Abstract
In this paper we consider axisymmetric black holes in supergravity and address the general issue of defining a first order description for them. The natural setting where to formulate the problem is the De Donder-Weyl-Hamilton-Jacobi theory associated with the effective two-dimensional sigma-model action describing the axisymmetric solutions. We write the general form of the two functions S m defining the first-order equations for the fields. It is invariant under the global symmetry group G (3) of the sigma-model. We also discuss the general properties of the solutions with respect to these global symmetries, showing that they can be encoded in two constant matrices belonging to the Lie algebra of G (3), one being the Nöther matrix of the sigma model, while the other is non-zero only for rotating solutions. These two matrices allow a G (3)-invariant characterization of the rotational properties of the solution and of the extremality condition. We also comment on extremal, under-rotating solutions from this point of view.
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Andrianopoli, L., D’Auria, R., Giaccone, P. et al. Rotating black holes, global symmetry and first order formalism. J. High Energ. Phys. 2012, 78 (2012). https://doi.org/10.1007/JHEP12(2012)078
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DOI: https://doi.org/10.1007/JHEP12(2012)078