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
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 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 gauge 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) gravity
: classical (super) gravity action
: boundary values of bulk fields
: dual CFT operators.
2) Gravity/Fluid correspondence
 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
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 GaussBonnet 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; expanded 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!