Mukund Rangamani

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 Einsteins 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.

A Covariant holographic entanglement entropy proposal

With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.

Heating up Galilean holography

We embed a holographic description of a quantum field theory with Galilean conformal invariance in string theory. The key observation is that such field theories may be realized as conventional superconformal field theories with a known string theory embedding, twisted by the R-symmetry in a light-like direction. Using the Null Melvin Twist, we construct the appropriate dual geometry and its non-extremal generalization. From the nonzero temperature solution we determine the equation of state. We also discuss the hydrodynamic regime of these non-relativistic plasmas and show that the shear viscosity to entropy density ratio takes the universal value �/s = 1/4� typical of strongly interacting field theories with gravity duals.

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).

Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm

We formulate a holographic Wilsonian renormalization group flow for strongly coupled systems with a gravity dual, motivated by the need to extract efficiently low energy behavior of such systems. Starting with field theories defined on a cut-off surface in a bulk spacetime, we propose that integrating out high energy modes in the field theory should correspond to integrating out a part of the bulk geometry. We describe how to carry out this procedure in practice in the classical gravity approximation using examples of scalar and vector fields. By integrating out bulk degrees of freedom all the way to a black hole horizon, this formulation defines a refined version of the black hole membrane paradigm. Furthermore, it also provides a derivation of the semi-holographic description of low energy physics.

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.

Causality & holographic entanglement entropy

A bstractWe identify conditions for the entanglement entropy as a function of spatial region to be compatible with causality in an arbitrary relativistic quantum field theory. We then prove that the covariant holographic entanglement entropy prescription (which relates entanglement entropy of a given spatial region on the boundary to the area of a certain extremal surface in the bulk) obeys these conditions, as long as the bulk obeys the null energy condition. While necessary for the validity of the prescription, this consistency requirement is quite nontrivial from the bulk standpoint, and therefore provides important additional evidence for the prescription. In the process, we introduce a codimension-zero bulk region, named the entanglement wedge, naturally associated with the given boundary spatial region. We propose that the entanglement wedge is the most natural bulk region corresponding to the boundary reduced density matrix.

Causal holographic information

A bstractWe propose a measure of holographic information based on a causal wedge construction. The motivation behind this comes from an attempt to understand how boundary field theories can holographically reconstruct spacetime. We argue that given the knowledge of the reduced density matrix in a spatial region of the boundary, one should be able to reconstruct at least the corresponding bulk causal wedge. In attempt to quantify the ‘amount of information’ contained in a given spatial region in field theory, we consider a particular bulk surface (specifically a co-dimension two surface in the bulk spacetime which is an extremal surface on the boundary of the bulk causal wedge), and propose that the area of this surface, measured in Planck units, naturally quantifies the information content. We therefore call this area the causal holographic information. We also contrast our ideas with earlier studies of holographic entanglement entropy. In particular, we establish that the causal holographic information, whilst not being a von Neumann entropy, curiously enough agrees with the entanglement entropy in all cases where one has a microscopic understanding of entanglement entropy.

Holographic Entanglement Entropy

The discussions in §2 and §3 make it rather clear that while we have to evaluate a sequence of functional integrals to compute the Renyi entropies, these are rather complex quantities which required us to work with QFTs on singular branched cover manifolds. Apart from the case of CFT2 discussed in §3, where the power of conformal invariance can be used to simplify the problem, this is a rather formidable task for interacting QFTs, in general.

Brownian Motion in AdS/CFT

We study Brownian motion and the associated Langevin equation in AdS/CFT. The Brownian particle is realized in the bulk spacetime as a probe fundamental string in an asymptotically AdS black hole background, stretching between the AdS boundary and the horizon. The modes on the string are excited by the thermal black hole environment and consequently the string endpoint at the boundary undergoes an erratic motion, which is identified with an external quark in the boundary CFT exhibiting Brownian motion. Semiclassically, the modes on the string are thermally excited due to Hawking radiation, which translates into the random force appearing in the boundary Langevin equation, while the friction in the Langevin equation corresponds to the excitation on the string being absorbed by the black hole. We give a bulk proof of the fluctuation-dissipation theorem relating the random force and friction. This work can be regarded as a step toward understanding the quantum microphysics underlying the fluid-gravity correspondence. We also initiate a study of the properties of the effective membrane or stretched horizon picture of black holes using our bulk description of Brownian motion.