# Fluid-gravity correspondence The **fluid-gravity correspondence** is a modern incarnation of the [[0229 Membrane paradigm|membrane paradigm]]. At the level of equations, one rewrites quantities such as the stress tensor, $T_{ab}$, and, when there is a global symmetry (for example), a conserved current, $J_a$, in terms of new variables and substitute them into equations of motion, giving rise to a set of equations for the new variables that resemble hydrodynamic equations for fluids. These "field redefinitions" are organised in terms of a gradient expansion. For example, without global charges, the leading order relation can be written as $ T_{a b}^{(0)}=\rho u_a u_b+p P_{a b}, $and the next order looks like $ T_{a b}^{(1)}=-2 \eta \sigma_{a b}-\zeta \theta P_{a b} ,$where $\theta$ and $\sigma_{ab}$ are quantities that contain one derivative, e.g., $\theta=\partial_a u^a$, and $\eta$ and $\zeta$ are shear and bulk viscosities. ## Refs - original - [[2007#Bhattacharyya, Hubeny, Minwalla, Rangamani]] - reviews - [[Rangamani2009Lectures]][](https://arxiv.org/abs/0905.4352) - [[Hubeny2010]] - relation to [[0181 AdS-BCFT|AdS/BCFT]] - [[MaganaMelnikovSilva2014]] - higher derivative gravity - [[ChandranathanABhattacharyyaPatraRoy2022]][](https://arxiv.org/pdf/2208.07856.pdf): the map does not generalise to GB gravity