We obtain the general static, spherically symmetric solution for the Einstein-Maxwell-dilaton system in four dimensions with a phantom coupling for the dilaton and/or the Maxwell field. This leads to new classes of black hole solutions, with single or multiple horizons. Using the geodesic equations, we analyse the corresponding Penrose diagrams revealing, in some cases, new causal structures.
Deep Dive into Phantom Black Holes in Einstein-Maxwell-Dilaton Theory.
We obtain the general static, spherically symmetric solution for the Einstein-Maxwell-dilaton system in four dimensions with a phantom coupling for the dilaton and/or the Maxwell field. This leads to new classes of black hole solutions, with single or multiple horizons. Using the geodesic equations, we analyse the corresponding Penrose diagrams revealing, in some cases, new causal structures.
Effective gravity actions emerging from string and Kaluza-Klein theories contain a rich structure where, beside the usual Einstein-Hilbert term, there is a scalar field, generically called a dilaton, coupled to an electromagnetic field. The asymptotically flat static black hole solutions for this Einstein-Maxwelldilaton (EMD) system [1,2] differ from the usual Reissner-Nordström solution of Einstein-Maxwell theory in that the inner horizon is singular for a non-vanishing dilaton coupling. Non-asymptotic flat static black hole solutions have also been obtained in [3] and further studied in [4,5,6]. Brane configurations, leading also to black hole solutions, have been largely studied in reference [7,8] using essentially the EMD theory.
The aim of the present work is to study the structures of the black holes of the EMD theory when a phantom coupling is considered. This is done by allowing the scalar field or the Maxwell field (or both) to have the “wrong” sign [9]. The importance of such extension of the normal (non-phantom) EMD theory is twofold. First, from the theoretical point of view, string theories admit ghost condensation, leading to phantom-type fields [10]. In principle, a phantom may lead to instability, mainly at the quantum level. But there are claims that these instabilities can be avoided [11]. The second motivation comes from the results of the observational programs of the evolution of the Universe, specially the magnitude-versus-redshift relation for the supernovae type Ia, and the anisotropy spectrum of the cosmic microwave background radiation: both observational programs suggest that the universe today must be dominated by an exotic fluid with negative pressure. Moreover, there is some evidence that this fluid can be phantom [12,13].
When the scalar field and/or the Maxwell field are allowed to contribute negatively to the total energy, the energy conditions (and specially the null energy condition ρ + p ≥ 0) can be violated, and the appearance of some new structures can be expected. This is the case of wormhole solutions to Einstein-scalar [14] or Einstein-Maxwell-scalar [15] theory with a phantom scalar field. Another instance is that of black hole solutions to Einstein theory minimally coupled to a free scalar field, which are forbidden by the no-hair theorem, but become possible if the kinetic term of the scalar field has the wrong sign (if the scalar field is phantom), as found both in 2+1 [16] and in 3+1 dimensions [17]. These phantom black holes have all the characteristics of the so-called cold black holes [18,19]: a degenerate horizon (implying a zero Hawking temperature) and an infinite horizon surface. Indeed, these cold black holes appeared in the context of scalar-tensor theory (e.g., in Brans-Dicke theory) in such circumstances that, after re-expressing the action in the Einstein frame, the scalar field comes out to be phantom.
The non-trivial dilatonic coupling of the scalar field with the electromagnetic term adds new classes of black hole solutions. In references [9,20] some investigations on phantom black holes in the context of EMD theory have been made, revealing some interesting new species of black holes. For example, in the case of a self-interacting scalar field in four dimensions, a phantom field may lead to a completely regular spacetime where the horizon hides an expanding, singularity-free universe [21]. Our goal here is to obtain the most general static black hole solutions when a phantom coupling is allowed for both the scalar and electromagnetic fields of the EMD theory.
We will classify the different possible black hole solutions coming out from the EMD theory for a phantom coupling of either the dilaton field, of the Maxwell field, or of both. It is remarkable that many of these new black holes have a degenerate horizon, hence a zero Hawking temperature. We will also analyse the causal structure of these black hole spacetimes. In some cases, the causal structure is highly unusual, such that no twodimensional Penrose diagram can be constructed. Another possibility is that of a spacetime with an infinite series of regular horizons separating successive non-isometric regions. Geodesically complete black hole spacetimes are also obtained.
This paper is organized as follows. In the next section we derive, following the procedure of [22], the general static spherically symmetric solutions (phantom and non-phantom) of the EMD theory. In section 3 the new black hole solutions are described in detail. The Penrose diagrams of these new solutions are constructed in section 4. In section 5 we present our conclusions.
Let us consider the following action:
which is the sum of the usual Einstein-Hilbert gravitational term, a dilaton field kinetic term, and a term coupling the Maxwell Lagrangian density to the dilaton, with the coupling constant λ real. The dilaton-gravity coupling constant η 1 can take either the value η 1 = 1 (dilaton) or η 1 = -1 (antidilaton).
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