Transport coefficients of charged Gauss-Bonnet black holes with arbitrary topology

TL;DR

This work constructs a new set of exact charged black holes in five-dimensional Gauss–Bonnet gravity with Thurston-horizon geometries, sourced by a nonlinear $H(P)$ electrodynamics. It analyzes energy conditions, derives the thermodynamics including a valid first law, and computes holographic DC conductivities for non-maximally symmetric horizons, uncovering topology-dependent transport and a critical temperature signaling phase-transition-like behavior. The results reveal that nonlinear electrodynamics plus horizon topology can produce anomalous transport (including sign changes) without scalar condensation, offering a novel holographic path to study non-equilibrium phenomena. Limitations arise for maximally symmetric horizons due to CS-point degeneracy, pointing to future work on extending beyond these backgrounds and clarifying the equivalent $L(F)$ formulation.

Abstract

In this study, we present a novel family of exact black hole solutions constructed in the context of five-dimensional Gauss-Bonnet gravity. These solutions add a non-linear charge to the Bañados-Teitelboim-Zanelli-like configurations known to exist with arbitrary Thurston horizon geometry. We establish constraints on the parameter space defining physically viable black holes, aligning with the standard energy conditions. An explicit proof of the first law of thermodynamics within our scenario is provided. We also employ holographic techniques to characterize the DC conductivities for the distinct horizon geometries, identifying a critical temperature indicative of phase transitions and exploring pertinent limits.

Figures (5)

• Figure 1: Left Panel: Representation of the location of the event horizon $r_h$ depending on the integration constant $\mu$. Here, from Table \ref{['tab:Thurston']}, the case $(i)$ ($K=-6$) is represented by a black curve, while cases $(ii)$ ($K=0$) and $(iii)$ ($K=6$) correspond to the blue curve and gray curve respectively. Right Panel: Representation of $r_h$ with respect $\mu$, where the case $(iv)$ ($K=-5/2$) is represented by a blue curve, while that cases $(v)$ ($K=-2$) and $(vi)$ ($K=-1/2$) correspond to the green curve and orange curve respectively. Finally, case $(vii)$ ($K=2$) is visualized via the red curve. Here, we have considered $\lambda=4 \alpha=1$ for our computations.
• Figure 2: The magenta region in the above plots displays the allowed parameter region where the WEC and NEC \ref{['eq:NEC']} are satisfied. The bordering darker lines in the triangular regions are admissible points. The vertical blue lines are constant $K$ regions (see Tab. \ref{['tab:Thurston']}).
• Figure 3: Electric permittivity as a function temperature $T$, for $\Phi_e=0.4$ and $\mathcal{Q}_{e}=1$. Positive and negative branches are present in this case, along with a vertical asymptote corresponding to $\Psi=0$. Here, we have considered $\lambda=4 \alpha=|\Omega_{3}|=\kappa=1$ for our computations.
• Figure 4: Specific heat as a function of the temperature $T$, where $\Phi_e=0.4$ and $\mathcal{Q}_{e}=1$. In this case, positive and negative branches are observed, as well as a vertical asymptote represented by $\Psi=0$. Here, we have considered $\lambda=4 \alpha=|\Omega_{3}|=\kappa=1$ for our computations.
• Figure 5: Illustration of two characteristic behaviors of the DC conductivities as a function of the temperature. On the left, we depict the behavior stemming from a configuration with an $\mathbb{S}^1\times \mathbb{H}^2$ horizon, while on the right, we consider the $\hbox{SL}_{2}(\mathbb{R})$ geometry. We introduce the dimensionless temperature $\tau = \lambda^2 T$, along with the test values of dimensionless quantities as $\alpha \lambda^4=1$ and $\mu=5$. This choice adheres to all energy conditions.