Abstract
On one hand the Geroch group allows one to associate spacetime independent matrices with gravitational configurations that effectively only depend on two coordinates. This class includes stationary axisymmetric four- and five-dimensional black holes. On the other hand, a recently developed inverse scattering method allows one to factorize these matrices to explicitly construct the corresponding spacetime configurations. In this work we demonstrate the construction as well as the factorization of Geroch group matrices for a wide class of black hole examples. In particular, we obtain the Geroch group SL(3, ℝ) matrices for the five-dimensional Myers-Perry and Kaluza-Klein black holes and the Geroch group SU(2, 1) matrix for the four-dimensional Kerr-Newman black hole. We also present certain non-trivial relations between the Geroch group matrices and charge matrices for these black holes.
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References
B. Julia, Application of supergravity to graviational theories, in Unified field theories of more than four dimensions, V. De Sabbata and E. Schmutze eds., World Scientific, Singapore (1983).
P. Breitenlohner, D. Maison and G.W. Gibbons, Four-dimensional black holes from Kaluza-Klein theories, Commun. Math. Phys. 120 (1988) 295 [INSPIRE].
D. Youm, Black holes and solitons in string theory, Phys. Rept. 316 (1999) 1 [hep-th/9710046] [INSPIRE].
D.D.K. Chow and G. Compère, Seed for general rotating non-extremal black holes of \( \mathcal{N} \) =8 supergravity, Class. Quant. Grav. 31 (2014) 022001 [arXiv:1310.1925] [INSPIRE].
D.D.K. Chow and G. Compère, Black holes in N =8 supergravity from SO(4, 4) hidden symmetries, Phys. Rev. D 90 (2014) 025029 [arXiv:1404.2602] [INSPIRE].
S. Tomizawa, Y. Yasui and A. Ishibashi, Uniqueness theorem for charged rotating black holes in five-dimensional minimal supergravity, Phys. Rev. D 79 (2009) 124023 [arXiv:0901.4724] [INSPIRE].
S. Hollands, Black hole uniqueness theorems and new thermodynamic identities in eleven dimensional supergravity, Class. Quant. Grav. 29 (2012) 205009 [arXiv:1204.3421] [INSPIRE].
G. Bossard, H. Nicolai and K.S. Stelle, Universal BPS structure of stationary supergravity solutions, JHEP 07 (2009) 003 [arXiv:0902.4438] [INSPIRE].
G. Bossard, Y. Michel and B. Pioline, Extremal black holes, nilpotent orbits and the true fake superpotential, JHEP 01 (2010) 038 [arXiv:0908.1742] [INSPIRE].
G. Bossard and C. Ruef, Interacting non-BPS black holes, Gen. Rel. Grav. 44 (2012) 21 [arXiv:1106.5806] [INSPIRE].
G. Bossard, Octonionic black holes, JHEP 05 (2012) 113 [arXiv:1203.0530] [INSPIRE].
G. Bossard and S. Katmadas, A bubbling bolt, JHEP 07 (2014) 118 [arXiv:1405.4325] [INSPIRE].
R.P. Geroch, A Method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (1971) 918 [INSPIRE].
R.P. Geroch, A method for generating new solutions of Einstein’s equation. 2, J. Math. Phys. 13 (1972) 394 [INSPIRE].
B. Julia, Infinite Lie algebras in physics, in the proceedings of Unified Field Theories and Beyond, Baltimore, U.S.A. (1981).
B. Julia, Group disintegrations, Conf. Proc. C 8006162 (1980) 331.
P. Breitenlohner and D. Maison, On the Geroch group, Annales Poincaré Phys. Theor. 46 (1987) 215 [INSPIRE].
P. Breitenlohner and D. Maison, Solitons in Kaluza-Klein theories, unpublished notes (June 1986).
H. Nicolai, Two-dimensional gravities and supergravities as integrable system, in the proceeding of Recent aspects of quantum fields, Schladming, Austria (1991), DESY-91-038 (1991).
I. Bakas, O(2, 2) transformations and the string Geroch group, Nucl. Phys. B 428 (1994) 374 [hep-th/9402016] [INSPIRE].
A. Sen, Duality symmetry group of two-dimensional heterotic string theory, Nucl. Phys. B 447 (1995) 62 [hep-th/9503057] [INSPIRE].
I. Bakas, Solitons of axion-dilaton gravity, Phys. Rev. D 54 (1996) 6424 [hep-th/9605043] [INSPIRE].
A.K. Das, J. Maharana and A. Melikyan, Duality, monodromy and integrability of two-dimensional string effective action, Phys. Rev. D 65 (2002) 126001 [hep-th/0203144] [INSPIRE].
V.A. Belinsky and V.E. Zakharov, Integration of the Einstein equations by the inverse scattering problem technique and the calculation of the exact soliton solutions, Sov. Phys. JETP 48 (1978) 985 [Zh. Eksp. Teor. Fiz. 75 (1978) 1953] [INSPIRE].
V.A. Belinsky and V.E. Sakharov, Stationary gravitational solitons with axial symmetry, Sov. Phys. JETP 50 (1979) 1 [Zh. Eksp. Teor. Fiz. 77 (1979) 3] [INSPIRE].
V. Belinski and E. Verdaguer, Gravitational solitons, Cambridge University Press, Cambridge U.K. (2001).
D. Rasheed, The Rotating dyonic black holes of Kaluza-Klein theory, Nucl. Phys. B 454 (1995) 379 [hep-th/9505038] [INSPIRE].
F. Larsen, Rotating Kaluza-Klein black holes, Nucl. Phys. B 575 (2000) 211 [hep-th/9909102] [INSPIRE].
R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [INSPIRE].
D. Katsimpouri, A. Kleinschmidt and A. Virmani, Inverse scattering and the Geroch group, JHEP 02 (2013) 011 [arXiv:1211.3044] [INSPIRE].
D. Katsimpouri, A. Kleinschmidt and A. Virmani, An inverse scattering formalism for STU supergravity, JHEP 03 (2014) 101 [arXiv:1311.7018] [INSPIRE].
C.N. Pope, Lectures on Kaluza-Klein theory, http://faculty.physics.tamu.edu/pope/.
S. Hollands and S. Yazadjiev, Uniqueness theorem for 5-dimensional black holes with two axial Killing fields, Commun. Math. Phys. 283 (2008) 749 [arXiv:0707.2775] [INSPIRE].
S. Giusto and A. Saxena, Stationary axisymmetric solutions of five dimensional gravity, Class. Quant. Grav. 24 (2007) 4269 [arXiv:0705.4484] [INSPIRE].
R. Emparan and A. Maccarrone, Statistical description of rotating Kaluza-Klein black holes, Phys. Rev. D 75 (2007) 084006 [hep-th/0701150] [INSPIRE].
T. Harmark, Stationary and axisymmetric solutions of higher-dimensional general relativity, Phys. Rev. D 70 (2004) 124002 [hep-th/0408141] [INSPIRE].
L. Andrianopoli, A. Gallerati and M. Trigiante, On extremal limits and duality orbits of stationary black holes, JHEP 01 (2014) 053 [arXiv:1310.7886] [INSPIRE].
W. Kinnersley, Generation of stationary Einstein-Maxwell fields, J. Math. Phys. 14 (1973) 651.
J.L. Hornlund and A. Virmani, Extremal limits of the Cvetič-Youm black hole and nilpotent orbits of G 2(2), JHEP 11 (2010) 062 [Erratum ibid. 1205 (2012) 038] [arXiv:1008.3329] [INSPIRE].
M. Cvetič and D. Youm, General rotating five-dimensional black holes of toroidally compactified heterotic string, Nucl. Phys. B 476 (1996) 118 [hep-th/9603100] [INSPIRE].
M. Cvetič and D. Youm, Entropy of nonextreme charged rotating black holes in string theory, Phys. Rev. D 54 (1996) 2612 [hep-th/9603147] [INSPIRE].
P. Figueras, E. Jamsin, J.V. Rocha and A. Virmani, Integrability of five dimensional minimal supergravity and charged rotating black holes, Class. Quant. Grav. 27 (2010) 135011 [arXiv:0912.3199] [INSPIRE].
R.G. Leigh, A.C. Petkou, P.M. Petropoulos and P.K. Tripathy, The Geroch group in Einstein spaces, arXiv:1403.6511 [INSPIRE].
L. Houart, A. Kleinschmidt, J. Lindman Hornlund, D. Persson and N. Tabti, Finite and infinite-dimensional symmetries of pure N =2 supergravity in D =4, JHEP 08 (2009) 098 [arXiv:0905.4651] [INSPIRE].
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Chakrabarty, B., Virmani, A. Geroch group description of black holes. J. High Energ. Phys. 2014, 68 (2014). https://doi.org/10.1007/JHEP11(2014)068
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DOI: https://doi.org/10.1007/JHEP11(2014)068