General Relativity and Gravitation

, Volume 42, Issue 3, pp 509–537 | Cite as

Multi-black holes from nilpotent Lie algebra orbits

  • Guillaume Bossard
  • Hermann Nicolai
Open Access
Research Article


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.


Supergravity Black holes Lie algebra 



We are grateful to Boris Pioline and Kelly Stelle for discussions and comments.

Open Access

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.AEI, Max-Planck-Institut für GravitationsphysikPotsdamGermany

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