Anholonomic Frames, Generalized Killing Equations, and Anisotropic Taub NUT Spinning Spaces

TL;DR

This paper develops a framework using anholonomic frames to construct and analyze anisotropic extensions of Taub-NUT spaces within (pseudo) Riemannian geometry, exploring solutions to Einstein's equations and properties of anisotropic spinning particles.

Contribution

It introduces a method to generate anisotropic Taub-NUT solutions using anholonomic frames and analyzes generalized Killing equations for anisotropic spinning spaces.

Findings

Derived conditions for 5D vacuum Einstein equations to admit anisotropic Taub-NUT solutions.

Expressed solutions of generalized Killing equations via anisotropically modified Killing-Yano tensors.

Applied the framework to four-dimensional locally anisotropic Taub-NUT manifolds with Euclidean signature.

Abstract

By using anholonomic frames in (pseudo) Riemannian spaces we define anisotropic extensions of Euclidean Taub-NUT spaces. With respect to coordinate frames such spaces are described by off-diagonal metrics which could be diagonalized by corresponding anholonomic transforms. We define the conditions when the 5D vacuum Einstein equations have as solutions anisotropic Taub-NUT spaces. The generalized Killing equations for the configuration space of anisotropically spinning particles (anisotropic spinning space) are analyzed. Simple solutions of the homogeneous part of these equations are expressed in terms of some anisotropically modified Killing-Yano tensors. The general results are applied to the case of the four-dimensional locally anisotropic Taub-NUT manifold with Euclidean signature. We emphasize that all constructions are for(pseudo) Riemannian spaces defined by vacuum soltions, with generic anisotropy, of 5D Einstein equations, the solutions being generated by applying the moving frame method.