Gluing Construction of Initial Data with Kerr–de Sitter Ends

Abstract

We construct initial data sets which satisfy the vacuum constraint equations of General Relativity with positive cosmologigal constant. More presilely, we deform initial data with ends asymptotic to Schwarzschild-de Sitter to obtain non-trivial initial data with exactly Kerr-de Sitter ends. The method is inspired from Corvino's gluing method. We obtain here a extension of a previous result for the time-symmetric case by Chruściel and Pollack in [10]. *

Hamiltonian dynamics in the space of asymptotically Kerr–de Sitter spacetimes

“…There is growing astrophysical evidence that spacetimes with positive cosmological constant provide physically correct models for cosmology. Large families of such non-compact, vacuum, general-relativistic models have been constructed (see [15,16,19,20,22,23,41] and references therein). In particular we have now large classes of initial data sets with one or more ends of cylindrical type, in which the metric becomes periodic when one recedes to infinity along half-cylinders.…”

“…Hence our interest in asymptotically Kerr-de Sitter metrics. We note that a large family of non-trivial vacuum spacetimes which are exactly Kerr-de Sitter outside the domain of dependence of a compact set has been constructed in [23]. Now, in the coordinate system of (3.1) below, all Killing vectors of Kerr-de Sitter metrics with rotation parameter a = 0 take the form…”

“…As such, we consider the collection of vacuum initial data which approach Kerr-de Sitter initial data in an asymptotically periodic end, together with first derivatives. (A large class of such initial data has been constructed in [15,20,23].) Our aim is to calculate θ b for such metrics.…”

A review of total energy–momenta in GR with a positive cosmological constant

“…In this subsection we review their work on asymptotically-Schwarzschild-de Sitter data and asymptotically-Kerr-de Sitter data. These are discussed in [60] and [61] respectively and these in turn rely on families of initial data for the Einstein equations shown to exist in [62] and [63] respectively. The Schwarzschild-de Sitter metric, sometimes called the Kottler solution, can be defined in space-time dimension n + 1 as…”

“…In [63], Cortier generalised the work in [62] to produce initial data which are deformations of that for Kerr-de Sitter (but only in space-time dimension 4, so the data surface has dimension 3). Again one has an infinite periodic metric but now with a particular nonzero periodic second fundamental form as well.…”