Near-Extremal Fluid Mechanics

TL;DR

The paper analyzes near-extremal black branes in AdS$_4$ with slowly varying boundary data $(T(x),\mu(x),u^\nu(x))$ in the regime $T\ll\omega,k\ll\mu$, introducing a double expansion in $\epsilon$ and $\tilde{T}=T/\omega$. It shows that the Einstein–Maxwell equations admit a systematic perturbative solution, with local constitutive relations at first order and non-local-in-time corrections at higher orders, and identifies four linear hydrodynamic-like modes plus attractor-type time-independent perturbations under driving. The analysis proceeds via a near-horizon AdS$_2$ throat and an outer region, employing inner/outer matching and holographic renormalization to extract the boundary stress tensor and current. At leading order, the boundary theory behaves like a relativistic charged fluid with viscosity $\eta= r_h^2/(16\pi G)$ and charge-transport coefficients $\chi_1,\chi_2$, while higher-order terms generate temporal non-localities that preserve a controlled derivative expansion. The work illuminates how near-extremal holographic fluids depart from conventional hydrodynamics through AdS$_2$-driven nonlocality and sets the stage for extensions to rotating or higher-dimensional black holes and related IR theories such as JT gravity.

Abstract

We analyse near-extremal black brane configurations in asymptotically $\mathrm{AdS}_4$ spacetime with the temperature $T$, chemical potential $μ$, and three-velocity $u^ν$, varying slowly. We consider a low-temperature limit where the rate of variation is much slower than $μ$, but much bigger than $T$. This limit is different from the one considered for conventional fluid-mechanics in which the rate of variation is much smaller than both $T$, $μ$. We find that in our limit, as well, the Einstein-Maxwell equations can be solved in a systematic perturbative expansion. At first order, in the rate of variation, the resulting constitutive relations for the stress tensor and charge current are local in the boundary theory and can be easily calculated. At higher orders, we show that these relations become non-local in time but the perturbative expansion is still valid. We find that there are four linearised modes in this limit; these are similar to the hydrodynamic modes found in conventional fluid mechanics with the same dispersion relations. We also study some linearised time independent perturbations exhibiting attractor behaviour at the horizon - these arise in the presence of external driving forces in the boundary theory.