Abstract
Charged Lifshitz black holes for the Einstein-Proca-Maxwell system with a negative cosmological constant in arbitrary dimension D are known only if the dynamical critical exponent is fixed as z = 2(D − 2). In the present work, we show that these configurations can be extended to much more general charged black holes which in addition exist for any value of the dynamical exponent z > 1 by considering a nonlinear electrodynamics instead of the Maxwell theory. More precisely, we introduce a two-parametric nonlinear electrodynamics defined in the more general, but less known, so-called ( $ \mathcal{H} $ , P )-formalism and obtain a family of charged black hole solutions depending on two parameters. We also remark that the value of the dynamical exponent z = D − 2 turns out to be critical in the sense that it yields asymptotically Lifshitz black holes with logarithmic decay supported by a particular logarithmic electrodynamics. All these configurations include extremal Lifshitz black holes. Charged topological Lifshitz black holes are also shown to emerge by slightly generalizing the proposed electrodynamics.
Highlights
Where x is a (D − 2)-dimensional vector
We remark that the value of the dynamical exponent z = D − 2 turns out to be critical in the sense that it yields asymptotically Lifshitz black holes with logarithmic decay supported by a particular logarithmic electrodynamics
We exhibit a family of two-parametric charged black hole solutions for a generic value of the dynamical exponent z > 1 and z = D − 2. We prove that this last value of the dynamical exponent z = D − 2 turns out to be critical in the sense that it yields asymptotically Lifshitz black holes with logarithmic decay supported by a particular logarithmic electrodynamics which can be obtained as a nontrivial limit of the initially proposed one
Summary
We are interested in extending the charged Lifshitz black holes obtained in [20] for a particular value of the dynamical exponent to much more general black hole configurations valid for any value z > 1. The resulting action turns out to be a functional of the conjugate antisymmetric tensor P μν in addition to the electromagnetic field Aμ, and is given by the last two terms of (2.1) The advantage of this formulation lies in the fact that the variation with respect to the electromagnetic field Aμ gives the nonlinear version of the Maxwell equations. We emphasize that Lifshitz asymptotic with generic anisotropy is apparently out of this category, since Lifshitz black holes charged in standard way [20] are known only if the dynamical critical exponent is fixed to z = 2(D − 2) Notice this is precisely the exponent for which Maxwell theory is recovered in our electrodynamics (2.2d) if the structural constants are chosen as β1 = 0 and β2 is appropriately fixed. An objective that we will be able to achieve
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