# Holographic shear viscosity and the KSS bound Einstein gravity predicts a relation between shear viscosity and entropy density for the boundary [[0429 Hydrodynamics|hydrodynamics]]: $ \eta=\frac{\hbar}{4 \pi k_{B}} s. $ The shear viscosity is set by the $k,\omega\to 0$ limit of the [[0473 Retarded Green's function|retarded Green's function]] of the stress tensor:$\eta=-\lim _{\omega \rightarrow 0} \frac{1}{\omega} \operatorname{Im} \tilde{G}_R^{x y, x y}(\omega, k=0).$In this limit, the radial evolution of the bulk perturbation is very simple and depends only on the near-horizon region of the gravitational solution. The Kovtun-Son-Starinets (KSS) bound states that this is a lower bound! ## Definition The one-point function due to a source $\phi_0$ is given by the thermal retarded correlator of $\mathcal{O}$:$\langle\mathcal{O}(\omega, \vec{k})\rangle_{\mathrm{QFT}}=-G^{R}(\omega, \vec{k}) \phi_{0}(\omega, \vec{k}).$ The low-frequency limit of this defines a transport coefficient:$\chi=-\lim _{\omega \rightarrow 0} \lim _{\vec{k} \rightarrow 0} \frac{1}{\omega} \operatorname{Im} G^{R}(\omega, \vec{k}) .$This means the low frequency limit response is just:$\langle\mathcal{O}\rangle_{\mathrm{QFT}}=-\chi \partial_{t} \phi_{0}(t).$ Some examples: - [[0430 Holographic shear viscosity|shear viscosity]], $\eta$: $\mathcal{O}=T_{xy}$, off-diagonal elements of the stress tensor; - DC conductivity, $\sigma$: take $\mathcal{O}=J_z$, a component of the electric current. ## Refs - pre-cursors - [[2001#Policastro, Son, Starinets]] ($\mathcal{N}=4$ SYM) and [[2002#Policastro, Son, Starinets]] - original - [[2004#Kovtun, Son, Starinets]]: also conjectures that this is a lower bound (KSS bound) - proofs - [[2008#Iqbal, Liu]]: proof using near-horizon analysis - [[Hod2009Essay]]: derivation using [[0445 Hod's universal relaxation bound|Hod's universal relaxation bound]] - explicit calculations - [[GubserPufuRocha2008]][](https://arxiv.org/abs/0806.0407) - [[BenincasaBuchelStarinets2005]][](https://arxiv.org/abs/hep-th/0507026) - relation to causality violation - [[2008#Brigante, Liu, Myers, Shenker, Yaida]] - breaking rotation invariance - [[2023#Baggioli, Cremonini, Early, Li, Sun]] - far-from-equilibrium extension - [[2020#Wondrak, Kaminski, Bleicher]] - [[2024#Wang, He, Li]] ## Theory dependence Higher derivative corrections do change the ratio $1/4\pi$. - review: [[2011#Cremonini (Review)]] - [[BuchelLiu2003]]: SUGRA - [[2004#Buchel, Liu, Starinets]] - [[BenincasaBuchel2005]][](https://arxiv.org/abs/hep-th/0510041) - [[0425 Gauss-Bonnet gravity|Gauss-Bonnet]]: - [[2007#Brigante, Liu, Myers, Shenker, Yaida]] - [[2016#Grozdanov, Kaplis, Starinets]] - curvature squared [[KatsPetrov2007]][](https://arxiv.org/abs/0712.0743) - [[2008#Brigante, Liu, Myers, Shenker, Yaida]] - tuning $\eta/s$ below (16/25)(1/4$\pi$) induces microcausality violation in the CFT, rendering the theory inconsistent! - general framework for higher-derivative corrections - [[2023#Buchel, Cremonini, Early]] - [[2024#Buchel]] - constrain theory from boundary data - [[2024#da Rocha]]