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

Abstract

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 with a cosmological constant. Among those, a distinct Petrov type D family was extracted and characterized by a topological product of two-dimensional constant-curvature spaces admitting the canonical form. This is a general family of spacetimes with constant curvature and it is derived and presented here in full detail. In addition, an algebraically general solution exhibiting the exact same non-zero spin coefficients is introduced. Beyond this, we introduce an algebraically general solution, obtained by imposing the same canonical Killing tensor form and applying a Lorentz transformation within the anti-symmetric null tetrad transformation. The resulting geometry describes a non-stationary, cylindrically symmetric spacetime in vacuum with cosmological constant. On this basis, we propose a new directive: by assuming the canonical forms of Killing tensors and implementing Lorentz transformations within the anti-symmetric null tetrad concept, a broader class of algebraically general solutions of Einstein's equations can be derived.