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Gravity and Hydrodynamics
Gravity and Hydrodynamics
太田, 昌宏
オオタ, マサヒロ
OHTA, Masahiro
We study relationships between gravitational theories and hydrodynamic systems with several approaches. There are at least three approaches which realize this idea: (i) the membrane paradigm, (ii) the AdS/CFT duality, (iii) the BKLS approach. In this thesis, we focus on each the AdS/CFT duality and the BKLS approach individually, and consider relationships among them and the membrane paradigm.
The oldest realization is (i) the membrane paradigm (Price et al. and Parikh et al.), which has been studied in the context of the black hole physics. They tried to map the dynamics of the black hole to hydrodynamics and focused on the black hole's event horizon. An observer outside the black hole cannot see the inside of the black hole but can see the surface of the black hole. Therefore, the black hole dynamics should be effectively described by a dynamical membrane on the stretched horizon, a timelike surface located slightly outside the true horizon. The membrane dynamics is described by the Einstein equation on the stretched horizon and the equation is the same as the Navier-Stokes equations, mathematically. However, the membrane paradigm has the unpleasant features as a fluid such as a negative bulk viscosity. In addition, the microscopic realization of the membrane paradigm is not clear. The membrane paradigm doesn't tell us the microscopic theory which the fluid is based on since just the transformation of the Einstein equation on the stretched horizon leads the hydrodynamic stress tensor.
On the other hand, (ii) the AdS/CFT duality, based on string theory, provides several explicit realizations of microscopic understanding of the corresponding hydrodynamics since the D-branes provides both the gravitational theory and the corresponding field theory. For example, D3-brane provides the AdS_5 x S^5 geometry and the strongly coupled N=4 Super Yang-Mills theory in the large-N_c limit. In addition, AdS/CFT duality is widely applied to real-world physics e.g., the quark-gluon plasma, the condensed matter physics and etc.
From the point of view of the holographic renormalization group, asymptotic AdS geometries correspond to field theories which approach conformal field theories in the UV limit. However, most of real world materials are not conformally invariant in the UV limit. Therefore, the formalism, which doesn't depend on asymptotics is needed. Recently, (iii) the BKLS approach is proposed by Bredberg, Keeler, Lysov, and Strominger. They introduced a timelike surface at arbitrary position for the ``boundary" where the fluid lives. This approach doesn't provide microscopic understanding of the fluid but should describe robust features of the correspondence between the gravity and hydrodynamics, instead.
First, we examine the AdS/CFT duality and the universality of the shear viscosity to the entropy density ratio eta/s for various holographic superfluids. In the study of the AdS/CFT duality, fluids corresponding to a large class of geometries ensure the universality eta/s=1/(4pi). The universality has been extensively studied, and this holds for all known examples which have been studied. We study three types of the holographic superfluids as yet another example of the universality: s-wave, p-eave and (p+ip)-wave holographic superfluids. They are characterized by the order parameter of the phase transition, i.e., in the bulk gravitational theories, the order parameter of the s-wave holographic superfluids is a scalar field, and the one of both p-wave and (p+ip) -wave holographic superfluids is a SU(2) gauge field. (The difference between the p-wave and (p+ip)-wave is condensing components of the gauge field.) For the s-wave case, the ratio has the universal value 1/(4pi) as in various holographic models. For the p-wave case, there are two shear viscosity coefficients because of the anisotropic boundary spacetime, and one coefficient has the universal value. For the other viscosity coefficient, the existing technique is not applicable since there is no tensor mode of metric perturbations which decouples from Yang-Mills perturbations. For the (p+ip)-wave case, the situation is the same as the case of the latter component in the p-wave. These results implies that p-wave and (p+ip)-wave holographic superfluids may not have the universality, and in fact, they are the first examples of the non-universal shear viscosity to the entropy density ratio. Our work triggered detailed studies of the non-universal shear viscosities.
Second, we study another realization, the BKLS approach, where the fluid is defined by the Brown-York tensor on a timelike surface at r=r_c in black hole backgrounds. We consider both Rindler space and the Schwarzschild-AdS (SAdS) black hole. The former describes an incompressible fluid, whereas the latter describes the vanishing bulk viscosity at arbitrary r_c. These two results, however, do not contradict with each other since the hydrodynamic regime used for the SAdS black hole ``differs" from the hydrodynamic regime used for Rindler space (when expressed in terms of the SAdS variables). We also find an interesting ``coincidence" with the black hole membrane paradigm which gives a negative bulk viscosity. In hydrodynamics, the velocity field is determined from the metric perturbations. Then, one can eliminate the velocity field completely in the hydrodynamic stress tensor. The resulting expression contains metric perturbations only, which is suitable to compare with the Brown-York tensor. In our approach, the velocity field is a consequence of metric perturbations. In addition, we obtain one of the second-order hydrodynamic transport coefficient tau_pi for the SAdS_5 black hole in the BKLS approach.
総研大甲第1587号
eng
thesis
https://ir.soken.ac.jp/records/4064
博士（理学）
2013-03-22
総合研究大学院大学
https://ir.soken.ac.jp/record/4064/files/甲1587_要旨.pdf
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2016-02-26
https://ir.soken.ac.jp/record/4064/files/甲1587_本文.pdf
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2016-02-17