arXiv subjects

Dionysios Kokkinos

Publications and source records attributed to Dionysios Kokkinos.

3 recordsLinked to original sources

arXiv2026

Killing tensors of Weyl's class

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.

gr-qc↗

arXiv2025

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

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.

gr-qc↗

arXiv2024

The Study of the Canonical forms of Killing tensor in vacuum with Λ

This paper is the initial part of a comprehensive study of spacetimes that admit the canonical forms of Killing tensor in General Relativity. The general scope of the study is to derive either new exact solutions of Einstein's equations that exhibit hidden symmetries or to identify the hidden symmetries in already known spacetimes that may emerge during the resolution process. In this preliminary paper, we first introduce the canonical forms of Killing tensor, based on a geometrical approach to classify the canonical forms of symmetric 2-rank tensors, as postulated by R. V. Churchill. Subsequently, the derived integrability conditions of the canonical forms serve as additional equations transforming the under-determined system of equations, comprising of Einstein's Field Equations and the Bianchi Identities (in vacuum with Λ), into an over-determined one. Using a null rotation around the null tetrad frame we manage to simplify the system of equations to the point where the geometric characterization (Petrov Classification) of the extracted solutions can be performed and their null congruences can be characterized geometrically. Therein, we obtain multiple special algebraic solutions according to the Petrov classification (D, III, N, O) where some of them appeared to be new. The latter becomes possible since our analysis is embodied with the usage of the Newman-Penrose formalism of null tetrads.

gr-qc↗