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Inverse Scattering Construction of Dipole Black Rings

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Progress in Mathematical Relativity, Gravitation and Cosmology

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 60))

Abstract

We describe an approach to systematically generate regular and asymptotically flat dipole black rings in a 5D Einstein–Maxwell-dilaton theory obtained from 6D vacuum gravity by Kaluza–Klein reduction. Our construction employs the inverse scattering method in six dimensions. We illustrate the scheme with the explicit construction of the singly-spinning dipole ring. These techniques can also be used to generate more general five-dimensional black ring solutions, displaying rotation along the two orthogonal planes, electric charge and magnetic dipole charge.

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Notes

  1. 1.

    Any solution of the vacuum Einstein equations in D dimensions possessing D − 2 commuting isometries can be written in the form (3), with G 0 and ν 0 only depending on (ρ, z) and with the constraint \(\det G_{0} = {-\rho }^{2}\) [5, 8].

  2. 2.

    The parameter counting is as follows: 4 from the rod endpoints (the solution is invariant under an overall shift in z so we subtract 1), 5 from the BZ parameters c i and b j , 8 from the coordinate mixing matrix S, and 1 from k.

  3. 3.

    This particular case requires no coordinate mixing to display asymptotic flatness.

  4. 4.

    The angular coordinates have periodicity Δ ϕ = Δ ψ = 2π k so this choice does not yield canonically normalized angular variables. Nevertheless, we adopt this convention to more easily compare our results with [4].

  5. 5.

    For general choices of parameters the solution (22), (23) is unbalanced. Avoidance of conical singularities is obtained by imposing a further constraint [4], yielding a final solution that is fully regular, balanced, and depending on three parameters, corresponding to mass, angular momentum and dipole charge.

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Acknowledgements

We thank the organizers and participants of the Spanish Relativity Meeting in Portugal, 3–7 September 2012, where this work was presented. J.V.R. is supported by Fundação para a Ciência e Tecnologia (FCT)-Portugal through contract no. SFRH/BPD/47332/2008. M.J.R. is supported by the European Commission—Marie Curie grant PIOF-GA 2010-275082. O.V. is supported in part by the Netherlands Organization for Scientific Research (NWO) under the VICI grant 680-47-603 and, at the time this work was done, by the Spanish Government research grant FIS2008-01980.

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Authors and Affiliations

  1. Centro Multidisciplinar de Astrofísica, Dept. de Física, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001, Lisboa, Portugal

    Jorge V. Rocha

  2. Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA, 02138, USA

    Maria J. Rodriguez

  3. Institute for Theoretical Physics and Spinoza Institute, Utrecht University, 3508 TD, Utrecht, The Netherlands

    Oscar Varela

  4. Institute of Physics, Sachivalaya Marg, Bhubaneswar, Odisha, India, 751005

    Amitabh Virmani

Authors
  1. Jorge V. Rocha
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  2. Maria J. Rodriguez
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  3. Oscar Varela
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  4. Amitabh Virmani
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Corresponding author

Correspondence to Jorge V. Rocha .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics, University of Minho, Braga, Portugal

    Alfonso García-Parrado

  2. Dept. of Mathematics, University of Minho, Braga, Portugal

    Filipe C. Mena

  3. Dept. of Mathematics, University of Minho, Braga, Portugal

    Filipe Moura

  4. Dept. of Mathematics, University of Minho, Braga, Portugal

    Estelita Vaz

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Rocha, J.V., Rodriguez, M.J., Varela, O., Virmani, A. (2014). Inverse Scattering Construction of Dipole Black Rings. In: García-Parrado, A., Mena, F., Moura, F., Vaz, E. (eds) Progress in Mathematical Relativity, Gravitation and Cosmology. Springer Proceedings in Mathematics & Statistics, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40157-2_8

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