Abstract
For \({\mathcal{N}\ge 2}\) supergravities, BPS black hole solutions preserving four supersymmetries can be superposed linearly, leading to well defined solutions containing an arbitrary number of such BPS black holes at arbitrary positions. Being stationary, these solutions can be understood via associated non-linear sigma models over pseudo-Riemannian spaces coupled to Euclidean gravity in three spatial dimensions. As the main result of this paper, we show that whenever this pseudo-Riemannian space is an irreducible symmetric space \({\mathfrak{G}/\mathfrak{H}^*}\), the most general solutions of this type can be entirely characterised and derived from the nilpotent orbits of the associated Lie algebra \({\mathfrak{g}}\). This technique also permits the explicit computation of non-supersymmetric extremal solutions which cannot be obtained by truncation to \({\mathcal{N}=2}\) supergravity theories. For maximal supergravity, we not only recover the known BPS solutions depending on 32 independent harmonic functions, but in addition find a set of non-BPS solutions depending on 29 harmonic functions. While the BPS solutions can be understood within the appropriate \({\mathcal{N}=2}\) truncation of \({\mathcal{N}=8}\) supergravity, the general non-BPS solutions require the whole field content of the theory.
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References
Papapetrou A.: A static solution of the equations of the gravitational field for an arbitrary charge distribution. Proc. R. Ir. Acad. A 51, 191 (1945)
Majumdar S.D.: A class of exact solutions of Einstein’s field equations. Phys. Rec. 72, 390 (1945)
Bates, B., Denef, F.: Exact solutions for supersymmetric stationary black hole composites [hep-th/0304094]
Ferrara S., Kallosh R., Strominger A.: \({\mathcal{N}=2}\) extremal black holes. Phys. Rev. D 52, 5412 (1995) [hep-th/9508072]
Ferrara S., Kallosh R.: Supersymmetry and attractors. Phys. Rev. D 54, 1514 (1996) [hep-th/9602136]
Breitenlohner P., Maison D., Gibbons G.W.: Four-dimensional black holes from Kaluza–Klein theories. Commun. Math. Phys. 120, 295 (1988)
Breitenlohner P., Maison D.: On nonlinear sigma-models arising in (super-)gravity. Commun. Math. Phys. 209, 785 (2000) [gr-qc/9806002]
Bossard, G., Nicolai, H., Stelle, K.S.: Universal BPS structure of stationary supergravity solutions, [hep-th/0902.4438]
Clement G., Galtsov D.V.: Stationary BPS solutions to dilaton-axion gravity. Phys. Rev. D 54, 6136 (1996) [hep-th/9607043]
Gunaydin M., Neitzke A., Pioline B., Waldron A.: BPS black holes, quantum attractor flows and automorphic forms. Phys. Rev. D 73, 084019 (2006) [hep-th/0512296]
Gaiotto D., Li W.W., Padi M.: Non-supersymmetric attractor flow in symmetric spaces. JHEP 0712, 093 (2007) [hep-th/0710.1638]
Cremmer E., Julia B.: The SO(8) supergravity. Nucl. Phys. B 159, 141 (1979)
Bossard, G.: The extremal black holes of \({\mathcal{N}=4}\) supergravity from \({\mathfrak{so}(8,2+n)}\) nilpotent orbits Gen. Relativ. Gravit. (in press) [hep-th/0906.1988]
Bellucci, S., Ferrara, S., Gunaydin, M., Marrani, A.: SAM lectures on extremal black holes in d = 4 extended supergravity, [hep-th/0905.3739]
Hotta K., Kubota T.: Exact solutions and the attractor mechanism in non-BPS black holes. Prog. Theor. Phys. 118, 969 (2007) [hep-th/0707.4554]
Gimon E.G., Larsen F., Simon J.: Black holes in supergravity: the non-BPS branch. JHEP 0801, 040 (2008) [hep-th/0710.4967]
Breitenlohner, P., Maison, D.: Solitons in Kaluza–Klein theories. In: Morris, H., Dodd, R. (eds.) Solitons in General Relativity (1986)
Stephani H., Kramer D., MacCallum M.A., Hoenselars C., Herlt E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2003)
Collingwood D.H., McGovern W.M.: Nilpotent Orbits in Semisimple Lie Algebra. Van Nostrand Reinhold Mathematics Series, New York (1993)
Günaydin M., Sierra G., Townsend P.K.: Exceptional supergravity theories and the magic square. Phys. Lett. B 133, 72 (1983)
Đoković D.Ž.: The closure diagram for nilpotent orbits of the split real form of E 8. CEJM 4, 573 (2003)
Ferrara S., Gimon E.G., Kallosh R.: Magic supergravities, \({\mathcal{N} = 8}\) and black hole composites. Phys. Rev. D 74, 125018 (2006) [hep-th/0606211]
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We are grateful to Boris Pioline and Kelly Stelle for discussions and comments.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bossard, G., Nicolai, H. Multi-black holes from nilpotent Lie algebra orbits. Gen Relativ Gravit 42, 509–537 (2010). https://doi.org/10.1007/s10714-009-0870-2
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DOI: https://doi.org/10.1007/s10714-009-0870-2