Understanding Einstein's Field Equations and Navier-Stokes Equations for Gravity and Fluid Dynamics

1. 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 ① ② 流体力学 热力学 dE =TdS 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, 2007 (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. 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:

7. 2+1+1 split of spacetime: 4-metric: 3-metric: 0 2-metric: 0

8. The Einstein-Hilbert action:

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

10. 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

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

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

13. Any dimensional case: 1107.1260 The event horizon H of a black hole in D dimensions is a (D-1)-dimensional null hyersurface generated by the null geodesic: The stretched horizon: generated by the time-like congruence: The unit normal to the stretched horizon is:

14. The induced metric on the stretched horizon: The induced metric on the (D-2)-dimensional space-like cross section of the stretched horizon: The extrinsic curvature of the stretched horizon:

15. The extrinsic curvature of the (D-2)-dimensional space-like cross section of the stretched horizon. Decompose this as The variation of the action:

16. 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: 0905.3638 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., 0712.2456, 0810.1545 M. Rangamani, 1107.5780 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 1101.2451: From Navier-Stokes to Einstein 1104.3281: Non-relativistic fluid dual to a asymptotically AdS gravity at finite cutoff surface 1202.4091: Holographic Forced Fluid Dynamics in Non-relativistic Limit 1208.0658: Incompressible Navier-Stokes Equations from Einstein Gravity Chern-Simons Term 1302.2016: 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: 1103.3022 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: