Finite Cutoff Surface Research Articles

Glue-on AdS holography for Toverline{T} -deformed CFTs

The Toverline{T} deformation is a solvable irrelevant deformation whose properties depend on the sign of the deformation parameter μ. In particular, Toverline{T} -deformed CFTs with μ < 0 have been proposed to be holographically dual to Einstein gravity where the metric satisfies Dirichlet boundary conditions at a finite cutoff surface. In this paper, we put forward a holographic proposal for Toverline{T} -deformed CFTs with μ > 0, in which case the bulk geometry is constructed by gluing a patch of AdS3 to the original spacetime. As evidence, we show that the Toverline{T} trace flow equation, the spectrum on the cylinder, and the partition function on the torus and the sphere, among other results, can all be reproduced from bulk calculations in glue-on AdS3.

Sonic velocity in holographic fluids and its applications**Supported by National Natural Science Foundation of China (NSFC) (11575083, 11565017, 11105004), the Fundamental Research Funds for the Central Universities (NS2015073), Shanghai Key Laboratory of Particle Physics and Cosmology (11DZ2260700), the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (Y5KF161CJ1). In addition,

Gravity/fluid correspondence acts as an important tool in investigating the strongly correlated fluids. We carefully investigate the holographic fluids at the finite cutoff surface by considering different boundary conditions in the scenario of gravity/fluid correspondence. We find that the sonic velocity of the boundary fluids at the finite cutoff surface is critical in clarifying the superficial similarity between the bulk viscosity and perturbation of the pressure for the holographic fluid, where we set a special boundary condition at the finite cutoff surface to explicitly express this superficial similarity. Moreover, we further take the sonic velocity into account to investigate a case with a more general boundary condition. In this more genaral case, although two parameters in the first order stress tensor of holographic fluid cannot be fixed, one can still extract the information about the transport coefficients by considering the sonic velocity seriously.

Rindler fluid with weak momentum relaxation

We realize the weak momentum relaxation in Rindler fluid, which lives on the time-like cutoff surface in an accelerating frame of flat spacetime. The translational invariance is broken by massless scalar fields with weak strength. Both of the Ward identity and the momentum relaxation rate of Rindler fluid are obtained, with higher order correction in terms of the strength of momentum relaxation. The Rindler fluid with momentum relaxation could also be approached through the near horizon limit of cutoff AdS fluid with momentum relaxation, which lives on a finite time-like cutoff surface in Anti-de Sitter(AdS) spacetime, and further could be connected with the holographic conformal fluid living on AdS boundary at infinity. Thus, in the holographic Wilson renormalization group flow of the fluid/gravity correspondence with momentum relaxation, the Rindler fluid can be considered as the Infrared Radiation(IR) fixed point, and the holographic conformal fluid plays the role of the ultraviolet(UV) fixed point.

Note on the Petrov-like boundary condition at finite cutoff surface in gravity/fluid duality

Previously it has been shown that imposing a Petrov-like boundary condition on a hypersurface may reduce the Einstein equation to the incompressible Navier-Stokes equation, but all these correspondences are established in the near horizon limit. In this note, we remark that this strategy can be extended to an arbitrary finite cutoff surface which is spatially flat, and the Navier-Stokes equation is obtained by employing a non-relativistic long-wavelength limit.

Petrov type I spacetime and dual relativistic fluids

The Petrov type I condition for the solutions of vacuum Einstein equations in both of the nonrelativistic and relativistic hydrodynamic expansions is checked. We show that it holds up to the third order of the nonrelativistic hydrodynamic expansion parameter, but it is violated at the fourth order even if we choose a general frame. On the other hand, it is found that the condition holds at least up to the second order of the derivative expansion parameter. Turn the logic around, through imposing the Petrov type I condition and Hamiltonian constraint on a finite cutoff surface, we show that the stress tensor of the relativistic fluid can be recovered with correct first order and second order transport coefficients dual to the solutions of vacuum Einstein equations.

Bulk viscosity of dual fluid at finite cutoff surface via gravity/fluid correspondence in Einstein–Maxwell gravity

Based on the previous paper arXiv:1207.5309, we investigate the possibility to find out the bulk viscosity of dual fluid at the finite cutoff surface via gravity/fluid correspondence in Einstein–Maxwell gravity. We find that if we adopt new conditions to fix the undetermined parameters contained in the stress tensor and charged current of the dual fluid, two new terms appear in the stress tensor of the dual fluid. One new term is related to the bulk viscosity term, while the other can be related to the perturbation of energy density. In addition, since the parameters contained in the charged current are the same, the charged current is not changed.

Holographic parity-violating charged fluid dual to Chern-Simons modified gravity

We discuss the ($2+1$)-dimensional parity-violating charged fluid on a finite cutoff surface ${\mathrm{\ensuremath{\Sigma}}}_{c}$, dual to the nondynamical and dynamical Chern-Simons (CS) modified gravities. Using the nonrelativistic long-wavelength expansion method, the field equations are solved up to $\mathcal{O}({\ensuremath{\epsilon}}^{2})$ in the nondynamical model. It is shown that there exists nonvortical dual fluid with shear viscosity $\ensuremath{\eta}$ and Hall viscosity ${\ensuremath{\eta}}_{A}$ on the cutoff surface ${\mathrm{\ensuremath{\Sigma}}}_{c}$. The ratio of shear viscosity over entropy density $\ensuremath{\eta}/s$ of the fluid takes the universal value $1/4\ensuremath{\pi}$, while the ratio of Hall viscosity over entropy density ${\ensuremath{\eta}}_{A}/s$ depends on the ${\mathrm{\ensuremath{\Sigma}}}_{c}$ and black brane charge $q$. Moreover, the nonvortical dual fluid obeys the magnetohydrodynamic (MHD) equation. However, these kinematic viscosities $\ensuremath{\nu}$ and ${\ensuremath{\nu}}_{A}$ related to $\ensuremath{\eta}$ and ${\ensuremath{\eta}}_{A}$ do not appear in this MHD equation due to the constraint condition ${\stackrel{\texttildelow{}}{\ensuremath{\partial}}}^{2}{\ensuremath{\beta}}_{j}=0$ for the ($2+1$)-dimensional dual fluid. Then, we extend our discussion to the dynamical CS modified gravity and show that the dual vortical fluid possesses another so-called Curl viscosity ${\ensuremath{\zeta}}_{A}$, whose ratio to entropy density ${\ensuremath{\zeta}}_{A}/s$ also depends on the ${\mathrm{\ensuremath{\Sigma}}}_{c}$ and $q$. Moreover, the value of $\ensuremath{\eta}/s$ still equals $1/4\ensuremath{\pi}$ and the result of ${\ensuremath{\eta}}_{A}/s$ agrees with the previous result under the probe limit of the pseudoscalar field at the infinite boundary in the charged black brane background for the dynamical CS modified gravity. This vortical dual fluid corresponds to the MHD turbulence equation in plasma physics.

Holographic charged fluid dual to third order Lovelock gravity

We study the dual fluid on a finite cutoff surface outside the black brane horizon in the third order Lovelock gravity. Using nonrelativistic long-wavelength expansion, we obtain the incompressible Navier-Stokes equations of dual fluid with external force density on the finite cutoff surface. The viscosity to entropy density ratio $\eta/s$ is independent of cutoff surface and does not get modification from the third order Lovelock gravity influence. The obtained ratio agrees with the results obtained by using other methods, such as the Kubo formula at the AdS boundary and the membrane paradigm at the horizon in the third order Lovelock gravity. These results can be related by Wilson renormalization group flow. However the kinematic viscosity receives correction from the third order Lovelock term. We show that the equivalence between the isentropic flow of the fluid and the radial component of the gravitational equation observed in the Einstein and Gauss-Bonnet gravities also holds in the third order Lovelock gravity. This generalization brings more understandings of relating the gravity theory to the dual fluid.

Petrov type I condition and dual fluid dynamics

Recently Lysov and Strominger [arXiv:1104.5502] showed that imposing Petrov type I condition on a $(p+1)$-dimensional timelike hypersurface embedded in a $(p+2)$-dimensional vacuum Einstein gravity reduces the degrees of freedom in the extrinsic curvature of the hypersurface to that of a fluid on the hypersurface, and that the leading-order Einstein constraint equations in terms of the mean curvature of the embedding give the incompressible Navier-Stokes equations of the dual fluid. In this paper we show that the non-relativistic fluid dual to vacuum Einstein gravity does not satisfy the Petrov type I condition at next order, unless additional constraint such as the irrotational condition is added. In addition, we show that this procedure can be inversed to derive the non-relativistic hydrodynamics with higher order corrections through imposing the Petrov type I condition, and that some second order transport coefficients can be extracted, but the dual "Petrov type I fluid" does not match the dual fluid constructed from the geometry of vacuum Einstein gravity in the non-relativistic limit. We discuss the procedure both on the finite cutoff surface via the non-relativistic hydrodynamic expansion and on the highly accelerated surface via the near horizon expansion.

Holographic charged fluid with anomalous current at finite cutoff surface in Einstein-Maxwell gravity

The holographic charged fluid with anomalous current in Einstein-Maxwell gravity has been generalized from the infinite boundary to the finite cutoff surface by using the gravity/fluid correspondence. After perturbing the boosted Reissner-Nordstrom (RN)-AdS black brane solution of the Einstein-Maxwell gravity with the Chern-Simons term, we obtain the first order perturbative gravitational and Maxwell solutions, and calculate the stress tensor and charged current of the dual fluid at finite cutoff surfaces which contains undetermined parameters after demanding regularity condition at the future horizon. We adopt the Dirichlet boundary condition and impose the Landau frame to fix these parameters, finally obtain the dependence of transport coefficients in the dual stress tensor and charged current on the arbitrary radical cutoff $r_c$. We find that the dual fluid is not conformal, but it has vanishing bulk viscosity, and the shear viscosity to entropy density ratio is universally $1/4\pi$. Other transport coefficients of the dual current turns out to be cutoff-dependent. In particular, the chiral vortical conductivity expressed in terms of thermodynamic quantities takes the same form as that of the dual fluid at the asymptotic AdS boundary, and the chiral magnetic conductivity receives a cutoff-dependent correction which vanishes at the infinite boundary.

Incompressible Navier-Stokes equations from Einstein gravity with Chern-Simons term

In (2+1)-dimensional hydrodynamic systems with broken parity, the shear and bulk viscosity is joined by the Hall viscosity and curl viscosity. The dual holographic model has been constructed by coupling a pseudo scalar to the gravitational Chern-Simons term in (3+1)-dimensional bulk gravity. In this paper, we investigate the non-relativistic fluid with Hall viscosity and curl viscosity living on a finite radial cutoff surface in the bulk. Employing the non-relativistic hydrodynamic expansion method, we obtain the incompressible Navier-Stokes equations with Hall viscosity and curl viscosity. Unlike the shear viscosity, the ratio of the Hall viscosity over entropy density is found to be cutoff scale dependent, and it tends to zero when the cutoff surface approaches to the horizon of the background spacetime.

Holographic forced fluid dynamics in non-relativistic limit

We study the thermodynamics and non-relativistic hydrodynamics of the holographic fluid on a finite cutoff surface in the Gauss–Bonnet gravity. It is shown that the isentropic flow of the fluid is equivalent to a radial component of gravitational field equations. We use the non-relativistic fluid expansion method to study the Einstein–Maxwell-dilaton system with a negative cosmological constant, and obtain the holographic incompressible forced Navier–Stokes equations of the dual fluid at AdS boundary and at a finite cutoff surface, respectively. The concrete forms of external forces are given.

Holographic RG flows and transport coefficients in Einstein-Gauss-Bonnet-Maxwell theory

We apply the membrane paradigm and the holographic Wilsonian approach to the Einstein-Gauss-Bonnet-Maxwell theory. The transport coefficients for a quark-gluon plasma living on the cutoff surface are derived in a spacetime of charged black brane. Because of the mixing of the Gauss-Bonnet coupling and the Maxwell fields, the vector modes/shear modes of the metric and Maxwell fluctuations turn out to be very difficult to decouple. We firstly evaluate the AC conductivity at a finite cutoff surface by solving the equation of motion numerically, then manage to derive the radial flow of DC conductivity with the use of the Kubo formula. It turns out that our analytical results match the numerical data in low frequency limit very well. The diffusion constant $D(u_c)$ is also derived in a long wavelength expansion limit. We find it depends on the Gauss-Bonnet coupling as well as the position of the cutoff surface.

Non-relativistic fluid dual to asymptotically AdS gravity at finite cutoff surface

Using the non-relativistic hydrodynamic expansion, we solve equations of motion for Einstein gravity and Gauss-Bonnet gravity with a negative cosmological constant within the region between a finite cutoff surface and a black brane horizon, up to the second order of the expansion parameter. Through the Brown-York tensor, we calculate the stress energy tensor of dual fluids living on the cutoff surface. With the black brane solutions, we show that for both Einstein gravity and Gauss-Bonnet gravity, the ratio of shear viscosity to entropy ensity of dual fluid does not run with the cutoff surface. The incompressible Navier-Stokes equations are also obtained in both cases.