Emergence of Gravity and RG Flow
01 Jan 2017-Vol. 187, pp 283-302
TL;DR: In this paper, Padmanabhan et al. reformulated Einstein's equations in AdS as a non-perturbative RG flow that further leads to a new approach towards constructing strongly interacting QFTs.
Abstract: This is a tribute to Padmanabhan’s works on the holographic principle which have consistently enunciated the profound philosophy that the classical equations of gravity themselves hold the key to understanding their holographic origin. I discuss how this can be realised by reformulating Einstein’s equations in AdS as a non-perturbative RG flow that further leads to a new approach towards constructing strongly interacting QFTs. For a concrete demonstration, I focus on the hydrodynamic limit in which case this RG flow connects the AdS/CFT correspondence with the membrane paradigm.
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TL;DR: In this paper, the authors present a collection of notes based on lectures given at IIT Madras in September 2019 and at IFT Madrid in November 2019 on applied holography and especially the analytic and numerical techniques involved.
Abstract: This is a collection of notes based on lectures given at IIT Madras in September 2019 and at IFT Madrid in November 2019. It is supposed to be a concise (and therefore not comprehensive) and pragmatic course on applied holography and especially the (basic) analytic and numerical techniques involved. The lectures are not focused on the large theoretical and fundamental background which can be found already in several places in the literature, but rather on concrete applications of Bottom-Up AdS-CFT to Hydrodynamics, QCD and Condensed Matter. The idea is to accompany the reader step by step through the various benchmark examples with a classmate attitude, providing details of the computations and open-source numerical codes in Mathematica, and sharing simple tricks and warnings collected during my research experience. At the end of this path, the reader will be in possess of all the fundamental skills and tools to learn by himself/herself more advanced techniques and to produce independent and novel research on the topic.
81 citations
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TL;DR: In this paper, the authors developed parallels between the holographic renormalization group in the bulk and the Wilsonian renormalisation group in dual field theory, and sharpened the analogy between the two sides by sharpening the analogy.
Abstract: We develop parallels between the holographic renormalization group in the bulk and the Wilsonian renormalization group in the dual field theory. Our philosophy differs from most previous work on the holographic RG; the most notable feature is the key role of multi-trace operators. We work out the forms of various single- and double-trace flows. The key question, `what cutoff on the field theory corresponds to a radial cutoff in the bulk?' is left unanswered, but by sharpening the analogy between the two sides we identify possible directions.
283 citations
TL;DR: A general ansatz for gravitational entropy can be provided using the criterion that any patch of area which acts as a horizon for a suitably defined accelerated observer must have an entropy proportional to its area as discussed by the authors.
Abstract: A general ansatz for gravitational entropy can be provided using the criterion that any patch of area which acts as a horizon for a suitably defined accelerated observer must have an entropy proportional to its area. After providing a brief justification for this ansatz, several consequences are derived. (i) In any static spacetime with a horizon and associated temperature β−1, this entropy satisfies the relation S = (1/2)βE where E is the energy source for gravitational acceleration, obtained as an integral of (Tab − (1/2)Tgab)uaub. (ii) With this ansatz of S, the minimization of Einstein–Hilbert action is equivalent to minimizing the free energy F with βF = βU − S where U is the integral of Tabuaub. We discuss the conditions under which these results imply S ∝ E2 and/or S ∝ U2 thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.
252 citations
TL;DR: In this article, a cutoff-dependent line-integral formula for the diffusion constant D (r = r�� c¯¯¯¯ outside the horizon in a general class of black hole geometries is derived.
Abstract: The problem of gravitational fluctuations confined inside a finite cutoff at radius r = r
c
outside the horizon in a general class of black hole geometries is considered. Consistent boundary conditions at both the cutoff surface and the horizon are found and the resulting modes analyzed. For general cutoff r
c
the dispersion relation is shown at long wavelengths to be that of a linearized Navier-Stokes fluid living on the cutoff surface. A cutoff-dependent line-integral formula for the diffusion constant D (r
c
) is derived. The dependence on r
c
is interpreted as renormalization group (RG) flow in the fluid. Taking the cutoff to infinity in an asymptotically AdS context, the formula for D(∞) reproduces as a special case well-known results derived using AdS/CFT. Taking the cutoff to the horizon, the effective speed of sound goes to infinity, the fluid becomes incompressible and the Navier-Stokes dispersion relation becomes exact. The resulting universal formula for the diffusion constant D(horizon) reproduces old results from the membrane paradigm. Hence the old membrane paradigm results and new AdS/CFT results are related by RG flow. RG flow-invariance of the viscosity to entropy ratio $ \frac{\eta }{s} $
is shown to follow from the first law of thermodynamics together with isentropy of radial evolution in classical gravity. The ratio is expected to run when quantum gravitational corrections are included.
238 citations
TL;DR: The leading singularity in the Borel transform of the hydrodynamic energy density with the lowest nonhydrod dynamic excitation corresponding to a 'nonhydrodynamic' quasinormal mode on the gravity side is identified.
Abstract: We utilize the fluid-gravity duality to investigate the large order behavior of hydrodynamic gradient expansion of the dynamics of a gauge theory plasma system. This corresponds to the inclusion of dissipative terms and transport coefficients of very high order. Using the dual gravity description, we calculate numerically the form of the stress tensor for a boost-invariant flow in a hydrodynamic expansion up to terms with 240 derivatives. We observe a factorial growth of gradient contributions at large orders, which indicates a zero radius of convergence of the hydrodynamic series. Furthermore, we identify the leading singularity in the Borel transform of the hydrodynamic energy density with the lowest nonhydrodynamic excitation corresponding to a ‘nonhydrodynamic’ quasinormal mode on the gravity side.
235 citations
TL;DR: In this article, dissipative test electromagnetic fields in a black-hole background were studied and they were shown to satisfy Ohm's law with a surface resistivity of $4.377$ ohms.
Abstract: We study dissipative test electromagnetic fields in a black-hole background. Quantities such as surface velocity, tangential electric field, normal magnetic induction, total surface current, and conduction surface current are introduced and are shown to satisfy Ohm's law with a surface resistivity of $4\ensuremath{\pi}\ensuremath{\simeq}377$ ohms. Associated with these currents there exists a "Joule heating". These currents can exist when the black hole is inserted in an external electric circuit, but they can exist even in the absence of external currents. In particular, we study the eddy currents induced by the rotation of a black hole in an oblique uniform magnetic field, and we show how the computation of the ohmic losses allows a very simple derivation of the torque exerted on the hole.
222 citations