Hydrodynamics of RN AdS$_4$ black hole and Holographic Optics

TL;DR

This work analyzes the vector sector of the charged RN–AdS$_4$ black hole to obtain momentum-dependent hydrodynamics and holographic Green functions for the dual field theory. By decoupling the bulk equations into master fields $\Phi_\pm$ and employing a hydrodynamic expansion, the authors derive diffusion-like behavior and extract the diffusion constant $D=\frac{1}{3\alpha(1+Q^2)}$ along with the corresponding Green functions and conductivities. They further compute the spectral functions and optical responses, showing that the low-frequency regime yields a negative refractive index $n_{DL}$ via the relations $\epsilon_T=1-\frac{4\pi e^2}{\omega^2}G_T(\omega,k)$ and $\mu(\omega)=[1+4\pi e^2G_T^{(2)}(\omega)]^{-1}$, with a static limit indicating zero permeability and superconductivity-like behavior. The results extend holographic optics to RN–AdS$_4$ and provide a framework for momentum-dependent response, with potential extensions to sound modes and magnetic-charge configurations.

Abstract

We consider the AdS$_4$ RN black hole and work out the momentum dependent hydrodynamic analysis for the vector modes. We also perform the spectral function calculation of the dual field theory. As an application, we consider the permittivity and permeability and find that for low frequency regime, the index of refraction is found to be negative, supporting the claim made in ref.\cite{amariti} for AdS$_5$. We also find that at static limit the medium has the zero permeability, a character of the superconductivity .

Figures (6)

• Figure 1: Spectral function of xt,xt component, Im ${\cal{G}}_{xtxt}$. Left : with fixed k=0.5 varying Q=0.5(thin), 1(thick), 1.5(dashed). Right : with fixed Q=1, varying k=0.2(thin), 0.5(thick), 1(dashed)
• Figure 2: Spectral function of xx component, $\hbox{Im} {\cal{G}}_{xx}$. Left : with fixed k=0.5 varying Q=0.5(thin), 1(thick), 1.5(dashed). Right : with fixed Q=1, varying k=0.2(thin), 0.5(thick), 1(dashed)
• Figure 3: Comparison between numerically (solid) and analytically computed in hydrodynamic limit (dashed) spectral function with Q=0.5. Left is for $G^{0}$(w) and right for $G^{0}$(w). These two functions are well matched in small w region but deviated with each other.
• Figure 4: Real(left) and imaginary(right) part of transverse electric permittivity, $\epsilon$(w) with $e$ =0.5 and Q=0.5 .
• Figure 5: Real(left) and imaginary(right) part of transverse magnetic permeability $\mu$(w) with $e$ =0.5 and Q=0.5.