Fluid%20dynamics%20from%20charged%20AdS%20Black%20holes

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Fluid dynamics from charged AdS Black holes

• Jin Hur, Kyung Kiu Kim and Sang-Jin Sin
• KIAS 2008

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Introduction
AdS / CFT AdS Black holes / Finite temperature
field theory Black holes Thermodynamics Variat
ion 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
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In Fluid dynamics derivatives -gt 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?
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There were many works and results about fluid
dynamics Our Approach follows arXiv0712.2456
Nonlinear Fluid Dynamics from Gravity Sayantani
Bhattacharyya, Veronika E Hubeny, Shiraz
Minwalla, Mukund Rangamani The black hole
solution without charge and angular
momentum arXiv0803.2526 Local Fluid Dynamical
Entropy from Gravity Sayantani Bhattacharyya,
Veronika E Hubeny, R. Loganayagam, Gautam Mandal,
Shiraz Minwalla, Takeshi Morita, Mukund
Rangamani, Harvey S. Reall arXiv0806.0006
Forced Fluid Dynamics from Gravity Sayantani
Bhattacharyya, R. Loganayagam, Shiraz Minwalla,
Suresh Nampuri, Sandip P. Trivedi, Spenta R.
Wadia Rotating black holes
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General construction
Construction of 0th order solution Boosted
solutions -gt Solution with parameters
(temperatures, velocity, charges,) Expand
parameters -gt 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
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Action Equations of motion Charged Black
Hole Solution
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Boosted solutions Einstein equation and
Maxwell equation operators Expand
to first
order Source terms are defined by
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Consider correction terms in metric and gauge
fields to find new solutions Source terms are
canceled by effects from correction terms
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For every order Maxwell equations Einstein
equations
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Constraints
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Physical quantities in Fluid dynamics Chemical
potential Boundary Stress Energy
Tensor Boundary Current
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Zeroth order solution
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First order solution
Source terms
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Metric
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Gauge fields
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Energy Momentum Tensor and Current
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Fluid dynamics from constraints
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Thermal Conductivity and Electrical Conductivity
from current
Coefficient of thermal conductivity and Thermal
conductivity Electrical conductivity
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Summary

• Charged black holes in AdS space / Fluid dynamics
  in Exterrnal Maxwell Fields
• Taking Limit Q 0 , Our solution reproduces
  BHMRs result
• -Taking Limit 0 , our current
  and thermal conductivity are same with recent
  works
• arXiv0809.2488 Fluid dynamics of
  R-charged black holes
• Johanna Erdmenger, Michael Haack, Matthias
  Kaminski, Amos Yarom
• arXiv0809.2596 Hydrodynamics from charged
  black branes
• Nabamita Banerjee, Jyotirmoy Bhattacharya,
  Sayantani Bhattacharyya, Suvankar Dutta, R.
  Loganayagam, P. Surówka
• -We obtained The electrical conductivity