The famous AdS/CFT calculation of the shear viscosity/entropy ratio for strongly coupled $N=4$ SYM relates the shear viscosity to the absorption cross section for fluctuations of the metric onto a black hole in AdS which in turn can be related to the absorption cross section for a massless scalar field. On the AdS side it seems clear that dissipation and relaxation back to an equilibrium configuration after a disturbance means that the disturbance has been absorbed by the black hole, and this would seem to imply that the disturbance propagates from small $z$ to large $z$ in Poincare coordinates where $z \rightarrow 0$ is the UV boundary. That is, the energy of the disturbance propagates from the UV to the IR where it is dissipated. First question: is this the right physical picture? Second question: this seems to imply on the CFT side that the dissipation of energy due to shear viscosity also involves a transfer of energy from the UV to the IR. Is this consistent with what is known in fluid mechanics for viscous flow? If so, can you provide a reference?
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$\begingroup$ I don't know the answer in general, but if you allow me quick stream of consciousness comment: I remember that in turbulent flow the physical picture is the opposite - energy is transferred from large scale structures to small ones, and is dissipated at the UV. $\endgroup$– user566Feb 15, 2011 at 5:02
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$\begingroup$ But turbulent flow in two (spatial) dimensions goes the other way, from UV to IR, right? (Doesn't seem directly relevant, but at least shows this direction can be physically sensible....) $\endgroup$– Matt ReeceFeb 15, 2011 at 5:34
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$\begingroup$ Just relying on vague and not so recent memory, you may be right. My comment at least is not directly relevant, the fluid flow modelled by AdS is not turbulent. Hopefully someone with experience in fluid dynamics will help. $\endgroup$– user566Feb 15, 2011 at 5:47
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$\begingroup$ Dear @Moshe, in your 1st comment, are you describing the Hawking radiation in the CFT variables? If the viscosity is linked to the absorption by a black hole, how could Jeff's translation go wrong? $\endgroup$– Luboš MotlFeb 15, 2011 at 6:52
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$\begingroup$ @Lubos: I was talking about the phenomenology of turbulent flow, as seen for example in wind tunnels. I don't see anything wrong with Jeff's reasoning, so I'm also puzzled - but there could be a simple and disappointing explanation. $\endgroup$– user566Feb 15, 2011 at 7:00
1 Answer
Perhaps I am just displaying my ignorance on the subject, but I will stick out my neck and try to answer - I am grateful for comments, especially if my picture is wrong.
ad first question: I would have pictured the dissipation as follows. By switching on some perturbation you bring the black hole into an excited state that can be treated perturbatively as long as it is a small enough excitation. This then leads to quasi-normal ringing of the black hole. The perturbed black hole state falls back to the unperturbed black hole state by emission of gravitational waves towards the AdS boundary. Thus, the dissipation process on the gravity side rather seems to be energy transported from the UV to the IR.
ad second question: By the UV/IR connection the quasi-normal ringing on the gravity side should on the fluid side correspond to energy in the IR being dissipated into the UV. I would have thought that this is precisely what happens in a viscous fluid: you have some long wavelength motion that gets decelerated due to friction, which is basically small wavelength excitations that then thermalize with the heat bath.
References on quasi-normal modes:
In AdS: hep-th/9909056 In general relativity: gr-qc/9909058
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$\begingroup$ This answer sounds physically plausible, but I don't see how to relate it to what people actually calculate. In the calculations of the viscosity I have looked at there is no mention of quasinormal modes. One works in Lorentzian signature and imposes boundary conditions at the horizon that only allow incoming modes, that is absorption of the perturbation. Perhaps there is a more complete physical description that includes some of what you say in this answer but this more complete description is not needed to compute the viscosity. $\endgroup$– phoFeb 16, 2011 at 3:05
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$\begingroup$ This paper explains the relation between quasinormal modes and holography: hep-th/0506184. It is based upon the discovery in hep-th/0112055 that the quasinormal frequencies determine the location of the poles of the retarded correlation function of the corresponding perturbation. BTW, quasinormal modes automatically require the imposition of the boundary conditions you mentioned (infalling bc's at the horizon and Dirichlet bc's at the AdS boundary). $\endgroup$ Feb 16, 2011 at 10:43
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$\begingroup$ Thanks @Daniel Grumiller. Those references look very helpful. $\endgroup$– phoFeb 16, 2011 at 13:44