Abstract
We consider a gravitating system consisting of a scalar field minimally coupled to gravity with a self-interacting potential and an U(1) electromagnetic field. Solving the coupled Einstein-Maxwell-scalar system we find exact hairy charged black hole solutions with the scalar field regular everywhere. We go to the zero temperature limit and we study the effect of the scalar field on the near horizon geometry of an extremal black hole. We find that except a critical value of the charge of the black hole there is also a critical value of the charge of the scalar field beyond of which the extremal black hole is destabilized. We study the thermodynamics of these solutions and we find that if the space is flat then at low temperature the Reissner-Nordstr\"om black hole is thermodynamically preferred, while if the space is AdS the hairy charged black hole is thermodynamically preferred at low temperature.
Highlights
JHEP11(2014)011 hair of neutral and charged scalar fields were derived
We go to the zero temperature limit and we study the effect of the scalar field on the near horizon geometry of an extremal black hole
We will investigate what is the effect of the scalar field to the near horizon geometry of the hairy black hole solutions we discussed in the previous section as the temperature goes to zero
Summary
We will review the general formalism discussed in [10] of a scalar field minimally coupled to curvature having a self-interacting potential V (φ), in the presence of an electromagnetic field. The Einstein-Hilbert action with a negative cosmological constant Λ = −6l−2/κ, where l is the length of the AdS which has been incorporated in the potential as Λ = V (0) (V (0) < 0) is. The resulting Einstein equations from the above action are. The energy momentum tensors Tμ(φν) and Tμ(Fν ) for the scalar and electromagnetic fields are. Where dσ is the metric of the spatial 2-section, which can have positive, negative or zero curvature, and Aμ = (At(r), 0, 0, 0) the scalar potential of the electromagnetic field.
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