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einsteins equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface.

Reduction of the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmüller space

In this paper the ADM (Arnowitt, Deser, and Misner) reduction of Einstein’s equations for three‐dimensional ‘‘space‐times’’ defined on manifolds of the form Σ×R, where Σ is a compact orientable surface, is discussed. When the genus g of Σ is greater than unity it is shown how the Einstein constraint equations can be solved and certain coordinate conditions imposed so as to reduce the dynamics to that of a (time‐dependent) Hamiltonian system defined on the 12g−12‐dimensional cotangent bundle, T*T(Σ), of the Teichmuller space, T(Σ), of Σ. The Hamiltonian is only implicitly defined (in terms of the solution of an associated Lichnerowicz equation), but its existence, uniqueness, and smoothness are established by standard analytical methods. Similar results are obtained for the case of genus g=1, where, in fact, the Hamiltonian can be computed explicitly and Hamilton’s equations integrated exactly (as was found previously by Martinec). The results are relevant to the problem of the reduction of the 3+1‐dimensi...

The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space

In this paper we complete the proof of global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space by showing that an appropriate norm of the solutions cannot blow up in a finite time. A key step in the proof is the demonstration that theL∞ norm of the curvature is boundeda priori. Our results apply to any compact guage group and to any invariant Higgs self-coupling which is positive and of no higher than quartic degree.

The global existence of Yang-Mills-Higgs fields in

In this paper and its sequel we shall prove the local and then the global existence of solutions of the classical Yang-Mills-Higgs equations in the temporal gauge. This paper proves local existence uniqueness and smoothness properties and improves, by essentially one order of differentiability, previous local existence results. Our results apply to any compact gauge group and to any invariant Higgs self-coupling which is positive and of no higher than quartic degree.

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Abstract In this work, we present results which support the Strong Cosmic Censorship Conjecture and some of the ideas of Belinskii, Khalatnikov, and Lifschitz. The results concern the behavior of the gravitational field in the neighborhood of singularities in the polarized Gowdy spacetimes. We rigorously show (using certain “energy” estimates) that as one approaches the singularity, the metric field of any of these vacuum solutions of Einsteins equations asymptotically approaches a solution of a truncated system of equations in which spatial curvature terms have been dropped. Hence, the gravitational fields, in a sense, spatially decouple near the singularity. Based on this asymptotic behavior, we also show that only in a very small subclass of the polarized Gowdy spacetimes is the curvature bounded near the singularity. Hence, the generic polarized Gowdy spacetime cannot be extended across a Cauchy horizon.

-dimensional Minkowski space. I. Local existence and smoothness properties

We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions).For the special case of a null surface diffeomorphic toT3 we characterize the degenerate vacuum solutions completely. These consist of an infinite dimensional family of “plane wave” spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic toT3 and prove the generality of the previously constructed non-degenerate solutions.We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einsteins equations are essentially an artefact of symmetry.

Asymptotic behavior of the gravitational field and the nature of singularities in gowdy spacetimes

Abstract We study the global properties of the Gowdy metrics generated by Cauchy data on the 3-torus. We show that the boundaries of the maximal Cauchy developments of Gowdy initial data sets are always “crushing singularities” in the sense of Eardley and Smarr. This means that each solution admits a slicing in which tr K ( t ) (the trace of the second fundamental form induced on the surface Σ t of the slicing) uniformly blows up as t approaches its limiting value. A theorem of Hawking shows that the maximal Cauchy development cannot extend beyond the boundary at which tr K blows up and our result shows that no singularities arise to halt the evolution until this boundary is reached. Thus each maximal Cauchy development is always as large as it can be, consistent with Hawkings theorem. We discuss the relevance of this result to the strong cosmic censorship conjecture and the question of when the crushing singularities are in fact curvature singularities.

Symmetries of cosmological Cauchy horizons

We formulate two global existence conjectures for the Einstein equations and discuss their relevance to the cosmic censorship conjecture. We argue that the reformulation of the cosmic censorship conjecture as a global existence problem renders it more amenable to direct analytical attack. To demonstrate the facilty of this approach we prove the cosmological version of our global existence conjecture for the Gowdy spacetimes onT3×R.

Global properties of Gowdy spacetimes with T3 × R topology☆

We prove that if a stationary, real analytic, asymptotically flat vacuum black hole spacetime of dimension n ≥ 4 contains a non-degenerate horizon with compact cross-sections that are transverse to the stationarity generating Killing vector field then, for each connected component of the black holes horizon, there is a Killing field which is tangent to the generators of the horizon. For the case of rotating black holes, the stationarity generating Killing field is not tangent to the horizon generators and therefore the isometry group of the spacetime is at least two dimensional. Our proof relies on significant extensions of our earlier work on the symmetries of spacetimes containing a compact Cauchy horizon, allowing now for non-closed generators of the horizon.

The global existence problem and cosmic censorship in general relativity

We present several results about the nonexistence of solutions of Einsteins equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.