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

OutlineA. 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 procedureB. The first order dual hydrodynamics via Gravity/Fluid correspondence in the Gauss-Bonnet gravityC. Chern-Simons effects in Gravity/Fluid correspondence: anomalous magnetic and vortical effectsD. Holographic charged fluid at finite cutoff surface via Gravity/Fluid correspondence E. Conclusion and discussion

A. Introduction

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 ga

uge theory on the boundary. ‘ t Hooft (‘93), Susskind (‘94)

AdS/CFT correspondence: 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) g

ravity

: classical (super) gravity action : boundary values of bulk fields : dual CFT operators.

2) Gravity/Fluid correspondence Long wavelength limit in AdS/CFT correspondence: any quant

um 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 boun

dary For example Black brane solution

Boosted black brane solution

Perturbative black brane solution

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

1) G-B gravity G-B gravity: action and equation

2) Solution: 5-dimensional black brane

3) Boosted black brane solution

: boundary coordinates : boost velocities; constants : projector onto spatial directions : raise and lower the indices in boundary

4) Perturbative solution (to first order) : slowly varying functions of the boundary coordinates; exp

anded around

Correction terms needed

Coefficients of the first order correction terms

Perturbative solution to the first order: around

Perturbative solution to the first order: global; covariant

5) Boundary stress-tensor in the G-B gravity Definition

Explicit expression:

Non-zero components: Inserting global perturbative metric

6) Stress tensor of the conformal field on the boundary

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 Action and solutions

Temperature and entropy density

Solution in the Eddington-Finkelstin coordinate system

Solution in the boosted frame

where Perturbative solution: Similar as the case without Maxwell field; changed; New contents

Stress tensor of the conformal field on the boundary

Conserved current of the conformal field on the boundary

C. Chern-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

Black brane Solution in the Eddington-Finkelstin coordinate

Boosted black brane solution

where Perturbative solution

Coefficients expressions

Stress tensor of the dual fluid on the boundary

Charged current of the dual fluid on the boundary

Notes: 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

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

: boundary coordinates : boost velocities; constants : projector onto spatial directions : raise and lower the indices in boundary

5) Perturbative solution (to first order)

6) The stress tensor and Charged current of dual fluid

7) Coefficients

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 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: 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

12) Entropy density and thermodynamic relations

13) The anomalous coefficients at the cutoff surface:

Comparison at the infinite boundary surface:

14) Non-conformal fluid, zero bulk viscosity?

Thanks!