Abstract
We show that there exists a class of charged BTZ-like black hole solutions in Lifshitz spacetime with a hyperscaling violating factor. The charged BTZ black hole is characterized by a charge-dependent logarithmic term in the metric function. As concrete examples, we give five such charged BTZ-like black hole solutions and the standard charged BTZ metric can be regarded as a special instance of them. In order to check the recent proposed universal relations between diffusivity and the butterfly velocity, we first compute the diffusion constants of the standard charged BTZ black holes and then extend our calculation to arbitrary dimension d, exponents z and θ. Remarkably, the case d = θ and z = 2 is a very special in that the charge diffusion Dc is a constant and the energy diffusion De might be ill-defined, but vB2τ diverges. We also compute the diffusion constants for the case that the DC conductivity is finite but in the absence of momentum relaxation.
Highlights
JHEP01(2018)068 late time behavior of the analytically continued partition function Z(β + it)Z(β − it) in holographic 2d CFTS
We obtain a class of black hole solutions analogous to charged BTZ black holes by considering d + 2-dimensional action with non-trivial Lifshitz dynamical exponent z and hyperscaling violating factor θ
Those BTZ-like black hole solutions can be realized because special combinations of d, z and θ lead to divergence of the mass, charges- and axionsrelated terms
Summary
In order to show how the BTZ-like black hole solution emerges, we first consider a general (d + 2)-dimensional action with an arbitrary Lifshitz dynamical exponent z and a hyperscaling violating factor θ. The result is that the metric function f in any d ≥ 1 dimension can be recast in a form similar to charged BTZ black holes in 2 + 1 dimensions. The metric function can recover that of charged BTZ black hole solution as β = 0. Critical black hole solutions at d + z − θ − 2 = 0 and d2 + 2θ − (z + θ)d = 0 The metric function and gauge fields in this case take their forms f (r). The metric function is a three-dimensional BTZ-like black hole solution with a Lifshitz dynamical exponent and a hyperscaling factor. Both dc electric conductivity σ22 driven by the real Maxwell field and the thermoelectric conductivities α1 and α2 are constants It is worth future investigating whether this black hole solution and its boundary dual has any more physical meaning. The hyperscaling violating factor modified BTZ black holes will be examined
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