text:
Domains @cea.fr @vub.ac.berobi.peschanski
Time-dependent AdS/CFT correspondence and the Quark-Gluon plasma
Alice Bernamonti\address Theoretische Natuurkunde, Vrije Universiteit Brussel, and International Solvay Institutes
Pleinlaan 2, B-1050 Brussels, Belgium and Robi Peschanski\address Institut de Physique Théorique, CEA, and CNRS (URA 2306), 91191 Gif/Yvette Cedex, France Aspirant FWO; Alice.Bernamonti@vub.ac.berobi.peschanski@cea.fr
Experiments on high-energy heavy-ion collisions reveal the formation and some intriguing properties of the Quark-Gluon Plasma (QGP), a new phase of matter predicted by Quantum Chromodynamics (QCD), the quantum field theory of strong interactions. The phenomenological success of relativistic hydrodynamic simulations with remarkably weak shear viscosity, modeling QGP as an almost perfect fluid, are in favor of the occurrence of a strongly-coupled QGP expanding and cooling during the reaction. A derivation of these features in QCD at strong coupling is still lacking and represents a very intricate theoretical challenge. As a quite unique modern tool to relate these dynamical features to a microscopic gauge field theory at strong coupling, time-dependent realizations of the AdS/CFT correspondence provide a fruitful way to study these properties in a realistic kinematic configuration. Relating a 4-dimensional Yang-Mills gauge theory with four supersymmetries (which is a conformal field theory, CFT4) with gravity in Anti-de Sitter space in five dimensions (AdS5), the AdS/CFT correspondence provides a useful "laboratory" to study yet unknown strong coupling properties of QCD. Besides the interest of revealing new aspects of the AdS/CFT correspondence in a dynamical set-up, the application to plasma formation leads to non trivial theoretical properties, as we will discuss in the lectures. The highlights of the present lectures are:
1. Emergence of an (almost) perfect hydrodynamic fluid at late proper-times after the collision.
2. Duality between an expanding 4-dimensional plasma and a black hole moving radially in the bulk.
3. Intimate link between conformal hydrodynamics and Einstein's equations in the asymptotically AdS5 space.
4. Possibility of studying the far-from-equilibrium stage of a gauge field theory at early collisional proper-times.
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Id:espcrc2.tex,v1.22004/02/2411:22:11speppingExp \runtitleTime-dependent AdS/CFT correspondence and the Quark-Gluon Plasma \runauthorA. Bernamonti and R. Peschanski
1 Heavy-Ion reactions and the QGP
One of the most striking lessons one may draw [1, 2] from experiments on heavy-ion collisions at high energy (e.g. at the RHIC accelerator) is that fluid hydrodynamics seems to be relevant for understanding the dynamics of the reaction (see, for instance, the reviews [2] ). Indeed, the elliptic flow [3] describing the anisotropy of the low transverse momentum particles produced in a collision at non zero impact parameter implies the existence of a collective flow of particles. It agrees with the picture of an hydrodynamical pressure gradient due to the initial eccentricity in the collision111 Note that a preliminary analysis of recent LHC data on the elliptic flow at the Alice detector seems to confirm the validity of the hydrodynamic approach [4] . . Moreover, the hydrodynamic simulations which are successful to describe this elliptic flow are consistent with an almost "perfect fluid" behavior, i.e. a small viscosity over entropy ratio η/s [5] . This ratio is particularly interesting since it depends mainly on fundamental features of the fluid, being in particular independent of the particle density (see later, eq.(1)).
The validity of an hydrodynamical description assuming a quasi-perfect fluid behavior has been nicely anticipated by Bjorken222The introduction of hydrodynamics in the description of high-energy hadronic collisions has been originally proposed by Landau [6] , assuming "full stopping" initial conditions which result in a non boost-invariant solution or Landau flow (see [7] for a unified description of Bjorken and Landau flows). However "full stopping" initial conditions do not seem to agree with present day data. in Ref. [8] . The so-called Bjorken flow is based on the approximation of an intermediate stage of the reaction process consisting in a boost-invariant quark-gluon plasma (QGP) behaving as a relativistically expanding fluid. It is formed after a (strikingly rapid) thermalization period and finally decays into hadrons. A schematic description of the process in light-cone kinematics is shown in Fig.1. Boost-invariance can be justified in the central region of the collision since the observed distribution of particles is flat, in agreement with the prediction of hydrodynamic boost-invariance, where (space-time) fluid and (energy-momentum) particle rapidities are proven to be equal [8] (see later the discussion in section 3).
Figure 1: Description of QGP formation in heavy ion collisions. The kinematic landscape is defined by τ=√x20−x21 ; η=12logx0+x1x0−x1 ; xT={x2,x3}. The coordinates along the light-cone are x0±x1, the transverse ones are {x2,x3} and τ is the proper time, while η is called the "space-time rapidity".
The Bjorken flow was instrumental for deriving and predicting many qualitative and even semi-quantitative features of the quark-gluon plasma formation in heavy-ion reactions. It was later confirmed and developed through numerical simulations where various effects have been included, up to the full 4-dimensional structure of the hydrodynamic flow. However, since hydrodynamics is an effective mean-field approach, it says only little on the relation with the microscopic gauge field theory, i.e., in the present case, Quantum Chromodynamics (QCD). Some important questions remain unsolved, such as the reason why the fluid behaves like a (almost) perfect fluid, what is the small amount of viscosity it may require, why and how fast thermalization proceeds, etc…
The problem is made even more difficult by the strong coupling regime of QCD, which is very probably required, since a perturbative description leads in general to high η/s ratio. In fact, large collision cross-sections with transversally cut-off momenta are required in order to explain a small macroscopic shear viscosity. The mean free path and the average transverse momentum induced by the gauge field medium should be small, in order to damp the nearby force transversal to the flow measuring shear viscosity. In a classical physics context, one may write
ηs=ρpart×λfreeׯpavρpart×kB [≥14πℏkB] , (1)
where ρpart, (resp. ¯pav) is the particle density, (resp. average transverse momentum) and λfree the mean free-path. Both small mean free-path and low average transverse momentum are expected in the non-perturbative regime of QCD and more generally from a gauge theory at strong coupling. We also quote in formula (1) the AdS/CFT value [9] 1/4π in terms of the fundamental constants' ratio ℏ/kB, corresponding to the large coupling limit of the conformal field theory. This AdS/CFT ratio appears to be very small compared to all known non-relativistic fluids, as suggested by the inequality between square brackets. Indeed, the AdS/CFT value has been proposed as an absolute theoretical lower bound [10] . A discussion is in progress to fully confirm this statement or to find counter-examples in microscopic theories (see [11] and references therein).
In any case, a very small phenomenological viscosity ratio is confirmed; not meaning that viscosity effects should be neglected . Observations on the elliptic flow primarily, and other observables, allow one to give estimates of η/s which are within range of the AdS/CFT value (see, e.g. [4, 5] ).
It is thus interesting to use our modern (while still in progress) knowledge of non-perturbative methods in quantum field theory to fill the gap between the macroscopic and microscopic descriptions of the quark-gluon plasma produced in heavy-ion collisions. Lattice gauge theory methods are very useful to analyze the static properties of the quark-gluon plasma, but they are still unable to describe the plasma in collision.
Hence we are led to rely upon the new tools offered by the Gauge/Gravity correspondence and in particular on the most studied and well-known of its realizations, namely the AdS/CFT duality [12] between N=4 supersymmetric Yang-Mills theory and type IIB superstring in the large Nc approximation. The properties of the gauge theory in (physical) Minkowski space in 3+1 dimensions are in one-to-one relation with properties of the bulk theory, in the 10D target space of strings. For our applications, strong coupling features will be related by duality to the geometric structure of the 5D metric with Minkowski boundary, which is built on top of vacuum AdS. At weak coupling, the physical space corresponds to the world-volume of a stack of infinitely many (Nc→∞) joint D3-branes. Note that the overall consistency of the correspondence scheme will be required, even if only geometrical properties will be used in practice.
One should be aware, when using AdS/CFT tools, that there does not yet exist a gravity dual construction for QCD. Also, the relation between perturbative (weak-coupling) and non-perturbative (strong-coupling) aspects of QCD are left over in the correspondence, which conveniently describes the strong coupling regime of gauge theories uniquely. However, the nice feature of the quark-gluon plasma is that it is a deconfined phase of QCD, characterized by collective degrees of freedom. Thus one may expect to get useful information from AdS/CFT. This has already been proved in the description of static geometries, by the evaluation of η/s [9] . Other aspects of the QGP can be studied via the duality, in a static plasma configuration at fixed temperature.
Our specific aim and the subject of the present lectures is the investigation of the Gauge/Gravity duality in a dynamical time-dependent setting using the AdS/CFT correspondence. The goal is to describe a collision process in a strongly coupled gauge theory, including in particular the cooling of temperature accompanying the proper-time evolution of the fluid.
In section 2, we introduce the theoretical tool of holographic renormalization and apply it, as a warm-up exercise, to a static uniform plasma. Sections 3 and 4 present, respectively, the late and early proper-time analysis of the plasma evolution, with particular emphasis on boost-invariant flows.
2 AdS/CFT and strong gauge interactions
In the previous section, we mentioned the ubiquity of hydrodynamic methods in the description of QGP produced at RHIC. Yet, despite their success in describing the data, we have to keep in mind that they are used as a phenomenological model, without a real derivation from gauge theory. This is understandable, since almost perfect fluid hydrodynamics is intrinsically a strong coupling phenomenon, for which one lacks a purely gauge theoretical method.333 Lattice QCD methods do not work well here, as this would require analytical continuation to Minkowski signature which is nontrivial in this context [13] .
On the other hand, there exists a wide class of gauge theories, which can be studied analytically at strong coupling. These are superconformal field theories with gravity duals. In the unifying context of string theory, using the AdS/CFT correspondence, one is able to map gauge theory dynamics (CFT) at strong coupling and large number of colors into solving Einstein's equations in asymptotically anti-de Sitter space (AdS). The theories with gravity duals can differ substantially from real world QCD at zero temperature. The best known example of such theories - N=4 super Yang-Mills (SYM) - is a superconformal field theory with matter in the adjoint representation of the gauge group SU(Nc). Because of the conformal symmetry at the quantum level this theory does not exhibit confinement. On the other hand, differences between N=4 SYM and QCD are less significant above QCD's critical temperature, when quarks and gluons are in the deconfined phase. Moreover, it was observed on the lattice that QCD exhibits a quasi-conformal trend [14] in a range of temperatures T>300 MeV. There the equation of state is reasonably approximated by the conformal relation ϵ=3p, corresponding to a traceless energy-momentum tensor. The above observations, together with experimental results suggesting that QGP is a strongly coupled medium, are an incentive to use the AdS/CFT correspondence as a tool to get insight into non-perturbative dynamics.
2.1 AdS/CFT correspondence
We will now describe how to set up an AdS/CFT computation for determining the space-time behavior of the gauge field plasma corresponding to a Bjorken boost-invariant flow. The problem is formulated in terms of the proper-time evolution of the matter energy-momentum tensor on the CFT boundary [15] . This method does not make any underlying assumption about local equilibrium or hydrodynamical behavior. We will indeed obtain the hydrodynamic expansion as a generic consequence of the late time behavior of the expanding and strongly coupled plasma.
Suppose we consider some macroscopic state of the plasma characterized by a space-time profile of the energy-momentum tensor
Tμν(xρ) ,μ,ν,ρ={0,⋯,3} . (2)
Then, since the AdS/CFT correspondence asserts an exact equivalence between gauge and string theories, such a state should have its counterpart on the gravity side of the correspondence. Typically, it will be given by a modification of the geometry of the original AdS5×S5 metric. This follows from the fact that operators in the gauge theory correspond to fields in supergravity (or string theory). When we consider a state with a nonzero expectation value of an operator, the dual gravity background will have the corresponding field modified from its 'vacuum' value. In the case of the energy-momentum tensor, the corresponding field is just the 5D metric. One then has to assume that the geometry is well defined, i.e. it is free from naked singularities - singularities not hidden by an event horizon -. This principle selects the allowed physical space-time profiles of the gauge theory energy-momentum tensor. Thus this becomes the main dynamical mechanism for selecting the appropriate strongly coupled gauge theory from the dual Einstein equations. In practice, the geometry will be obtained from a "holographic renormalization" procedure starting from the boundary. The boundary conditions serve as initial conditions for the construction of the bulk features, the fifth dimension acting as a renormalization scale. Let us describe in detail this construction.
The Gauge/Gravity duality can be described as an "holographic" correspondence between the 4-dimensional physical space where the gauge theory lives and the 5-dimensional space where the supergravity (weak curvature) approximation of the 10-dimensional string theory is valid. It means qualitatively that the whole information should be the same on both sides of the correspondence, despite the difference in dimensionality. In practice, this notion of "holography" has a precise realization in terms of the "holographic renormalization" program [16] . Let us illustrate the holographic renormalization by simple examples, which will in fact be sufficient for the applications we have in mind.
Suppose that, when considering the presence of matter on the 4-dimensional physical space, our corresponding 5-dimensional geometry is parameterized by
ds2=gμν(xρ,z)dxμdxν+dz2z2≡g5Dαβdxαdxβ, (3)
where we adopt the Fefferman-Graham definition [17] of the 5D metric. The flat case gμν=ημν parametrizes AdS5 in Poincaré coordinates. The conformal boundary of space-time is at z=0.
Considering the general metric (3) from the point-of-view of the AdS/CFT correspondence, the following questions are in order:
i) What are the constraints imposed on gμν(xρ,z)?
ii) What is the corresponding energy-momentum profile ⟨Tμν(xρ)⟩?
The metric (3) has to be a solution of 5D Einstein's equation with negative cosmological constant444 One can show that such solutions lift to 10D solutions of ten dimensional type IIB supergravity. The effective 5D negative cosmological constant comes from the 5-form field in 10D supergravity.
Rαβ=12gαβ(R−12) . (4)
The expectation value of the energy momentum tensor may be recovered by expanding the metric near the boundary z=0. The "holographic renormalization" procedure [16] , for a Minkowskian boundary metric, gives555 In principle, in (5) also additional logarithmic terms appear (see [16] ). They are absent in all the cases we are going to examine.
gμν(xρ,z)=ημν+z4g(4)μν(xρ)+… , (5)
⟨Tμν(xρ)⟩=N2c2π2⋅g(4)μν(xρ) . (6)
The relations (5,6) can be used in two ways. Firstly, given a solution of Einstein's equations, we may read off the corresponding gauge theoretical energy-momentum tensor. Secondly, given a traceless and conserved energy-momentum profile, one may integrate Einstein's equations into the bulk in order to obtain the dual geometry666 This can be done order by order in z2, which is a near-boundary expansion. However potential singularities are hidden deep in the bulk, thus this power series needs to be resummed. . Then the criterion of non singularity of the geometry will determine the allowed space-time evolution of the plasma. Let us note that this formulation is in fact quite far from a conventional initial value problem.
2.3 Example: static uniform plasma
Before moving to the case of an expanding plasma, it is convenient to consider the simple situation of a static uniform plasma with a constant energy-momentum tensor. This will provide a useful exercise introducing the main tools for further applications.
Starting from a constant diagonal energy-momentum tensor
with p∥=p⊥ and energy density and pressure related by ϵ=3p, one has to solve the 5D Einstein's equations for an Ansatz
ds2=−ea(z)dt2+eb(z)d→x2+dz2z2 (8)
obeying the above-mentioned boundary conditions (5,6). Inserting (8) into the Einstein's equations (4) and splitting them into components, one gets
R11⇒−8a′+za′2−6b′+3za′b′+2za′′=0 ,
R55⇒−2a′+za′2−6b′+3zb′2+2z(a′′+3b′′)=0 , R22⇒−2a′−12b′+za′b′+3zb′2+2zb′′=0 .
R11−R55⇒a′=zb′2+2zb′′zb′−2, (9)
while combining it with the expression from R22 for a′ gives
−3b′2+zb′2+zb′′=0. (10)
From those last two equations, it is easy to obtain independent equations for a(z) and b(z) and their solutions. With appropriate normalizations at z=0, one gets
b ≡ log(1+z4z40)
a ≡ 2log(1−z4z40)−log(1+z4z40) , (11)
where z0 is the integration constant. Hence, Einstein's equations can be exactly solved analytically in this case and we find [15] that the exact dual geometry of such a system is
ds2=−(1−z4z40)2(1+z4z40)dt2z2+(1+z4z40)d→x2z2+dz2z2. (12)
This metric may look at first glance unfamiliar, but a change of coordinates
~z≡z√1+z4z40 (13)
transforms it to the standard metric form of the AdS-Schwarzschild static black hole777 The identification of the metric (12) as the dual of a static fluid at fixed temperature has previously been made in [18] using the approach 'from bulk to boundary'. This is the reverse way w.r.t. holographic renormalization which proceeds 'from boundary to bulk'. The interest of holographic renormalization is that it allows a construction of the dual geometry starting only from 4-dimensional physical data.
~z2ds2=−(1−~z4~z40)dt2+d→x2+11−~z4~z40d~z2, (14)
with ~z0=z0/√2 being the location of the horizon. Before we proceed further, let us note here one crucial thing: the fact that the dual geometry of a gauge theory system with constant energy density is a black hole was not an assumption, but rather an outcome of a computation.
The Hawking temperature
T=1π~z0≡√2πz0 (15)
is then identified with the gauge theory temperature, and the entropy with the Bekenstein-Hawking black hole entropy
S=N2c2π~z30=π22N2cT3 (16)
which is 3/4 of the entropy at zero coupling. This identification between the black hole characteristics and the equilibrium thermodynamics of the plasma reveals a striking correspondence between the near-horizon and boundary properties of the gauge/gravity system.
To finish our discussion of the static black hole, we note that the Fefferman-Graham coordinates cover only the part of space-time lying outside of the horizon. It will appear useful to introduce different coordinates allowing one to go inside the horizon, such as the Eddington-Finkelstein's coordinate system. Through the time coordinate redefinition
u≡t−14~z0(2arctan~z~z0+log~z0+~z~z0−~z) , (17)
the metric (14) becomes:
~z2ds2=−(1−~z4~z40)du2+d→x2+2dud~z . (18)
Now the metric is well defined at ~z=~z0 and the horizon can be smoothly crossed. Note that the time coordinate gets an infinite shift near the horizon, here being harmless, but which reveals to be quite subtle for the time-dependent backgrounds we will consider now.
3 AdS/CFT and late time quark-gluon plasma flow
In this part, we describe applications of the methods introduced in section 2 to the analysis of plasma dynamics in strongly coupled gauge theories. The time dependence of the system introduces considerable computational difficulties and it is in general a hard task to achieve a detailed quantitative description of the plasma evolution. Nevertheless, significant information on the dynamics can be extracted working in simplified and symmetric, but realistic setups. We shall consider specific regimes of the problem in exam, but some general lessons can still be drawn about the highly nonlinear regime of the plasma expansion.
Throughout the next sections we will model the products of heavy-ion collisions with a N=4 SYM plasma at finite temperature. We mainly focus on the results obtained in Ref. [15] concerning the late proper-time behavior of the quark-gluon plasma. Recent and detailed reviews on the subject are also Refs. [19] and [20] .
The structure of the sections is organized in the following way. In section 3.1 we review briefly the "Bjorken flow" description of the central rapidity region of heavy-ion collisions. This serves as an introduction to section 3.2, where we present the gauge theory setup which has been used in [15] to model the products of heavy-ion reactions. The description is in fact inspired by the hydrodynamic description of heavy-ion collisions by Bjorken [8] and based on the central assumption of boost invariance. This hypothesis seems experimentally to be valid in the central rapidity region of the heavy-ion collision process. In section 3.3, we review the computation of the holographic dual geometry of the plasma system in the asymptotic limit of large proper-time. The properties of this class of time-dependent space-times are investigated in section 3.4. The requirement of the absence of naked singularities in the bulk selects uniquely the gravity dual of a perfect fluid. This non-singular solution can be interpreted as a black hole moving off from the AdS boundary in the radial direction, as a function of proper-time. We conclude with sections 3.5 and 3.6, where we present attempts of going beyond the assumptions of perfect hydrodynamics and boost invariance.
3.1 Hydrodynamic Bjorken flow
The introduction of relativistic hydrodynamics in the description of high-energy hadronic reactions is due to Landau in a remarkable premonitory paper [6] . However, the experimental evidence of a connection between heavy-ion collisions and relativistic hydrodynamics has been first modeled by Bjorken in [8] . In this description, boost-invariance of the plasma arises as an outcome of the assumption of hydrodynamics, in a semi-classical approximation where the fluid follows ballistic trajectories or, in other terms, the momentum-space rapidity is equal to the space-time one (see Fig.2). The model is therefore well suited to parametrize the central region of ultra-relativistic collisions, for which the particle distribution is nearly flat as a function of rapidity.
Figure 2: Semi-classical fluid trajectories: the In-Out cascade. The rapidity y is equal to the space-time rapidity η=12logx+x−.
In this section, we review a number of aspects of this picture. The summary is intended to be useful in the follow-up to motivate the assumptions of the AdS/CFT models we discuss and to interpret the physical results we obtain.
For the discussion that follows, it is convenient to parametrize flat four-dimensional space-time of coordinates (t,x1,x⊥), where x⊥=(x2,x3), in terms of the light-cone coordinates (x+,x−,x⊥). The latter are defined by x±≡t±x1, in which the flat space-time metric reads
ds2=−dx+dx−+dx2⊥. (19)
The proper-time τ and space-time rapidity η of the fluid are related to the light-cone coordinates by x±≡τe±η.
In the first stages of the collision, a rather dense interacting medium is created and the individual partonic or hadronic degrees of freedom can be neglected to a good approximation. This observation justifies the treatment of the medium as a fluid and, moreover, allows the assumption of local equilibrium. In the simplest perfect fluid approximation, the energy-momentum tensor reads
Tμν=(ϵ+p)uμuν−pημν, (20)
where ϵ, p and uμ are the energy density, pressure and 4-velocity of the fluid, respectively; while ημν is the flat metric of expression (19). We further assume an equation of state of the form
ϵ=gp, (21)
where 1/√g is the sound velocity of the liquid. In the case of a conformal theory, the traceless condition Tμμ=0 implies g=3 in four dimensions. The continuity equation ∂μTμν=0 for a perfect fluid of energy density and pressure related by (21) reduces to two equations
g∂+(lnp) = −(1+g)22∂+p−g2−12e−2y∂−y
g∂−(lnp) = (1+g)22∂−p−g2−12e2y∂+y (22)
for the pressure and the rapidity
y≡12ln(ϵ+pϵ−p) (23)
of the fluid. The other thermodynamical quantities can be derived from the equation of state (21) and the standard thermodynamical identities
p+ϵ=Ts,dϵ=Tds, (24)
where T and s are the temperature and entropy density of the fluid and we have assumed, for simplicity, vanishing chemical potential.
The result is
p=ϵg=p0Tg+1,s=s0Tg∝pgg+1, (25)
for constant p0 and s0.
The key observation is that the expansion of the fluid can be described by a semi-classical picture. In fact, the high occupation numbers of the system allow to assume that the fluid components follow quasi-classical trajectories in space-time, here straight-line trajectories starting at the origin (see Fig.2), and given by
y=η. (26)
Plugging the Ansatz (26) into (3.1), one obtains
g∂+(lnp)=−1+g2x+,g∂−(lnp)=−1+g2x−, (27)
from which it easily follows
lnpp0=1+g2logx+x− ⇒ p(τ)=p0τ1+gg . (28)
The simple Ansatz (26) thus leads to a boost-invariant system. Such a physical description is suitable to model the central rapidity region of highly relativistic heavy-ion collisions, where a central plateau in the distribution of particles is detected.
Specifically, in the perfect fluid four-dimensional case, the behavior is
p(τ)=ϵ(τ)3=p0τ43∼T4(τ),s∼T3(τ) . (29)
Note that the entropy density verifies s=s0τ−1. Hence the total entropy S per unit of rapidity (and transverse area) is constant [8] (as expected from a perfect fluid) since
dS≡∫sd3x=sτ∫d2x⊥dy . (30)
3.2 Boost-invariant flow
Inspired by Bjorken's analysis, the methods provided by the AdS/CFT correspondence have been applied to study the dynamics of boost-invariant strongly interacting gauge theory matter in [15] . There are technical reasons for making this symmetry assumption, due to the complication of the Einstein's equations in the gravity dual. On a more theoretical level, using the N=4 SYM theory as a substitute to QCD seems at first sight highly problematic. Moreover, experimentally the assumption of boost invariance is not optimal, since the observed multiplicity distribution of particles in heavy-ion collisions is rapidity-flat only in the central region. In terms of the hydrodynamic modelization, this means that the entropy is rapidity-independent, and thus obeying boost-invariance, only in the same region.
All these warning remarks have to be certainly taken into account, when using the AdS/CFT correspondence in a physical context. However, there are convincing arguments to show that for appropriate observables and by using AdS/CFT methods, one obtains a first fruitful approach to the behavior of QCD at strong coupling. From a phenomenological point-of-view, boost-invariance is widely used in modeling the relativistic heavy-ion collisions at RHIC in the central region of rapidity, where the hydrodynamic approach seems successful in describing the data and the occurrence of the quark-gluon plasma. Moreover, this plasma is a non-confining phase of QCD for which no specific hadronic scale seems to be relevant. In a more theoretical language, in this regime QCD appears to be nearly conformal. The above arguments suggest that it is reasonable to apply boost-invariance, at least in a first approximation, and to study the dynamics of the N=4 SYM plasma under consideration.
In a boost-invariant setup, it is natural to parametrize the flat four-dimensional boundary in terms of the coordinates (τ,η,x⊥), where τ is the proper-time, η the space-time rapidity and x⊥=(x2,x3) denotes the transverse directions. We already introduced these coordinates in the previous section, but it is worth being more explicit here. In terms of the Minkowski coordinates (t,x1,x⊥), the definition of τ and η reads
t=τcoshη,x1=τsinhη, (31)
and the metric has the form
ds2=−dτ2+τ2dη2+dx2⊥. (32)
For simplicity, further assume independence on the transverse coordinates x⊥, corresponding to the limit of infinitely large nuclei colliding.
The energy-momentum tensor Tμν of the gauge theory can then be written in diagonal form and has only three non-vanishing components: Tττ, Tηη and Tx2x2=Tx3x3, which depend on proper-time. The conservation condition DνTμν=0 and, since we are dealing with a conformal theory, the tracelessness condition Tμμ=0 further restrict the form of the energy-momentum tensor. It can be shown that all components can be expressed in terms of a single function ϵ=ϵ(τ):
Tμν=diag(ϵ,−τ2(τϵ′+ϵ),ϵ+τϵ′2,ϵ+τϵ′2). (33)
The function ϵ(τ) can be interpreted as the energy density of the plasma at mid-rapidity (i.e. at x1=0) as a function of proper-time. The last kinematic constraint on the stress-energy tensor is the positive energy condition. For any time-like vector tμ, the energy density in the reference frame whose time-like direction is specified by tμ should be non-negative
Tμνtμtν≥0. (34)
For the Ansatz (33), it implies
ϵ(τ)≥0,ϵ′(τ)≤0,τϵ′(τ)≥−4ϵ(τ). (35)
The explicit expression of ϵ specifies the dynamics of the theory and it is the information we aim to determine through the AdS/CFT correspondence.
In the following, we restrict the analysis to the class of gauge theories whose late proper-time evolution is specified by a power-law parameter s, such that
ϵ(τ)∝1τs. (36)
The positive energy condition (34) fixes the range of possible powers s to 0≤s≤4.
Notice that the family of gauge theories described by (36), includes a number of cases of physical interest. If the longitudinal pressure Tηη vanishes, see (33), the plasma is in a free streaming phase which is expected to be well suited to the case of a weakly coupled plasma [21] . Solving for ϵ in (33) leads to ϵ(τ)∝τ−1. If the gauge theory plasma behaves, instead, as a perfect fluid, the energy-momentum tensor has to be of the form (20) with ϵ=3p. Under the assumption of boost invariance, uμ=(1,0,0,0) and the comparison with (33) leads to the Bjorken solution of section 3.1: ϵ(τ)∝τ−4/3. The case ϵ(τ)∝τ0 is also of interest, it describes a fully anisotropic medium with negative longitudinal pressure p∥≡Tηη/τ2=−ϵ=−p⊥ that might be relevant for the early times dynamics at strong coupling, as advocated in [22] . As we shall see in section 4, the status of early times dynamics in the AdS/CFT correspondence seems to be more complex.
While we will deal in detail with the dual description of the early-time flow in the following, in the present part we restrict the analysis to the range 0<s<4. In what follows, we will determine which, if any, of the above behaviors is relevant for the description of the large proper-time evolution.
3.3 Large proper-time behavior of the gravity dual
In this section, following the procedure of holographic renormalization introduced in section 2.2, we repeat the construction of [15] of the dual geometries to the boost invariant gauge theories with energy densities of the form (36). Since the bulk metric shares the same symmetries of the plasma system, the most general Ansatz in Fefferman-Graham coordinates reads
z2ds2 = −ea(τ,z)dτ2+τ2eb(τ,z)dη2 (37)
+ec(τ,z)dx2⊥+dz2.
According to the holographic dictionary, to determine the metric components a(τ,z), b(τ,z) and c(τ,z), Recalling Einstein equations with a negative cosmological constant
Rαβ=12gαβ(R−12) (38)
one has to solve them in a power series expansion in the radial coordinate z. The indices α,β in (38) denote the bulk coordinates (τ,z,η,x⊥) and the metric components are subject to the boundary conditions
a(τ,z)=−z4ϵ(τ)+z6a6(τ)+z8a8(τ)+… (39)
around z=0 and similar ones for b(τ,z) and c(τ,z). By studying the explicit form of the Fefferman-Graham expansion, it can be shown that the large τ asymptotics of the exact solutions can be obtained analytically (see [15] for the details of the computation). In fact, introducing the scaling variable
v=zτs4, (40)
the exact solutions have an expansion of the form
a(τ,z)=a(v)+O(1τα), (41)
for some positive power α. The asymptotic equations, obtained from (38) taking the limit τ→∞, while keeping v fixed, can then be solved exactly. The solution is
a(v) = A(v)−2m(v)
b(v) = A(v)+(2s−2)m(v) (42)
c(v) = A(v)+(2−s)m(v)
A(v) = 12ln(1−Δ2(s)v8)
m(v) = 14Δ(s)ln1+Δ(s)v41−Δ(s)v4 (43)
Δ(s) = √3s2−8s+824.
3.4 Selection of the perfect fluid
Let us now examine in more detail the geometry we obtained. The dual asymptotic metric of the boost-invariant plasma has a potential singularity for v=Δ−1/4(s), where the argument of the logarithm in (3.3) vanishes.
As usual, to distinguish a physical singularity from a coordinate one, it is sufficient to construct a scalar invariant formed out of the Riemann tensor which diverges at the singularity location. While the curvature scalar is everywhere regular, the simplest non-trivial scalar that can be considered is the square of the Riemann tensor (or Kretschmann scalar)
R2≡RαβγδRαβγδ. (44)
In the asymptotic limit τ→∞ with v fixed, the general structure of this expression becomes
R2=N(v,s)(1−Δ2(s)v8)4+O(1τβ), (45)
for a positive β. The explicit expression of the numerator N(v,s) is quite complicated and can be found in [15] . For generic s, the above asymptotic expression for R2 diverges at v=Δ−1/4(s) and the singularity is a physical naked one. It turns out that only for s=4/3 the fourth order pole of the denominator gets cancelled by an identical contribution in the numerator leaving a bounded scalar invariant. Only the physical singularity in the dual geometry of a perfect fluid (which will be found at z=∞, see later on) can be hidden behind an event horizon.
The above analysis suggests a criterion to select the profile ϵ(τ) in (33). The proposal of [15] is that a physical boundary energy-momentum tensor has to be dual to a geometry satisfying the cosmic censorship. It then turns out that the requirement of the absence of naked singularities in the bulk uniquely selects perfect hydrodynamics as the large proper-time dynamics of the strongly coupled plasma.
The scaling variable (40) and the energy density for a perfect relativistic fluid are
v=zτ13,ϵ(τ)=e0τ43, (46)
where we have reinstated a dimensionful parameter e0 in ϵ. In this case, the metric coefficients simplify considerably and the asymptotic geometry can be written in the compact Fefferman-Graham form
z2ds2 = −(1−e03z4τ4/3)21+e03z4τ4/3dτ2+dz2
+(1+e03z4τ4/3)(τ2dy2+dx2⊥).
This geometry is analogous to the static AdS black brane solution of section 2.2 and thus the potential singularity is indeed a coordinate one. Here, however, the horizon position depends on proper-time
z0=(3e0)14τ13. (48)
Although a precise notion of temperature and entropy still lacks in a dynamical setup, in order to obtain a qualitative estimate one could naively generalize the static formula to infer
T=√2πz0=(e03)14√2πτ13 (49)
for the temperature and the total entropy S∝sτ=const., see (30). Observe that these estimates are in agreement with the "Bjorken flow" expressions of equation (29). Since the temperature of the black hole coincides with the one of the plasma, the gravity dual of the plasma cooling off during expansion is in terms of a black hole which moves off in the radial direction of the bulk, from the boundary towards the interior. A similar picture had already been qualitatively proposed in [23] .
3.5 In-flow viscosity and relaxation time
It should be stressed that the results of section 3.4 were obtained in the scaling limit. The gravity dual selects perfect hydrodynamics only in the strict asymptotic limit of infinite proper-time. Indeed, as shown for instance in [19] and [20] , it is straightforward to repeat the computation of the square of the Riemann tensor up to the first subleading corrections in the metric and check that it is always singular for perfect fluid dynamics. The evaluation of R2 leads to
R2=R0(v)+1τ43R2(v), (50)
where R0(v) is finite, but R2(v) develops a 4th order singularity. From the gravity side, it is then clear that the energy density in (46) needs to be corrected. It turns out that these corrections are due to viscous hydrodynamics, but it is instructive to see how the result arises using purely AdS/CFT methods. Assume that
ϵ(τ)=1τ43(1−2η0τr) (51)
for a generic positive exponent r. Solving Einstein's equations with this boundary condition and computing the Kretschmann scalar yields a result of the form
R2=R0(v)+1τrR1(v) + 1τ2r~R2(v) (52)
+ 1τ43R2(v),
where the last two terms are always singular. To obtain a bounded Kretschmann scalar, the only possibility is to make those two terms proportional to each other and therefore to set r=2/3. This is exactly the scaling of a viscosity correction to Bjorken hydrodynamics with shear viscosity η=η0/τ (which follows from η∝T3 in the N=4 SYM case). Moreover, in [24] it has been shown that, to implement the cancellation of the potentially singular terms, the coefficient η0 has to be tuned to η0=2−1/23−3/4. This specific value corresponds to a shear viscosity to entropy ratio
ηs=14π. (53)
It is remarkable that the above result, obtained in a non-linear and dynamical setup coincides with the shear viscosity value obtained in [9] , studying the response of a static plasma at fixed temperature to small perturbations.
The procedure can be further generalized to higher orders to compute the other hydrodynamics coefficients. In general, the metric components a(τ,z), b(τ,z) and c(τ,z) in (37) have an expansion of the form
a(τ,z)=∑nan(v)1τ2n3, (54)
while the one of the curvature invariant is
R2=∑nRn1τ4n3. (55)
The 2nd order formalism is obtained truncating the sums above at n=3 and has been worked out in [25] (with a correction given in Ref. [26] , see further on). It results into a relaxation time
τr=2−ln22πT, (56)
which is about thirty times shorter than the one estimated on the basis of Boltzmann's kinetic theory. The value (56) has been corrected by matching the results for the Bjorken flow of Ref. [25] with the full second-order formalism for hydrodynamics in conformal field theories at finite temperature derived in [26] .
Surprisingly, at this order, a new feature arises, namely the presence of a leftover logarithmic singularity in the expansion of R2. In [25] , it has been argued that such a divergence could be cured considering the coupling of a dilaton to the bulk metric. A non vanishing dilaton profile turns on, on the gauge theory side, a non-zero expectation value of TrF2, meaning that electric and magnetic modes are no longer equilibrated. The tuning of the dilaton field to achieve a non-singular Kretschmann scalar, leads to a negative expectation value ⟨TrF2⟩<0, implying that electric modes dominate.
However, the logarithmic divergences that arise at NNLO in the energy density expansion, have been considered more extensively in [27] and [28] (see also [29] and [30] ). The conclusion is that the logarithmic singularity origins in a pathology of the Fefferman-Graham coordinates. There exists in fact a singular transformation to Eddington-Finkelstein coordinates, which yields a regular and smooth metric from the boundary up to the standard black brane singularity. In this coordinate system, the late proper-time expansions of curvature invariants is regular at the horizon. In order to proceed to higher orders, one would then have to perform the analysis in Eddington-Finkelstein coordinates.
3.6 Duality beyond boost-invariance
The efficacy of the approach that we presented in the previous sections resides in that no assumption about the dynamics of the plasma has to be made. It is the gravity dual analysis alone that selects the physically relevant evolution. However, in order to solve the full system of Einstein's equations one is in general constrained to work in highly symmetric setups and in specific regimes of interest. In particular, it is very hard to relax the assumption of boost invariance of the gauge theory matter. Significant progress in this direction has been achieved through a different approach in [31] , where it has been shown in general how hydrodynamics arises from the gravity side.
The starting point of the analysis is the five-dimensional static AdS black brane which is dual to a strongly coupled plasma at rest (i.e. with flow vector uμ=(1,0,0,0)) and at temperature T. In incoming Eddington-Finkelstein coordinates, the metric reads
ds2 = −r2[1−(Tπr)4]dv2+2dvdr (57)
+r2ηijdxidxj,
where the coordinate r is related to the radial coordinate z by r=1/z. The conformal boundary of space-time is at r=∞ and the horizon at r=T/π. This choice of the coordinate system is consistent with the comments of the end of section 3.5; these coordinates are well defined on the horizon and extend all the way from the boundary to the black brane singularity.
By performing a boost, one can obtain the dual geometry to a uniformly moving plasma with 4-velocity uμ
ds2 = −r2[1−(Tπr)4]uμuνdxμdxν
−2uμdxμdr+r2(ημν+uμuν)dxμdxν.
The main idea of [31] is to promote T and uμ to be slowly varying functions of the boundary Minkowski coordinates. In this way, the geometry ceases to be an exact solution of Einstein's equations because of the appearance of gradients of the temperature and of the velocity. The metric (3.6) can be corrected order by order in a derivative expansion. The integration constants that arise at each order can be fixed requiring regularity of the metric at the horizon. The energy-momentum tensor of the plasma can then be read off from the corrected geometry and it is expressed, as the metric itself, in terms of T, uμ and their derivatives. Explicit results have been worked out up to second order and read
Tμν = 116πG5[(πT)4(ημν+4uμuν)
−2(πT)3σμν+(πT2)(log2Tμν2a
+(2−log2)(13Tμν2c+Tμν2d+Tμν2e))],
where G5 is the five dimensional Newton's constant and the rather lengthy expressions of the quantities σμν, Tμν2a, etc. can be found in the original paper [31] . The first term in Tμν is simply the perfect fluid energy-momentum tensor, the second term involves the shear viscosity, while the third is the contribution of second order hydrodynamics.
The conservation equation ∂μTμν=0 is equivalent, by definition, to the relativistic Navier-Stokes equation. Therefore, the dual geometry of every solution of the hydrodynamic equations is given by (3.6) plus correction terms up to the same order in the gradient expansion.
We should note, however, that this construction is adequate exclusively for near the equilibrium setups. Starting from the boosted black brane metric, in fact, one implicitly assumes the existence of a hydrodynamical description, in terms of flow velocity and energy density. As we explain in section 4, when dealing with processes that do not admit such a description, as the early stages of heavy-ion collisions, one has to recur to different methods.
4 AdS/CFT and early time quark-gluon plasma flow
One of the main theoretical challenges of the physics of quark-gluon plasma is to describe its thermalization process.
The plasma is created in the initial stages of heavy-ion collisions, during which the system is certainly out-of-equilibrium. However, simulations based on the RHIC data suggest that an hydrodynamical description becomes reliable at times <1 fm/c after the collision event [32, 33, 34] . Perturbative QCD calculations [33, 35, 36] lead to a much longer thermalization time τtherm≳2.5 fm/c, pointing in the direction of a strongly coupled scenario. As we saw, holographic methods are, up to now, one of the few tools of investigation available in this context, as well as a very promising one.
In this section, we describe some steps forwards towards a description of the thermalization process of QGP, modeled by a strongly coupled N=4 SYM plasma at finite temperature. Since it is a developing subject on a settled theoretical question dealing with far-from-equilibrium dynamics at strong coupling, we will discuss three different approaches.
i) In section 4.1, we presents estimates of the thermalization response-time computed through the quasi-normal modes of small perturbations away from the thermal background. This explores the information one can get on thermalization and isotropization from linearized solutions, coupling non-hydrodynamic perturbations to a moving black hole background.
ii) Section 4.2 is devoted to the discussion of the fully non-linear thermalization dynamics of a boost-invariant plasma, evolving from an anisotropic out-of-equilibrium early stage towards the hydrodynamic regime. The results point towards a dependence on initial conditions but also to the indication of a fast (if not complete) isotropization as characteristic of strongly coupled dynamics.
iii) In section 4.3 we mention the results obtained in modeling the Lorentz-contracted relativistic nuclei of a heavy-ion collision through initial shock waves and look for a solution of the dual gravitational field after the collision. This provides a realistic scenario, at least from the kinematic point of view.
A warning is in order to remind the following unsolved problem. The preferred phenomenological scenario for the QGP formation starts from initial conditions which may be dominated by weak QCD coupling, due to the high density of partons in a fast boosted nucleus. Hence the problem remains open to describe on a microscopic basis the transition from a far-from-equilibrium weakly coupled system towards a hydrodynamic one at strong coupling. At least the AdS/CFT correspondence allows to make a few first (and quite unique at present) steps on the understanding of strongly coupled dynamics.
4.1 Thermalization response-time
4.1.1 Quasi-normal modes
The quasi-normal modes of a black hole describe the response of the system after a small perturbation. Since the field that sources the perturbation can fall into the black hole or radiate to infinity, the modes of oscillations decay, their frequencies being complex. The geometry therefore undergoes damped oscillations that produce an exponential decay of the perturbation. The frequencies and damping of these oscillations are entirely fixed by the black hole, and are independent of the initial perturbation. For black holes in asymptotically flat space-times, the exponential decay of the quasi-normal modes is followed by a power-law decrease [37] . The decay is instead purely exponential in the case of black holes in AdS [38] .
In the AdS/CFT framework, perturbing a static black hole in AdS corresponds to perturbing the approximately thermal dual state, and the decay of the perturbations describes relaxation back to thermal equilibrium. Computing the quasi-normal modes of the equations of motion linearized around the background allows therefore to estimate the thermalization response-time of the strongly coupled dual gauge theory after a small deviation from equilibrium.
The behavior of quasi-normal modes should be contrasted with the one of hydrodynamical excitations that go to zero at small transverse momentum and have a slower power-law decay. In the static case, for instance, both quasi-normal modes and viscosity calculation correspond to poles of specific retarded propagators, respectively for a scalar and metric deformation, through Fourier transform in the time variable. However, in the limit of zero transverse momentum, the poles corresponding to the viscosity case vanish. The "non-hydrodynamic" nature of the scalar quasi-normal modes allows instead to obtain a finite thermalization response-time in this limit.
The quasi-normal modes analysis has been carried out in [39] for black holes in four, five and seven dimensional global AdS and perturbations produced by a minimally coupled scalar field excitation. See also [40, 41, 42] for later developments. In asymptotically AdS space times, quasi-normal modes are defined as solutions of the wave equation that are purely ingoing near the horizon and vanish at the boundary. For a minimally coupled scalar perturbation, they are obtained solving the wave equation for a massless scalar field ∇2Φ=0 in the gravitational background, with absorbing boundary conditions at the horizon z=z0 and Dirichlet conditions at the boundary z=0. One writes
∇2Φ≡1√−g∂α(√−ggαβ∂βΦ)=0 (60)
for the standard covariant coupling of the scalar field Φ to the background metric gαβ of determinant g.
4.1.2 Static black hole
Since the main example throughout the notes is N=4 SYM in four-dimensional Minkowski space, in the following we refer the results obtained in the case of a static planar black hole in AdS5. To facilitate the comparison with the results of [43] discussed later in the section, we work in Fefferman-Graham coordinates.
Recall that in these coordinates, see (12), the metric of a five-dimensional black brane reads
where (t,→x) are the boundary coordinates, z is the AdS radius and z0 is the location of the event horizon in the bulk.
In the case of the static metric (61), Eq.(60) writes
− 1z3(1−z4z40)21+z4z40 ∂2tΦ(t,z) (62)
+ ∂z[1z3(1−z8z80)∂zΦ(t,z)]=0 .
Using the separation of variables
Φ(t,z)=eiωt ϕ(z), (63)
the wave equation (62) leads to a Heun equation in the bulk variable [40]
ϕ′′ + 1−~z2~z(1−~z)(2−~z)ϕ′
+(z0ω)28~z(1−~z)(2−~z)ϕ=0
where ~z≡(1−(z/z0)2)2/(1+(z/z0)4), z0=√2(πT)−1, cf. (15), and the prime denotes the derivative with respect to ~z.
The quasi-normal modes analysis then proceeds through a combination of analytic and numerical methods (see [39] for details). Note that quasi-normal modes appear also as poles in the complex frequency plane of the Fourier transform in time t of the retarded Green's function (see e.g. [44] ).
Interestingly enough, for a homogeneous scalar excitation, the dominant decay mode at large time is given by a quasi-normal mode whose frequency acquires a non zero imaginary part and thus corresponds to a minimal exponential decay mode with temperature. Technically, by matching Eq.(4.1.2) with a Shrödinger equation
∂2z∗ϕ(z∗)+ √8 e12z∗sinh3/2(8z∗)(ωπT)2ϕ(z∗)=0 (65)
through the change of variable
z→z∗≡14 [tanh−1(z4/z40)]4,
one gets quantized "energy states". The dominant exponential decay mode, corresponds to the state with minimal energy in the Shrödinger equation (65). Denoting ωk the eigenvalues of Eq. (65), one finds
ωkπT=ak−ibk, (66)
⋯>bk>⋯>b1=2.74667, (67)
where T denotes both the plasma and black brane temperature. The corresponding response timescale is thus given by the first eigenvalue ω1 and leads to τresp=1/Imω1∼0.116/T.
4.1.3 Moving black hole
A similar analysis was carried out in [43] for a black hole moving away in the radial bulk direction. This geometry was discussed in section 3.4 and is the gravity dual of a boost-invariant expanding perfect fluid in N=4 SYM at large proper-time. Knowledge of the quasi-normal modes of this black hole gives an estimate of the thermalization response-time of the strongly coupled gauge theory, that is the relaxation time after a perturbation of the local thermal equilibrium due to the coupling to a scalar field.
The background geometry describing a black hole moving off from the boundary in the z direction was given in (3.4). In terms of the scaling variable v=zτ−1/3, it writes
z2ds2 = −(1−v4)2(1+v4)dτ2+ (68)
+(1+v4)(τ2dy2+dx2⊥)+dz2.
Consider now the wave equation of a canonically coupled scalar field Φ in the boost-invariant setting. In the scaling limit v=const and τ→∞, the corresponding Klein-Gordon equation (60) is now expressed in terms of the two variables τ and z. Through the change of variables (τ,z)→(τ,v), so that
∂z→τ−13∂v,∂τ→∂τ−13τ−43∂v, (69)
and neglecting the non diagonal terms in the large τ expansion, one gets
− 1v3(1+v4)21−v4∂2τΦ(τ,v) (70)
+ τ−23 ∂v(1v3(1−v8)∂vΦ(τ,v))=0 ,
which is similar to (62) but with a τ-dependent coefficient. Performing the separation of variables
Φ(τ,z)=f(τ)ϕ(v), (71)
it reduces to two decoupled equations
∂2τf(τ)+^ω2τ−23f(τ) =0 (72)
+ ^ω2(1−v4)2v3(1+v4) ϕ(v)=0.
where we introduced an a priori arbitrary parameter ^ω, whose allowed values will be fixed by the (quantized) solutions of (LABEL:QNMs2).
Equation (72) is solved by linear combinations of the Bessel functions
√τJ±34(32^ωτ23), (74)
with relevant large τ behavior
f(τ)∼τ16e32i(^ω1)τ23 (75)
where Im ^ω1=2.74667 is the same as in (67). Indeed, the equation in the v variable (LABEL:QNMs2) is formally identical to the Heun equation (4.1.2) of the static black hole and therefore leads to the same numerical values for the quasi-normal modes and frequencies. However, the variables v and τ of (71) are different from the variables z and t relevant in the static case. Moreover, the proper-time dependence (75) has a non trivial scaling compared to the plane wave dependence of the static solution.
Noting that the moving horizon is at z0=(3e0)14τ13=√2πT, cf. (48,49), it was suggested in [43] , that this result could be understood in terms of an adiabatic approximation of the expanding case, with locally fixed temperature. Indeed,
|f(τ)|∼e−32Im(^ω1) τ23∼e−πb1Tτ , (76)
where ω1=πT(a1−ib1) is the dominant decay mode in the static case (67).
Plugging in the numerical value of the dominant decay mode (67) in (75) leads to a damping factor of the form
exp(−32⋅2.7466⋅τ23)∼e−8.3Tτ . (77)
The above results were extended in [41] to include also vector and tensor quasi-normal modes.
In the limits to which this analysis can be considered quantitatively relevant for the thermalization in heavy-ion reactions, the fast decay modes of frequency (67) damp out to 1/e of their original amplitude in a time no greater than
τresp∣∣T=Tpeak=0.116T∣∣T=Tpeak≈0.08 fm/c, (78)
for a typical initial peak gauge theory temperature Tpeak≈300 MeV, which is commonly considered as reasonable in phenomenology. Given the highly anisotropic momentum space distribution expected in the early stages of a RHIC collision, one tentatively estimates a certain finite number n of e-foldings of the "response thermalization process" have to elapse before hydrodynamic approximations can be used. The estimated thermalization time becomes
τtherm∼n×τresp∣∣T=Tpeak, (79)
leading to τtherm≈0.3 fm/c following the guess of [41] of about n=4 iterations. The above estimate compares in order of magnitude with the simulations predictions on real experiments. Nevertheless, it should be clear that it is extrapolated from the quasi-normal modes analysis, that, by definition, only applies to the late-time stages of the whole thermalization process. Moreover, the thermalization response time has been obtained in a linearized approximation around the black hole background through AdS/CFT methods.
As a final remark on this response time studies, it is worth noting that the same properties are valid for quasi-normal modes dominating the transverse space diagonalization of the metric after a non diagonal perturbation. Let us indeed introduce a non diagonal transverse component gx1,x2 using its contravariant expression
gx1x2(z,τ)≡z21+v4gx1x2 . (80)
It verifies (with gx1x2(z,τ)=f(τ) gx1x2(v)) the same equations (72,LABEL:QNMs2); such as
∂v(1v3(1−v8)∂vgx1x2(v)) + ^ω2v3(1+v4)21−v4gx1x2(v) (81)
= 0 .
In some sense, isotropization and thermalization response-times are equal at large proper-times. We now will turn to the study of the non-linear problem of isotropization and thermalization when starting from far-from-equilibrium initial conditions.
4.2 Thermalization of boost-invariant plasma
In the present section we describe three different approaches to the study of the thermalization process of a strongly coupled boost-invariant plasma. The analysis are to be intended as complementary and point towards a fast isotropization/thermalization. However, as we shall discuss, this represents only an early stage for a full understanding of far-from-equilibrium dynamics leading to the QGP formation.
4.2.1 Solution with "full anisotropy"
One of the first holographic estimates of the isotropization/thermalization time of the quark-gluon plasma appeared in [22] . The setup of this analysis is similar to the one considered in [15] , but with differences which we will comment at the end of this subsection.
The starting proposal of the paper is that Bjorken hydrodynamics at late times can be singled out, in the holographic renormalization program, relaxing the assumptions to only require that the metric tensor is a real and single-valued function of the coordinates everywhere in the bulk, without imposing any constraint on the curvature invariants. Applying the same strategy to early time dynamics [22] leads to infer the existence of a solution corresponding to the fully anisotropic case with constant energy density ϵ. In the notation of section 3.2, it corresponds to the solution with s=0 in equation (36), and thus to "full anisotropy" at initial time, namely ε=p⊥=−p∥. The system is initially anisotropic, with negative longitudinal pressure, and is expected to evolve at late times to ideal Bjorken hydrodynamics, with longitudinal and transverse pressure components equal and positive. Isotropization must take place at some intermediate proper-time.
Assuming a smooth transition between full anisotropy and full isotropy, the isotropization time τ


text:
Cdl. Burke Warns of Vatican Push for One-World Government
News: World News
by Jules Gomes • ChurchMilitant.com • January 1, 2020
'Synodal way' is Trojan horse to protestantize Catholic Church
ROME (ChurchMilitant.com) - Pope Francis' call for a global educational pact at the Vatican in May 2020 is potentially an event to promote a one-world government, a leading traditionalist cardinal is warning.
Cardinal Raymond Burke, former prefect of the Apostolic Signatura, is also excoriating the Amazon Synod's motif of "ecological conversion" as a masonic subterfuge to advance a one-world government and the jargon of "synodal conversion" in Amazon and German synods as a deception to protestantize the Catholic Church.
The outspoken prelate is calling for more orthodox lay apostolates who can defend Catholicism because of "a tremendous spread of confusion and error in the Church" and the silence of many bishops.
Cdl. Marx and German bishops: Protestantizing the Church
Burke told The Wanderer in a Dec. 26 column that "'ecological conversion' is being used as an argument for a one-world government," which is the "idea of completely secularized people who no longer recognize that the governance of the world is in the hands of God, who entrusts it to individual governments, nations, and groupings of people according to nature itself," comparing it to the Tower of Babel.
The idea of a one-world government is fundamentally the same phenomenon that was displayed by the builders of the Tower of Babel who presumed to exercise the power of God on earth to unite heaven with earth, which is simply incorrect. What we truly need is a religious conversion, in other words, a strong teaching and practice of faith in God and obedience to the order with which He has created us.
Professor Yoram Hazony, author of The Virtue of Nationalism, affirmed Burke's comments, saying to Church Militant: "Hebrew Scripture is utterly hostile to the idea of world empire, seeing it as usurping the place of God. Christians who have not abandoned the Old Testament understand this well. A biblical worldview is one that envisions a world of independent nations, each approaching God according to its own inheritance."
Hazony's groundbreaking book marshals extensive biblical evidence to refute the idea of a one-world government.
The cardinal, who in 2017 urged Francis to clarify the apostolic exhortation Amoris Laetitia, said he is grateful the Amazon Synod's final document wasn't explicitly promoting "the most egregious apostasy in the working document, namely the denial of the Lordship of Our Lord Jesus Christ and the promotion of a form of pantheism, the worship of so-called 'Mother Earth.'"
"With regard to 'ecological conversion,' what I see behind this is a push for worship of 'Mother Earth.' In truth, our mother is not the earth — our mother is the Blessed Virgin Mary in the sense that she gave birth to our Savior. We do not have another mother, certainly not a pagan idol like the Pachamama," Cdl. Burke said.
Nevertheless, "this kind of thinking was in the mind of many of the synod fathers," and hence the Church needs "to be extremely cautious that no element of this pernicious apostasy enters into any kind of official document that follows," he forewarned.
Sounding alarm bells over the methodology of "synodal conversion," the prelate said he fears the infiltration of a "very Protestant idea of the Church," where democratic meetings take precedence over apostolic tradition.
We do not have another mother, certainly not a pagan idol like the Pachamama.
"No synod is binding — this is a contradiction in terms. A synod has no legislative power in the Church. The purpose of a synod is to assist the pope in teaching the Church's doctrine more effectively and applying its discipline more fully," he said, cautioning that a decision made by a local synod could be subversively imposed on the universal church.
Dr. Gavin Ashenden, a recent convert from Anglicanism to the Catholic Church, told Church Militant that Burke is "fully justified" in his "apprehension about a dangerous precedent of the 'Synodal Way' set by the Amazon and German synods."
The lay theologian said the "creeping revisionism" to which Burke was alerting Catholics "was exactly the strategy that the progressives successfully pursued to undermine what remaining orthodoxy subsisted within Anglicanism."
He warned:
Ecological conversion as a push for Mother Earth worship?
The technique is as simple as it is deceptive and corrosive. In Anglican terms, innovation was sold on the basis that there would be a period of discernment. The synodal process masked what was, in fact, a political campaign and not a spiritual charism. The intention of the progressives was to lie to their traditionalist opponents to lull them into a false trust and spiritual somnolence.
As happened within Anglicanism, Burke is warning that deaconesses would be used as a Trojan horse to ordain women priests: "One of the women who served on the papal commission regarding this question said openly that she and those she represents are not interested in the diaconate, but the priesthood."
Ordinatio Sacerdotalis, Pope John Paul II's apostolic letter limiting priestly ordination to men, "was an exercise of the papal Magisterium," said Burke. "It is firmly rooted in the unchanging tradition of the Church and continues to have doctrinal force. Women cannot be ordained to orders at any level, including the diaconate, the priesthood, and the episcopate."
He explained further:
If one studies Church history, it is clear that the Church has never ordained deaconesses. Deaconesses were women who assisted at certain rites, for example, the baptism of women. Women never received the Sacrament of Orders for assisting at such rites. Ordination of women simply cannot be — it represents a great defect in the final document of the Pan-Amazon Synod.
Attacking Francis' claim that "God willed a plurality of religions" as "heretical," Burke said he is waiting for Pope Francis to clarify these remarks following the signing of the Abu Dhabi declaration. "In the meantime, Catholic universities have been ordered to teach this declaration and so the confusion continues to spread," he lamented.
The creeping revisionism is exactly the strategy that progressives pursued to undermine Anglicanism.
Burke insisted that the Church's teaching on the death penalty has not changed despite Pope Francis' insertion of a revision in the Catechism: "The Catechism of the Catholic Church is not a proper instrument to introduce such matters. It is not a tool for the proposal of new teachings."
Moreover, "there are no new 'ecological sins,'" he stressed. "The same Ten Commandments that the Lord God gave us on Mount Sinai are in force today."
In his recent message for the 2020 launch of the Global Compact on Education, Pope Francis uses the word "global" six times, urging "more inclusive education." The pope calls for an "educational village" and a "global agreement about an education" that includes ecological concern as well as "dialogue between religions."
A Tale of Two Popes: Nationalism or Globalism?
By Jules Gomes • 0
Amazon Synod Proposes New Episcopal Body to Implement Pope's Decrees
'Miraculous' Novena to Protect From Papal Confusion
By Anita Carey • 0

redpajama_set_name: RedPajamaCommonCrawl


text:
Home / Magazine / What Is Kundalini & How Can You Access it?
What Is Kundalini & How Can You Access it?
If you've ever taken a Kundalini Yoga class you'd notice it's vastly different than other types of Yoga. This is no goat, flying upside down, or naked yoga. It's in a class all its own. There's something graceful yet mind-expanding about this type of practice. You might even call it hypnotic.
(Image: Tamara Muth-King of Brianna Rose)
Kundalini refers to an extremely powerful energy (linked to the divine feminine archetype) coiled up at the base of the spine. Once envisioned by ancient Hindus as a serpent-like entity, they believed this energy capable of rising within the body. It's said that Kundalini helps deepen creativity, intuition, and your own natural gifts and abilities. In a way, Kundalini Yoga practice acts like a snake charmer, propelling this magnetic force to awaken and fill you with its intensity. Part of the beauty of Kundalini is, it's not about looking good, while striking the perfect pose. This is a deep, soul-searching practice; one that involves accessing your subtle body's innate wisdom. Even though all yoga aims for increased awareness, Kundalini is known as "the yoga of awareness." The sequences help you tap into your consciousness in the hopes that you'll be able to access your own dormant power.
Where does the style come from? On the surface Kundalini Yoga can appear trendy, because it's unique. But it's ancient. Kundalini dates back to one of the oldest Hindu texts in existence, "the Upanishads." Considered a form of Raja Yoga, Kundalini prescribes to the 8 limbs that the sage Patanjali, referred to in the "Yoga Sutras." While Kundalini originated in these ancient texts, the Kundalini practiced in the West, has been altered from its version in the Upanishads; it was make anew by innovative Gurus such as Yogi Bhajan, perhaps the most well known teacher of Kundalini in the states; Yogi Bhajan brought this unique type of yoga, melded with traditions from his Sikh culture to the states in 1968. Intrigued by its trippy vibe, Kundalini Yoga was quickly embraced by the hippies. You'll often hear the term, "The golden chain," in Kundalini class; this refers to the lineage or chain or teachers that came before you. Certain traditions associated with Kundalini Yoga, such as white clothing, said to expand one's auras, and wearing a head covering to harness your energetic flow, are carryovers from Yogi Bhajan's Sikh culture, incorporated into his teachings.
Brianna Rose is a Kundalini Yoga teacher (taught through the lineage of Yogi Bhajan) also a Kundalini Business Coach for women entrepreneurs, CEO of the Light Leader Collective. Brianna calls Kundalini Yoga "the owner's manual to the human body." Passionate about Kundalini's teachings, Brianna says, "It teaches you how to manage your energy and create a life you love." How does it work? Brianna says, this energetic Kundalini force, "once awakened, moves through your entire chakra system beginning with the root chakra and ascending to your crown. This provides deep healing and cleanses away impurities or blocks in your energy centers." In a practical sense, will this affect your daily life? Brianna explains that, "once clear, you are able to have a deeper connection to yourself and take action on your soul's mission. It makes you unshakeable."
If the hope of awakening new energy sounds enticing... be aware that Kundalini Yoga is not easy. Asanas can be held for extremely long periods of time. Initially, sequences may look and sound unusual, bordering on strange. And you might feel self-conscious doing them. Brianna says, "The mantras, postures, and breath work can all be challenging." So what should you anticipate when taking a class? Brianna gives the examples, "Sometimes you're holding your arms up over your head for 62 minutes. Other times you're in an awkward position breathing with your tongue sticking out and panting like a dog."
Just because the work takes determination, doesn't mean you should shy away from the practice. In fact, Brianna says, "that's the beauty of Kundalini. You are pushing yourself. You become the guru." Working in this way helps you build courage and inner resolve. Brianna says, "When you've pushed yourself like that, it's embodied and engrained in every cell, fiber, and tissue of your being." Taking the concept of a comfort zone and turning it inside out, Brianna says, "Kundalini pushes your physical body to new limits, as it creates a mental strength to carry you through any 'test' or 'difficulties' life might throw at you."
What sets Kundalini apart from other types of yoga and meditation? Even though sequences can be physically strenuous and involve familiar asanas and breathing exercises, at its root, Kundalini seems to be structured as an internal journey.
Eyes are usually closed, or partially closed during class, adding a trance-like vibe. And you will be given focal points for the eyes.
Expect chanting. And plenty of it. "Sat, Nam." is one of the most popular chants in Kundalini Yoga and meditation, meaning "truth," and "name or identity." Chanting is intrinsic for the meaning behind the words, also the vibrational effects on the tongue that permeate throughout the body and mind.
That snake-like energy coiled up at the base of your spine isn't the only colorful visual... You'll be consistently required to call upon your imagination to visualize throughout class. Often your mental powers will be directed to your chakras, in order to align with their energetic resources.
Bandhas are different kinds of locks: they involve tightening and squeezing different muscles in the body, in order to gain control of life force energy (prana).
Ready to go within, and experience Kundalini Yoga? Just like with any yoga or meditation, you can begin at home or in a class setting. Eventually it's advisable to practice daily, or many times a week. Although, Kundalini is a little different. It involves lots of energetic work and detailed instructions. So when you start, it's worthwhile to research teachers in your area and explore a class. You'll find that practicing with others lends a meaningful element of shared energy that deepens the experience.
To further explore Kundalini's teachings, listen to this illuminating interview with Hari Kaur, senior teacher and author of, "A Woman's Book of Yoga," and "A Woman's Book of Yoga," led by Yoga instructor and international Author, Elena Brower. And if you'd like to watch a class in action, check out this video with world renowned Kundalini Yoga Master, Gurmukh, author of "The Eight Human Talents."


Mazda Talk Forums • View topic - RX8 RP-F1 ...silver or SBC Finish Wheel Pictures!
can someone post more pics of the rpf1's with the sbc finish? i saw a yellow rx-8 elsewhere in this forum but it was only one pic and not the greatest shot..
BigIreland wrote: can someone post more pics of the rpf1's with the sbc finish? i saw a yellow rx-8 elsewhere in this forum but it was only one pic and not the greatest shot..
The SBC finish is new on that 18x9.5 and perhaps only three customers even have it so far. Thus, don't expect much response to this.
Re: RX8 RP-F1 ...silver or SBC Finish Wheel Pictures!
Got a few sets more of the SBC finish!
FedEx just dropped mine off today, Thanks Super Speedy delivery.. Ill post pics of them on my Winning blue 8 so you have both finishes with Winning Blue 8's .. ! Thanks again Brian. They look so much better in person.


Discussion in 'Windows 7' started by majnoo, Dec 8, 2009.
with Spyware Doctor i removed the Spyware but My Search is not working now.
it is enabled from Windows Features and there at the right top of my computer or any folder like it has to be but whatever i write in it won't even start search.
pass sharing is disabled, ip is same, work group is same but cannot acces any system on network.


a transit guy, at a highway conference?
Here's a seemingly odd pairing: the nation's top transit advocate speaking at a highway conference.
That's what we'll have here in Des Moines this week, when Bill Millar, the president of the American Public Transportation Association, will be in town for the 2010 annual meeting of the Mississippi Valley Conference of the American Association of State Highway and Transportation Officials (AASHTO). He will be speaking Thursday morning on the future of public transportation.
It isn't that odd, when you think about it. Transit isn't the enemy of roads any more than sidewalks or bike paths or airports are. They're all a part of the same transportation network, and they are stronger as a whole than as individual parts.
Without roads, transit couldn't use park-and-ride lots to increase ridership. A bus system couldn't even exist without roads. But without buses, roads would have to accommodate thousands more cars every day, increasing carbon monoxide emissions and the wear and tear of the roads, to say nothing of the money needed to maintain or widen existing roads.
So really, it's in everyone's best interest if the highway and transit folks play nice.
Millar will also be speaking Friday morning at the Greater Des Moines Partnership. Check back here for a report on his remarks.
William Millar is the president of the American Public Transportation Association (APTA). Since coming to APTA in 1996, Bill has sought to expand APTA's reach and effectiveness, guiding it to legislative victories and dramatically increasing federal investment in public transportation.
Bill is a longtime advocate for high-speed rail; under his leadership, the High-Speed Ground Transportation Association became part of APTA in 2006. APTA has created the Center for High-Speed Rail to bring together the expertise and resources of its members to realize the vision for high-speed rail in the United States.
Prior to APTA, Bill served 19 years at the Port Authority of Allegheny County, the principal transit operator serving Pittsburgh, PA. As its executive director from 1983-1996, he oversaw the development and operation of bus, busway, light rail, paratransit and inclined plane service. He is the founder of Pittsburgh's award-winning ACCESS paratransit service.
From 1973-77, Bill worked for the Pennsylvania DOT, where he developed and managed Pennsylvania's Free Transit Program for Senior Citizens and led the Penn DOT's rural public and community transit efforts. He began his career as the county transportation planner in Lancaster, PA.
Mr. Millar is a strong supporter of transportation research and is the recipient of the Founding Father Award for his leadership in establishing the Transit Cooperative Research Program (TCRP). He has been a member of the executive committee of the Transportation Research Board for many years and served as its chair in 1992. He also serves on advisory committees of several university transportation research institutes.
A well-known expert in the field of public transportation and transportation policy, Bill has published numerous articles and has often testified before the U.S. Congress. He is a frequent speaker and lecturer at conferences and seminars and is an adjunct professor in the School of Public Policy at George Mason University.
Mr. Millar is the recipient of many awards, including APTA's Jesse Haugh Award for Transit Manager of the Year (1987), the Transportation Research Board's W. N. Carey, Jr. Distinguished Service Award (1999); Pattison Partnership Award from the Intermodal Passenger Institute (2001); and Railway Age's Graham Claytor Award (2006).
Bill has a BA from Northwestern University and an MA from the University of Iowa majoring in urban transportation planning and policy analysis. He lives in Falls Church, VA with his wife and two children and commutes to work on Washington's Metrorail.


text:
Home USA Politics The Far Left Eat Their Own
The Far Left Eat Their Own
Fri July 21, 2017
America's most "tolerant" intolerant have turned their back on one of their own. World famous liberal attorney, professor, and TV commentator Alan Dershowitz has done the unthinkable in the view of the far left; he provided an opinion outside of the box.
The feisty former Harvard Law Professor and defender of O.J. Simpson spoke out of turn with the far left agenda and provided his opinion that differs from the non-stop Trump-hating mantra of the "tolerate" intolerants. That does not sit well with the machine-like 24/7 conspiracy-driven hard left and its compliant media brothers.
Thus, Dershowitz's name is mud with his usual saddle pals at The New York Times. God forbid any lefty speaks out of turn before reviewing that day's talking points on Russian collusion and the next big idea for impeachment of the president. According to the Washington Examiner, the Times aren't speaking with Alan.
It seems Dershowitz is doing the unthinkable. He is actually looking at the flimsy legal arguments of the left in terms of Trump's predicament on a variety of fronts. He is not being sidetracked by hysteria and goose-stepping dictums from his liberal buddies. There is no ideological emotion.
Dershowitz is violating political correctness. He recently tried to publish an opinion piece in The New York Times about the hated President Trump. The opinion was not in sync with the Times editorial staff, thus not worthy of their "balanced" editorial page. The professor dared to suggest the president likely did not attempt to obstruct justice by firing former FBI Director James Comey.
Professor Alan Dershowitz.
That is blasphemy in the world of political lemmings. Dershowitz told the Washington Examiner over the phone that the Times had "no response" to his submission. He provided his usual disclaimer that he is still an ardent supporter of Hillary Clinton, just to keep the lights on in liberal-land. But he felt compelled to respond to an op-ed in the Times that argued Trump could be charged for criminal conduct in Comey's firing.
He suggested the unthinkable to the "tolerant." He thought that maybe the readers of the Times might like to hear the other side of the argument. What?
The Professor Emeritus of Harvard Law School said, "And I really do think The New York Times does not want its readers to hear an alternative point of view on the issue of whether or not Trump administration is committing crimes."
For a man of his brains and background, this is a revelation?
left talking points


text:
Home » TV & Movies » Curvy Seashell House from Grand Designs is on the market for £1.45million
Curvy Seashell House from Grand Designs is on the market for £1.45million
Dream of living in a home featured on Grand Designs?
Well, in some cases, you can't – like this home in Brighton, whose developer loved it so much he couldn't bear to sell it.
The good news is that there are other Grand Designs properties that you can get your paws on.
Case in point: the famous 'Curvy Seashell House' that was on the show in 2005.
The four-bed home received a lot of attention at the time due to its pale pink curving walls (hence the curvy seashell name) and corkscrew staircase tower.
It was built by Pat Becker in the garden of her home in Shaldon, Devon, and inspired by its seafront location.
Since appearing on Grand Designs, the property remained in the same family – until now, when it's been listed for sale for the first time ever, at an asking price of £1.45million.
Estate Agents Bradleys of Shaldon said: 'As featured in the popular Channel 4 series of Grand Designs.
'The house was built in 2005, echoing a seashell design, this modern coastal property is situated in a prime Shaldon location, in the exclusive Horse Lane neighbourhood.
'Nestled between the extensive Botanical Gardens and the Ness Lawns, The Garden House offers fabulous estuary and coastal views.'
In the Grand Designs episode Kevin McCloud said: 'It's like being in Miami or the Mediterranean.
'This place is so quirky – there are bits I really like and others I do not.'
Even if you're not a diehard Grand Designs fan (gasp), there's still a lot to like about this home.
With 2,104 sw ft of space spread across two floors, it would make a dreamy family home for anyone dreaming of seaside life.
On the ground floor is a cloakroom, an open-plan sitting and dining room (with double-height windows to let you take in the view, and an exposed wooden ceiling), a sunny conservatory, a kitchen and utility room, and the main bedroom, complete with an en suite bath and shower room and access to vaulted loft space.
Go down the stairs to the lower ground floor and you'll find a guest suite, two more double bedrooms, a large family bathroom, and a workshop or store room.
In terms of outdoor space, you've got multiple balconies, plus walled gardens accessed by a cobbled path.
And just a walk away is The Ness, along with Shaldon's beaches.
It really does feel like villa living, just in the UK instead of sunny Mallorca.
If you fancy putting in an offer, you can do so through the Bradleys website.
Get in touch by emailing [email protected]
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Next Post:A Place in the Sun host mortified as guest bluntly admits 'I would've preferred Jasmine' before property tour


text:
var config = require('config-lite')(__dirname);
var Mongolass = require('mongolass');
var mongolass = new Mongolass();
mongolass.connect(config.mongodb);

exports.User = mongolass.model('User', {
  name: { type: 'string' },
  password: { type: 'string' },
  avatar: { type: 'string' },
  gender: { type: 'string', enum: ['m', 'f', 'x'] },
  bio: { type: 'string' }
});
exports.User.index({ name: 1 }, { unique: true }).exec();// 根据用户名找到用户,用户名全局唯一

var moment = require('moment');
var objectIdToTimestamp = require('objectid-to-timestamp');

// 根据 id 生成创建时间 created_at
mongolass.plugin('addCreatedAt', {
  afterFind: function (results) {
    results.forEach(function (item) {
      item.created_at = moment(objectIdToTimestamp(item._id)).format('YYYY-MM-DD HH:mm');
    });
    return results;
  },
  afterFindOne: function (result) {
    if (result) {
      result.created_at = moment(objectIdToTimestamp(result._id)).format('YYYY-MM-DD HH:mm');
    }
    return result;
  }
});

exports.Post = mongolass.model('Post', {
  author: { type: Mongolass.Types.ObjectId },
  title: { type: 'string' },
  content: { type: 'string' },
  pv: { type: 'number' }
});
exports.Post.index({ author: 1, _id: -1 }).exec();// 按创建时间降序查看用户的文章列表

exports.Comment = mongolass.model('Comment', {
  author: { type: Mongolass.Types.ObjectId },
  content: { type: 'string' },
  postId: { type: Mongolass.Types.ObjectId }
})

exports.Comment.index({ postId: 1, _id: 1 }).exec();// 通过文章 id 获取该文章下所有留言,按留言创建时间升序
exports.Comment.index({ author: 1, _id: 1 }).exec();// 通过用户 id 和留言 id 删除一个留言

redpajama_set_name: RedPajamaGithub


SearchDocs XML is now available.
Any searchstring can be used, single words or a sequence of words (like "move the insertion point"), which is not possible with the filter and search routines of the Rev Documentation.
The "Dictionary"-, "Faq"-, "Topics"- and "Glossary"-folders of the Revolution "Help" directory can be searched. The results are displayed in the field on the right with the searchstring colored and the XML-file addresses inserted as links. Clicking on such a file link displays the complete article in the left field - again with the searchstring colored. If the article itself contains links these are again displayed for further inspection.
If one of the links points to to a stack in the "reference" folder, this stack will be opened when clicking on the link.
Select a folder for the search by checking one of the radio buttons.
One option of the latest version is to search folders "Dictionary", "Faq", and "Topics" at the same time. This will slightly increase the search time - to about 160 milliseconds on the average for the 2000 XML files together (measured on a WindowsXP computer with 2 GHz).
Folder "Glossary" is not included in this combined search as it only fulfils the function of a lexicon of "Transcript" terms, but it can be searched separately.
Average search-and-display-time is less than 100 milliseconds (for the total 1496 files of the "Dictionary") on a WindowsXP computer with 2 GHz and somewhat more - but still below 200 milliseconds - on a Mac G4 with 667 MHz.
New "copy" feature: Text can be selected in the fields on the right and left using the mouse ("traversalon" is set to true for this two fields). Then apply the usual copy-and-paste procedures from the Menubar or through keyboard shortcuts, e.g. CTRL-C and CTRL-V on Windows.
Metacard user need to copy the Revolution Help folder into their main folder of the MC IDE. For cosmetic reasons - to be able to display some of the inserted Rev icons - one should also copy stack "revicons.rev" into the Metacard folder.
Revolution users must place their version of "SearchDocs XML.rev" in the same folder with "Revolution.exe" (Revolution.app in MacOS) to work properly.
1. When the learner types into the input field, the typed letters disappear from the scramble field. Only letters contained in the scramble field can be typed into the input field, otherwise a "warning" will appear.
2. If the user deletes letters from the input field, they reappear in the scramble field; the learner can move the cursor inside the input field using the arrow keys, then press the backspace key to remove the letter on the left of the cursor.
3. You can re-scramble the word to possibly get a better idea what the word could be.
4. Pressing "Help" shows the first and last letter of the sought word and displays dashes as placeholders for the remaining letters in between.
5. Button "more letters", which appears after first pressing button "Help", adds more - randomly selected - letters to the help field. The last two dashes in a word however remain, the user has to find out them on his/her own.
"Simple" as it is, the exercise card of the stack needs 25 controls to achieve the described basic functionality and 8 of them contain scripts. The longest script is that of the input field, which controls the features 1. and 2. explained above. This script makes use of the "offset" function, "returninfield" and "rawkey" handlers, and the "selectedchunk" function, the last one to determine the place of the insertion point in the input field. A special problem comes up when you have to deal with special "national" characters, because the "rawkey" values in this case are different from the "numtochar" values; therefore I included some script lines to take care of the German "Umlaute" (ä, ö, ü).
b) to add the possibility to export the lexicon to an external text file and to import from a choice of external files.


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Posted by cath in leftovers, Recipes, specials, variations.
Just got too much haggis leftover from last night's Burns supper? How about gravy (yes I like gravy with haggis, neeps and tatties), got any noodles? If so, try my noodles in gravy with haggis. A Chinese style leftovers dish.
Cook some large, flat noodles, drain and set aside.
Heat up some gravy in a shallow frying pan, crumble in some leftover cooked haggis, heat through. Pour over noodles and serve.
Water from boiling potato and neeps, red wine, lamb stock, tomato puree, garlic or better roasted garlic, bay leaf, herbs, mustard.
Boil up the liquid ingredients, add herbs, garlic, tomato and mustard, boil again. Leave until needed then reheat and strain before serving. Thicken if you prefer, with cornflour and cold water mixed to a paste, or butter and flour mixed together. Add the thickener and boil up again.
Haggis can be used much like mince so I've heard of and tried a few haggis alternatives. Haggis samosas, pakoras, wontons are quite common in Scotland. A few years ago some friends also brought haggis burritos to Edinburgh, so if you've got no leftovers at home and are in the area… try them at Los Cardos, Leith Walk. They sell their haggis all the time, not just Burns night, and their other options are also great.
Mmmm. Enjoy your burns night leftovers.
Posted by cath in comfort food, freeze-friendly, Recipes, specials, variations.
Leftover Haggis and the works from Burns night? Make sure to make the most of your leftovers. Here's an idea for eating up leftover Burns supper ingredients.
Just like fish and potato cakes, similar Haggis, neeps and tatties cakes are great.
Add crumbled haggis to leftover mash, I think it helps the binding to keep the mixture at least 50% potato, but any leftover neeps or other root mash can also be added. You shouldn't need seasoning as the leftovers will have been made with salt and pepper already. Mix everything well.
Shape the mixture into small, flat cakes. Pat each side with white plain flour. Leave, shaped and floured, in the fridge for at least an hour to firm up making them easier to fry.
I'm making these tonight so the finished dish photo will come later, here are the haggis cakes ready to go in the fridge.
To cook, shallow fry in a little oil (haggis and mash are already fatty so you don't need a lot) until browned on both sides. Flip just once if you can, by giving the first side a good 5 minute sizzle. Flip over carefully as they can break up… this is where a rest in patty form can help.
Serve up with some fresh rocket leaves for a fancy supper or just some ketchup or brown sauce for real comfort food!


Hey T, how much for that i5 chip?
I just saw this - how about $85 shipped?
I did see your WIFI USB stick, but I would rather have one that's N. It's going in my sister's gaming rig because she doesn't want wires all over the place.
how do you measure holes for a fan mont!
He sold all his stuff, so you'd have to do some more looking, but you can probably find a similar deal, especially when Sandy Bridge E comes out.
$165 for the CPU, the same GTX450 for $117, get a P67 board from NewEgg for about $100, and you have $18 left over for some ram. I know Kenkicker had some 2GB DDR3 kits he was selling for $20 shipped.
Hey t_ski, I know you're a intel lover, and I was wondering if a Q6600, GTX 450, 8 gigs of 800 and some HP board is worth $400. I'm thinking of getting my first intel rig!
Yo homey, where do you work? Just curious, I worked for Sprint over the summer and saw all just one of their regional data centers. It was epic, thousands of fiber lines, millions of $'s of equipment, tons of goodies. Not much storage more router based equip, but I did see a single data closet filled with hdd's, prolly 20+ in just one cab.
can we chat on MSN or something about this QX?
How much you want for the E8400?


text:
Bader deserved award for fielding prowess
St. Louis outfielder tied with Inciarte for most Outs Above Average, +21
By Mike Petriello
Harrison Bader wasn't one of the three finalists for the National League Gold Glove in center field, won for the third straight year by Ender Inciarte. He didn't take the Wilson Defensive Player of the Year Award in center either, somehow won by Kevin Kiermaier despite playing only 88 games
Harrison Bader wasn't one of the three finalists for the National League Gold Glove in center field, won for the third straight year by Ender Inciarte. He didn't take the Wilson Defensive Player of the Year Award in center either, somehow won by Kevin Kiermaier despite playing only 88 games in the outfield, 30 fewer than Bader did.
On some level, this is understandable, because Bader began the year as a fourth outfielder behind Marcell Ozuna, Tommy Pham and William Fowler, not ascending to an everyday role until early August. But by nearly every defensive metric we can think of, Bader was the best defensive outfielder in the game on a per-play basis in 2018 -- and he surely passed the eye test as well.
So since Bader isn't going to come away from his rookie season with any hardware, let's do it for him. He was the most impressive outfielder we saw all year, and if any player was snubbed in the postseason award balloting, it was him.
We investigated this back in August, not long after Bader was installed as the Cardinals' everyday center fielder. He then started all but four games in center over the final two months, so let's update the numbers to show just how strong of a fielding season he ended up having.
No outfielder saved more outs in 2018
Our preferred measure of outfield range is Outs Above Average, a cumulative metric that takes the difficulty and probability of each individual outfield opportunity being made and aggregates it into a season-long total. In 2017, for example, the range went from +29 OAA (Byron Buxton) to negative-18 OAA (Matt Kemp).
In 2018, there was a tie atop the leaderboard. Sure, Inciarte is fantastic and deserves all of the accolades he receives. But he didn't finish by himself at number one. He was tied with Bader.
Most Outs Above Average in 2018
+21 -- Bader, Cardinals
+21 -- Inciarte, Braves
+19 -- Lorenzo Cain, Brewers
+17 -- Adam Engel, White Sox
+16 -- Billy Hamilton, Reds
It's not just Statcast™, either. If you're more comfortable with Defensive Runs Saved, Bader's +19 was fourth-best among outfielders. If you like Ultimate Zone Rating, Bader's +10.7 was also fourth best.
That's all very impressive, and the conversation could easily be ended right there. But there's something else we need to take into account here: playing time. All of those metrics mentioned above are counting stats, meaning that you need opportunities to accumulate them. Bader's 915 outfield innings were 56th most in the Major Leagues; he played 426 fewer innings than Inciarte did, for example.
If Bader tied Inciarte in Outs Above Average despite not playing nearly as many innings, that must mean that he was doing more with the opportunities he was given. As it turns out, he was.
Bader added more value than any other qualified outfielder
In order to see how an outfielder did on a rate basis, independent of playing time, we ask a much simpler question: How much value did they add compared to how much an an average outfielder would have added?
We can do it that way, too. Based on the difficulty of the opportunities Bader received, an average outfielder would have made the catch 85 percent of the time. But in reality, he converted 93 percent of those chances, meaning that he added +8 percentage points of value over an average outfielder.
That's a big gap. It's the largest of any qualified outfielder, by a large gap.
Most Catch Percentage Added in 2018
+8 -- Bader, Cardinals (85 percent expected, 93 actual)
+5 -- Hamilton, Reds (85, 90)
+5 -- Engel, White Sox (87, 92)
+5 -- Inciarte, Braves (88, 93)
It's many of the same names, of course, but that's to be expected. (For comparison, Kiermaier, who compiled +7 OAA when he was healthy, added three points of value by catching 93 percent of the balls hit his way when an average outfielder would have converted 90 percent.)
Now, it's important to remember what these numbers are expressing. They're entirely about range and plays made, and they do not include arm strength. We point that out because it's possible that this actually hurts Bader, who also showed off a cannon. Of the 25 hardest throws tracked by Cardinals fielders this year, 17 of them were from Bader, topping out at 99.2 mph when he just barely missed throwing out Ryan Zimmerman at the plate on a sacrifice fly on Sept. 5.
Through July 31, Bader started just 14 of the 107 games for St. Louis in center field. Pham started nearly all of the others, with spot appearances from Tyler O'Neill and Yairo Munoz. From August 1 on, with Pham traded to Tampa Bay, Bader started 51 of the remaining 55 games in center.
"What you need to be asking is, 'When did everybody else learn that I was fast?'" said Bader to MLB.com's Joe Trezza in June. "I've been fast for a really long time. These new metrics are helping everyone see that."
Bader is not wrong. His rookie year was a tremendous success, as he proved that he's immediately in the conversation for the best outfield defender in baseball. In 2018, Bader might have actually been that player, awards or not. Given a full season of play in 2019, we might be having a very different conversation at this time next year.
Harrison Bader