1. Einstein Field Equations & Navier-Stokes Equations
The 2013 Cross-Strait Meeting on particle Physics and Cosmology
NCU, March 29-April 2.
Einstein Field Equations & Navier-Stokes Equations
Rong-Gen Cai （蔡荣根）
Institute of Theoretical Physics
Chinese Academy of Sciences

2. 引力理论 流体力学 热力学 dE =TdS ① ② ① Rindler Horizon: T. Jacobson, 1995
Black Hole Horizon: T. Padamanabhan, 2002
Apparent Horizon: R.G. Cai and S.P. Kim, 2005
② T. Damour, 1978, R. Price and K. Thorne, 1986
S. Bhattacharyya et al, (gravity/fluid correspondence)
I. Bredberg et al, 2011

3. Einstein’s Field Equations:
Incompressible Navier-Stokes Equations:

4. Navier-Stokes Equation:
Waves follow our boat as we meander across the lake, and
turbulent air currents follow our flight in a modern jet.
Mathematicians and physicists believe that an explanation for
and the prediction of both the breeze and the turbulence can
be found through an understanding of solutions to the
Navier-Stokes equations. Although these equations were
written down in the 19th century, our understanding of them
remains minimal. The challenge is to make substantial
progress toward a mathematical theory which will unlock the
secrets hidden in the Navier-Stokes equations.

5. Outline of the Talk:
1、Black hole: Membrane Paradigm
2、 AdS/CFT: Incompressible fluid in AdS boundary
3、Gravity holography: Incompressible fluid in a finite
cutoff time-like hypersurface

6. Black hole is a window to quantum gravity
Thermodynamics of black hole：
（S.Hawking, 1974, J. Bekenstein, 1973)

7. Entropy in a system with surface area A: S<A/4G
Holography of Gravity
Entropy in a system with surface area A: S<A/4G
           
(G‘t Hooft)
(L. Susskind)
The world is a hologram？

8. AdS/CFT correspondence (J. Maldacena, 1997)
IIB superstring theory on AdS5 x S5
N=4 SYM Theory
“Real conceptual change
in our thinking about Gravity.”
(E. Witten, Science 285 (1999) 512)

9. 1、Black Hole： Membrane Paradigm gr-qc/9712077
Stretched Horizon:
a time-like surface just
outside the true horizon
On the true horizon:
null generator:
On the stretched horizon:

10. 2+1+1 split of spacetime:
4-metric:
3-metric:
2-metric:

11. The Einstein-Hilbert action:

12. The stretched horizon can be thought of as a fluid membrane,
obeying the Navier-Stokes equation.
To see this, take the limit:

13. Decompose this into a traceless part and a trace
This is just the equation for the stress of a two-dimensional
viscous Newtonian fluid with

14. Damour-Navier-Stokes equation:
The flux of the momentum into black hole
The focusing equation for a null geodesic congruence .

15. This equation can be explained as the heat transfer equation
for a two-dimensional fluid

16. In any dimension D case:
The membrane stress tensor:
This can be regarded as a (D-2)-dimensional viscous fluid:

17. The entire stress-tensor on the stretched horizon can be
expressed as:
The evolution of the energy density:
From the stretched horizon to the true horizon:
here some regularization is needed.

18. In this case:
Further generalization, for example, Gauss-Bonnet gravity.

19. From Damour-Navier-Stokes to Navier-Stokes: 1012.0119
In DNS: Lie derivative
In NS: convective derivative
Padmanabhan argued that it is better to study the structure
of gravitational field equations in freely falling frame rather
than in arbitrary coordinates and showed that in a locally
inertial frame, the DNS equation becomes identical to NS.

20. The incompressible NS equations from black hole
membrane dynamics:
Incompressible NS equation:

21. Consider a (d+2)-dimensional bulk with coordinates:
The metric:
Let H is a (d+1)-dimensional null hypersurface characterized
by the null normal vector:
Define the null hypersurface in the bulk by
with coordinates:

22. In this coordinates, one has:
One may choose the normalization:
Then identify the remaining components with the fluid
velocity:
The black brane in AdS spacetime:
The black brane has a horizon at:

23. Now consider slowly varying perturbations of the static solution,
and we are working to leading order in derivative expansion:
This is equivalent to the non-relativistic motion limit:
To the leading order, the solution becomes:

24. On the horizon:
At zero order:
Consider the Lie transport of the horizon along the null
normal vector n:
Decompose it into a trace
part and trace free part:

25. To the leading order, one has
Extrinsic curvature:
Parameterize the surface gravity as
P(x): the pressure of the fluid

26. Now consider the Einstein equation:
One has the focusing equation:
In the vacuum case, the leading order of the above equation:

27. Momentum conservation equation:
In the vacuum case, the leading order terms are at

28. 2、AdS/CFT: Incompressible fluid in AdS boundary
S. Bahttacharyya ,et al., ,
M. Rangamani,
AdS/CFT: fluid at AdS boundary
The dynamics of relativistic hydrodynamics

29. The dynamics of AdS gravity
The dynamics of nearly equilibrated systems
Given the AdS/CFT correspondence, in the long wavelength
regime, Einstein equations must be reduced to the equations
of hydrodynamics.

30. Relativistic fluid dynamics:

31. The Navier-Stokes scaling limit:
The equations of relativistic (or any other compressible) fluid
dynamics reduce to a universal form under a combined low
amplitude and long wavelength scalings.
1) The fluid is non-relativistic, v<< c,
2) The fluid is incompressible; v<< c_s
3) The temporal component
4) The spatial components NS equation:

32. The perturbative construction of gravity solutions:
The seed solution: AdS_{d+1} Schwarzschild black hole
The global thermal equilibrium state. The boundary coordinates:
One can generate a d-parameter family of solutions by
boosting the solution along the translationally invariant
spatial directions y^i, leading to a solution parametreized
by a normalized timelike velocity field

33. The boosted AdS black brane in ingoing coordinates:
We cannot specify any slowly varying T(x) and u(x). A true
solution to Einstein’s equation is obtained only when the
functions T(x) and u(x) in addition to being slowly varying
satisfy a set of equations which happens to be precisely the
conservation equations of fluid dynamics.
The corrected solution up to second
order in derivatives is available .

34. Stress tensor of dissipative fluid:
Transport coefficients:

35. 3、Gravity holography: incompressible fluid at a finite cutoff
time-like hypersurface
: From Navier-Stokes to Einstein
: Non-relativistic fluid dual to a asymptotically
AdS gravity at finite cutoff surface
: Holographic Forced Fluid Dynamics in
Non-relativistic Limit
: Incompressible Navier-Stokes Equations from
Einstein Gravity Chern-Simons Term
: Petrov I Condition and Dual Fluid Dynamics

36. The hydrodynamical limit and the epsilon expansion
Scaling symmetry:
Higher order

37. (p+2)-dimensional Einstein equations and
(p+1)-dimensional incompressible Navier-Stokes Equations
Ingoing Rindler coordinates:
Consider a timelike hypersurface
at r=r_c, its intrinsic metric is flat

38. Bulk solution:

39. Dual fluid:
The induce metric on the cutoff surface:
The extrinsic curvature:

40. To solve the Einstein equations:
The first nontrivial equation appears at order e^2:
Take this to be the case, at order e^3:

41. Summary:
Consider the portion of Minkowski spacetime between
a flat hypersurface Σ_c, given by the equation,
X^2-T^2=4r_c, and its future horizon H^+, the null
surface X=T.
2) Keep the intrinsic metric flat, and the effects of finite
perturbations of the extrinsic curvature of Σ _c can
be studied.
3) A regular Ricci flat metric exists provided that the Brown-
York stress tensor on Σ _c is that of an incompressible
Navier-Stokes fluid.

42. 4) More precisely, they work in a hydrodynamic
non-relativistic limit and construct the bulk metric
up to third order in the hydrodynamic expansion.
5) This provides a potential example of a holographic
duality involving a flat spacetime.

43. Key results:
a) The metric up to order e^2 with constant velocity and
pressure fields is actually flat space in disguise: it can be
obtained from the Rindler metric by a linear coordinate
transformation combining a boost .
b) To extend the solution to next order, promote the velocity
and pressure to be spacetime dependent quantities, subject
to the requirement that the Einstein equations hold up to e^3.
c) To satisfy this requirement, one needs introduce new terms
of order e^3 in the metric. One finds the following holds.

44. d) Einstein equations can in fact be satisfied to arbitrarily
high order in epsilon, by adding appropriate terms to the
metric and modifying the NS equations and the
incompressibility condition by specific higher derivative
corrections. E gets corrected at even powers of e and
E_i get modified at odd powers of e.

45. AdS gravity ：Non-relativistic case， 1104.3281
The motivation is two-fold: 1) if there exists a bulk stress
tensor; 2) AdS/CFT correspondence
In order to consider a (p+1)-dimensional fluid in a flat
spacetime, consider a (p+2)-dimensional bulk
Induced metric:

46. Rescaling to Minkowski spacetime
Consider two finite diffeomorphism transformations

47. The second one:
here k(r) is a linear function as k(r)= br +c
This metric still solves the corresponding gravitational field
equations, but if we promote v_i and delta k(r)=k(r)-r
=(b-1)r +c to be dependent on the coordinates x^a, then the
resulted metric is no longer an exact solution.

48. In order to solve the gravity equations, we take the so-called
hydrodynamics expansion and non-relativistic limit:
We demand both (b-1) and c scale as e^2, then up to e^2,

49. The resulted metric: to order e^2

50. If we take
then …..
Now we consider
Consider the case:

51. The seed metric only solves the gravity equations at O(e^1).
At the order O(e^2)
should be added to the bulk metric. To be regular at the
horizon, one needs F(r_c)=0.

52. Then the final metric at O(e^2)

53. The Einstein gravity with a negative cosmological constant
The black brane solution
At the order O(e^2)

54. It solves the gravity equations at order e^2. take an example,
consider p=3.
The BY stress tensor:

56. At order e^2,
At order e^3,
When r_c infinity, one takes

58. From T^(2)

59. The Gauss-Bonnet case
The black brane metric:

60. In this case
The BY stress tensor:

62. At order e^2
At next order

64. Thank You !