1. Gravity/Fluid Correspondence and Its Application on Bulk Gravity with U(1) Gauge Field
Ya-Peng Hu
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing
(南京航空航天大学)
Reference: Y.P Hu and J.H Zhang, Adv.High Energy Phys (2014) [arXiv: hep-th]

2. Mainly based on:
S.Bhattacharyya, V.E.Hubeny, S.Minwalla and M.Rangamani, JHEP 0802, 045 (2008)
M.Rangamani, Class. Quant.Grav.26, (2009)
J.Erdmenger, M.Haack, M.Kaminski and A.Yarom, JHEP 0901, 055 (2009) [arXiv: [hep-th]].
J. Hur, K. K. Kim and S. J. Sin, JHEP 0903, 036 (2009)
N.Banerjee, et al. JHEP 1101, 094 (2011)
D.T.Son and P.Surowka, Phys. Rev. Lett. 103, (2009) [arXiv: [hep-th]].
Y.P.Hu, P.Sun and J.H.Zhang, Phys. Rev. D 83, (2011)
Y. P. Hu and C. Y. Park, Phys. Lett. B 714, 324 (2012)

3. Outline A. Introduction 1) Gravity/Fluid correspondence
2) Fundamental algorithm and procedure in
Gravity/Fluid correspondence
B. Applications of Gravity/Fluid correspondence on the Bulk Gravity with 𝑈(1) Gauge Field
1) The simplest case
2) In the presence of an external field
3) Adding 𝑈(1) Chern-Simons term in the bulk
C. Conclusion and discussion

4. A. Introduction 1) 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/ )
Stress tensor of viscous fluid with conformal symmetry broken
Stress tensor of conformal viscous fluid

5. 2) 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

6. Perturbative black brane solution
Boundary stress tensor (asymptotical AdS5)
Stress tensor of the dual fluid on the boundary

7. B. Applications on the Bulk Gravity with 𝑈(1) Gauge Field: The simplest case
1) Einstein-Maxwell gravity
2) RN-AdS black brane solution: in Eddington-Finkelstin coordinate system

8. 3) Boosted RN-AdS black brane solution
: boundary coordinates
: boost velocities; constants
: projector onto spatial directions
: raise and lower the indices in boundary

9. 4) Perturbative solution (to first order)
5) The stress tensor and Charged current of dual fluid
Stress tensor of dual fluid:

10. Charged current of dual fluid:

11. B. Applications on the Bulk Gravity with 𝑈(1) Gauge Field: In the presence of an external field
1) Key point: a constant external field added into the boosted black brane solution
2) Peturbative solution: coefficients changed by the external field
3) Results: stress tensor same; charged current changed

12. B. Applications on the Bulk Gravity with 𝑈(1) Gauge Field: Adding 𝑈(1) Chern-Simons term in the bulk
2) Results: stress tensor same; charged current changed

13. 3) External field chosen: two cases

14. C. Conclusion and discussion
1) Gravity/Fluid correspondence: long wavelength limit in
AdS/CFT correspondence
2) Fundamental algorithm and procedure in Gravity/Fluid
correspondence
3) Stress tensor of Conformal viscous fluid deduced
4) Applications of Gravity/Fluid correspondence on the Bulk Gravity with 𝑈(1) Gauge Field: stress tensor; Charged current of dual fluid
5) Three cases: The simplest case; In the presence of an external field; Adding 𝑈(1) Chern-Simons term in the bulk

15. 6) Results: stress tensor same; charged current changed 7) Anomalous effects (related to Chern-Simons terms): anomalous magnetic effect and vortical effect; 8) Other Chern-Smions effects, i.e. gravitational Chern- Simons term in the bulk 9) Holographic fluid at finite cutoff surface

16. Thanks!