Applications of holography to strongly coupled Hydrodynamics

Resumo

Let us summarize the main results of this work and the some possible future research directions related to them. We have seen that the quasinormal spectra of black holes, and thus of the poles of corresponding retarded correlators, are related to geometric properties of the black holes, in particular to its causal structure. In the large frequency limit the location of the singularities of thermal correlators can be explained in terms of null geodesics bouncing in the singularities and the boundaries of an eternal AdS black hole. This computation is related to the Schwinger-Keldish formalish up to identification of the fields living in different boundaries with insertion of operators in different pieces of the SK path. An interesting extension of this work is to study whether the geometric analysis of null geodesics can be generalized to other backgrounds like rotating or charged black holes, or to other geometries like asymptotically flat spaces where there is no boundary like in aAdS, so the space does not act like a box. It can also be interesting to study the effects of quantum corrections to the background, what implies to go beyond the large N limit or equivalently to introduce higher curvature corrections. We have seen that the response to small external perturbations is completely determined by the poles of the retarded propagators and their associated residues and used it to study the linear response of the plasma phase of the strongly coupled N = 4 SYM theory. This allows us to explore the validity of the hydrodynamic regime, based on integrating out all the modes but the hydrodynamic ones, in two different ways: defining a hydrodynamic time and legth scales that measure from when on the contribution of the hydrodynamic modes becomes dominant and by analyzing in which range of wavelengths and frequencies the hydrodynamic modes alone give a reliable description of the system. The very short hydrodynamic times obtained indicate that the perturbed plasma thermalises extremely fast, a result that can be used as an indication of fast thermalization at RHIC. The breakdown of the hydrodynamic approximation is somehow built into the theory: when decreasing the wavelength, more and more quasinomal modes have to be taken into account to describe the system. The weight of collective excitations in the plasma depends crucially on the value of the residues. For the shear and R-charge diffusion modes, it shows an oscillatory decaying behavior, signaling their decoupling at small momentum. It would be interesting to examine if that behavior is universal of hydrodynamic diffusion modes. A priori there is an indication that something drastic happens at short wavelength coming from the observation that causality is preserved but at the same 114 Summary and Outlook time first order hydrodynamic (that is acausal) is reproduced at long wavelengths. At RHIC there are processes taking place that cannot be described within the linear response theory. Therefore it would be of high interest to go beyond this approximation in order to have a more meaningful definition of the hydrodynamic scale that might be relevant for RHIC experiments in which the scale of thermalization is crucial. All the discussion only contains adjoint matter. To really have a predictive model relevant for real world experiments it would be nice to repeat the analysis including fundamental matter, achieved by the addition of new sets of branes in the string background. Finally, we have studied the hydrodynamic behavior of the holographic superfluid given by an abelian gauge model with a massive scalar field in a fixed AdS black hole background. The lowest quasinormal mode of the charged scalar field becomes tachyonic at a certain value of the temperature. This instability indicates that the scalar field condenses and the system undergoes a second order phase transition. At the phase transition this scalar mode is massless and its susceptibility diverges, so it can be identified with the Goldstone boson appearing at the SSB. Below the critical temperature this mode is identified with the second sound. Using holography, we have been able to compute reactive transport coefficients as the speed of second sound, computed directly from thermodynamic considerations in other works, and also non-thermodynamic quantities like absorptive transport coefficients, as is the case of attenuation of the second sound or the diffusion constant. As a side result, we have developed a method to compute the physical quasinormal modes of coupled systems in terms of the non-gauge invariant variables. In this analysis the backreaction due to the presence of the scalar and gauge fields has been neglected. An obvious extension is to consider the backreacted model in which metric fluctuations are allowed and the dynamics of the system is richer. In that model even for a neutral scalar field there exist a phase transition at a finite temperature due to the instability of the background. For small charge of the scalar two competing mechanisms responsible of the condensation are present, the coupling to the gauge field and the instability of the metric. It will be interesting to find which mode is the order parameter of the phase transition for each case and what happens in the broken phase in the second case. It is also interesting the study of the interplay between normal and second sound. The complete investigation of the backreacted case hopefully will shed some light on the origin of the conductivity gap in these models. Another natural extension of this model is to consider high curvature corrections to examine whether or not the phase transition takes place and how the phase diagram is modified. Another extension of interest would be the explicit breaking of conformal invariance introducing for instance a mass deformation and study how the transition is affected.