Existence of non-trivial, vacuum, asymptotically simple spacetimes

Abstract

We construct non-trivial vacuum space-times with a global I + . The construction proceeds by proving extension results for initial data sets across compact boundaries, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon. * Supported in part by a grant of the Polish Research Foundation KBN. † Supported in part by the ACI program of the French Ministry of Research. 1 Here we mean the metric induced by the Schwarzschild metric on the usual t = 0 hypersurface in Schwarzschild space-time; we will make such an abuse of terminology throughout.

Static Vacuum Solutions from Convergent Null Data Expansions at Space-Like Infinity

“…Another reason for giving such a characterization results from the work by Corvino ( [5], [6]), Corvino and Schoen ( [7]), and Chruściel and Delay ( [3], [4]). These authors deform given asymptotically flat vacuum data outside prescribed compact sets to vacuum data which are exactly static or stationary near or asymptotically static or stationary at space-like infinity and use such data to discuss the existence of null geodesically complete solutions which have a smooth asymptotic structure at null infinity.…”

A note on asymptotically flat metrics on ℝ³ which are scalar-flat and admit minimal spheres

“…We begin with the following proposition from [CD1]. We recall the proof of this, both for completeness and to amplify some of the details.…”

“…This requires an analysis both of the geometry of the initial data as well as of the spacetime evolution of this data. Such an analysis has been carried out in Chruściel and Mazzeo [CM], from which it follows, for example, that for small masses, the data produced by [CD1] in Prop. 3.1 has as its outermost minimal spheres with respect to the end M \ B(0, r) precisely the union of the Schwarzschild horizons.…”

“…In this note we combine a construction by Miao [Mi] with a construction by Chruściel and Delay [CD1] to produce asymptotically flat metrics on R 3 which have zero scalar curvature and multiple minimal spheres (Thm. 3.2).…”

Asymptotically hyperbolic metrics on the unit ball with horizons

“…See [4], [12], [8], and [14] for examples, and for existence of many blackholes, please see [5]. From the point of view of general relativity, these are examples of globally regular and asymptotically flat initial data for the Einstein vacuum equations containing a trapped surfaces.…”