标题： 基于规范 Killing 张量形式获得爱因斯坦方程代数一般解的指令 显示英文标题

标题： A Directive for Obtaining Algebraically General Solutions of Einstein Equations Based on the Canonical Killing Tensor Forms

摘要： 这项工作延续了早期的研究，在这些研究中，规范Killing张量形式的存在以及一般零四维变换的应用导致了在具有宇宙常数的真空时空中的多种解（Petrov类型D、III、N）。其中，识别出一个独特的Petrov类型D族，其特征是二维常曲率空间的拓扑乘积和一个无剪切、不发散、非测地的零共形，该共形接受第二规范形式。在此详尽地推导并介绍了这一族。此外，还引入了一个代数一般的解，具有曲率奇点，描述了一个非稳态的圆柱对称时空，同时表现出无剪切、不发散和非测地的零共形。该解源自相同的规范形式，通过在对称零四维框架内应用洛伦兹变换而得出。基于此，我们提出一个新的指导原则：通过假设Killing张量的规范形式，并在对称零四维概念内实施洛伦兹变换，可以推导出爱因斯坦方程的更广泛的代数一般解。 显示英文摘要

摘要： This work follows earlier investigations in which the existence of canonical Killing tensor forms and the application of general null tetrad transformations led to a variety of solutions (Petrov types D, III, N) in vacuum spacetimes with a cosmological constant. Among those, a distinct Petrov type D family was identified, characterized by a topological product of two-dimensional constant-curvature spaces and a shear-free, non-diverging, non-geodesic null congruence admitting the second canonical form. This family is derived and presented here in full detail. In addition, an algebraically general solution with a curvature singularity, describing a non-stationary cylindrically symmetric spacetime, also exhibiting a shear-free, non-diverging, and non-geodesic null congruence, is introduced. This solution arises from the same canonical form, developed through the application of Lorentz transformations within the framework of symmetric null tetrads. On this basis, we propose a new directive: by assuming the canonical forms of Killing tensors and implementing Lorentz transformations within the symmetric null tetrad concept, a broader class of algebraically general solutions of Einstein's equations can be derived.