Abstract
We consider four dimensional stationary and axially symmetric spacetimes for conformally coupled scalar-tensor theories. We show that, in analogy to the Lewis-Papapetrou problem in General Relativity (GR), the theory at hand can be recast in an analogous integrable form. We give the relevant rod formalism, introduced by Weyl for vacuum GR, explicitly giving the rod structure of the black hole of Bocharova et al. and Bekenstein (BBMB), in complete analogy to the Schwarzschild solution. The additional scalar field is shown to play the role of an extra Weyl potential. We then employ the Ernst method as a concrete solution generating example to obtain the Taub-NUT version of the BBMB hairy black hole, with or without a cosmological constant. We show that the anti-de Sitter hyperbolic version of this solution is free of closed timelike curves that plague usual Taub-NUT metrics, and thus consists of a rotating, asymptotically locally anti-de Sitter black hole. This stationary solution has no curvature singularities whatsoever in the conformal frame, and the NUT charge is shown here to regularize the central curvature singularity of the corresponding static black hole. Given our findings we discuss the anti-de Sitter hyperbolic version of Taub-NUT in four dimensions, and show that the curvature singularity of the NUT-less solution is now replaced by a neighboring chronological singularity screened by horizons. We argue that the properties of this rotating black hole are very similar to those of the rotating BTZ black hole in three dimensions.
Highlights
JHEP05(2014)039 work of Carter [7], or an electromagnetic field, but one still does not know, for example, the equivalent of the charged rotating solution of the Kerr-Newman black hole in higher dimensions.1 some of the properties of the Weyl metrics survive in higher dimensions [10], it is found that the cosmological constant spoils the integrability properties of the Weyl and Papapetrou problems [11, 12] by introducing a non-trivial curvature scale in the action
In analogy to the LewisPapapetrou problem in General Relativity (GR), the theory at hand can be recast in an analogous integrable form
We show that the anti-de Sitter hyperbolic version of this solution is free of closed timelike curves that plague usual Taub-NUT metrics, and consists of a rotating, asymptotically locally anti-de Sitter black hole
Summary
As mentioned in the introduction, a notable solution to the equations of motion (1.3)– (1.5), when Λ = 0 and αc = 0, is the BBMB solution [16, 17] It is a static and spherically symmetric configuration, given by the metric. The rod structure of the BBMB solution bears an interesting analogy with ordinary GR This is manifest if we recast its metric (2.15) in the form (2.5). We can point out that all spherically symmetric and static solutions of (2.1), given in [35] and parametrized by three constants (m, ǫ, δ), can be generated using this rod structure with the following densities σω cos ǫ 2 and σγ si√n ǫ 23 and the same lengths. Can be obtained by adding a semi-infinite ω-rod of density 1/2 to the previous rod structure of the BBMB solution
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