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We consider a gravitating system of vanishing cosmological constant consisting of an electromagnetic field and a scalar field coupled to the Einstein tensor. A Reissner-Nordström black hole undergoes a second-order phase transition to a hairy black hole of generally anisotropic hair at a certain critical temperature which we compute. The no-hair theorem is evaded due to the coupling between the scalar field and the Einstein tensor. Within a first-order perturbative approach, we calculate explicitly the properties of a hairy black hole configuration near the critical temperature and show that it is energetically favourable over the corresponding Reissner-Nordström black hole. © 2012 IOP Publishing Ltd.
Kolyvaris et al. (Tue,) studied this question.