Holographic interpretation of acoustic black holes

PHYSICAL REVIEW D 92, 084052 (2015)

Holographic interpretation of acoustic black holes Xian-Hui Ge,2,* Jia-Rui Sun,1,6,† Yu Tian,3,6,‡ Xiao-Ning Wu,4,6,7,§ and Yun-Long Zhang5,∥ 1

Institute of Astronomy and Space Science, Sun Yat-Sen University, Guangzhou 510275, China 2 Department of Physics, Shanghai University, Shanghai 200444, China 3 School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China 4 Institute of Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China 5 Department of Physics and Center of Advanced Study in Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan 6 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China 7 Hua Loo-Keng Key Laboratory of Mathematics, CAS, Beijing 100190, China (Received 16 August 2015; published 26 October 2015) With the attempt to find the holographic description of the usual acoustic black holes in fluid, we construct an acoustic black hole formed in the d-dimensional fluid located at the timelike cutoff surface of a neutral black brane in asymptotically AdSdþ1 spacetime; the bulk gravitational dual of the acoustic black hole is presented at the first order of the hydrodynamic fluctuation. Moreover, the Hawking-like temperature of the acoustic black hole horizon is showed to be connected to the Hawking temperature of the real anti–de Sitter (AdS) black brane in the bulk, and the duality between the phonon scattering in the acoustic black hole and the sound channel quasinormal mode propagating in the bulk perturbed AdS black brane is extracted. We thus point out that the acoustic black hole appearing in fluid, which was originally proposed as an analogous model to simulate Hawking radiation of the real black hole, is not merely an analogy, it can indeed be used to describe specific properties of the real AdS black holes, in the spirit of the fluid/gravity duality. DOI: 10.1103/PhysRevD.92.084052

PACS numbers: 04.70.Dy, 11.10.Wx, 11.25.Tq

I. INTRODUCTION Searching for the relationship between gravity and fluid has a long history. The original study dates back to the late 1970s, during which the black hole membrane paradigm was developed [1] (see also [2,3]). It was shown that in the membrane paradigm formalism, the black hole can be regarded as a viscous fluid living on the null or timelike surface (membrane) on or outside its horizon, while the membrane actually acts as a boundary or a cutoff surface of the black hole spacetime with appropriate boundary conditions and it captures information which can be used as an effective description of the physics inside itself. Later on, Unruh showed that, for the nonrelativistic, irrotational inviscid moving fluid, the equation of motion (EOM) governing the dynamics of the sound mode (phonons) can be expressed as a massless Klein-Gorden equation in an effective spacetime background containing the sonic horizon when the local fluid velocity exceeds the speed of sound, which resembles the real black hole. Consequently, a Hawking-like temperature can be defined for the sonic horizon analogous to the real black hole, so it was named as the acoustic black hole [4]. However, although the acoustic *

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1550-7998=2015=92(8)=084052(13)

black hole possesses many characteristics that resemble the real black hole system, it seems that its dynamics, governed by the Euler or the Navier-Stokes equation and equation of continuity (for relativistic fluid, the EOMs are conservation equations), has nothing to do with that of the latter, determined by the Einstein equation (plus dynamical equations of the background matter fields). Therefore, the acoustic black hole was merely regarded as an analogous gravitational model to mimic the phenomena in real gravitational systems, and the testing of Hawking-like radiation in acoustic black holes does not mean the detecting of the Hawking radiation from real black holes. Nevertheless, topics on analogous gravity still received much attention during the past years, both theoretically and experimentally, with the attempt to obtain some insights for studying the real gravitational systems, for example, the emergence of acoustic black holes from the Bose-Einstein condensation [5,6], the superfluid helium-3 and other cold bosonic systems [7], and from the superconductors [8,9], e.g., see a nice up-todate review in [10] and references therein. An interesting question is the following: can we add new interpretations to the word “analogous;” e.g., can we really gain the information of a real black hole from an acoustic black hole made in the laboratory? The answer is probably yes. When taking the holographic principle into account, it is natural to expect that there exists a bulk holographic description of the acoustic black hole that emerged from the fluid on the boundary in the context of the fluid/gravity

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GE et al.


correspondence [11–13] (which is the low frequency and long wavelength version of the gauge/gravity duality [14–18]). More specifically, we can ask what will happen in the bulk black hole when the supersonic phenomenon appears in its dual fluid system (or any finite temperature interacting field theory in the long time and long wavelength limit) on the boundary of the asymptotically anti–de Sitter (AdS) spacetime? According to the fluid/gravity correspondence, the fluid located at the asymptotic boundary has the same temperature and entropy with those of its dual bulk black hole. Furthermore, it was shown that the Navier-Stokes equation of the boundary fluid is also dual to the long wavelength behavior of the Einstein equation of the bulk gravity [19,20]. Hence, when an acoustic black hole forms in the boundary fluid, besides matching the fluid temperature and the entropy with those of its corresponding bulk black hole (note that the temperature of the acoustic horizon, determined by the gradient of the fluid velocity at the acoustic horizon, is different with the temperature of the fluid, while their relation is determined by the EOMs of the fluid), its dynamics can also, in principle, be fully determined from the dynamics of the bulk gravitational theory, and the linearized normal mode fluctuations of the boundary fluid (which are described by the sound mode, shear mode, and the transverse traceless mode) correspond to the linearized perturbations of the dual bulk gravity and gauge fields (which are the scalar mode, vector mode, and the tensor mode in the long time and long wavelength limit together with the quasinormal mode boundary condition), respectively [21]. Thus, for a compressible fluid on the boundary or cutoff surface of the AdS black brane, there should exist a phonon/scalar quasinormal mode correspondence. Now that the phonon also propagates into the acoustic black hole emerging from the fluid, it is expected that the acoustic black hole can probably be related to or mapped to a real black hole in the asymptotically AdS spacetime. In this paper, we give the derivation to construct a d-dimensional acoustic black brane formed in the fluid located at the finite timelike cutoff surface in a neutral black brane in asymptotically AdSdþ1 spacetime,1 in the spirit of the Wilsonian approach to the fluid/gravity correspondence [23–26]. We show that the acoustic black hole geometry can be obtained from the correspondence between the dynamics which govern the fluid and the gravity, namely, the equivalence between the conservation equation of fluid at the cutoff surface and the constraint equations of the Einstein equation of the bulk AdS black brane. Besides, the Hawking-like temperature of the acoustic black hole horizon can indeed be connected to the real Hawking temperature of its dual bulk AdS black brane, which may give strong supports to various studies on detecting the Hawking-like radiation from the acoustic black hole.

Furthermore, we also find that the normal mode excitation in the acoustic black hole, the phonon, is dual to the sound channel of the quasinormal mode in the bulk AdS black brane with first order hydrodynamic fluctuations. Based on these results, the acoustic black hole can indeed be used to study certain bulk gravitational (plus the possible bulk matter fields) perturbations, i.e., the scalar quasinormal mode perturbation, and the appearance of the supersonic phenomenon might introduce a testable effect in its dual bulk AdS black hole. In this sense, the acoustic black hole is no longer just an analogous model of the real black hole. The rest parts of this paper is organized as follows. In Sec. II, we give a brief review of the original derivation of acoustic black holes from the general d-dimensional nonrelativistic inviscid and viscous fluid, respectively. In Sec. III, we obtain the acoustic black holes in the relativistic fluid with first order dissipations. Then in Sec. IV, we present a holographic realization of the d-dimensional acoustic black hole formed in fluid at the finite timelike cutoff surface in asymptotically AdSdþ1 spacetime, based on the fluid/gravity correspondence. Various properties of the holographic acoustic black hole are studied. Especially, we show its dual bulk gravitational geometry, determine the relation between the temperature of the acoustic horizon and that of the dual real black hole horizon, and show there is a duality between the phonon in the acoustic black hole and the scalar quasinormal mode of the bulk AdS black brane. The conclusions and discussions are drawn in Sec. V. In Appendices A and B, we list the explicit form of the first order metric corrections to the bulk AdS black brane and a three-dimensional holographic relativistic rotating acoustic black hole at the cutoff surface, respectively. II. ACOUSTIC METRIC FROM NONRELATIVISTIC FLUID A. Inviscid acoustic black brane For the nonrelativistic d-dimensional convergent locally vorticity free (irrotational) neutral and inviscid fluid moving in the Minkowski spacetime, its dynamics is governed by the Euler equation and the equation of continuity ρð∂ t v~ þ v~ · ∇~vÞ ¼ −∇p − ρ∇Φ; ∂ t ρ þ ∇ · ðρ~vÞ ¼ 0;

where Φ is the potential field of the external force such as the Newtonian gravitational field. Since ∇ × v~ ¼ 0, the velocity can be described by the gradient of a potential field, i.e., v~ ¼ ∇ψ. Considering small hydrodynamic fluctuations of the background fluid up to linear order while keeping the external force potential Φ fixed2 2

1

For a preliminary attempt on the related problem see [22], where an acoustic black hole in four-dimensional conformal ideal fluid at the AdS boundary was analyzed.

ð1Þ

Note that this linearized perturbation is different with the derivative expansion of the hydrodynamic variables, which will not alter the configuration of the background fluid, namely, the background fluid is still the ideal fluid.

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HOLOGRAPHIC INTERPRETATION OF ACOUSTIC BLACK …

ξ ¼ ξ¯ þ δξ

and ψ ¼ ψ¯ þ δψ;

ð2Þ

where ξ ¼ ln ρ. Then the EOM of the sound wave (phonon) is just the Klein-Gorden equation of a massless scalar field pffiffiffiffiffiffi 1 pffiffiffiffiffiffi ∂ μ ð −¯gg¯ μν ∂ ν δψÞ ¼ 0 −¯g

ð3Þ

propagating in an effective acoustic geometry background [4] ds2ac ¼


becomes equal to cs at z ¼ zsh . In the linear order approximation, we can expand v¯ z as ∂ v¯ z v¯ z ¼ cs þ ðz − zsh Þ ≡ cs − κðz − zsh Þ; ð9Þ ∂z zsh requiring κ > 0; thus, when z ≤ zsh , v¯ z ≥ cs , consequently, phonons cannot escape from the region z ≤ zsh and zsh is the location of the sonic horizon (where we have chosen the fluid velocity to be along the direction of −z). Then the near acoustic horizon geometry of the acoustic black brane is

2 ρ¯ d−2 ð−ðc2s − v~¯ 2 Þdt2 − 2¯vi dxi dt þ dxi dxi Þ; ð4Þ cs

ds2ac

−2 pffiffiffiffiffiffi d ~¯ 2 ¼ v¯ i v¯ j δij , cs is the speed of where −¯g ¼ ρ¯ d−2 cd−2 s , v sound, and

g¯ μν

1 ¼ pffiffiffiffiffiffi −¯g

− cρ¯2

− cρ¯2 v¯ i

s

− cρ¯2 v¯ j s

s

ρ¯ c2s

ðc2s δij − v¯ i v¯ j Þ

! :

Furthermore, making the coordinate transformation Z v¯ i dxi ; t¼τ− c2s − v~¯ 2

ð5Þ

ð6Þ

2 v¯ i v¯ j ρ¯ d−2 2 2 ~ 2 i j i −ðcs − v¯ Þdτ þ 2 ¼ dx dx þ dx dxi cs cs − v~¯ 2 2 2 c ρ¯ d−2 − 2s dτ2 þ Psij dxi dxj ; ð7Þ ¼ cs γs

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where γ s ¼ 1= 1 − v~¯ 2 =c2s , u¯ i ¼ γ s v¯ i =cs , and the projecting operator is Psij ¼ δij þ u¯ i u¯ j . In the spatially flat case, we can choose the coordinates such that the fluid velocity only has the zth component, i.e., v¯ i ¼ v¯ z , where z ≡ xd−1 , then we have ds2ac

ð10Þ

which resembles the near horizon geometry of the real black hole and can be further written in the Rindler spacetime. The temperature of the acoustic black hole is just κ 1 ∂ v¯ z T sh ¼ ¼ : ð11Þ 2π 2π ∂z zsh B. Viscous acoustic black brane

where the vorticity free condition ensures that d2 acting on both sides of Eq. (6) give the same results, i.e., the zeros. The effective background, Eq. (4), becomes ds2ac

2 ρ¯ d−2 −2κcs ðz − zsh Þdτ2 ¼ cs cs 2 a dz þ dx dxa ; þ 2κðz − zsh Þ

2 ρ¯ d−2 v¯ 2z 2 2 2 2 i −ðcs − v¯ z Þdτ þ 2 ¼ dz þ dx dxi cs cs − v¯ 2z 2 c2 ρ¯ d−2 −ðc2s − v¯ 2z Þdτ2 þ 2 s 2 dz2 þ dxa dxa ; ¼ cs cs − v¯ z ð8Þ

where the spatial index a runs from 1 to d − 2. The acoustic black brane appears when the local fluid velocity v¯ z exceeds the speed of sound cs . Let us further require that the fluid velocity only depends on the coordinate z and it

The acoustic black hole can also be generalized into the viscous fluid, see [27], in which the four-dimensional fluid with shear viscosity was discussed. Here we consider the general d-dimensional viscous fluid with both the shear and bulk viscosities. The dynamics of such fluid is described by the Navier-Stokes and the conservation equations d−3 2 ρð∂ t v~ þ v~ · ∇~vÞ ¼ −∇p þ η∇ v~ þ η þ ζ ∇ð∇ · v~ Þ d−1 − ρ∇Φ; ∂ t ρ þ ∇ · ðρ~vÞ ¼ 0:

ð12Þ

where η is the shear viscosity and ζ is the bulk viscosity of the fluid. Similar to the inviscid fluid case, the EOM of the sound mode can be derived from taking the linearized hydrodynamic perturbations, Eq. (2), to the fluid, then pffiffiffiffiffiffi 1 □δψ ¼ pffiffiffiffiffiffi ∂ μ ð −¯gg¯ μν ∂ ν δψÞ −¯g 2 2d − 4 cs d−2 1 ηþζ ¼− ð∂ t þ v~¯ · ∇Þ∇2 δψ; d−1 ρ¯ c2s ð13Þ which is a modified Klein-Gorden equation with higher derivative corrections, where ∇2 ¼ ηij ∂ i ∂ j . However, the

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effective acoustic spacetime in which the sound modes propagate is the same as Eq. (4). In the Eikonal approximation δψ ¼ aðxÞ expð−iωt þ ~ ik · x~Þ [in which aðxÞ varies slowly with respect to x], the dispersion relation of the sound mode obtained from the viscous acoustic black brane is

relativistic hydrodynamics, see, e.g., the cases for ideal fluid in [28,29]. We will extend the discussion into relativistic fluid with dissipations. For the d-dimensional neutral relativistic viscous fluid flowing in a curved spacetime, its stress tensor (in the first order expansion of the temperature and velocity fields) is

2

ω2 − 2k~ · v~¯ ω þ ðc2s k~ − ðk~ · v~¯ Þ2 Þ 2ðd − 2Þ 2 ν þ μ ðk~ · v~¯ − ωÞk~ ¼i d−1 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðd − 2Þ μ 2 2 νþ ⇒ ω ¼ k~ · v~¯ c2s k~ − ðk~ Þ2 d−1 2 i 2ðd − 2Þ 2 − ν þ μ k~ ð14Þ 2 d−1 i 2ðd − 2Þ 2 ¼ k~ · v~¯ cs k − ν þ μ k~ 2 d−1 1 ðd − 2Þ μ 2 ~2 νþ ∓ ð15Þ k k þ Oðk4 Þ; 2cs d − 1 2 pffiffiffiffiffi 2 where k ¼ j k~ j, and ν ¼ ρη¯ and μ ¼ ρζ¯ are the kinematic viscosities.

T μν ¼ ϵuμ uν þ pPμν − 2ησ μν − θζPμν ; where σ

μν

¼ P P ∇ðα uβÞ − μα

νβ

θ P d − 1 αβ

and θ ¼ ∇λ uλ ð17Þ

are the shear tensor and expansion associated with the velocity fields uα and Pμν ¼ uμ uν þ gμν . The EOMs are obtained by projecting the conservation equation ∇μ T μν ¼ 0 along the longitudinal direction (equation of continuity) ∇μ ðϵuμ Þ þ p∇μ uμ þ ηððuα ∇α uν Þðuβ ∇β uν Þ þ ð∇μ uα Þð∇α uμ Þ − uν ∇μ ∇μ uν Þ 2 η ð∇μ uμ Þ2 ¼ 0 þ ζ− d−1

III. ACOUSTIC BLACK BRANE FROM RELATIVISTIC FLUID The above acoustic black hole description of nonrelativistic fluid can be accordingly generalized into the

ð16Þ

ð18Þ

and in the transverse direction (dynamical equation)

ðϵ þ pÞuμ ∇μ uλ þ Pμλ ∇μ p − ηðð∇μ uμ Þðuα ∇α uλ Þ − ðuα ∇α uν Þðuβ ∇β uν Þuλ þ ð∇μ uλ Þðuα ∇α uμ Þ þ ðuμ ∇μ uα Þð∇α uλ Þ þ uμ uα ∇μ ∇α uλ þ ∇μ ∇μ uλ þ ∇μ ∇λ uμ þ ðuν ∇μ ∇μ uν Þuλ þ ðuν ∇μ ∇ν uμ Þuλ Þ 2 η ð∇λ ∇α uα þ ðuμ ∇μ uλ Þð∇α uα Þ þ ðuν ∇ν ∇α uα Þuλ Þ ¼ 0: − ζ− d−1

It is easy to see that in the nonrelativistic limit, namely, uα ¼ ð1; v~ Þ, j~vj ≪ 1, p ≪ ϵ, v~ dp dt ≪ ∇p, and ϵ ¼ ρ þ 1 2 v þ ρε → ρ (ε is the internal energy density and we 2 ρ~ have set the speed of light c ¼ 1), together with taking the flat spacetime limit gμν → ημν , Eqs. (18) and (19) reduce to the Navier-Stokes equation and the equation of continuity, Eq. (12), of the nonrelativistic fluid. For the conformal fluid flowing in the conformally flat spacetime we have

ds2ac

d−2 2 d−2 T¯ ¼ ð−c2s u¯ μ u¯ ν þ Pμν Þdxμ dxν : cs

ð19Þ

ð21Þ

ð20Þ

Like the situation in the nonrelativistic fluid, the presence of viscosities will not alter the acoustic geometry, Eq. (21). Instead, they will break the Lorentz symmetry of the fluid. The corresponding EOM for the phonon is pffiffiffiffiffiffi 1 □δψ ¼ pffiffiffiffiffiffi ∂ μ ð −¯gg¯ μν ∂ ν δψÞ −¯g 2 2d − 4 η cs d−2 1 μ ¼− u¯ ∂ μ ð∂ λ ∂ λ δψÞ; ð22Þ d − 1 T¯ s¯ T¯ d−2 c2s

and the corresponding acoustic metric is the same as that for ideal fluid (up to an numerical conformal factor) [28,29]

where ∂ μ ψ ¼ huμ ∝ Tuμ , with h the enthalpy density and then in the plane wave approximation the dispersion relation is

ζ ¼ 0;

p¼

ϵ ; d−1

ϵ ¼ σT d and c2s ¼

1 ; d−1

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HOLOGRAPHIC INTERPRETATION OF ACOUSTIC BLACK … 2

ω ¼ cs k − iΓs k~ þ Oðk3 Þ;

ð23Þ

where we have set the background fluid velocity u~¯ ¼ 0 in the above equation, and d−2 η d−2 η ¼ ð24Þ Γs ¼ ¯ d − 1 T s¯ d − 1 ϵ¯ þ p¯ is called the attenuation constant which characterize the dissipation of the fluid. The dispersion relation, Eq. (23), is in accord with the result obtained from doing linearized perturbation in the static conformal fluid [30]. IV. HOLOGRAPHIC DERIVATION OF ACOUSTIC BLACK HOLES Although the acoustic black hole formed from the supersonic phenomena in hydrodynamics discussed in Sec. II and Sec. III shared similar properties as those of the real black hole, it can only be treated as a black hole analogy since their dynamic origins seem to have no relationship with each other. However, as we will show in the rest of the paper, the acoustic black hole formed in the fluid can indeed be mapped to a real black hole in an asymptotically AdS spacetime, based on the fluid/gravity duality. Let us consider the bulk d þ 1 dimensional boosted asymptotic AdS black brane with constant d-velocities, uμ ¼ γð1; v~ Þ ¼ ðu0 ; u~ Þ and ημν uμ uν ¼ −1, ds2 ¼ H2 ðrÞdr2 þ H1 ðrÞð−fðrÞuμ uν þ Pμν Þdxμ dxν ;


via the coordinate transformation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2 ðrÞ dxμ → dx0μ ¼ dxμ uμ dr ¼ dxμ uμ dr; ð29Þ H1 ðrÞfðrÞ in which r is the tortoise coordinate, and “þ” indicates the outgoing while “−” corresponds to the ingoing coordinates. Since our purpose is to construct the holographic acoustic black hole on the membrane moving between the bulk black brane horizon or stretched horizon and the asymptotical boundary, in the following analysis we can choose the boosted black brane metric either in Eq. (25) or in Eq. (28) as the bulk background. From the fluid/gravity duality, the temperature T and entropy S of the fluid are identical to the Hawking temperature T H and entropy SBH of the dual bulk black brane, which are, respectively, pffiffiffiffiffiffi f 0 H1 T ¼ T H ¼ pffiffiffiffiffiffiffiffiffi and 4π fH2 rh Z ðH ðr ÞÞd−1 dd−1 x: ð30Þ S ¼ SBH ¼ 1 h 4Gdþ1 At the cutoff surface r ¼ rc , the reduced fluid velocity contains a redshift factor as [31] (note that when the hydrodynamic fluctuations are taken into account, the fluid velocity can become slowly varying functions with respect to the spacetime) u~ μ ðrc Þ ¼

ð25Þ

pffiffiffiffiffiffiffiffiffiffiffi fðrc Þuμ ;

ð31Þ

where the horizon is located at r ¼ rh in which fðrh Þ ¼ 0 is satisfied and Pμν ¼ uμ uν þ ημν is the projecting operator. The boosted black brane solution is obtained by making the Lorentz boost transformation x0μ ¼ Lμν xν to the original static black brane

and the temperature measured by local observers on the membrane r ¼ rc is

ds2 ¼ H 2 ðrÞdr2 þ H1 ðrÞð−fðrÞdt2 þ dx2i Þ;

For the asymptotical neutral AdSdþ1 black brane we have

ð26Þ

TH ffi ≥ TH: T c ¼ pffiffiffiffiffiffiffiffiffiffi fðrc Þ

where L00 ¼ γ ≡ u0 ;

H1 ðrÞ ¼

L0i ¼ γβi ≡ ui ;

and H2 ðrÞ ¼

βi βj ui uj Lij ¼ ðγ − 1Þ 2 þ δij ¼ ðγ − 1Þ 2 þ δij ; u~ β~ 2 β~ ¼ βk βk :

ð27Þ

To remove the coordinate singularity at the horizon, Eq. (25) can be written in the Eddington-Finkelstein coordinate as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ 2 H1 ðrÞH2 ðrÞfðrÞuμ dxμ dr

r2 ; L2

fðrÞ ¼ 1 −

1 ; fðrÞH1 ðrÞ

ð32Þ

rdh rd ð33Þ

then the temperature and the entropy volume density of the dual boundary field theory are TH ¼

rh d rd−1 1 4πT H L d−1 h and s ¼ ¼ ; d 4πL2 4Gdþ1 Ld−1 4Gdþ1 ð34Þ

2

þ H1 ðrÞð−fðrÞuμ uν þ Pμν dxμ dxν Þ

ð28Þ

respectively. In addition, from the first law of thermodynamics

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dϵ ¼ Tds;

ð35Þ

and the first law of thermodynamics, δϵc ¼ T c δsc ;

the energy density of the dual boundary field theory is ϵ¼

ðd − 1Þrdh ; 16πGdþ1 Ldþ1

ð36Þ

and the Euler relation ϵ þ p ¼ Ts with

p¼

rdh

16πGdþ1 Ldþ1

ð37Þ

are still held at the cutoff surface r ¼ rc when the fluid is isentropic, where the entropy density sc of fluid at the cutoff surface is the same as that of the fluid at the AdS boundary, i.e., sc ¼ s and the variation δ is acting on the horizon radius rh . Then the speed of sound of the fluid at the cutoff surface can be computed directly via cˆ 2s

is satisfied at the asymptotical boundary of the AdSdþ1 spacetime. A. Stress tensor of the fluid at cutoff surfaces The renormalized holographic stress tensor T cμν on the cutoff surface Σc ðr ¼ rc Þ can be obtained from the BrownYork formalism [32–34]; for the spacetime background in Eq. (25), it is d γˆ μν H12 T cμν ¼ − Kˆ cμν − γˆ μν Kˆ c þ ðd − 1Þ þ ; ð38Þ 8πGdþ1 L

ð43Þ

δpc 1 d rdh þ ¼ ¼ δϵc r¼rc d − 1 2ðd − 1Þ rdc fðrc Þ ¼ c2∞ þ

d 1 − fðrc Þ ; 2ðd − 1Þ fðrc Þ

ð44Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c∞ ¼ 1=ðd − 1Þ is the value of the speed of sound of the d-dimensional conformal fluid. It can be seen that the speed of sound is a monotonically decreasing function of rc which runs along the radial direction from thepvalue cˆ s ¼ ∞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi at the black hole horizon, rc ¼ rh to cˆ s ¼ 1=ðd − 1Þ at rc → ∞, which indicates that the fluid is incompressible when the cutoff surface or the membrane is chosen at the horizon while it becomes compressible when rc > rh .

where the induced extrinsic curvature B. Acoustic black hole from fluid at the cutoff surface Kˆ cμν

1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ r γ μν ðrc Þ; ¼ 2H1 ðrc Þ H2 ðrc Þ

ð39Þ

and γ μν ðrc Þ ¼ H1 ðrc Þˆγ μν is the induced metric on the μα νβ timelike cutoff surface and T μν c ¼ γ γ T cαβ . Note that “ ” represents higher derivative terms constructed from the induced metric in order to cancel the UV divergences and these higher curvature terms vanish for the spatially flat case considered here. Let us consider the bulk neutral black AdSdþ1 brane, using Eq. (33); then the holographic stress tensor at the cutoff surface can be expressed as that for the ideal fluid [35] T cμν ¼ ϵc u~ μ u~ ν þ pc Pμν ;

ð40Þ

where energy density and pressure at the cutoff surface are, respectively, ϵc ¼ and

pffiffiffiffiffiffiffiffiffiffiffi ðd − 1Þrdc ð1 − fðrc ÞÞ dþ1 8πGdþ1 L

pc ¼ −ϵc þ

rdh d 16πGdþ1 Ldþ1

pffiffiffiffiffiffiffiffiffiffiffi ; fðrc Þ

ð41Þ

which means that the Euler relation T c sc ¼ ϵc þ pc

ð42Þ

From the fluid/gravity duality, the conservation equation of the fluid at the cutoff surface is equivalent to the constraint equation of rμ components of the bulk Einstein equation. Then the acoustic metric of the fluid at the cutoff surface can be constructed from perturbing the longitudinal mode of the conservation equation u~ ν ∇cμ T μcν ¼ 0, ∇cμ ðϵc u~ μ Þ þ pc ∇cμ ðu~ μ Þ ¼ 0;

ð45Þ

and the transverse mode Pνλ ∇cμ T μcν ¼ 0, i.e., Pμλ ∇cμ pc þ ðϵc þ pc Þu~ μ ∇cμ u~ λ ¼ 0;

ð46Þ

where ∇cμ is the covariant derivative compatible with the induced metric γ μν at the cutoff surface and we have used the normalized condition, γ μν u~ μ u~ ν ¼ −1. In addition, we have required that the stress tensor T μν c at the cutoff surface still have the form of the ideal fluid, although u~ ν , ϵc , and pc became slowly varying functions of xα . Generally speaking, from the bulk gravity side, this kind of hydrodynamic fluctuations will cause T μν c to be modified by the dissipative terms. However, as we have discussed in Sec. III, the presence of viscosities will only modify the EOM for the phonon by the third derivative terms and thus break the Lorentz symmetry of the fluid, while the acoustic geometry remains the same as that of the ideal fluid. Thus at this step,

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we can still use the EOMs of the ideal fluid, i.e., Eqs. (45) and (46), to determine the acoustic geometry. Following the similar steps as those in the relativistic irrotational fluid [36], namely, defining ∇cμ ψ u ¼ uμ dxμ ≡ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxμ ; αβ −γ ∇cα ψ∇cβ ψ

ð47Þ

ds2ac ¼

dt ¼ dτ − ð48Þ

which reduces to the nonrelativistic one when the velocity does not depend on time apparently. Perturbing the fluid up to the linearized order as ψ ¼ ψ¯ þ δψ;

ϵc ¼ ϵ¯ c þ δϵc and pc ¼ p¯ c þ δpc ;

ð49Þ

then the EOM for the phonon, i.e., the perturbation of the velocity potential, is ∂μ

1 ¯μ ¯ν n¯ 2c μν − 2 u~ u~ þ P ∂ ν δψ ¼ 0: ϵ¯ c þ p¯ c cˆ s

ð50Þ

¼

2 d−2 n¯ 2c ð−ˆc2s u¯~ μ u¯~ ν þ Pμν Þdxμ dxν ; cˆ s ð¯ϵc þ p¯ c Þ

ð51Þ

where n¯ c is the zeroth order particle number density of the fluid which is proportional to the fluid entropy density and Z dϵc nc ¼ n0 exp ; ð52Þ ϵc þ pc where n0 ≡ nc ðpc ¼ 0Þ. When approaching the asymptotical boundary rc → ∞, we have pffiffiffiffiffiffiffiffiffiffiffi ðd − 1Þrdc ðd − 1Þrdh ϵ∞ ¼ ð1 − fðr Þ Þ → ; c 8πGdþ1 Ldþ1 16πGdþ1 Ldþ1 rc →∞ rdh d rdh pffiffiffiffiffiffiffiffiffiffiffi → ; p∞ ¼ −ϵc þ 16πGdþ1 Ldþ1 16πGdþ1 Ldþ1 fðrc Þ rc →∞ ð53Þ then Eq. (51) reduces to the form in the flat spacetime case ds2ac

2 n20 d−2 ¼ ϵ∞ 3 ð−c2∞ u¯ μ u¯ ν þ Pμν Þdxμ dxν c∞ d 2 d−2 2 n0 σ d d−2 2 ¼T ð−c2∞ u¯ μ u¯ ν þ Pμν Þdxμ dxν ; c3∞ d 2 d

ð54Þ

d−1 as in Eq. (21), where σ ¼ 4G1dþ1 ð4πL . More explicitly, d Þ the acoustic geometry, Eq. (51), can be written in a form as

ðˆc2s − fðr1 c ÞÞu¯~ 0 u¯~ i 1 þ γ 2 ðˆc2s fðrc Þ − 1Þ

dxi :

ð56Þ

Without loss of generality, one can choose the coordinate to let the fluid flowing along the xd−1 ≡ z direction, namely, u¯ μ ¼ ðγ; 0; …; 0; u¯ z Þ ¼ γð1; 0; …; 0; v¯ z Þ. Thus, Eq. (55) reduces to 2 d−2 n¯ 2c 2 dsac ¼ −ðfðrc Þˆc2s − u¯~ 2z α2c Þdτ2 cˆ s ð¯ϵc þ p¯ c Þ fðrc Þˆc2s 2 a ð57Þ þ dz þ dx dxa ; fðrc Þˆc2s − u~¯ 2z α2c where we have defined α2c ≡

Consequently, the relativistic acoustic metric at the cutoff surface can be obtained ds2ac

2 d−2 n¯ 2c −ð1 − γ 2 ð1 − cˆ 2s fðrc ÞÞÞdτ2 cˆ s ð¯ϵc þ p¯ c Þ ðfðr1 c Þ − cˆ 2s Þu¯~ i u¯~ j i j ð55Þ þ δij þ dx dx ; 1 − γ 2 ð1 − cˆ 2s fðrc ÞÞ

in which

the vorticity free condition can be expressed as ~ u~ ¼ 0; u∧d


1 ðd − 2Þ 1 1þ − cˆ 2s ¼ : fðrc Þ 2ðd − 1Þ fðrc Þ

ð58Þ

Then it is straightforward to check that supersonic phenomena appears or the acoustic black hole forms when fðrc Þˆc2s − u¯~ 2z α2c ≤ 0; which gives pffiffiffiffiffiffiffiffiffiffiffi fðrc Þcˆ s u¯ z v¯ z ffi ≥ cˆ s : or u¯~ z ¼ pffiffiffiffiffiffiffiffiffiffiffi ≥ pffiffiffiffiffiffiffiffiffiffi u~¯ z ≥ αc fðrc Þ fðrc Þ

ð59Þ

ð60Þ

For example, when the cutoff surface is taken to the AdS boundary, from Eq. (59) we have 1 u¯ z ≥ pffiffiffiffiffiffiffiffiffiffiffi d−2

or

1 v¯ z ≥ pffiffiffiffiffiffiffiffiffiffiffi ¼ c∞ : d−1

ð61Þ

Furthermore, we can write the acoustic metric, Eq. (57), in a more explicit form by requiring u¯~ z ¼ u¯~ z ðzÞ to be a monotonically increasing function of z (with direction along −z) and it reaches pffiffiffiffiffiffiffithe ffi critical value at the acoustic ~ ˆs fðr Þ c c horizon, u¯~ z ðzsh Þ ¼ ≡ cˆ s . Then, expand u¯~ z ðzÞ αc

αc

around the spacetime point ðt ¼ 0; …; 0; zsh Þ up to the linear order in a covariant form, namely, u¯~ z ðzÞ ¼ u¯~ z ðzsh Þ þ ðz − zsh Þ∂ z u¯~ z jzsh , Eq. (57) becomes 2 d−2 n¯ 2c 2 dsac ¼ −2αc c~ˆ s j∂ z u¯~ z jðz − zsh Þdτ2 cˆ s ð¯ϵc þ p¯ c Þ c~ˆ s 2 a ð62Þ dz þ dx dxa þ 2αc j∂ z u¯~ z jðz − zsh Þ

084052-7

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and the temperature of the acoustic horizon is T sh ¼

αc j∂ u¯~ j ; 2π z z zsh

ð63Þ

which is related to the acceleration of the background fluid at the acoustic horizon z ¼ zsh as

uμ ðxα Þ ds2 ¼ −2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxμ dr fðrc ;rh ðxα ÞÞ r2 fðr;rh ðxα ÞÞ α α μ ν 1− ðx Þu ðx Þþη u þ 2 ν μν dx dx fðrc ;rh ðxα ÞÞ μ L ð1Þ

ð1Þ

þgrr ðr;xα Þdr2 Þ;

a¯~ z jzsh ¼ u¯~ α ∇cα u¯~ z jzsh ¼−

pffiffiffiffiffiffiffiffiffiffiffi 2π cˆ s T sh p ffiffiffiffiffiffiffiffiffiffi ffi þ ∂ ln fðrc Þjzsh : z α2c fðrc Þ

ð64Þ

For convenience, we will omit the “bar” index for the background hydrodynamic variables in the following subsections. C. Perturbations from the bulk side Let us study the holographic dual of the acoustic black hole in Eq. (62) from the perturbations of the bulk black brane in Eq. (25) caused by small hydrodynamic fluctuations at the cutoff surface, and we will focus on the linearized perturbation. Recall that hydrodynamics describes states close to the thermal equilibrium, which allows the local velocity fields as well as the local temperature of the fluid to be slowly varying functions of the spacetime coordinates xα , namely [19,37], uμ → uμ ðxα Þ and T → Tðxα Þ ðor rh → rh ðxα ÞÞ:

ð1Þ

þðgμν ðr;xα Þdxμ dxν þ2gμr ðr;xα Þdxμ dr

ð65Þ

These hydrodynamic fluctuations can be viewed as the variation of source terms from the fluid, which will in turn cause backreaction to the background geometry, which indicates that the acoustic black hole formed in the fluid at the cutoff surface corresponds to the bulk asymptotic AdS black brane with first order corrections. In order to make the perturbed geometry satisfy the original bulk Einstein ð1Þ equation, additional metric corrections gAB should be added to the perturbed metric. To solve the fluctuations, it is more convenient to use the metric in the ingoing EddingtonFinkelstein coordinate and make affi scaling transformation pffiffiffiffiffiffiffiffiffiffi for the time coordinate t→t= fðrc Þ; then Eq. (28) becomes

ð66Þ

where we have chosen the background to be the AdSdþ1 black brane. To compare Eq. (66) with its holographic counterpart in Eq. (62), we will rotate the coordinates to let the fluid move in the −z direction and further require that the local velocity fields and the local temperature of the fluid only be the function of the coordinate z. Then the vorticity free condition u∧du ¼ 0 is automatically satisfied. In addition, since we are interested in the phenomena in the near acoustic horizon region, we will expand uz ðzÞ and rh ðzÞ at the location of the acoustic horizon ðt ¼ 0; …; 0; z ¼ zsh Þ without loss of generality; this is similar with the usual treatments in the fluid/gravity duality in which the hydrodynamic variables are expanded at xα ¼ 0. Therefore, the nonvanishing metric components ðoÞ [denoted by gAB ] in the first and second lines of Eq. (66) are r2 2 fðr; rh ðzÞÞ ðoÞ gtt ¼ 2 ut ðzÞ 1 − −1 ; fðrc ; rh ðzÞÞ L r2 fðr; rh ðzÞÞ ðoÞ ðoÞ gtz ¼ gzt ¼ 2 ut ðzÞuz ðzÞ 1 − ; fðrc ; rh ðzÞÞ L ðoÞ

r2 δab ðwith a; b ≠ zÞ; L2 r2 2 fðr; rh ðzÞÞ ¼ 2 uz ðzÞ 1 − þ1 ; fðrc ; rh ðzÞÞ L ut ðzÞ ðoÞ ffi; ¼ gtr ¼ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðrc ; rh ðzÞÞ

gab ¼ ðoÞ

gzz

ðoÞ

grt

uz ðzÞ ðoÞ ðoÞ ffi; grz ¼ gzr ¼ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðrc ; rh ðzÞÞ

ð67Þ

where

2πT sh dˆcs rd−1 cˆ s h uz ðzÞ ¼ þ ðz − zsh Þ − pffiffiffiffiffiffiffiffiffiffiffi þ ∂ z rh jzsh ; αc αc fðrc Þ 2αc fðrc Þrd 1 2π cˆ s T sh dˆc2s rd−1 h pffiffiffiffiffiffiffiffiffiffiffi ∂ z rh jzsh ; þ þ ðz − zsh Þ − ut ðzÞ ¼ αc fðrc Þ αc 2αc fðrc Þrd rh ðzÞ ¼ rh ðzsh Þ þ ðz − zsh Þ∂ z rh jzsh ≡ rh þ ðz − zsh Þ∂ z rh jzsh ; drd−1 drd−1 h ðz − zsh Þ∂ z rh jzsh h ðz − zsh Þ∂ z rh jzsh ≡ fðrÞ − ; d r rd drd−1 ðz − zsh Þ∂ z rh jzsh drd−1 h ðz − zsh Þ∂ z rh jzsh fðrc ; rh ðzÞÞ ¼ fðrc ; rh Þ − h ≡ fðr Þ − ; c d rc rdc fðr; rh ðzÞÞ ¼ fðr; rh Þ −

084052-8

ð68Þ


which contain both the zeroth and the first order derivative terms of hydrodynamic fluctuations. On the other hand, the first order metric corrections in the third and fourth lines of Eq. (66) can be decomposed into the SOðd − 1Þ algebraically symmetric forms; after ð1Þ further imposing the radial gauge grA ¼ 0, they can be expressed into four independent parts as ð1Þ


since the reduced metric on the cutoff surface is flat in the rescaled coordinates. The functions si ðrÞ (i ¼ 1, 2), vðrÞ, and tðrÞ belong to the scalar, vector, and tensor channels, respectively, and will decouple with each other in the linearized Einstein equation. To determine the first order perturbed bulk geometry, we need to substitute Eqs. (67) and (69) into Eq. (66) and solve the linearized bulk Einstein equation

ð1Þ

grr ðrÞ ¼ gμr ðr; xα Þ ¼ 0; ð1Þ gμν ðr; xα Þ

θ P s ðrÞ ¼ θuμ uν s1 ðrÞ þ d − 1 μν 2 þ 2aðμ uνÞ vðrÞ þ σ μν tðrÞ;

δRAB ¼ − ð69Þ

in which θ ¼ ∂ μ uμ ;

aμ ¼ uν ∂ ν uμ

σ μν ¼ Pαμ Pβν ∂ ðα uβÞ −

and

θ P d − 1 μν

ð70Þ

are, respectively, the expansion, acceleration, and shear tensor associated with the velocity field uμ of the dual fluid, and the index is lowered and raised by ημν and ημν ,

d δgAB : L2

ð71Þ

A natural boundary condition for solving the metric corrections are the Dirichlet boundary condition at the cutoff surface, i.e., s1 ðrc Þ ¼ s2 ðrc Þ ¼ vðrc Þ ¼ tðrc Þ ¼ 0. What is more, an additional gauge should be adopted to solve Eq. (71), and we use the Landau frame, in which the first order correction of the fluid stress tensor is transverse at the timelike cutoff surfaces, i.e., ð1Þ

u~ μ T cμν ¼ 0:

ð72Þ

The general solution has been solved out in [26] as

r rd−1 rd−1 c c −d 1 − þ ðd − 2ÞfðrÞ þ 2 − d fðr Þ c ; rd−1 rd−1 ðd − 1Þfðrc Þ3=2 2r r ; s2 ðrÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi 1 − rc fðrc Þ d r rc rd−1 c r d − 1 þ rc fðrc Þ 1 − d−1 ; vðrÞ ¼ − r r ðd − 1Þˆc2s rc fðrc Þ3=2 2 pffiffiffiffiffiffiffiffiffiffiffi 2 1 fðrÞ rh 1 1 rdh tðrÞ ¼ and hðrÞ ¼ 2 F1 1; ; 1 þ ; d ; fðrc Þr hðrÞ − hðrc Þ þ ln rh d fðrc Þ d d r r s1 ðrÞ ¼

ð1Þ

and the nonvanishing components of gμν will be listed in Appendix A. Besides, the first order corrections of the stress tensor at the finite cutoff surface r ¼ rc is given by μνð1Þ

Tc

¼ −2ηc σ μν þ ζ c θPμν ;

ηc ¼

rd−1 h ; 16πGdþ1 Ld−1

ζ c ¼ 0;

ð74Þ

where ηc and ζ c are the shear and bulk viscosities at the cutoff surface, respectively. Note that the shear viscosity and the entropy density of the dual fluid at the rescaled cutoff surface with xi → xi L=rc are ηðrc Þ ¼ rd−1 h 16πGdþ1 rd−1 c

rd−1

and sðrc Þ ¼ 4G h rd−1 , respectively, which both dþ1 c indicate that the shear viscosity over the entropy density of the fluid

ηðrc Þ ηc 1 ¼ ¼ sðrc Þ sc 4π

ð73Þ

ð75Þ

will not vary as the cutoff surface r ¼ rc moving from the black brane horizon to the asymptotical boundary of the AdS spacetime. D. Temperature of acoustic black hole vs temperature of real black hole Recall that one of the original motivations for studying acoustic black holes was to mimic the Hawking radiation, e.g., detecting the Hawking-like temperature of the acoustic horizon. Nevertheless, the detection of the former cannot indicate the observation of the Hawking temperature of real black holes, in the absence of further evidences such as the dynamical origins. Now, using the holographic construction, the temperature of the acoustic black hole can

084052-9

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indeed be connected to that of its dual AdS black brane—a real black hole. Explicitly, the conservation equation ∇cμ T μcν ¼ 0, together with the Euler relation, Eq. (42), and the first law of thermodynamics, Eq. (43), of the fluid at the cutoff surface r ¼ rc can be combined into a single equation as

where the subleading second order derivative terms have been ignored. Equation (79) has a simple expression when the cutoff surface moves to the AdS boundary rc → ∞, then ∂ z ln T H jzsh ¼ 2πT sh cs :

~ μÞ ∇cμ ln sc ¼ ðθ~ u~ μ − cˆ −2 s a 1 α þ ðu~ u~ α þ cˆ −2 ~ βα ; s Pμ Þ∇cβ σ 2πT c μ

∇cμ ln rh ¼ ∇cμ ln T H ¼

1 ~ ~ μÞ ðθu~ μ − cˆ −2 s a d−1 1 α ðu~ u~ α þ cˆ −2 ~ βα ; þ s Pμ Þ∇cβ σ 2πðd − 1ÞT c μ

ð77Þ

~ α Þ ¼ ∇cμ u~ μ ðxα Þ and the where the expansion is θðx acceleration of the fluid velocity u~ μ ðxα Þ is a~ μ ðxα Þ ¼ u~ ν ðxα Þ∇cν u~ μ ðxα Þ. Since the second part in Eq. (76) or Eq. (77) is the second order derivative terms coming from the first order dissipations corrections to the fluid, when the fluid velocity is slowly varying (which requires that the fluid at the cutoff surface is close to thermal equilibrium), namely, its acceleration is small, these subleading terms can be ignored. Besides, using ∇cμ u~ ν ðxα Þ ¼ ∂ μ u~ ν ðxα Þ þ Γνcμλ u~ λ ðxα Þ (where Γνcμλ is the Christoffel connection associated with the induced metric γ μν and it will not be modified up to the first order metric corrections due to the Dirichlet boundary condition at r ¼ rc ), it is straightforward to check that they are respectively related to their counterparts in Eq. (70) as θðxα Þ ~ α Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; θðx fðrc ; rh ðxα ÞÞ a~ μ ðxα Þ ¼ aμ ðxα Þ þ Pνμ ∂ ν ln

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðrc ; rh ðxα ÞÞ;

σ βα ffi: σ~ βα ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðrc ; rh ðxα ÞÞ

ð78Þ

For the holographic acoustic black hole metric, Eq. (62), we obtain the relationship between the temperature of the acoustic black hole T sh and the dual bulk black brane Hawking temperature T H at the acoustic horizon z ¼ zsh as ∂ z ln T H jzsh ¼

4π

pffiffiffiffiffiffiffiffiffiffiffi fðrc Þð1 − cˆ 2s Þ

2 2ðd − 1Þfðrc Þα2c − dð1 − fðrc ÞÞðˆc2s þ αcˆ 2c Þ s

E. The sound mode/scalar quasinormal mode duality

ð76Þ

which can be further written as

T sh ; cˆ s ð79Þ

ð80Þ

Recall that when the spacetime background is the static AdS black brane (or in the locally static frame), the components of metric variation δgμν ðr; xα Þ [which is the normal mode perturbation with respect to the original unperturbed background geometry, Eq. (26)], if we are letting the mode propagate along the zcoordinate, are δgtt , δgtz , δgzz , δgrr , δgtr , δgrz , and δgaa , which will compose the longitudinal channel of the quasinormal mode [which is the SOðd − 2Þ scalar mode] of the bulk gravitational perturbation; this requires the SOðd − 2Þ gauge invariant decomposition of the bulk linearized gravitational equations. Then the dual operators on the AdS boundary corresponding to the rest bulk gravitational perturbations are T ttc , T tz c, a , from the field/operator correspondence in T zz , and T c ca the gauge/gravity duality. In the long wavelength and low frequency limit, the bulk scalar channel of the quasinormal modes corresponds to the sound mode fluctuation of the fluid on the AdS boundary [21]. While in the present case, the bulk gravitational background is the first order perturbed geometry with the hydrodynamic fluctuation, Eq. (66), then the quasinormal mode will be obtained by further perturbing the perturbed geometry, Eq. (66). However, the quasinormal mode perturbation is different with the hydrodynamic fluctuation from the bulk in Sec. IV C, in which the quasinormal mode perturbation (the Lie derivative on metric) can be viewed as the probe field propagating in the unperturbed geometry, while the bulk hydrodynamic perturbation (the partial or covariant derivative on hydrodynamic variables) will cause the original geometry to be corrected by higher derivative terms. To determine the bulk scalar quasinormal mode dual to the phonon (sound normal mode) scattering in the acoustic black hole, Eq. (57), formed in the fluid at the cutoff surface r ¼ rc , note that the normal mode perturbation of the fluid at the cutoff surface is listed in Eq. (49), where δϵc and δpc are acting on rh , i.e., the temperature of the fluid. From the bulk quasinormal mode perturbation side, the corresponding metric perturbations at r ¼ rc and at the acoustic horizon ðt ¼ 0; …; 0; z ¼ zsh Þ can be obtained from normal mode variation on the perturbed geometry Eq. (66), which is

084052-10


δgrt ¼

δut ðzÞjzsh du rd−1 þ d t h 3=2 δrh ; fðrc Þ 2rc fðrc Þ

δuz ðzÞjzsh du rd−1 þ d z h 3=2 δrh ; fðrc Þ 2rc fðrc Þ θ δgtt ¼ θu2t s01 ðrc Þ þ P s0 ðr Þ d − 1 tt 2 c þ 2at ut v0 ðrc Þ þ σ tt t0 ðrc Þ δrh ;


νμ ∂μ P ∂ ν δψðt; zÞ Zc ðrc ; t; zÞ μ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ −u ∂ μ ∂ μ ln rh þ : θs02 ðrc Þrh d−1 −∂ α ψ∂ α ψ

ð85Þ

δgrz ¼

Furthermore, the dual operator of Zc on the asymptotical ˆ combined boundary rc → ∞ is also a scalar operator O from the components of the boundary stress tensor T tt, T tz , T zz , and T aa . In addition, the boundary stress tensor couples to the source part of the boundary metric via Z Z ð0Þ ˆ 0; T μν γ μν ∼ ð86Þ OZ

δgtz ¼ δgzt ¼ θut uz s01 ðrc Þ þ

θ u u s0 ðr Þ d−1 t z 2 c 0 0 þ ðat uz þ az ut Þv ðrc Þ þ σ tz t ðrc Þ δrh ; θ δgzz ¼ θu2z s01 ðrc Þ þ P s0 ðr Þ d − 1 zz 2 c 0 0 þ 2az uz v ðrc Þ þ σ zz t ðrc Þ δrh ; δgab ¼ ðs02 ðrc Þ − t0 ðrc ÞÞ

θ δ δr ; d − 1 ab h

∂M

ð0Þ

where Z0 is the source term of Zc and γ μν is the source term of the induced metric on the AdS boundary, respectively. Therefore, there is a duality between the sound channel quasinormal mode Zc propagating in the bulk perturbed AdS black brane and the phonon δψ scattering in the acoustic black hole geometry formed from the fluid on the boundary or cutoff surface.

ðwith a; b ≠ t; zÞ; ð81Þ

where 0 is the partial derivative with respect to rh , and δrh ¼ δrh ðrc ; t; zÞ. The above variables can be combined into X δgaa Zc ¼ u2z δgtt þ u2t δgzz − 2ut uz δgtz þ a

¼ θs02 ðrc Þδrh ; or; Zc ¼ −

2rdc fðrc Þ3=2 θs02 ðrc Þ t ðu δgrt þ uz δgrz Þ; drd−1 h

ð82Þ

which form a gauge invariant scalar field Zc ¼ Zc ðrc ; t; zÞ—the sound channel of the bulk quasinormal mode [thanks to the remaining SOðd − 2Þ rotational symmetry of the background spacetime]. Together with normal mode perturbation of the longitudinal part of Eq. (77), when omitting the subleading second order derivative terms, it is ∂μ δrh ðrc ; t; zÞ μ μ ∂ μ ln rh þ δu ðt; zÞ ¼ −u ∂ μ rh d−1 ð83Þ and a variation of Eq. (47), the phonon field δψðt; zÞ is Pνμ ∂ ν δψðt; zÞ ffi δuμ ðt; zÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −∂ α ψ∂ α ψ

∂M

ð84Þ

for μ ¼ t, z. Then the one-to-one map between Zc and the phonon δψðt; zÞ is

V. CONCLUSIONS AND DISCUSSIONS In this paper, we realized the holographic interpretation of the acoustic black hole based on the formalism of the fluid/ gravity correspondence. An acoustic black hole geometry formed in the fluid at the finite timelike cutoff surface (membrane) in a neutral boosted black brane in asymptotically AdS spacetime was constructed, based on the matching between the conservation equation of fluid stress tensor and the constraint equations of the bulk Einstein equation. Besides, it was shown that the bulk dual of the acoustic black hole is the AdS black brane corrected with first order hydrodynamic fluctuation. Moreover, we determined the connection between the temperature of the acoustic black hole and the Hawking temperature of the real AdS black brane in the bulk. What is more, we showed that the phonon field, which comes from the normal mode excitation of the fluid at the cutoff surface and scatters in the acoustic black hole geometry, is dual to the scalar field—the sound channel of quasinormal modes propagating in the bulk perturbed AdS black brane. Therefore, we pointed out that from the viewpoint of the fluid/gravity duality, the acoustic black hole formed in the fluid is no longer just an analogous model of the real black hole. The remarkable connection between the two seemly different systems sheds light on the study of the analogous gravitational models, which aim to acquire insights for studying various phenomena in the presence of gravity, such as the Hawking radiation. According to our results, the appearance of the acoustic black hole in the fluid located on the finite timelike cutoff surface indeed corresponds to a modification or perturbation to the geometry of the bulk AdS black brane, and the detecting of the Hawkinglike temperature of the acoustic horizon can indeed give us some information about the Hawking temperature of the real

084052-11

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black brane, at least in the asymptotically AdS or Lifshitz spacetime cases. There are many interesting related problems to explore such as more on the duality between the acoustic black hole and its dual bulk AdS black brane, e.g., comparing the scattering of phonons by the acoustic black hole and the same process of the sound channel quasinormal mode in the bulk AdS black brane, studying the acoustic black hole in the fluid with anomalies, more about the experimentally testable effect of the temperature of the acoustic black hole on its dual bulk black hole, the supersonic phenomena in the quark-gluon plasma, finding the effective action (such as in [38]) to describe the holographic acoustic black holes, analyzing acoustic black holes in many other condensed matter systems, etc.

X. H. G. was supported by the NSFC (No. 11375110). J. R. S. was supported by the National Natural Science Foundation of China under Grant No. 11205058 and the Open Project Program of the State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (No. Y5KF161CJ1). Y. T. was partially supported by the NSFC with Grant No. 11475179. X. N. W. was partially supported by the NSFC with Grant No. 11175245. Y. L. Z. thanks CASTS (No. 104R891003) at NTU for support, MOST (Grant No. 104-2811-M-002-080), and SKLTP (No. 09KL141Y31) for support during the KITPC workshop.

ACKNOWLEDGMENTS

APPENDIX A: FIRST ORDER METRIC CORRECTIONS

We would like to thank Rong-Gen Cai, Giuseppe Policastro, and Hong-Bao Zhang for useful discussions.

The nonvanishing components of the first order metric corrections in Eq. (69) are

θ θ 2 P s ðrÞ þ 2at ut v2 ðrÞ þ ut at − P tðrÞ ; ¼ θut s1 ðrÞ þ d − 1 tt 2 d − 1 tt θ 1 θ ð1Þ ð1Þ gtz ðr; zÞ ¼ gzt ðr; zÞ ¼ θut uz s1 ðrÞ þ P s ðrÞ þ ðat uz þ az ut Þv2 ðrÞ þ ðu a þ ∂ z ut þ ut az Þ − P tðrÞ ; d − 1 tz 2 2 z t d − 1 tz θ θ ð1Þ 2 gzz ðr; zÞ ¼ θuz s1 ðrÞ þ P s ðrÞ þ 2az uz v2 ðrÞ þ uz az þ ∂ z uz − P tðrÞ ; d − 1 zz 2 d − 1 zz θ ð1Þ gab ðr; zÞ ¼ ðs2 ðrÞ − tðrÞÞ δ ðwith a; b ≠ t; zÞ; ðA1Þ d − 1 ab ð1Þ gtt ðr; zÞ

ds2ac ∼ −ðˆc2s fðrc Þ − α2c u¯~ 2i Þdt2 þ 2α2c u~¯ 0 ðu~¯ ϱ dϱ þ u¯~ φ dφÞdt

where, at the acoustic horizon z ¼ zsh , we have ut ¼

1 pffiffiffiffiffiffiffiffiffiffiffi ; αc fðrc Þ

θ ¼ ∂ z uz ¼ −

uz ¼

cˆ s ; αc

þ α2c ðu~ 0ϱ dϱ þ u~ 0φ dφÞ2 þ ϱ2 dφ2 þ dϱ2 ;

2πT sh pffiffiffiffiffiffiffiffiffiffiffi þ ∂ z rh jzsh ; αc fðrc Þ 2αc fðrc Þrd dˆcs rd−1 h

at ¼ uz ∂ z ut ¼ −

2π cˆ 2s T sh dˆc3 rd−1 h ffi ∂ z rh jzsh ; þ 2 psffiffiffiffiffiffiffiffiffiffi 2 αc 2αc fðrc Þrd

az ¼ uz ∂ z uz ¼ −

2π cˆ s T sh dˆc2s rd−1 h p ffiffiffiffiffiffiffiffiffiffi ffi þ ∂ z rh jzsh : α2c fðrc Þ 2α2c fðrc Þrd

ðA2Þ

APPENDIX B: THREE-DIMENSIONAL RELATIVISTIC ROTATING ACOUSTIC BLACK HOLE

pffiffiffiffiffiffiffiffiffiffiffi with u¯~ μ ¼ fðrc Þγð1; − aϱ ; bÞ, where a and b are constants. Equation (B1) is just the relativistic counterpart of the three-dimensional rotational acoustic black hole formed in the draining bathtub. When putting the cutoff surface to the AdS boundary and taking the nonrelativistic limit, Eq. (B1) reduces to the known result ds2ac

For simplicity we focus on three-dimensional irrotational fluid and adopting the polar coordinates; then Eq. (51) is written as (omitting the conformal factor)

ðB1Þ

1 a2 þ b2 a 1− dt2 − dϱdt þ bdφdt ∼− 2 ϱ ϱ2 þ ϱ2 dφ2 þ dϱ2

that was studied in [27].

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ðB2Þ

HOLOGRAPHIC INTERPRETATION OF ACOUSTIC BLACK … [1] T. Damour, Black hole eddy currents, Phys. Rev. D 18, 3598 (1978). [2] K. S. Thorne, R. H. Price, and D. A. Macdonald, Black Holes: The Membrane Paradigm (Yale University Press, New Haven, 1986). [3] R. H. Price and K. S. Thorne, Membrane viewpoint on black holes: Properties and evolution of the stretched horizon, Phys. Rev. D 33, 915 (1986). [4] W. G. Unruh, Experimental Black Hole Evaporation, Phys. Rev. Lett. 46, 1351 (1981). [5] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Sonic Analog of Gravitational Black Holes in Bose-Einstein Condensates, Phys. Rev. Lett. 85, 4643 (2000). [6] C. Barcelo, S. Liberati, and M. Visser, Towards the observation of Hawking radiation in Bose-Einstein condensates, Int. J. Mod. Phys. A 18, 3735 (2003). [7] G. E. Volovik, Black hole horizon and metric singularity at the brane separating two sliding superfluids, Pis’ma Zh. Eksp. Teor. Fiz. 76, 296 (2002) [JETP Lett. 76, 240 (2002)]. [8] X.-H. Ge and S.-J. Sin, Acoustic black holes for relativistic fluids, J. High Energy Phys. 06 (2010) 087. [9] X.-H. Ge, S.-F. Wu, Y. Wang, G.-H. Yang, and Y.-G. Shen, Acoustic black holes from supercurrent tunneling, Int. J. Mod. Phys. D 21, 1250038 (2012). [10] C. Barcelo, S. Liberati, and M. Visser, Analogue gravity, Living Rev. Relativity 8, 12 (2005); 14, 3 (2011). [11] G. Policastro, D. T. Son , and A. O. Starinets, From AdS=CFT correspondence to hydrodynamics, J. High Energy Phys. 09 (2002) 043. [12] G. Policastro, D. T. Son, and A. O. Starinets, From AdS=CFT correspondence to hydrodynamics. 2. Sound waves, J. High Energy Phys. 12 (2002) 054. [13] D. T. Son and A. O. Starinets, Viscosity, black holes, and quantum field theory, Annu. Rev. Nucl. Part. Sci. 57, 95 (2007). [14] G.’t Hooft, Dimensional reduction in quantum gravity, arXiv:gr-qc/9310026. [15] L. Susskind, The world as a hologram, J. Math. Phys. 36, 6377 (1995). [16] J. M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]. [17] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428, 105 (1998). [18] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998). [19] S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, Nonlinear fluid dynamics from gravity, J. High Energy Phys. 02 (2008) 045.

PHYSICAL REVIEW D 92, 084052 (2015) [20] I. Bredberg, C. Keeler, V. Lysov, and A. Strominger, From Navier-Stokes to Einstein, J. High Energy Phys. 07 (2012) 146. [21] P. K. Kovtun and A. O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72, 086009 (2005). [22] S. R. Das, A. Ghosh, J.-H. Oh, and A. D. Shapere, On dumb holes and their gravity duals, J. High Energy Phys. 04 (2011) 030. [23] I. Bredberg, C. Keeler, V. Lysov, and A. Strominger, Wilsonian approach to fluid/gravity duality, J. High Energy Phys. 03 (2011) 141. [24] T. Faulkner, H. Liu, and M. Rangamani, Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm, J. High Energy Phys. 08 (2011) 051. [25] D. Brattan, J. Camps, R. Loganayagam, and M. Rangamani, CFT dual of the AdS Dirichlet problem: Fluid/gravity on cut-off surfaces, J. High Energy Phys. 12 (2011) 090. [26] N. Pinzani-Fokeeva and M. Taylor, Towards a general fluid/ gravity correspondence, Phys. Rev. D 91, 044001 (2015). [27] M. Visser, Acoustic black holes: Horizons, ergospheres, and Hawking radiation, Classical Quantum Gravity 15, 1767 (1998). [28] V. Moncrief, Stability of stationary, spherical accretion onto a Schwarzschild black hole, Astrophys. J. 235, 1038 (1980). [29] N. Bilic, Relativistic acoustic geometry, Classical Quantum Gravity 16, 3953 (1999). [30] R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, J. High Energy Phys. 04 (2008) 100. [31] R. Emparan and M. Martinez, Black branes in a box: Hydrodynamics, stability, and criticality, J. High Energy Phys. 07 (2012) 120. [32] M. Henningson and K. Skenderis, The holographic Weyl anomaly, J. High Energy Phys. 07 (1998) 023. [33] V. Balasubramanian and P. Kraus, A stress tensor for anti–de Sitter gravity, Commun. Math. Phys. 208, 413 (1999). [34] S. de Haro, S. N. Solodukhin, and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS=CFT correspondence, Commun. Math. Phys. 217, 595 (2001). [35] R. Emparan, V. E. Hubeny, and M. Rangamani, Effective hydrodynamics of black D3-branes, J. High Energy Phys. 06 (2013) 035. [36] M. Visser and C. Molina-Paris, Acoustic geometry for general relativistic barotropic irrotational fluid flow, New J. Phys. 12, 095014 (2010). [37] P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45, 473001 (2012). [38] P. Kovtun, G. D. Moore, and P. Romatschke, Towards an effective action for relativistic dissipative hydrodynamics, J. High Energy Phys. 07 (2014) 123.

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