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Near-Extremal Fluid Mechanics

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.