Navier-Stokes Equations on Black Hole Horizons and DC Thermoelectric Conductivity

TL;DR

Using AdS/CFT, the authors show that the DC thermoelectric conductivities of a strongly coupled CFT with a holographic lattice can be obtained from solving linearised, time-independent forced Navier–Stokes equations for an incompressible charged fluid on the black hole horizon. The horizon currents are read off from horizon data, with conductivities expressed in terms of a horizon-geometry matrix $L$ as $\bar{\kappa}^{ij}=(L^{-1})^{ij}4\pi sT$, $\alpha=\bar{\alpha}=(L^{-1})^{ij}4\pi \rho$, and $\sigma=(L^{-1})^{ij}4\pi \rho^2/s$, and with $\eta_H = s_H/(4\pi)$ for the shear viscosity of the horizon fluid. The method applies to generic holographic lattices breaking translations in all directions and does not rely on a hydrodynamic limit, unifying and extending previous results for special lattices. In perturbative coherent metals, the leading-order horizon analysis yields Drude-like DC peaks and a Wiedemann–Franz-type relation, providing a universal horizon-level description of DC transport.

Abstract

Within the context of the AdS/CFT correspondence we show that the DC thermoelectric conductivity can be obtained by solving the linearised, time-independent and forced Navier-Stokes equations on the black hole horizon for an incompressible and charged fluid.