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Taub-NUT as Bertrand spacetime with magnetic fields. (English) Zbl 1378.37097

This paper further elucidates the enigmatic Taub-NUT metric, a metric once described as “a counterexample to almost anything” (C. W. Misner, 1967). In particular, it studies Euclidean Taub-NUT metric and relates it, through its symmetries, to a the Euclidean Bertrand metric with magnetic monopoles and dipoles, and, in the process, it allows the understanding of both.
The paper’s outline is as follows. After a brief introduction, conserved quantities are considered using both a dynamical systems description of the Taub-NUT metric and a Holten algorithm description. Next Bertrand spacetimes with magnetic fields are described together with their Kepler-oscillator duality. The Taub-NUT metric is then derived from the self-dual Bianchi type-IX metric described by the Darboux-Halphen system. A discussion follows of curvature, anti-self-duality, and topological invariants. The Taub-NUT metric is interpreted as a gravitational instanton, allowing its study as an integrable system. An easy derivation and comparison between the spatial Killing-Yano tensors deduced from first integrals and the corresponding hyper-Kähler structure is given together with a verification of the existence of a graded Lie algebra structure via Schouten-Nijenhuis brackets.
Researchers in both general relativity and string theory will find this article enlightening.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
83C50 Electromagnetic fields in general relativity and gravitational theory
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
81R12 Groups and algebras in quantum theory and relations with integrable systems
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