On the Construction of a Geometric Invariant Measuring the Deviation from Kerr Data
On the Construction of a Geometric Invariant Measuring the Deviation from Kerr Data
复制标题
测量克尔数据偏差的几何不变量的构造
DOI：
10.1007/s00023-010-0063-2
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
J. A. Valiente Kroon
中科院分区：
文献类型：
--
作者：
Thomas Bäckdahl;J. A. Valiente Kroon
本文详细而严格地证明了爱因斯坦真空场方程的初始数据集的几何不变量的构造。当且仅当初始数据集对应于克尔时空的数据时，这个几何不变量才消失，因此，它表征了这种类型的数据。所提出的结构是有效的，提升和非提升的初始数据集，在某种意义上，渐近Schwarzschildean。作为一个初步的步骤，以建设的几何不变，分析的克尔时空的特征方面的杀戮旋量进行。空间旋量分裂的（时空）Killing旋量方程进行，以获得一组三个条件，确保存在的初始数据集的发展的Killing旋量。为了构造几何不变量，我们引入了近似Killing旋量的概念。这些旋量是初始超曲面所固有的对称价2旋量，并且满足一定的二阶椭圆方程--近似Killing旋量方程。该方程作为非负积分泛函的欧拉-拉格朗日方程出现。这个泛函构成了我们的几何不变量的一部分，然而，整个泛函并不是来自变分原理。研究了Killing旋量方程解的渐近性态，给出了一个解的存在性定理。
This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation—the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant—however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.