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R Soc Open Sci. 2018 May 9;5(5):170826. doi: 10.1098/rsos.170826. eCollection 2018 May.

Comparison of vacuum static quadrupolar metrics.

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School of Physics and Space Research Center, University of Costa Rica, San José, Costa Rica.
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, AP 70543, Ciudad de México 04510, Mexico.
Dipartimento di Fisica and ICRA, Università di Roma 'La Sapienza', 00185 Roma, Italy.
Institute of Experimental and Theoretical Physics, Al-Farabi Kazakh National University, Almaty, Kazakhstan.


We investigate the properties of static and axisymmetric vacuum solutions of Einstein equations which generalize the Schwarzschild spherically symmetric solution to include a quadrupole parameter. We test all the solutions with respect to elementary and asymptotic flatness and curvature regularity. Analysing their multipole structure, according to the relativistic invariant Geroch definition, we show that all of them are equivalent up to the level of the quadrupole. We conclude that the q-metric, a variant of the Zipoy-Voorhees metric, is the simplest generalization of the Schwarzschild metric, containing a quadrupole parameter.


naked singularities; quadrupole moment; vacuum metrics

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