Initial Data Engineering

Abstract

We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.

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

“…The constraint equations admit many solutions (permitting the specification of different initial conditions) and it is important to understand the structure of the set of all possible initial data on a given manifold. Various approaches have been given for constructing solutions including parabolic methods [Ba93] and gluing constructions [Co00] [CIP05]. From the point of view of classifying the set of all possible solutions, the most fruitful technique has been the conformal method initiated by Lichnerowicz [Li44] and extended by Choquet-Bruhat and York [CBY80].…”

A gluing construction for non-vacuum solutions of the Einstein-constraint equations

“…Except for the case of generic non-gravitational fields described entirely by an energy density function ρ and a current density vector field J (satisfying a strict energy condition ρ > |J|), the Corvino-Schoen techniques have not yet been generalized away from the vacuum case. It is, however, expected that this can be done; it would then follow that the gluing theorems obtained for vacuum data in [9] would extend to non-vacuum data.…”

General relativity and gravitation: a centennial perspective

“…Connected sum gluing starts with a pair of smooth solutions of the constraints, chooses a point in each, and produces a new solution of the constraints on the connected sum manifold 6 which is identical to the original solutions away from the chosen gluing points. This gluing technique [67,68] works for any pair of initial data sets-compact, AE, or AH-so long as a certain non degeneracy condition (see [68]) holds at the gluing points. It also works for most coupled-in matter source fields.…”