Fluid dynamics from charged AdS Black holes Jin Hur, Kyung Kiu Kim and Sang-Jin Sin KIAS 2008

Introduction AdS / CFT AdS Black holes / Finite temperature field theory Black holes ~ Thermodynamics Variation of Thermodynamics with very small derivatives ~ Fluid dynamics Deformation of AdS Black Holes with small derivatives ~ Conformal Fluid dynamics Fluid dynamics ~ Effective theory of CFT

In Fluid dynamics; derivatives -> very small Main contributions come from low momentum and low energy modes Many theorists hope that the Conformal fluid dynamics and fluid dynamics have same universal features. Can we explain fluid dynamics systems by black hole physics?

There were many works and results about fluid dynamics Our Approach follows arXiv: arXiv: : Nonlinear Fluid Dynamics from Gravity Sayantani BhattacharyyaSayantani Bhattacharyya, Veronika E Hubeny, Shiraz Minwalla, Mukund Rangamani – The black hole solution without charge and angular momentumVeronika E HubenyShiraz MinwallaMukund Rangamani arXiv: arXiv: : Local Fluid Dynamical Entropy from Gravity Sayantani BhattacharyyaSayantani Bhattacharyya, Veronika E Hubeny, R. Loganayagam, Gautam Mandal, Shiraz Minwalla, Takeshi Morita, Mukund Rangamani, Harvey S. ReallVeronika E HubenyR. Loganayagam Gautam MandalShiraz MinwallaTakeshi MoritaMukund Rangamani Harvey S. Reall arXiv: arXiv: : Forced Fluid Dynamics from Gravity Sayantani BhattacharyyaSayantani Bhattacharyya, R. Loganayagam, Shiraz Minwalla, Suresh Nampuri, Sandip P. Trivedi, Spenta R. Wadia – Rotating black holesR. LoganayagamShiraz MinwallaSuresh NampuriSandip P. TrivediSpenta R. Wadia

General construction Construction of 0 th order solution Boosted solutions -> Solution with parameters (temperatures, velocity, charges,…) Expand parameters -> This is not a solution of equations of motion (Einstein equation, Maxwell equation, …) Corrections in fields( metric, gauge fields,…) Finding new solution for a given derivative order

Action Equations of motion Charged Black Hole Solution

Boosted solutions Einstein equation and Maxwell equation operators Expand to first order Source terms are defined by

Consider correction terms in metric and gauge fields to find new solutions Source terms are canceled by effects from correction terms

For every order Maxwell equations Einstein equations

Constraints

Physical quantities in Fluid dynamics Chemical potential Boundary Stress Energy Tensor Boundary Current

Zeroth order solution

First order solution Source terms

Metric

Gauge fields

Energy Momentum Tensor and Current

Fluid dynamics from constraints

Thermal Conductivity and Electrical Conductivity from current Coefficient of thermal conductivity and Thermal conductivity Electrical conductivity

Summary -Charged black holes in AdS space / Fluid dynamics in Exterrnal Maxwell Fields -Taking Limit Q = 0, Our solution reproduces BHMR’s result -Taking Limit = 0, our current and thermal conductivity are same with recent works : arXiv: : Fluid dynamics of R-charged black holesarXiv: : Johanna Erdmenger, Michael Haack, Matthias Kaminski, Amos YaromJohanna ErdmengerMichael HaackMatthias KaminskiAmos Yarom arXiv: arXiv: : Hydrodynamics from charged black branes : Nabamita Banerjee, Jyotirmoy Bhattacharya, Sayantani Bhattacharyya, Suvankar Dutta, R. Loganayagam, P. SurówkaNabamita BanerjeeJyotirmoy BhattacharyaSayantani BhattacharyyaSuvankar DuttaR. LoganayagamP. Surówka -We obtained The electrical conductivity