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\((2+1)\)-Dimensional charged black holes with scalar hair in Einstein–Power–Maxwell Theory

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Abstract

In \((2+1)\)-dimensional AdS spacetime, we obtain new exact black hole solutions, including two different models (power parameter \(k=1\) and \(k\ne 1\)), in the Einstein–Power–Maxwell (EPM) theory with nonminimally coupled scalar field. For the charged hairy black hole with \(k\ne 1\), we find that the solution contains a curvature singularity at the origin and is nonconformally flat. The horizon structures are identified, which indicates the physically acceptable lower bound of mass in according to the existence of black hole solutions. Later, the null geodesic equations for photon around this charged hairy black hole are also discussed in detail.

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Notes

  1. We thank the referee for pointing mathematical and physical inconsistencies about energy conditions for three dimensional Einstein–Power–Maxwell (EPM) theory in [24]. Consider the nonlinear electrodynamics term \(\mathcal {|F|}^k\) solely, the energy density is given by \(\rho _{\mathcal {|F|}^k}=-T^{t}_{t}=(2k-1)\mathcal {|F|}^k\). In order to make the energy conditions holding in gravity with usual Maxwell source (\(k=1\)) or conformal invariant Maxwell source (\(k=\frac{3}{4}\)), i.e. \(k>\frac{1}{2}\), we choose the Maxwell terms in the action as \(+\mathcal {|F|}^k\). This also leads to vanishing electric field at large r for the cases with \(k>\frac{1}{2}\).

  2. In fact, the determination of the mass in presence of a scalar field is subtle problem, as has been discussed in the literature (see [42] and [47]).

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Acknowledgements

Wei Xu was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11505065 and No. 91636111, and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. CUG150630). De-Cheng Zou is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11605152, and Natural Science Foundation of Jiangsu Province under Grant No. BK20160452.

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  1. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China

    Wei Xu

  2. School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Wei Xu

  3. Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou, 225009, China

    De-Cheng Zou

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Xu, W., Zou, DC. \((2+1)\)-Dimensional charged black holes with scalar hair in Einstein–Power–Maxwell Theory. Gen Relativ Gravit 49, 73 (2017). https://doi.org/10.1007/s10714-017-2237-4

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