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An introduction to the Gravity/Fluid correspondence and its applications. Ya-Peng Hu College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China 2013.03.28 @ICTS, hefei. Outline. A. Introduction 1) AdS/CFT correspondence : a direct realization of the

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An introduction to the gravity fluid correspondence and its applications

An introduction to the Gravity/Fluid correspondence and its applications

Ya-Peng Hu

College of Science, Nanjing University of Aeronautics and Astronautics,

Nanjing 210016, China

2013.03.28 @ICTS, hefei


Outline
Outline applications

A. Introduction

1) AdS/CFT correspondence : a direct realization of the

Holography principle

2) Gravity/Fluid correspondence: long wavelength limit in

AdS/CFT correspondence; fundamental algorithm and procedure

B. The first order dual hydrodynamics via Gravity/Fluid

correspondence in the Gauss-Bonnet gravity

C. Chern-Simons effects in Gravity/Fluid correspondence:

anomalous magnetic and vortical effects

D. Holographic charged fluid at finite cutoffsurface via

Gravity/Fluid correspondence

E. Conclusion and discussion


A introduction
A. Introduction applications

1) AdS/CFT correspondence : a direct realization of the Holography principle

  • Entropy of Black hole: S= f(A); S=A/4

  • Holography principle: Gravity in the bulk is equivalent to a gauge theory on the boundary. ‘ t Hooft (‘93), Susskind (‘94)


  • AdS/CFT correspondence: applications

    Type IIB string theory on AdS5 × S5 dual to

    N = 4 Super-Yang-Mills theory in 3+1 dimensions

  • General formalism: In the approximation of classical (super) gravity

    : classical (super) gravity action

    : boundary values of bulk fields

    : dual CFT operators.


2) Gravity/Fluid correspondence applications

  • Long wavelength limit in AdS/CFT correspondence: any quantum field effectively described by hydrodynamics; CFT becomes conformal fluid

  • Gravity/Fluid correspondence application: conformal viscous fluid stress tensor can be deduced from the dual gravity (hep-th/0712.2456)

  • Stress tensor of viscous fluid with conformal symmetry broken

  • Stress tensor of conformal viscous fluid


  • Fundamental algorithm and procedure in Gravity/Fluid correspondence:

    Black brane solution  Boosted black brane solution 

    Perturbative black brane solution 

    Boundary stress tensor (or other quantities) in the bulk

    Stress tensor (or other quantities) of the dual fluid on the boundary

  • For example

    Black brane solution

    Boosted black brane solution


Perturbative black brane solution correspondence:

Boundary stress tensor (asymptotical AdS5)

Stress tensor of the dual fluid on the boundary


B the first order hydrodynamics via gravity fluid correspondence in the gauss bonnet gravity
B. The first order hydrodynamics via Gravity/Fluid correspondence in the Gauss-Bonnet gravity

1) G-B gravity

  • G-B gravity: action and equation


2) Solution: 5-dimensional black brane correspondence in the Gauss-Bonnet gravity


3) Boosted black brane solution correspondence in the Gauss-Bonnet gravity

: boundary coordinates

: boost velocities; constants

: projector onto spatial directions

: raise and lower the indices in boundary


4) Perturbative solution (to first order) correspondence in the Gauss-Bonnet gravity

  • : slowly varying functions of the boundary coordinates; expanded around

  • Correction terms needed

  • Coefficients of the first order correction terms



5) Boundary stress-tensor in the G-B gravity correspondence in the Gauss-Bonnet gravity

  • Definition


  • Explicit expression: correspondence in the Gauss-Bonnet gravity

  • Non-zero components: Inserting global perturbative metric


6) Stress tensor of the conformal field on the boundary correspondence in the Gauss-Bonnet gravity

  • Background metric: the dual conformal filed

  • Expectation value of the stress tensor of the dual conformal field

  • Stress tensor of the dual theory

  • Pressure and viscosity


7) Adding the Maxwell field correspondence in the Gauss-Bonnet gravity

  • Action and solutions

  • Temperature and entropy density




C chern simons effects in gravity fluid correspondence
C. correspondence in the Gauss-Bonnet gravityChern-Simons effects in Gravity/Fluid correspondence

1) Maxwell Chern-Simons effect: anomalous magnetic and vortical effects (based on my two papers arXiv:1103.3773 Phys.Rev.D83:126003,2011 and arXiv:1112.4227)

  • G-B gravity with Maxwell Chern-simons: action and equations


  • Black brane solution correspondence in the Gauss-Bonnet gravity

  • Black brane Solution in the Eddington-Finkelstin coordinate



Coefficients expressions correspondence in the Gauss-Bonnet gravity



  • Notes: correspondence in the Gauss-Bonnet gravity

    Anomalous magnetic and vortical effects:

    Charged current non-conserved: triangle anomalies

    New term: general case; physical or unphysical?

    The other special case:


D holographic charged fluid with anomalous current at finite cutoff surface
D. Holographic charged fluid with anomalous correspondence in the Gauss-Bonnet gravity current at finite cutoffsurface

1) Einstein-Maxwell gravity with Chern-Simons term

2) RN-AdS black brane solution


3) RN-AdS black brane solution in Eddington-Finkelstin coordinate system

New Coordinate Transformation:


4) Boosted RN-AdS black brane solution coordinate system

: boundary coordinates

: boost velocities; constants

: projector onto spatial directions

: raise and lower the indices in boundary


5) Perturbative solution (to first order) coordinate system

6) The stress tensor and Charged current of dual fluid


7) Coefficients coordinate system

Fixed by considering three conditions:


8)Final expressions of the stress tensor and charged current of dual fluid at finite cutoff surface


E conclusion and discussion
E. of dual fluid at finite cutoff surfaceConclusion and discussion

1) Gravity/Fluid correspondence: long wavelength limit in

AdS/CFT correspondence

2) Fundamental algorithm and procedure in Gravity/Fluid

correspondence

3) Indeed deduce conformal viscous fluid in G-B gravity

4) The ratio of viscosity to entropy density: same as

other methods

5) Entropy density: same as black brane solution

6) Considering Maxwell field in the bulk: dual charged current

in the boundary


7) Chern-Simons effects in Gravity/Fluid correspondence: of dual fluid at finite cutoff surface

Anomalous magnetic and vortical effects;

Hall viscosity;

………

8) The anomalous magnetic and vortical effects

9) New term physical or unphysical?

10) Underlying relation between Topological insulator in

condensate matter and Chern-Simons term?

(PRB78,195424,2008)


11) Near the infinite boundary case: large of dual fluid at finite cutoff surface

12) Entropy density and thermodynamic relations


13) The anomalous coefficients at the cutoff surface: of dual fluid at finite cutoff surface

Comparison at the infinite boundary surface:

14) Non-conformal fluid, zero bulk viscosity?


Thanks
Thanks! of dual fluid at finite cutoff surface


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