Rong-Gen Cai （ 蔡荣根 ）

1. 基金委暗能量及其基本理论高级研讨班，2012.4.7-17，杭州基金委暗能量及其基本理论高级研讨班，2012.4.7-17，杭州 Einstein Equations & Navier-Stokes Equations Rong-Gen Cai （蔡荣根） Institute of Theoretical Physics Chinese Academy of Sciences

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

3. AdS/CFT 和七个千禧年问题: Clay Mathematics Institute (http://www.claymath.org/millennium) The seven Millennium Prize Problems (US$7 million): • Birch and Swinnerton-Dyer Conjecture • 2) Hodge Conjecture • 3) Navier-Stokes equations • 4) P vs NP • 5) Poincare Conjecture • 6) Riemann Hypothesis • 7) Yang-Mills theory

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: • Nonrelativistic case: vacuum Einstein gravity • 2. Nonrelativistic case: AdS gravity • 3. Relativistic case

6. 1. Vacuum Einstein Gravity Refs: 1101.2451: From Navier-Stokes to Einstein 1103.3022: The holographic fluid dual to vacuum Einstein gravity 1105.4482: Higher curvature gravity and the holographic fluid dual to flat spacetime

7. a) The hydrodynamical limit and the epsilon expansion Scaling symmetry: Higher order

8. (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

9. Consider a Minkowski space in Cartesian chart The following region is often called Rindler wedge Defining a coordinate transformation Then in the Rindler chart, the Minkowski space turns to be:

10. The Rindler coordinate chart has a coordinate singularity at x = 0. The acceleration of the Rindler observers diverges there. As we can see from the figure illustrating the Rindler wedge, the locus x = 0 in the Rindler chart corresponds to the locus in the Cartesian chart, which consists of two null half-planes. Unruh temperature: for a uniformly accelerating observer with acceleration a

11. Bulk solution: 1). Leading order

12. 2) Nonlinear solution in the epsilon expansion Consider the metric:

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

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

15. Summary: • Consider the portion of Minkowski spacetime between • a flat hypersurface Sigma_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 Sigma_c can • be studied. • 3) A regular Ricci flat metric exists provided that the Brown- • York stress tensor on Sigma_c is that of an incompressible • Navier-Stokes fluid.

16. 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.

17. A systematic construction of the bulk solution to all orders In 1103.3022, Skenderis et al present an algorithm for systematically reconstructing a solution of the (d+2) dimensional vacuum Einstein equations from a (d+1) dimensional fluid, extending the non-relativistic hydro- dynamical expansion of Bredberg et al in 1101.2451.

18. 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

19. 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.

20. 1) Equilibrium configurations Consider a class of Ricci flat spacetime with i) A hyprosurace with a flat induced metric ii) The Brown-York tensor on Sigma_c: takes the form of perfect fluid, and iii) they are stationary with respect to tau, and homogeneous in space

21. Ingoing Rindler metric There are only two infinitesimal diffeomorphisms yielding Metrics satisfying the above conditions (i),(ii), (iii) . Consider the following two finite diffeomorphisms: (i) The first is a constant boost: (ii) A linear shift and a rescaling

22. Then, one has The Brown-York stress tensor: And thermodynamics:

23. 2) Seed metric for near-equilibrium configurations View It solves the Einstein equations up to O(e^3), provided one imposes incompressibility

24. associated Brown-York stress tenor Expanding the stress tensor up to order e^2 :

25. 3) Constructing the solution to all orders The details are neglected! Sorry! Take an example to order O(e^3) The bulk solutions to order (O^5) are given. And the Correction to the BY tensor

26. at O(e^4)

27. at O(e^5)

28. Stress tensor:

29. Corrections to NS and incompressibility e^2: incom. Con. e^3: NS eq. The conservation of the BY tensor  Conservation of the BY stress tensor at order e^n is required In order to construct the bulk metric at the same order.

30. at order e^4: The incompressibility gets modified: at order e^5: The NS equations get corrected

31. Characterizing the dual fluid Energy density: Relativistic hydrodynamics for vanishing equilibrium energy density The constraint

32. Up to second order in gradients where The energy density in local rest frame

33. Determining the transport coefficients Perform the expansion of PI_{ab} in e to (e^6), and compare to the BY tensor given from the bulk solution Note that c_5 and c_6 both vanish at O(e^6), thus four of six coefficients can be determined.

34. Interesting questions: • 1) the correspondence extends beyond the hydrodynamics • regime? • string embedding? • How this correspondence changes if one adds a bulk • stress tensor or consider higher derivative corrections to • Einstein gravity? Will such changes modify the • properties of the dual fluid? • relativistic construction of the bulk metric? • 5) curved space? • 6) Should not be limited to flat or black hole spacetime, any • spacetime should also work by the equivalent principle.

35. Higher derivative gravity: 1105.4482, Chirco et al.)

36. General setup Induced metric: Bulk metric: Vacuum solution with ingoing Rindler metric:

37. Dual fluid: Near-equilibrium configurations: Two diffeomorphisms, namely a boost and the translation

38. Expanding the metric to O(e^2), one has the seed metric This metric is the unique singulaity-free solutions the vacuum Einstein equation up to O(e^3), provided

39. In GR, the momentum constrain equation on the surface can be expressed in terms of the BY tensor as

40. Consider the Gauss-Bonnet gravity Many of second order transport coefficients appear At order O(e^4).

41. The bulk solution is found to order O(e^5).

42. 2. Nonrelativistic case: AdS gravity (arXiv: 1104.3281, Cai et al) 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 (p+2)-dimensional bulk Induced metric:

43. Rescaling to Minkowski spacetime Consider two finite diffeomorphism transformations

44. 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 solutions.

45. 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,

46. The resulted metric: to order e^2

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

48. Then 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.

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

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