Abstract
We present the derivation, for these vacuum metrics, of the Painlevé VI equation first obtained by Christodoulakis and Terzis, from the field equations for both minkowskian and euclidean signatures. This allows to give the precise connection with some old results due to Kinnersley. The hyperkähler metrics are shown to belong to the Multi-Centre class and for the cases exhibiting an integrable geodesic flow the relevant Killing tensors are given. We conclude by the proof that for the Bianchi B family, excluding type III, there are no hyperkähler metrics.
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Åman, J.E.: In: Bonnor, W.B., Islam, J.N., MacCallum, M.A.H. (eds.) Classical General Relativity (Proceedings of the 1983 London Conference), pp. 1–4, Cambridge University Press, Cambridge (1983)
Cahen M., Defrise L.: Commun. Math. Phys. 11, 56 (1968)
Christodoulakis T., Terzis P.A.: J. Math. Phys. 47, 102502 (2006)
Christodoulakis T., Terzis P.A.: Class. Quant. Grav. 24, 875 (2007)
Collinson C.D.: Int. J. Theor. Phys. 15, 311 (1976)
Cosgrove C.M., Scoufis G.: Stud. Appl. Math. 88, 25 (1993)
Duval C., Valent G.: J. Math. Phys. 46, 053516 (2005)
Geroch R.: J. Math. Phys. 12, 918 (1971)
Gibbons G.W., Ruback P.J.: Commun. Math. Phys. 115, 267 (1988)
Gromak V.I., Laine I., Shimomura S.: Painlevé Differential Equations in the Complex Plane. Walter de Gruyter, Berlin, New York, NY (2002)
Keane A.J., Tupper B.O.: Class. Quant. Grav. 27, 245011 (2010)
Kinnersley W.: J. Math. Phys. 10, 1195 (1969)
Lorenz-Petzold D.: Acta Phys. Polon. B 15, 117 (1984)
MacCallum M.A.H., Moussiaux A., Tombal P., Demaret J.: J. Phys. A Math. Gen. 15, 1757 (1982)
Moussiaux A., Tombal P., Demaret J.: J. Phys. A Math. Gen. 14, L277 (1981)
Okamoto K.: Ann. Mat. Pura Appl. 146, 337 (1987)
Perelomov A.M.: Integrable Systems of Classical Mechanics and Lie Algebras. Birkhauser Verlag, Basel, Boston, Berlin (1990)
Siklos S.T.C.: J. Phys. A Math. Gen. 14, 395 (1981)
Stewart J.M., Ellis G.F.R.: J. Math. Phys. 9, 1072 (1968)
Terzis P.A., Christodoulakis T.: Gen. Relativ. Gravit. 41, 469 (2009)
Terzis, P.A., Christodoulakis, T.: arXiv:gr-qc/1007.1561
Valent G.: Commun. Math. Phys. 244, 571 (2004)
Valent G., Ben Yahia H.: Class. Quantum Grav. 24, 255 (2007)
Walker M., Penrose R.: Commun. Math. Phys. 18, 265 (1970)
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Valent, G. Further results on non-diagonal: Bianchi type III vacuum metrics. Gen Relativ Gravit 44, 1103–1127 (2012). https://doi.org/10.1007/s10714-012-1340-9
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DOI: https://doi.org/10.1007/s10714-012-1340-9