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
Einstein’s Field Equations:
R
1
 g  R  8 GT
2
Incompressible Navier-Stokes Equations:
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):
1) 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
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.
Outline of the Talk:
1. Nonrelativistic case: vacuum Einstein gravity
2. Nonrelativistic case: AdS gravity
3. Relativistic case
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
a) The hydrodynamical limit and the epsilon expansion
Scaling symmetry:
Higher order
(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
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:
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
Bulk solution:
1). Leading order
2) Nonlinear solution in the epsilon expansion
Consider the metric:
Dual fluid:
The induce metric on the cutoff surface:
The extrinsic curvature:
To solve the Einstein equations:
The first nontrivial equation appears at order e^2:
Take this to be the case, at order e^3:
Summary:
1) 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 BrownYork stress tensor on Sigma_c is that of an incompressible
Navier-Stokes fluid.
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.
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 hydrodynamical expansion of Bredberg et al in 1101.2451.
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
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.
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
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
Then, one has
The Brown-York stress tensor:
And thermodynamics:
2) Seed metric for near-equilibrium configurations
View
It solves the Einstein equations up to O(e^3), provided
one imposes incompressibility
associated Brown-York stress tenor
Expanding the stress tensor up to order e^2 :
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
at O(e^4)
at O(e^5)
Stress tensor:
Corrections to NS and incompressibility
The conservation of the BY tensor 
e^2: incom. Con.
e^3: NS eq.
Conservation of the BY stress tensor at order e^n is required
In order to construct the bulk metric at the same order.
at order e^4: The incompressibility gets modified:
at order e^5: The NS equations get corrected
Characterizing the dual fluid
Energy density:
Relativistic hydrodynamics for vanishing
equilibrium energy density
The constraint
Up to second order in gradients
where
The energy density in local rest frame
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.
Interesting questions:
1) the correspondence extends beyond the hydrodynamics
regime?
2) string embedding?
3) 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?
4) 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.
Higher derivative gravity: 1105.4482, Chirco et al.)
General setup
Induced metric:
Bulk metric:
Vacuum solution with ingoing Rindler metric:
Dual fluid:
Near-equilibrium configurations:
Two diffeomorphisms, namely a boost and the translation
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
In GR, the momentum constrain equation on the surface
can be expressed in terms of the BY tensor as
Consider the Gauss-Bonnet gravity
Many of second order transport coefficients appear
At order O(e^4).
The bulk solution is found to order O(e^5).
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:
Rescaling to Minkowski spacetime
Consider two finite diffeomorphism transformations
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.
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,
The resulted metric: to order e^2
If we take
then …..
Now we consider
Consider the case:
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.
Then the final metric at O(e^2)
The Einstein gravity with a negative cosmological constant
The black brane solution
At the order O(e^2)
It solves the gravity equations at order e^2. take an example,
consider p=3.
The BY stress tensor:
At order e^2,
At order e^3,
When r_c infinity, one takes
From T^(2)
The Gauss-Bonnet case
The black brane metric:
In this case
The BY stress tensor:
At order e^2
At next order
3. Relativistic case: vacuum Einstein gravity
Refs:
1201.2678 : The relativistic fluid dual to vacuum Einstein gravity
1201.2705 : The relativistic Rindler hydrodynamics
The stress tensor up to second order in gradients
The fluid dual to vacuum Einstein gravity with vanishing
Equilibrium energy density :
At first order in gradients
two coefficients: shear viscosity and bulk viscosity
At second order in gradients
For vacuum Einstein gravity:
Entropy current:
A general feature of system away from equilibrium is that
they posses an entropy current with non-negative divergence.
At equilibrium:
In hydrodynamics regime with vanishing energy density:
Five parameters
In the case with fluid dual to vacuum Einstein gravity
The entropy current:
with non-negative divergence.
Relativistic construction of near-equilibrium solutions
(1) Seed metric
Induced metric on the hypersurface
To obtain the zero-order seed metric, one does
Second, one performs the boost:
Associated BY tensor:
Integration scheme
We start with the zero order seed emtric
At the first order
from which one has eta=1, and xi’=0
At second order
Energy-momentum tensor
Entropy current
The constraint for
necessary positive
condition.
Thank You !