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

TL;DR

This work proposes a directive for deriving algebraically general solutions to Einstein's equations by enforcing canonical Killing-tensor forms and applying anti-symmetric null tetrad transformations that preserve the Killing-tensor structure. It yields two principal outcomes from the same K^2_{} framework with λ7=0: (i) a Petrov type D, constant-curvature family in vacuum with Λ≠0, and (ii) a Petrov type I, non-stationary, cylindrically symmetric vacuum solution that can exist for both Λ>0 and Λ<0. The type D solution exhibits a 2-space product geometry with constant curvature; geodesics admit separability (Carter constant) and the Weyl scalars satisfy Ψ0Ψ4=9Ψ2^2 with Ψ2=Λ, while the type I solution represents a more general, time-dependent spacetime that can be interpreted as a spinning cosmic-string–like geometry in certain static limits. The analysis highlights how the choice of null-tetrad transformation affects algebraic type and solvability, offering a methodological pathway to generate and classify hidden-symmetry spacetimes across cosmological constant regimes with potential connections to accelerating black-hole-like configurations in extended theories.

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.