Einstein Constraint Equations Research Articles

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.

Scalar Curvature Deformation and a Gluing Construction for the Einstein Constraint Equations

On a compact manifold, the scalar curvature map at generic metrics is a local surjection [F-M]. We show that this result may be localized to compact subdomains in an arbitrary Riemannian manifold. The method is extended to establish the existence of asymptotically flat, scalar-flat metrics on ℝn (n≥ 3) which are spherically symmetric, hence Schwarzschild, at infinity, i.e. outside a compact set. Such metrics provide Cauchy data for the Einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically Schwarzschild near spatial infinity.

Asymptotically hyperbolic non-constant mean curvature solutions of the Einstein constraint equations

We describe how the iterative technique used by Isenberg and Moncrief to verify the existence of large sets of non-constant mean curvature solutions of the Einstein constraints on closed manifolds can be adapted to verify the existence of large sets of asymptotically hyperbolic non-constant mean curvature solutions of the Einstein constraints.

A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds

We prove that a certain class of conformal data (open in the set of all conformal data) maps to solutions of the Einstein constraint equations with nonconstant mean curvature. The method of proof is constructive, and it should apply to larger classes of conformal data as well.

Constant mean curvature solutions of the Einstein constraint equations on closed manifolds

We prove in detail a theorem which completes the evaluation and parametrization of the space of constant mean curvature (CMC) solutions of the Einstein constraint equations on a closed manifold. This theorem determines which sets of CMC conformal data allow the constraint equations to be solved, and which sets of such data do not. The tools we describe and use here to prove these results have also been found to be useful for the study of non-constant mean curvature solutions of the Einstein constraints.

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 Teichmüller 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-dimensional Einstein equations (formulated on circle bundles over Σ×R and with a spacelike Killing field tangent to the fibers of the chosen bundle) and to the recent discussion by Witten of the possible exact solvability of the ‘‘topological dynamics’’ associated with Einstein’s equations in 2+1 dimensions.

Entropy extremum of relativistic self-bound systems: A geometric approach

Conditions for entropy extremum of self-bound relativistic systems of particles, in stationary axisymmetric motion, are obtained under the following constraints: (i) The system is kept isolated in a geometrical sense, implying that the total mass-energy and total angular momentum, defined by the asymptotic behavior of the metric, are kept constant, (ii) the total number of particles is kept constant, and (iii) Einstein's constraint equations are imposed on a spacelike hypersurface. It is shown that if the system is in mechanical equilibrium, the total entropy is an extremum for all trial nonequilibrium configurations that satisfy the constraints and respect the symmetry, if and only if (i) Einstein's dynamical equations are satisfied, (ii) the temperature and the chemical free energy, as seen from infinity, are constant, and (iii) the system is rigidly rotating. The proof does not depend on a particular functional expression for the mass or for the angular momentum; consequently, no related Lagrange multipliers have been used.

A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature

We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a so-called global supersolution but does not require a global subsolution. As a consequence, we construct a class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, extending a recent result of Holst, Nagy, and Tsogtgerel [HNT07] which constructed similar solutions in the presence of matter. We give a second proof of this result showing that vacuum solutions can be obtained as a limit of [HNT07] non-vacuum solutions. Our principal existence theorem is of independent interest in the near-CMC case, where it simplifies previously known hypotheses required for existence.