Mars M, Peón-Nieto C
Universidad de Salamanca, Salamanca, Spain.
Charles University of Prague, Czech Republic.
Philos Trans A Math Phys Eng Sci. 2024 Mar 4;382(2267):20230047. doi: 10.1098/rsta.2023.0047. Epub 2024 Jan 15.
The Kerr and Kerr-de Sitter metrics share remarkable local geometric properties in four dimensions. Gibbons found a generalization of the Kerr-de Sitter metric to higher dimensions, to which the local characterization above cannot be applied. One viable approach to characterize this family is to understand the behaviour of these metrics at future null infinity. We review Friedrich's and Fefferman-Graham formalisms to discuss the asymptotic initial value problem of ([Formula: see text])-vacuum spacetimes in arbitrary dimensions and study their properties: geometric identification and conformal equivalence of data, Killing initial data and conformal equivalence of boundary conformal Killing vectors (CKV). These results are used to review a recent characterization of Kerr-de Sitter in terms of its asymptotic data, namely conformal flatness at [Formula: see text] together with a canonical TT tensor constructed from specific CKV at [Formula: see text]. Allowing for arbitrary CKV defines the (larger) Kerr-de Sitter-like class. All these metrics can be obtained explicitly as limits or analytic extensions of Kerr-de Sitter. The Kerr-de Sitter-like class is also characterized by the property of being Kerr-Schild and fulfilling a certain falloff condition. In addition, in five dimensions, this class corresponds to all algebraically special metrics with non-degenerate optical matrix. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.
克尔（Kerr）度规和克尔 - 德西特（Kerr - de Sitter）度规在四维空间中具有显著的局部几何性质。吉本斯（Gibbons）发现了克尔 - 德西特度规到更高维度的一种推广，上述局部特征不适用于此。刻画这个族的一种可行方法是了解这些度规在未来零无穷远处的行为。我们回顾弗里德里希（Friedrich）和费弗曼 - 格雷厄姆（Fefferman - Graham）形式体系，以讨论任意维度下（[公式：见原文]）真空时空的渐近初值问题，并研究它们的性质：数据的几何识别和共形等价、 Killing 初值数据以及边界共形 Killing 向量（CKV）的共形等价。这些结果被用于回顾最近根据其渐近数据对克尔 - 德西特的一种刻画，即在[公式：见原文]处的共形平坦性以及由[公式：见原文]处的特定 CKV 构造的规范 TT 张量。允许任意 CKV 定义了（更大的）类克尔 - 德西特类。所有这些度规都可以明确地作为克尔 - 德西特的极限或解析延拓得到。类克尔 - 德西特类还具有克尔 - 希尔德（Kerr - Schild）性质并满足一定的衰减条件。此外，在五维空间中，这个类对应于所有具有非退化光学矩阵的代数特殊度规。本文是“广义相对论中渐近、共形方法与分析的界面”讨论会议文集的一部分。
