Boundary Derivative Expansion Research Articles

Holographic KMS relations at finite density

We extend the holographic Schwinger-Keldysh prescription introduced in [1] to charged black branes, with a view towards studying Hawking radiation in these backgrounds. Equivalently we study the real time fluctuations of the dual CFT held at finite temperature and finite chemical potential. We check our prescription using charged Dirac probe fields. We solve the Dirac equation in a boundary derivative expansion extending the results in [2]. The Schwinger-Keldysh correlators derived using this prescription automatically satisfy the appropriate KMS relations with Fermi-Dirac factors.

Chiral vortical effect from the compactified D4-branes with smeared D0-brane charge

By using the boundary derivative expansion formalism of fluid/gravity correspondence, we study the chiral vortical effect from the compactified D4-branes with smeared D0-brane charge. This background corresponds to a strongly coupled, nonconformal relativistic fluid with a conserved vector current. The presence of the chiral vortical effect is induced by the addition of a Chern-Simons term in the bulk action. Except that the non-dissipative anomalous viscous coefficient and the sound speed rely only on the chemical potential, most of the other thermal and hydrodynamical quantities of the first order depend both on the temperature and the chemical potential. According to our result, the way that the chiral vortical effect coefficient depends on the chemical potential seems irrelevant with whether the relativistic fluid is conformal or not. Stability analysis shows that this anomalous relativistic fluid is stable and the doping of the smeared D0-brane charge will slow down the sound speed.

Fluid/gravity correspondence: second order transport coefficients in compactified D4-branes

We develop the boundary derivative expansion (BDE) formalism of fluid/gravity correspondence to nonconformal version through the compactified, nearextremal black D4-brane. We offer an explicit calculation of 9 second order transport coefficients, i.e., the τπ , τπ∗, τΠ, λ1,2,3 and ξ1,2,3 for the strongly coupled, uncharged and nonconformal relativistic fluid which is the holographic dual of compactified, near extremal black D4-brane. We also show that the nonconformal fluid considered in this work is free of causal problem and admits the Haack-Yarom relation 4λ1 − λ2 = 2ητπ.

Fluid/gravity correspondence: A nonconformal realization in compactified D4 branes

We develop the framework of boundary derivative expansion (BDE) formalism of fluid/gravity correspondence in a compactified D4-brane system, which is a nonconformal background used in top-down holographic QCD models. Such models contain the D4-D6 model and the Sakai-Sugimoto (SS) model, with the background of the compactified black D4 branes under the near-horizon limit. By using the dimensional reduction technique, we derive a 5D Einstein gravity minimally coupled with three scalar fields from the 10D D4-brane background. Following the BDE formalism of fluid/gravity correspondence in the conformal background, we directly derive all the first order transport coefficients for nonconformal gluonic matter. The results of the ratio of the bulk to shear viscosity and the sound speed agree with those obtained from the Green-Kubo method. This agreement guarantees the validity of the BDE formalism of fluid/gravity duality in the nonconformal D-brane background, which can be used to calculate the second order transport coefficients in nonconformal background.

Forced fluid dynamics from gravity in arbitrary dimensions

We consider long wavelength solutions to the Einstein-dilaton system with negative cosmological constant which are dual, under the AdS/CFT correspondence, to solutions of the conformal relativistic Navier-Stokes equations with a dilaton-dependent forcing term. Certain forced fluid flows are known to exhibit turbulence; holographic duals of forced fluid dynamics are therefore of particular interest as they may aid efforts towards an explicit model of holographic steady state turbulence. In recent work, Bhattacharyya et al. have constructed long wavelength asymptotically locally AdS5 bulk space-times with a slowly varying boundary dilaton field which are dual to forced fluid flows on the 4–dimensional boundary. In this paper, we generalise their work to arbitrary space-time dimensions; we explicitly compute the dual bulk metric, the fluid dynamical stress tensor and Lagrangian to second order in a boundary derivative expansion.

Moving quark in a viscous fluid

To study the rate of energy and momentum loss of a heavy quark in QGP, specifically in the hydrodynamic regime, we use fluid/gravity duality and construct a perturbative procedure to find the string solution in gravity side. We show that by this construction the drag force exerted on the quark can be computed perturbatively, order by order in a boundary derivative expansion. At ideal order, our result is just the drag force exerted on a moving quark in thermal plasma with thermodynamics variables promoted to become local functions of space and time. Furthermore, we apply this procedure to a transverse quark in Bjorken flow and compute the first-derivative corrections, namely the viscous corrections, to the drag force.

Hydrodynamics from charged black branes

We extend the recent work on fluid-gravity correspondence to charged black-branes by determining the metric duals to arbitrary charged fluid configuration up to second order in the boundary derivative expansion. We also derive the energy-momentum tensor and the charge current for these configurations up to second order in the boundary derivative expansion. We find a new term in the charge current when there is a bulk Chern-Simons interaction thus resolving an earlier discrepancy between thermodynamics of charged rotating black holes and boundary hydrodynamics. We have also confirmed that all our expressions are covariant under boundary Weyl-transformations as expected.

Gravity and hydrodynamics: lectures on the fluid-gravity correspondence

We discuss recent developments in the hydrodynamic description of strongly coupled conformal field theories using the AdS/CFT correspondence. In particular, we review aspects of the fluid-gravity correspondence which provides a map between a class of inhomogeneous, dynamical, black hole solutions in asymptotically AdS spacetimes and arbitrary fluid flows in the strongly interacting boundary field theory. We explain how the geometric duals to the fluid dynamics are constructed in a boundary derivative expansion and use the construction to extract the hydrodynamic transport coefficients. In addition, we also describe the recent developments extending the correspondence to incorporate matter fields and to non-relativistic systems. Based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge Theories, Geneva, Switzerland (February 2009).

Forced fluid dynamics from gravity

We generalize the computations of [1] to generate long wavelength, asymptotically locally AdS5 solutions to the Einstein-dilaton system with a slowly varying boundary dilaton field and a weakly curved boundary metric. Upon demanding regularity , our solutions are dual, under the AdS/CFT correspondence, to arbitrary fluid flows in the boundary theory formulated on a weakly curved manifold with a prescribed slowly varying coupling constant. These solutions turn out to be parameterized by four-velocity and temperature fields that are constrained to obey the boundary covariant Navier Stokes equations with a dilaton dependent forcing term. We explicitly evaluate the stress tensor and Lagrangian as a function of the velocity, temperature, coupling constant and curvature fields, to second order in the derivative expansion and demonstrate the Weyl covariance of these expressions. We also construct the event horizon of the dual solutions to second order in the derivative expansion, and use the area form on this event horizon to construct an entropy current for the dual fluid. As a check of our constructions we expand the exactly known solutions for rotating black holes in global AdS5 in a boundary derivative expansion and find perfect agreement with all our results upto second order. We also find other simple solutions of the forced fluid mechanics equations and discuss their bulk interpretation. Our results may aid in determining a bulk dual to forced flows exhibiting steady state turbulence.

Local fluid dynamical entropy from gravity

Spacetime geometries dual to arbitrary fluid flows in strongly coupled N=4 super Yang Mills theory have recently been constructed perturbatively in the long wavelength limit. We demonstrate that these geometries all have regular event horizons, and determine the location of the horizon order by order in a boundary derivative expansion. Intriguingly, the derivative expansion allows us to determine the location of the event horizon in the bulk as a local function of the fluid dynamical variables. We define a natural map from the boundary to the horizon using ingoing null geodesics. The area-form on spatial sections of the horizon can then be pulled back to the boundary to define a local entropy current for the dual field theory in the hydrodynamic limit. The area theorem of general relativity guarantees the positivity of the divergence of the entropy current thus constructed.

Higher derivative corrections to locally black brane metrics

In this paper we generalize the construction of locally boosted black brane space time to higher derivative gravities. We consider the Gauss-Bonnet term (with coefficient $\alpha'$) as a toy example. We find the solution to the $\alpha'$ corrected Einstein equations to first order in the boundary derivative expansion. This allows us to find the $\alpha'$ corrections to the boundary stress tensor in the presence of the Gauss-Bonnet term in the bulk action. We therefore obtain the ratio of shear viscosity to entropy which agrees with other methods of computation in the literature.

Nonlinear fluid dynamics from gravity

Black branes in AdS5 appear in a four parameter family labeled by their velocity and temperature. Promoting these parameters to Goldstone modes or collective coordinate fields—arbitrary functions of the coordinates on the boundary of AdS5—we use Einstein's equations together with regularity requirements and boundary conditions to determine their dynamics. The resultant equations turn out to be those of boundary fluid dynamics, with specific values for fluid parameters. Our analysis is perturbative in the boundary derivative expansion but is valid for arbitrary amplitudes. Our work may be regarded as a derivation of the nonlinear equations of boundary fluid dynamics from gravity. As a concrete application we find an explicit expression for the expansion of this fluid stress tensor including terms up to second order in the derivative expansion.