Weyl类的Killing张量

中文摘要 AI 辅助

本工作是构建用于识别代数一般（Petrov I型）时空隐藏对称性的系统框架的初始步骤。受此启发，我们聚焦于Weyl类，该类提供了一类简单的代数一般静态轴对称几何解。利用秩2 Killing张量的标准型，我们系统推导了此类时空容许真实隐藏对称性所需满足的约束条件。我们的分析对Weyl类子类（包括真空和电真空情形）中可积性的阻碍给出了解析表征。特别地，我们发现Weyl类中无额外Killing矢量的成员（无论是真空还是电真空情形），均不具备能提供完全可积性所需额外运动积分的不可约Killing张量。而Weyl类中容许第三个额外Killing矢量等距（即Levi-Civita）的成员，仅通过其显式对称性就已完全可积，仅容许可约Killing张量。该结果为此前的解析和数值研究提供了解析补充，建立了隐藏对称性缺失与完全可积性破坏之间的系统关联。除此处获得的具体分类外，这些结果还证明了我们基于Killing张量标准型的方法，作为探测Petrov I型时空隐藏对称性的系统工具的潜力。

英文摘要

This work represents an initial step towards a systematic framework for identifying hidden symmetries in algebraically general (Petrov type I) spacetimes. Motivated by this, we focus on Weyl's class, which provides a simple yet a class of algebraically general solutions of static, axisymmetric geometries. Employing the canonical forms of rank-2 Killing tensors, we systematically derive the constraints that must be satisfied for these spacetimes to admit genuine hidden symmetries. Our analysis yields an analytical characterization of the obstructions to integrability within subclasses of Weyl's class, both in vacuum and electrovacuum. In particular, we find that no two-Killing-vector member of Weyl's class, in vacuum or electrovacuum, admits an irreducible Killing tensor capable of supplying the additional integral of motion required for complete integrability. Members of Weyl's class admitting a third additional Killing-vector isometry (i.e. Levi-Civita) are already completely integrable through their manifest symmetries alone admitting only reducible Killing tensors. This result provides an analytic complement to previous analytical and numerical investigations, and establishes a systematic connection between the absence of hidden symmetries and the breakdown of complete integrability. Beyond the specific classification obtained here, these results demonstrate the potential of our approach regarding the canonical forms of Killing tensor as a systematic tool for probing hidden symmetries in Petrov type I spacetimes.

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25 pages