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Maciejewski, Andrzej J.; Przybylska, Maria; Valent, Galliano

The geodesic flow of the BGPP metric is Liouville integrable. (English) Zbl 1527.83039

Classical Quantum Gravity 40, No. 15, Article ID 155007, 13 p. (2023).
Summary: We prove that the geodesics equations corresponding to the BGPP metric are integrable in the Liouville sense. The \(\mathrm{SO}(3, \mathbb{R})\) symmetry of the model allows to reduce the system from four to two degrees of freedom. Moreover, solutions of the reduced system and its degenerations can be given explicitly or reduced to a certain quadrature. In degenerated cases BGPP metric coincides with the Eguchi-Hanson metric and for this case the mentioned quadrature can be calculated explicitly in terms of elliptic integrals.

MSC:

83C40 Gravitational energy and conservation laws; groups of motions
53C22 Geodesics in global differential geometry
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

BGPP matric; integrability; gravitational instantons
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References:

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