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**Existence of non-trivial, vacuum, asymptotically simple spacetimes.**
*(English)*
Zbl 1005.83009

Classical Quantum Gravity 19, No. 9, L71-L79 (2002); erratum ibid. 19, No. 12, 3389 (2002).

In the paper [Commun. Math. Phys. 214, 137-189 (2000; Zbl 1031.53064)], J. Corvino presented a gluing construction for scalar flat metrics, leading to the striking result of the existence of non-trivial scalar flat metrics which are exactly Schwarzschild at large distances.

The method consists of gluing an asymptotically flat metric \(g\) with a Schwarzschild metric on an annulus \(B(0,2R_0)\setminus B(0, R_0)\). The authors show that if \(R_0\) is large enough, then the gluing can be performed so as to preserve the time-symmetric scalar constraint equation \(R(g) = 0\), where \(R(g)\) is the Ricci scalar of \(g\). The above method is used to construct vacuum spacetimes which admit conformal compactifications at null infinity with a high degree of differentiability and with a global \(\mathcal J^+\). 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 the existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon. The Corvino-Schoen technique is used to add Einstein-Rosen bridges to manifolds satisfying \(R = 0\), assuming in addition a local parity condition. The deformation of the metric related to the addition of the bridge preserves the condition \(R = 0\) and modifies the metric only in an arbitrarily small neighbourhood of the point at which the bridge is added. This allows one to connect pairs of time-symmetric vacuum initial data without perturbing the metric away from a small neighbourhood of the bridges, or to create wormholes within a given initial data set.

The method consists of gluing an asymptotically flat metric \(g\) with a Schwarzschild metric on an annulus \(B(0,2R_0)\setminus B(0, R_0)\). The authors show that if \(R_0\) is large enough, then the gluing can be performed so as to preserve the time-symmetric scalar constraint equation \(R(g) = 0\), where \(R(g)\) is the Ricci scalar of \(g\). The above method is used to construct vacuum spacetimes which admit conformal compactifications at null infinity with a high degree of differentiability and with a global \(\mathcal J^+\). 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 the existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon. The Corvino-Schoen technique is used to add Einstein-Rosen bridges to manifolds satisfying \(R = 0\), assuming in addition a local parity condition. The deformation of the metric related to the addition of the bridge preserves the condition \(R = 0\) and modifies the metric only in an arbitrarily small neighbourhood of the point at which the bridge is added. This allows one to connect pairs of time-symmetric vacuum initial data without perturbing the metric away from a small neighbourhood of the bridges, or to create wormholes within a given initial data set.

Reviewer: Serguey M.Pokas (Odessa)

### MSC:

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

83C30 | Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory |