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On the Construction of a Geometric Invariant Measuring the Deviation from Kerr Data

机译：关于测量Kerr数据偏差的几何不变量的构造

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摘要

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.
机译：本文包含有关爱因斯坦真空场方程初始数据集的几何不变量构造的详细而严格的证明。当且仅当初始数据集对应于Kerr时空的数据时，此几何不变性才消失，因此，它表征了这种类型的数据。在某种意义上，渐近式Schwarzschildean表示的构造对于增强和非增强的初始数据集有效。作为构造几何不变量的初步步骤，我们根据Killing旋子对Kerr时空的特征进行了分析。执行（时空）Killing Spinor方程的空间Spinor拆分以获得一组三个条件，以确保存在初始数据集发展的Killing Spinor。为了构造几何不变式，我们引入了近似Killing旋子的概念。这些自旋是初始超曲面固有的对称价2自旋，并且满足某个二阶椭圆方程-近似的Killing自旋方程。此方程式为非负积分函数的Euler-Lagrange方程式。该函数构成了我们的几何不变式的一部分，但是，整个函数并非来自变分原理。研究了近似Killing spinor方程解的渐近行为，并给出了一个存在性定理。

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