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Charged black rings from inverse scattering

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Abstract

The inverse scattering method of Belinsky and Zakharov is a powerful method to construct solutions of vacuum Einstein equations. In particular, in five dimensions this method has been successfully applied to construct a large variety of black hole solutions. Recent applications of this method to Einstein–Maxwell-dilaton (EMd) theory, for the special case of Kaluza–Klein dilaton coupling, has led to the construction of the most general black ring in this theory. In this contribution, we review the inverse scattering method and its application to the EMd theory. We illustrate the efficiency of these methods with a detailed construction of an electrically charged black ring.

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Notes

  1. In Ref. [30] a further reduction to 2D was considered, and the fruitful intertwining of this solution-generating technique with the ISM was studied. The ISM from the point of view of two dimensional symmetries has also been recently investigated in [33]. Hidden symmetries have also been employed in [34, 35] to obtain charged black rings in other theories, following the pioneering work [36].

  2. As we explain in Sect. 3 the conformal factor is completely specified by the Killing metric. Hence the information contained in the rod structure is enough to fully reconstruct the line element.

  3. The operation of removing a (anti-)soliton is the inverse of adding a (anti-)soliton. For a diagonal seed it is easy to show that removing a (anti-)soliton \(\widetilde{\mu }_i\) with a trivial BZ vector of the form \((m_0)_a=\delta _{ab}\) simply results in multiplying \((G_0)_{bb}\) by \(-\widetilde{\mu }_i^2/\rho ^2\), leaving the remaining components invariant.

  4. However, note that to correctly generate the dipole ring from the seed of Fig. 3 one must remove and add anti-solitons at \(z=a_0\) and \(z=a_2\), whereas for the construction we present here we will apply a solitonic transformation at \(z=a_0\) and \(z=a_4\).

  5. Generically, after such solitonic transformations one is not guaranteed to obtain a final metric with standard orientation, i.e., with the semi-infinite rods coinciding with the \(\phi \) and \(\psi \) directions. This can be remedied by making a coordinate transformation, \(\mathbf G \rightarrow \tilde{\mathbf{G }}=\mathbf S ^T \mathbf G \, \mathbf S \) with \(\mathbf S \in SL(4,\mathbb R )\). However, the construction discussed in this paper requires no coordinate mixing and so the matrix \(\mathbf S \) is trivial.

  6. The solution is invariant under an overall shift in \(z\) so we subtract 1 from the total number of \(a_i\)’s.

  7. We adopt the standard definition for an electric charge in \(D\) dimensions, normalizing by the area of the unit \((D-2)\)-sphere. For \(D=5\) we get \(Q_e=\frac{1}{2\pi ^2}\int _{S_\infty ^3}*F\).

  8. The explicit transformation of the coordinates and parameters that takes our solution (54–56) to the Kunduri-Lucietti charged ring is given by

    $$\begin{aligned} t&= \hat{t}\,, \quad (\psi ,\phi )=\frac{1+\hat{\nu }(1-k^2)}{\sqrt{1+\hat{\lambda }}}(\hat{\psi },\hat{\phi })\,, \quad x=\frac{\hat{x}-\hat{\lambda }}{1-\hat{\lambda }\hat{x}}\,, \quad y=\frac{\hat{y}-\hat{\lambda }}{1-\hat{\lambda }\hat{y}}\,,\end{aligned}$$
    (57)
    $$\begin{aligned} \lambda&= \hat{\lambda }\,, \quad \nu =\frac{\hat{\lambda }-\hat{\nu }(1-k^2)}{1-\hat{\lambda }\hat{\nu }(1-k^2)}\,, \quad \gamma =\frac{k^2 \hat{\lambda }}{1-k^2}\,, \quad R=\frac{\sqrt{(1+\hat{\lambda })\left( 1-\hat{\lambda }\hat{\nu } (1-k^2)\right) }}{1+\hat{\nu } (1-k^2)}\hat{R}\,, \end{aligned}$$
    (58)

    where hatted quantities correspond to coordinates and parameters employed in Ref. [41]. We note in passing that Eqs. (39) and (40) of Ref. [41] contain small typos.

  9. The family of solutions obtained in Ref. [40] generically have non-vanishing \(S^2\) angular momentum and dipole charge. Setting these quantities to zero corresponds to taking the limit \(\mu \rightarrow 0\) while keeping \(a_1\) finite, in the notation of [40]. The resulting solution is parametrized by three numbers \((k, c, a_3)\) and the comparison with the explicit solution constructed in this paper may be done employing the following relations between parameters:

    $$\begin{aligned} R^2 = \frac{2}{1+c^2}k^2\,, \qquad \gamma = \frac{8 a_3^2 c^3}{(1+c^2)[(1-c^2)^2-4 a_3^2 c^2]}\,, \qquad \nu = c\,. \end{aligned}$$
    (67)

    Special care must be taken when comparing the formulas for the electric charge and potential: the normalization adopted in [40] differs from ours by a multiplicative factor of \(\pi /16\).

References

  1. Horowitz, G.T.: Black Holes in Higher Dimensions. Cambridge University Press, Cambridge (2012)

  2. Casalderrey-Solana, J., Liu, H., Mateos, D., Rajagopal, K., Wiedemann, U.A.: Gauge/String Duality, Hot QCD and Heavy Ion Collisions, arXiv:1101.0618 [hep-th]

  3. Rangamani, M.: Gravity and Hydrodynamics: lectures on the fluid-gravity correspondence. Class. Quant. Grav. 26, 224003 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  4. Hartnoll, S.A.: Lectures on holographic methods for condensed matter physics. Class. Quant. Grav. 26, 224002 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  5. Chrusciel, P.T., Costa, J.L., Heusler, M.: Stationary black holes: uniqueness and beyond. Living Rev. Rel. 15, 7 (2012)

    Google Scholar 

  6. Emparan, R., Reall, H.S.: Black holes in higher dimensions. Living Rev. Rel. 11, 6 (2008)

    Google Scholar 

  7. Rodriguez, M.J.: On the black hole species (By Means of Natural Selection). In: Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, Review Talk 523 (2012)

  8. Emparan, R., Reall, H.S.: A rotating black ring solution in five-dimensions. Phys. Rev. Lett. 88, 101101 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  9. Kerr, R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Tangherlini, F.R.: Schwarzschild field in n dimensions and the dimensionality of space problem. Nuovo Cim. 27, 636 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  11. Myers, R.C., Perry, M.J.: Black holes in higher dimensional space–times. Ann. Phys. 172, 304 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Belinsky, V.A., Zakharov, V.E.: Stationary gravitational solitons with axial symmetry. Sov. Phys. JETP 50, 1–9 (1979)

    ADS  Google Scholar 

  13. Pomeransky, A.A.: Complete integrability of higher-dimensional Einstein equations with additional symmetry, and rotating black holes. Phys. Rev. D 73, 044004 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  14. Tomizawa, S., Morisawa, Y., Yasui, Y.: Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method. Phys. Rev. D 73, 064009 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  15. Tomizawa, S., Nozawa, M.: Vacuum solutions of five-dimensional Einstein equations generated by inverse scattering method. II. Production of black ring solution. Phys. Rev. D 73, 124034 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  16. Pomeransky, A.A., Sen’kov, R.A.: Black ring with two angular momenta, hep-th/0612005 (2006)

  17. Elvang, H., Figueras, P.: Black Saturn. JHEP 0705, 050 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  18. Elvang, H., Rodriguez, M.J.: Bicycling black rings. JHEP 0804, 045 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  19. Herdeiro, C.A.R., Rebelo, C., Zilhão, M., Costa, M.S.: A double Myers–Perry black hole in five dimensions. JHEP 0807, 009 (2008)

    Article  ADS  Google Scholar 

  20. Evslin, J., Krishnan, C.: The black di-ring: an inverse scattering construction. Class. Quant. Grav. 26, 125018 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  21. Emparan, R., Reall, H.S.: Generalized Weyl solutions. Phys. Rev. D 65, 084025 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  22. Alekseev, G.A.: Thirty years of studies of integrable reductions of Einstein’s field equations, arXiv:1011.3846 [gr-qc] (2010)

  23. Belinsky, V.A.: L-A pair of a system of coupled equations of the gravitational and electromagnetic fields. JETP Lett. 30(1), 32–35 (1979)

    ADS  Google Scholar 

  24. Belinsky, V., Ruffini, R.: On axially symmetric soliton solutions of the coupled scalar vector tensor equations in general relativity. Phys. Lett. B 89, 195 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  25. Rocha, J.V., Rodriguez, M.J., Virmani, A.: Inverse scattering construction of a dipole black ring. JHEP 1111, 008 (2011)

    Article  ADS  Google Scholar 

  26. Emparan, R.: Rotating circular strings, and infinite nonuniqueness of black rings. JHEP 0403, 064 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  27. Yazadjiev, S.S.: Completely integrable sector in 5-D Einstein-Maxwell gravity and derivation of the dipole black ring solutions. Phys. Rev. D 73, 104007 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  28. Yazadjiev, S.S.: Solution generating in 5D Einstein–Maxwell-dilaton gravity and derivation of dipole black ring solutions. JHEP 0607, 036 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  29. Bouchareb, A., Clément, G., Chen, C.-M., Gal’tsov, D.V., Scherbluk, N.G., Wolf, T.: G(2) generating technique for minimal D=5 supergravity and black rings. Phys. Rev. D 76, 104032 (2007) [Erratum-ibid. D 78, 029901 (2008)]

    Google Scholar 

  30. Figueras, P., Jamsin, E., Rocha, J.V., Virmani, A.: Integrability of five dimensional minimal supergravity and charged rotating black holes. Class. Quant. Grav. 27, 135011 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  31. Gal’tsov, D.V., Scherbluk, N.G.: Generating technique for U(1)**3 5D supergravity. Phys. Rev. D 78, 064033 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  32. Gal’tsov, D.V., Scherbluk, N.G.: Three-charge doubly rotating black ring. Phys. Rev. D 81, 044028 (2010)

    Article  ADS  Google Scholar 

  33. Katsimpouri, D., Kleinschmidt, A., Virmani, A.: Inverse scattering and the Geroch group. JHEP 1302, 011 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  34. Elvang, H.: A charged rotating black ring. Phys. Rev. D 68, 124016 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  35. Hoskisson, J.: A charged doubly spinning black ring. Phys. Rev. D 79, 104022 (2009)

    Article  ADS  Google Scholar 

  36. Hassan, S.F., Sen, A.: Twisting classical solutions in heterotic string theory. Nucl. Phys. B 375, 103 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  37. Elvang, H., Emparan, R., Figueras, P.: Non-supersymmetric black rings as thermally excited supertubes. JHEP 0502, 031 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  38. Rocha, J.V., Rodriguez, M.J., Varela, O.: An electrically charged doubly spinning dipole black ring. JHEP 1212, 121 (2012)

    Article  ADS  Google Scholar 

  39. Chen, Y., Hong, K., Teo, E.: A doubly rotating black ring with dipole charge. JHEP 1206, 148 (2012)

    Article  ADS  Google Scholar 

  40. Feldman, A.L., Pomeransky, A.A.: Charged black rings in supergravity with a single non-zero gauge field. JHEP 1207, 141 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  41. Kunduri, H.K., Lucietti, J.: Electrically charged dilatonic black rings. Phys. Lett. B 609, 143 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Harmark, T.: Stationary and axisymmetric solutions of higher-dimensional general relativity. Phys. Rev. D 70, 124002 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  43. Iguchi, H., Izumi, K., Mishima, T.: Systematic solution-generation of five-dimensional black holes. Prog. Theor. Phys. Suppl. 189, 93–125 (2011)

    Article  ADS  Google Scholar 

  44. Belinski, V., Verdaguer, E.: Gravitational Solitons. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  45. Izumi, K.: Orthogonal black di-ring solution. Prog. Theor. Phys. 119, 757 (2008)

    Article  ADS  MATH  Google Scholar 

  46. Rocha, J.V., Rodriguez, M.J., Varela, O., Virmani, A.: Inverse scattering construction of dipole black rings. In: Proceedings of the Spanish Relativity Meeting in Portugal, ERE (to appear, 2012)

  47. Tomizawa, S., Yasui, Y., Ishibashi, A.: Uniqueness theorem for charged dipole rings in five-dimensional minimal supergravity. Phys. Rev. D 81, 084037 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  48. Gauntlett, J.P., Myers, R.C., Townsend, P.K.: Black holes of D = 5 supergravity. Class. Quant. Grav. 16, 1 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. Kunz, J., Maison, D., Navarro-Lérida, F., Viebahn, J.: Rotating Einstein–Maxwell-dilaton black holes in D dimensions. Phys. Lett. B 639, 95 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Acknowledgments

We thank James Lucietti for bringing Ref. [41] to our attention, and Axel Kleinschmidt for a comment on the draft. 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 Organisation for Scientific Research (NWO) under the VICI grant 680-47-603. A.V. thanks IUCAA Pune for hospitality where part of this work was done.

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Authors and Affiliations

  1. Centro Multidisciplinar de Astrofísica, Departamento de Física, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 , Lisbon, 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 , 751005, India

    Amitabh Virmani

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This article belongs to the Topical Collection: Progress in Mathematical Relativity with Applications to Astrophysics and Cosmology.

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Rocha, J.V., Rodriguez, M.J., Varela, O. et al. Charged black rings from inverse scattering. Gen Relativ Gravit 45, 2099–2121 (2013). https://doi.org/10.1007/s10714-013-1586-x

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