标题： 基于正则Killing张量形式的爱因斯坦方程代数一般解的获取指令 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，发现了一类典型的Killing张量形式的存在性，并应用广义的零格拉斯曼变换得出了一系列真空时空（具有宇宙学常数）中的解（Petrov类型D、III、N）。其中，一个独特的Petrov类型D族被识别出来，该族的特点是由二维常曲率空间的拓扑积以及一个无剪切、非发散、非测地的零 congruence 组成，并允许第二种典型形式。这一族在此被完全详细地推导和呈现。此外，还引入了一个具有曲率奇异性且描述非定态圆柱对称时空的代数一般解，此解同样表现出无剪切、非发散、非测地的零 congruence。此解源于相同的典型形式，通过在对称零格拉斯曼框架内应用洛伦兹变换而发展出来。基于此，我们提出一个新的方向：假设Killing张量的典型形式并在对称零格拉斯曼概念中实施洛伦兹变换，可以衍生出爱因斯坦方程更广泛的代数一般解类。 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.