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On the symmetry orbits of black holes in non-linear sigma models · Around

2011

On the symmetry orbits of black holes in non-linear sigma models

Hörnlund, Josef Lindman

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Breitenlohner, Maison and Gibbons claimed some time ago that all bona-fide four dimensional asymptotically flat non-degenerate black holes are in a symmetry orbit of the Schwarzschild/Kerr black hole in a large set of theories of gravity and matter.

Their argument involved reducing the theory on a time-like Killing vector field and analysing the resulting three dimensional sigma model of maps to a symmetric space $G/H$.
In the construction of their proof, they conjectured the existence of a suitable $H$-transformation that always remove the electromagnetic charges of the four dimensional black hole solution.
We show in this short note that such a transformation does not exist in general, and discuss a set of boundary conditions on the horizon for the scalar fields in the sigma model that yield black holes for which the result by Breitenlohner, Maison and Gibbons can be applied.

Built on

M. Berger, “Les espaces symétriques noncompacts,” Ann. Sci. École Norm. Sup
1957
Earlier work this paper cites.
PhD thesis, University of Hamburg, 1957
J. Ehlers, Konstruktionen und Charakterisierung von Lösungen der Einsteinschen Gravitationsfeldgleichungen · 1957
Earlier work this paper cites.
R. Geroch, “A method for generating solutions of Einstein’s equations,” J. Math. Phys
1971
Earlier work this paper cites.
T. Matsuki, “The orbits of affine symmetric spaces under the action of minimal parabolic subgroups,” J. Math. Soc. Japan
1979
Earlier work this paper cites.

Similar

W. Simon, “A simple proof of the generalized electrostatic israel theorem,” General Relativity and Gravitation
1985
Cited alongside, same era.
Springer Berlin / Heidelberg, 1985
D. Maison, “Stationary solutions of the einstein-maxwell equations,” in Non-Linear Equations in Classical and Quantum Field Theory · 1985
Cited alongside, same era.
P. Breitenlohner, D. Maison, and G. W. Gibbons, “Four-Dimensional Black Holes from Kaluza-Klein Theories,” Commun. Math. Phys
1988
Cited alongside, same era.

Then

D. Maison, “Duality and hidden symmetries in gravitational theories,” Lect. Notes Phys
2000
Later among the works it cites.
G. Bossard, H. Nicolai, and K. S. Stelle, “Universal BPS structure of stationary supergravity solutions,” JHEP
Original
2009
Later among the works it cites.
J. Lindman Hörnlund and A. Virmani, “Extremal limits of the Cvetic-Youm black hole and nilpotent orbits of G 2 ( 2 ) G_{2(2)} ,” JHEP
Original
2010
Later among the works it cites.

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