PHYSICAL REVIEW D 102, 104009 (2020)

Geometry outside of acoustic black holes in (2 + 1)-dimensional spacetime

Qing-Bing Wang 1 and Xian-Hui Ge1,2,* 1Department of Physics, Shanghai University, Shanghai 200444, China 2Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China

(Received 21 July 2020; accepted 12 October 2020; published 4 November 2020)

Analogue black holes, which can mimic the kinetic aspects of real black holes, have been proposed for many years. The growth of the radial momentum toward the acoustic horizon is calculated for acoustic black holes in flat and curved spacetimes. Surprisingly, for a freely infalling vortex approaching the acoustic , the Lyapunov exponent of the growth of the momentum at the horizon saturates the chaos bound ΛLyapunov ≤ 2πT. We investigate the orbits of test vortices and sound wave rays in the (2 þ 1)-dimensional “curved” spacetime of an acoustic black hole. We show that the vortices orbit, the sound wave orbit, and the time delay of sound are similar to those famous effects of general relativity. These effects can be verified experimentally in future experiments.

DOI: 10.1103/PhysRevD.102.104009

I. INTRODUCTION Acoustic black holes for relativistic fluids were derived – Analogue black holes have recently been a hot topic from the Abelian Higgs model [6 9]. Since the Abelian as they can provide interesting connections between Higgs model describes high energy physics, the result in astrophysical phenomena with tabletop experiments. [8] implies that acoustic black holes might be created in Remarkably, the very recent experiments have reported high energy physical processes, such as quark matters that the thermal and the corresponding and neutron stars. The quasi-one-dimensional supersonic temperature were observed in a Bose-Einstein condensate flow of a Bose Einstein condensate (BEC) in Laval nozzle system [1] (see [2] for the experiment on stimulated the (convergent divergent nozzle) is considered in [10] in Hawking radiation in an optical system). In the seminal order to find out which experimental settings can amplify paper of Unruh [3], the idea of using hydrodynamical flows the effect of acoustic Hawking radiation and provide as analogous systems to mimic a few properties of black observable signals. The acoustic black holes with non- hole physics was proposed. In this model, sound waves like extremal black D3-brane were considered in [11].The light waves, cannot escape from the horizon, and therefore particle production near the horizon of the acoustic black it is named “acoustic (sonic) black hole” (ABH). A moving hole was studied by [12]. The acoustic black hole fluid with speed exceeding the local sound velocity through geometry in a viscous fluid with a dissipation effect a spherical surface could in principle form an acoustic black was considered in [13]. Recently, measures of operator complexity have received hole. The is located on the boundary between considerable attention in studies of information scrambling subsonic and supersonic flow regions. The horizon, ergo- and quantum complexity in holographic systems [14,15].The sphere, and Hawking radiation of (3 þ 1)-dimensional conjectured momentum/complexity duality states that the rate static and rotating acoustic black holes were later studied of operator complexity growth corresponds to appropriately in [4]. The spherical singular hypersurface in static super- defined radial component of momentum for a test system fluid was studied in [5]. It was shown that these shells form falling into a black hole [14,15]. For the Schwarzschild-AdS acoustic lenses analog to the ordinary optical lenses. black hole, a test particle’s momentum grows at a maximal rate [14] and the Lyapunov exponent obtained saturates the Λ ≤ 2π chaos bound Lyapunov T proposed in [16]. It seems to *Corresponding author. be a universal property because all the near horizon regions [email protected] with a nonzero temperature are locally the Rindler-like. We are going to examine the momentum growth of an infalling Published by the American Physical Society under the terms of vortex toward acoustic black holes and check whether the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to chaos bound is saturated for analogue black holes. This might the author(s) and the published article’s title, journal citation, be realizable because acoustic black holes would be realized and DOI. Funded by SCOAP3. in tabletop experiments.

2470-0010=2020=102(10)=104009(14) 104009-1 Published by the American Physical Society QING-BING WANG and XIAN-HUI GE PHYS. REV. D 102, 104009 (2020)

ρ More than that, we further explore the geodesic motion direction. Assuming 0 and cs are two constants and ρ0 outside a (2 þ 1)-dimensional acoustic black hole. We coordinates independent, absorbing the factor c into the “ ” s will concentrate on predicting the orbits of test particles line element on the left and setting cs ¼ 1, we can write (i.e., vortices) in fluids and sound rays in the “curved” acoustic spacetime. We will exhibit the Lyapunov exponent ds2 ¼ −ð1 − v2Þdt2 þð1 − v2Þ−1dr2 of a particle falling radially near the event horizon and þ r2ðdθ2 þ sin2 θdϕ2Þ: some well-known effects analogous to those in general relativity—the vortices orbits, the sound wave ray deflec- For a spatially two-dimensional fluid model, we can take tion, the time delay of sound waves for acoustic black θ ¼ π, so that dθ2 ¼ 0, sin2θ ¼ 1. The metric rewritten in a holes. These results can be verified by two-dimensional 2 spherically symmetric superfluid experiments. matrix form is 0 1 The paper is organized as follows: In Sec. II, we briefly −ð1 − 2Þ 00 review the geometry of acoustic black holes. In Sec. III, v B 2 −1 C considering an infalling vortex towards acoustic black gμν ¼ @ 0 ð1 − v Þ 0 A: ð2Þ holes, we calculate the growth of the radial momentum. 00r2 In Sec. IV, we further consider the Lyapunov exponent of radial motion of vortices toward the event horizon It is also known from fluid continuity and incompress- of the acoustic black hole which is embedded in the ibility ∇ · v ¼ 0 (for outside the central area), the speed v Schwarzschild-anti–de Sitter (AdS) geometry. We then should satisfy v ¼ −λ=r, where λ is a positive constant and study the orbit of vortices and the corresponding stability the negative sign means that the direction of the fluid in Sec. V. The sound wave trajectory is then studied in velocity points to the center of the acoustic black hole (see Sec. VI in which the deflection and the time delay of the Fig. 1). Then the (2 þ 1)-dimensional acoustic black hole sound wave prorogation are studied. The conclusion and metric can be written as [14,20] discussions are provided in the last section. 1 ds2 ¼ −fðrÞdt2 þ dr2 þ r2dϕ2; II. THE METRIC OF ACOUSTIC BLACK HOLES IN fðrÞ FLAT AND CURVED SPACETIMES λ2 fðrÞ¼1 − : ð3Þ In this section, we mainly consider three different kinds r2 of acoustic black holes: the line element first given by ¼ λ Unruh [3,4], the metric obtained from the Gross-Pitaeskii The acoustic event horizon locates at r0 . The Hawking equation in curved spacetime [6] and the metric obtained temperature of the acoustic black hole is given by [3,4,21] – from general relativistic fluids [17 19]. The reason to 0 f ðrÞj ¼λ 1 consider these three different types of metric is for the T ¼ r ¼ : ð4Þ purpose of examining the universality of the relation 4π 2πλ Λ ≤ 2π Lyapunov T. Later, we will focus on the geodesic motion in acoustic black hole in flat spacetime which is more realizable in experiments.

A. Acoustic black holes in Minkowski spacetime We first consider the general acoustic black hole metric given by Unruh, which is derived from the fluid continuity equation in a uniform fluid medium [3] (see also [4]),

2 2 2 2 2 ds ¼ g dt þ g dr þ gθθdθ þ gϕϕdϕ tt rr 2 ρ0 c ¼ −ðc2 − v2Þdt2 þ s dr2 c s c2 − v2 s s 2 2 2 2 þ r ðdθ þ sin θdϕ Þ : ð1Þ FIG. 1. Schematic view of a two-dimensional acoustic black hole. The arrow represents the direction of fluid flow. The black circle is the acoustic horizon where the fluid velocity reaches the Here we assume the flow is a static spherically symmetric sound velocity. The region in the ring represents the acoustic conservation flow. In this formula, cs refers to the speed black hole. For the region in the central black circle, we can of sound in the fluid medium, and ρ0 is the fluid density. regard it as a sink leading to the high-dimensional space from The velocity v is the flow rate of the fluid along the radial which the fluid flows to the third dimensional space.

104009-2 GEOMETRY OUTSIDE OF ACOUSTIC BLACK HOLES IN … PHYS. REV. D 102, 104009 (2020)

B. Acoustic black holes from Gross-Pitaeskii (2 þ 1)-dimensional acoustic black hole metric embedded equation in curved spacetime in AdS-Schwarzschild spacetime as follows: In curved spacetime, the Gross-Pitaeskii theory is 2 ¼ð GR ABHÞ μ ν ¼ G 2 þ G 2 þ G ϕ2 given by ds gμν gμν dx dx ttdt rrdr ϕϕd : Z   ð8Þ pffiffiffiffiffiffi ¼ 4 − j∂ φj2 þ 2jφj2 − b jφj4 ð Þ S d x g μ m ; 5 2 2 GR 2 By setting c ¼ 1 and cs ¼ 1=3 and taking the sign of gμν ABH as ημν and the sign of gμν as δμν, we can write the metric φ where is a complex scalar order parameter. The function components as φ is taken as a probe and does not backreact on the background geometry. The acoustic black hole metric can 1 G ¼ − ð Þ ð Þ be derived by considering perturbations around the back- tt 3 fABH r fGR r ; ground ðρ0; θ0Þ: ρ ¼ ρ0 þ ρ1 and θ ¼ θ0 þ θ1. The acous- 1 G ¼ G ¼ 4 ð Þ tic black hole metric then can be written as [6] rr ; ϕϕ r ; 9 fABHðrÞfGRðrÞ 2 GR ABH μ ν ds ¼ðgμν gμν Þdx dx where the blacken factors are given by c   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis 2 r GR 2 2 2 ¼ ðcs − vrv Þgtt dt 3λ 2 μ 2 r0 c − vμv f ðrÞ¼1 − ;fðrÞ¼r 1 − : ð10Þ s ABH r2 GR r2 2 μ c − vμv pffiffiffi þ 2 s GR 2 þð 2 − μÞ GR ϑ2 cs 2 r grr dr cs vμv gϑϑ d Note that the horizon 3λ of the acoustic black hole is cs − vrv required to be larger than the event horizon r0 of the GR 2 μ 2 G þ g ðc − vμv Þdϕ ; ð6Þ black hole. Inside the event horizon tt is less than 0 ϕϕ s G andpffiffiffi rr is greater than 0; between the two horizons, i.e., 3λ >r>r0, the opposite is true, G > 0 and G < 0. ¼ −θ_ ¼ ∂ θ ð ¼ θ ϕÞ tt rr where vt , vi i , i r; ; , cs the sound of In the curved spacetime, the Hawking temperature at the speed, m denotes a parameter related to the particle mass acoustic horizon is [6] and b is a parameter. Intriguingly, this is an analogue black sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi ! hole embedded in the spacetime governed by general 1 ABH − GR GR gtt 0GR gtt 0ABH relativity. Among them, gμν is the spacetime metric of T ¼ pffiffiffiffiffiffiffi − g þ g − GR tt ABH tt pffiffi ABH 4π Grr gtt gtt gravity and gμν describes the geometry of the acoustic r¼ 3λ μ ν ¼ θ ϕ 2 2 black hole, where ; t; r; ; . The metric tensor used 3λ − r0 GR ¼ ; ð11Þ to raise or lower indices of vμ is by gμν . The dispersion 6πλ μ 2 2 relation is given by v vμ ¼ m − cs.Asm ¼ 0, we can μ 2 where the prime represents derivative with respect to r. recover the normalized four-velocity relation v vμ ¼ −c . s ABH δ For a Schwarzschild black hole in the background of If gμν and cs are taken as μν and 1 respectively, the metric ¼ 2 2 (8) reduces to the form (7) of the AdS-Schwarzschild thermal radiation, the event horizon is at r0 M=c . GR ¼ 6 2 spacetime. On the other hand, if we take gμν and cs to be At rs pffiffiffiM=c , the sound speed of thermal radiation is ημν and 1 respectively, then the metric (8) returns to the c ¼ c= 3 with c the speed of light. In the region of s metric given in (3). r

104009-3 QING-BING WANG and XIAN-HUI GE PHYS. REV. D 102, 104009 (2020)

w ¼ −∂ ψ ð Þ η_ μ uμ μ ; 13 _μ þ Γμ α β − u ¼ 0 ð Þ u αβu u 2η ; 18 and the specific enthalpy w ¼ðp þ ρÞ=n. By making the μ ABH μ ν acoustic perturbations on u , w, n, ψ, where η ¼ −gμν u u > 0 and the dot represents the derivative with respect to τ. Note that the acoustic black μ μ μ u → u þ δu ; w → w þ δw; hole metric in general does not satisfy the Einstein n → n þ δn; ψ → ψ þ δψ; ð14Þ equation. The canonical momentum can be obtained by β δ gαβu we can obtain the wave equation, ¼ S ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ pα δ α μ ν : 19     u −gμνu u pffiffiffiffiffiffiffiffiffiffiffi w ∂ GR n μν n ν μ ∂μ −g g þ 1 − u u ∂νδψ ¼ 0: w GR n ∂w We choose the gauge τ ¼ t and take the ansatz r ¼ rðtÞ, ϕ ¼ constant. Then, Eqs. (18) and (19) reduce to ð15Þ   grrg2 1=2 By some transformations and taking c2 ¼ n ∂w j , the r_ ¼ − −grrg − tt ; ð20Þ s w ∂n s=n tt A2 form of metric can be recast as (see [17–19] for details) ¼ð− 2 tt − Þ1=2 ð Þ 2 ˜ μ ν GR 2 μ ν pr A g grr grr ; 21 ds ¼ Gμνdx dx ¼½gμν þð1 − csÞuμuνdx dx : ð16Þ where r<_ 0 means the vortex falls into the acoustic black In fact, there is a common coefficient factor in front of pffiffiffiffiffiffiffiffiffiffi ¼ 2 η 0 each metric component, which is related to the speed of hole and A gtt= > is an integral constant. We – sound, the density of fluid particles, the density of fluid substitute Eq. (3) into (20) (21) and obtain energy and the pressure of the fluid. In order to simplify the   f3 1=2 calculation, we set this coefficient be 1. There are some r_ ¼ − f2 − ; ð22Þ differences between formula (6) and formula (16). The A2 former is obtained from the Gross-Pitaevskii theory, while 2 −2 −1 1=2 the latter is obtained from fluid mechanics. pr ¼ðA f − f Þ : ð23Þ

III. FALLING INTO THE (2 + 1)-DIMENSIONAL We evaluate the growth rate of the momentumpffiffiffi in the ρ ∼ 1 ACOUSTIC BLACK HOLES IN MINKOWSKI Rindler coordinate [29,30]. Since d =dr = f, the SPACETIME: VORTEX MOTION AND CHAOS Rindler momentum pρ near the acoustic horizon is given by pffiffiffi In this section, we study the momentum growth of an 2 −1 1=2 pρ ∼ fp ∼ ðA f − 1Þ : ð24Þ infalling vortex toward an acoustic black hole. Unstable r orbits and the momentum growth can be quantified by their We can find ðr − λÞ ∝ e−4πTt in Eq. (22) near the acoustic Lyapunov exponents [23]. We use Lyapunov exponent to black hole horizon, where T ¼ 1=ð2πλÞ is the Hawking describe the motion of a vortex near the acoustic black hole. temperature of the acoustic black hole. Near the acoustic It was proposed that vortices can behave as relativistic horizon, the radial momentum pρ can be approximated as particles with their dynamics governed by the fluid metric

[24,25] and their stability ensured by a topological number. −1=2 2πTt pρ ∝ ðr − λÞ ∼ e : ð25Þ Vortices with mass m0 given by the Einstein’s relation ¼ 2 E m0c [26,27] cannot propagate at velocities faster than 2πTt We then obtain pρ ∼ e . Therefore, the growth of the the sound speed. Let us consider a vortex with mass m0 ¼ 1 freely falling into the acoustic black hole along the radial radial momentum near the acoustic horizon is described by Λ axis. The acoustic metric is given in Eq. (3). The action of the Lyapunov exponent Lyapunov, the geodesic motion of the infalling vortex is [28] Λ ¼ 2π ð Þ Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lyapunov T: 26 ¼ − − α β τ ð Þ S gαβu u d ; 17 It is interesting to note that the chaos bound is satisfied by the momentum growth of the “test” particle in the geometry where uα ¼ dxα=dτ is tangent to the world line and xα is the of acoustic black holes. This shows that acoustic black spacetime coordinate, τ is an arbitrary parameter of the holes are similar to the real black hole, which provides the vortex world line. The equation of motion from the above possibility for the experimental simulation of the chaotic action is behavior near the black hole horizon.

104009-4 GEOMETRY OUTSIDE OF ACOUSTIC BLACK HOLES IN … PHYS. REV. D 102, 104009 (2020)

IV. VORTEX MOTION AND CHAOS of acoustic black holes, the growth of radial momentum FOR ACOUSTIC BLACK HOLES still satisfies the chaotic bound. IN CURVED SPACETIME In this section we further consider vortex motion for B. Acoustic black holes from general acoustic black holes in curved spacetime. From the field relativistic fluids theory and fluid mechanics, we will consider the motion of From the perspective of fluid mechanics, in the vortices as probe particles falling along the radial direction (2 þ 1)-dimensional Schwarzschild background, uμ ¼ in Schwarzschild-AdS and Schwarzschild space-time ð−1;− 1 ð2MÞ1=2;0Þ, Eq. (16) is considered as respectively, so as to further test whether the momentum f0 r 2 ˜ μ ν growth chaos bound to the falling particles in the bulk are ds ¼ Gμνdx dx still satisfied in the curved space-time background.   pffiffiffiffiffiffiffi 2 2 4 2M ¼ − f0 dt˜ þ pffiffiffi dtdr 3 3f0 r A. Acoustic black holes from Gross-Pitaevskii equation   4M 1 In a Schwarzschild-AdS spacetime, through the þ þ dr2 þ r2dϕ2; ð Þ 3 2 32 Hadamard product, we consider that the metric (8) of f0r f0 2 þ 1 the acoustic black hole in the curved space-time of ( )- 2M where M is the mass of the black hole and f0 ¼ 1 − . dimensional Schwarzschild-AdS background. r By considering the coordinate transformation dt ¼ Similarly, the action in curved spacetime is considered as pffiffiffiffiffi dt˜ þ 2 2pMffiffi dr, we can obtain Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3f0 r ¼ − −G α β τ ð Þ 2 μ ν S αβu u d : 27 ds ¼ Gμνdx dx   2 3 þ 4 − 2 ¼ − 2 − f0r M r 2 þ 2 ϕ2 ð Þ We can obtain the radial momentum, f0 dt 2 dr r d : 33 3 2f0r − 3f0r   2 2 3 3 f f f f 1=2 For a null hypersurface FðxμÞ¼0, it satisfies the r_ ¼ − ABH GR − ABH GR ; ð28Þ 3 9A2 definition,   ∂F ∂F f−1 f−1 1=2 Gμν ¼ 0: ð34Þ p ¼ A 3f−2 f−2 − ABH GR ; ð Þ ∂xμ ∂xν r ABH GR A2 29 F pffiffiffiffiffiffiffiffiffiffiffiffi For steady-state spacetime, should be time independent, 2 and spacetime should be symmetric along ϕ direction, that where A ¼ Gtt=η1 > 0 is the integral constant with the μ ν F timelike condition η1 ¼ −Gμνu u > 0. is, is a function of r. Therefore, the acoustic horizon can pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ 6M Since dρ=dr ¼ 1= f f , the Rindler momentum be calculated as . And from the metric (32), we can ABH GR find that the Hawking temperature of the acoustic balck pρ is hole is T ¼ p1ffiffi .   24 3πM R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3A2 1=2 α β Analogously, from the action S ¼ − −Gαβu u dτ, pρ ¼ − 1 : ð30Þ fABHfGR we obtained the radial coordinate change rate and radial momentum are respectively, Near the acoustic horizon, the Rindler momentum pρ can   be approximated as G G2 1=2 r_ ¼ − − tt − tt ; ð35Þ pffiffiffi G A2G 2 1=2  pffiffiffi  rr rr 3 3A −1=2 pρ ≈ pffiffiffi pffiffiffi ∼ r− 3λ : ð31Þ   2 1=2 2ð3λ− 3r0Þðr− 3λÞ A Grr AGrr pr ¼ − − Grr ¼ − r;_ ð36Þ Gtt Gtt By calculating the expression (28), we can obtain pffiffiffi 3λ2− 2 −4πTt r0 where A is an integral constant and greater than 0. Through ðr − 3λÞ ∝ e , where T ¼ 6πλ . We can find that the Lyapunov exponent Λ of the stable acoustic the approximate calculation near the acoustic horizon, we Lyapunov _ ∼ ð − 6 Þ ∼ −4πTt black hole in AdS-Schwarzschild space-time (i.e., the obtained r r M e from Eq. (35). The Rindler temperature T is a constant at the acoustic horizon) is momentum can be expressed as 2π Λ ¼ 2π equal to T and still satisfies the relation Lyapunov T. 1=2 −1 2 AG This is consistent with the results of pure acoustic black ¼ G = ¼ − rr _ ∼ 2πTt ð Þ pρ rr pr G r e ; 37 holes obtained in Eq. (26). It shows that for different types tt

104009-5 QING-BING WANG and XIAN-HUI GE PHYS. REV. D 102, 104009 (2020)   which shows the momentum growth bound still holds at the 1 l2 l2 V ðrÞ¼ −v2 þ − v2 ; ð45Þ horizon. eff 2 r2 r2 This demonstrates that not only near the real black hole, but also for the general nonrotating acoustic black hole, the and v ¼ −λ=r. We obtain the expression of the effective exponent of the momentum growth of the Rindler co- potential, Λ ≤ 2π ordinate near the event horizon satisfy Lyapunov T. 1 V ðrÞ¼ ½ðl2 − λ2Þr2 − λ2l2: ð46Þ V. VORTICES ORBITS eff 2r4 In the above two sections, we consider the momentum For a concrete acoustic black hole, we know that λ is a growth of the vortex as the test particle near the horizon of the 2 positive constant, and l ≥ 0. If the first order derivative acoustic black hole. In this section, we will study the 0 ð Þ¼0 of the effective potential Veff r , the critical radiuspffiffi trajectory of vortices near acoustic black holes. We will 2 2 2 2λ l 2 2 pffiffiffiffiffiffiffiffi2λl satisfies r ¼ 2 2. When l > λ , we can obtain r ¼ , consider the falling trajectory of vortices with different initial l −λ l2−λ2 energy and angular momentum, and numerically calculate and the second derivative of the effective potential, t 00 4λ2l2 the radial momentum of particles growth with time . V ðrÞ¼− 6 < 0, which means that r is the radius eff r of the unstable orbit. When l2 ≤ λ2, we can find V0 ðrÞ > 0 A. Effective potential eff and V00 ðrÞ < 0, which means that the effective potential ϕ eff Since the metric (2) is independent of t and , the V ðrÞ increases with the increase of r and there is no stable ξ u η u u eff quantities · and · are conserved, where is the orbit [36,37]. “ ” ξα ¼ð1 0 0Þ ηα ¼ three-velocity of the vortex, ; ; and As shown in Fig. 2, the effective potential energy V ðrÞ ð0 0 1Þ eff ; ; are killing vectors. The conserved energy per unit does not have a minimum value, so there is no stable of static mass is [31,32] circular orbit for the acoustic black hole of this type. dt Figure 3 shows four types orbits for values of l and e.As ϵ ¼ −ξ · u ¼ð1 − v2Þ : ð38Þ dτ shown in the first row in Fig. 3, when the conservation angular momentum of the test particle is 0, the radial The conservation angular momentum per unit of rest mass is potential function of the particle increases monotonically dϕ with r and the particle always falls straight into the acoustic l ¼ η · u ¼ r2 : ð39Þ dτ black hole. The second and the third row in Fig. 3 with the same potential but has different value of e (l=λ ¼ 1.4), “ ” The three-velocity vectors satisfy the relation, which means the different orbits. The value of e2=λ2 is 1.16, α β u · u ¼ gαβu u ¼ −1: ð40Þ and the vortex will fall into the acoustic black hole as shown in the second row in Fig. 3. When the value of Substituting (2) into (40), we arrive at e2=λ2 ¼ 1.08, the vortex will approach but escape the acoustic black hole as shown in the third row in Fig. 3, 2 t 2 2 −1 r 2 2 ϕ 2 −ð1 − v Þðu Þ þð1 − v Þ ðu Þ þ r ðu Þ ¼ −1: ð41Þ which also demonstrates the critical unstable orbit with the ratio of e2=λ2 is about 1.12. Writing ut ¼ dt=dτ, ur ¼ dr=dτ, and uϕ ¼ dϕ=dτ, and substituting (38) and (39) into (41) to eliminate dt=dτ and dϕ=dτ, we obtain 0.10   2 dr 2 l2 −ð1 − v2Þ−1ϵ2 þð1 − v2Þ−1 þ ¼ −1: ð42Þ 1.5 dτ r2 0.05 1 The square of the conservation energy per unit of rest mass 0 can be obtained as

    5 10 15 20 dr 2 l2 ϵ2 ¼ þ þ 1 ð1 − v2Þ: ð43Þ dτ r2 0.05 By defining the effective energy [33–35],   ϵ2 − 1 1 2 ε ≡ ¼ dr þ ð Þ ð Þ FIG. 2. We plot the effective potential V as a function of r=λ Veff r ; 44 eff 2 2 dτ by setting l=λ ¼ 0, 1, 1.5, 2. It can be seen from the figure that there is no minimum point for the effective potential, that is, there where the effective potential is is no stable circular orbit for the vortex.

104009-6 GEOMETRY OUTSIDE OF ACOUSTIC BLACK HOLES IN … PHYS. REV. D 102, 104009 (2020)

0.10 10

0.05 5

0 0.00

5 0.05

10 02468101214 10 50 510

0.10 10

0.05 5

0 0.00

5 0.05

10 02468101214 10 50 510

0.10 10

0.05 5

0 0.00

5 0.05

10 02468101214 10 50 510

FIG. 3. From the first row, it can be seen that when l ¼ 0, the potential function increases monotonically with r, and the vortex always falls straight into the acoustic black hole. The case of l2 < λ2 is similar. The value l=λ is 1.4 in other parts. For l2 > λ2, if the vortex energy is large enough, it will fall into the acoustic black hole, as shown in the second row. If the vortex energy is low, when the vortex is near the acoustic black hole, it will escape from the acoustic black hole, as shown in the third pair of pictures, where the dotted line on the right is the position of unstable orbit, corresponding to the maximum potential energy in the left figure.

B. Radial infalling orbit where r decreases with the increase of τ in which the “three- ” τ ¼ 0 From the effective energy equation (44) and the effective velocity component is given by dr=d v< . Taken together with the time component dt=dτ givenbyEq.(38), potential equation (45), we can study the radial motion of the “ ” ϵ¼1 ¼0 the three-velocity is vortices. From the Eq. (44) with and l ,thereis   λ2 −1 λ α ¼½ð1 − 2Þ−1 0¼ 1 − − 0 ð Þ u v ;v; 2 ; ; : 48     r r 1 2 1 1 2 1 λ2 0 ¼ dr − 2 ¼ dr − ð Þ λ 0 v 2 ; 47 Considering that > ,wetakeanegativesign.Rewritten 2 dτ 2 2 dτ 2 r (47) in the form,

104009-7 QING-BING WANG and XIAN-HUI GE PHYS. REV. D 102, 104009 (2020)

1 dr ¼ dτ; ð49Þ v 250   2 −1 λ 2 r 200 − dr ¼ − dr ¼ dτ; ð50Þ r2 λ 150 the integral result is 100 1=2 1=2 1=2 rðτÞ¼½−2λðτ − τÞ ¼ð2λÞ ðτ − τÞ ; ð51Þ 50 where τ is an integral constant. Computing dt=dr from Eq. (38) with ϵ ¼ 1 and Eq. (47),weobtain 3.0 3.5 4.0 4.5 5.0   dt λ2 −1 r r3 ¼ð1 − v2Þ−1v−1 ¼ − 1 − ¼ − : FIG. 4. In the figure, the blue points are the values of pρ and t dr r2 λ λðr2 − λ2Þ calculated by numerical solution, and the green line is the curve of exponential fitting of data. Setting t ¼ 0, λ ¼ 1, which means ð52Þ 0.9997t 2πT ¼ 1, and r near the horizon, the fitting result is pρ ∼ e . After integration, we have Z 2 þ λ2 ð 2 − λ2Þ provide a theoretical basis for simulating the chaotic ¼ ð1 − 2Þ−1 −1 ¼ − r ln r ð Þ t v v dr t 2λ ; 53 behavior of particles falling into the real black hole. where t is another integral constant. From (53),when VI. SOUND WAVE ORBITS r → ∞;t→ −∞ , the vortex is falling from infinity. In addition to the computation of the Lyapunov expo- From Eq. (51) we can see that for any fixed r1 value nent, the stability of the sound wave orbits and the Shapiro outside the horizon, it takes only a finite amount of time delay of sound propagation near acoustic black hole proper time τ to reach r2 ¼ λ, while Eq. (53) shows that it takes an infinite quantity of coordinate time t.Thisis deserve further investigations. In what follows, we first ¼ λ calculate the deflection angle and then the time delay of just a sign that the coordinates are flawed at r .This 2 þ 1 shows that for the falling object A itself, the time from sound wave propagation in a ( )-dimensional acoustic black hole background in flat spacetime. r1 to r2 is limited. For the observer B with fixed coordinates, he observed that A keeps approaching the horizon r2. The observation here refers to the sound A. Sound wave deflection ω waves received by B from A. Replacing ∞, v∞ with The calculation of sound wave orbits in the acoustic ω v ω v ω v r1 , r1 and replacing , with r2 , r2 in (A9),we geometry is analogous to the calculation of vortices orbits, ω ¼ 0 then obtain r1 . The sound waves from r2 are but with some important differences. For vortices, the 1 infinitely redshift. B receives A’s sound frequency is unstable critical orbit does not necessarily exist, and the ’ becoming lower and lower, and finally A s voice seems critical radius of the orbit is related to angular momentum, to solidify at r2. while the unstable critical radius of the acoustic wave is We can also use formula (53) to verify the relationship fixed and smaller. The world line of sound waves can be between the radial momentum of vortex falling andp time.ffiffiffiffiffi described by the coordinates xα as function of a family of From Eqs. (20), (21) and (24), we obtain pρ ∼−r=_ ð fηÞ. affine parameters χ. The null vector uα ¼ dxα=dχ is tangent Further considered the formula (53), the relationship to the world line. Since the acoustic metric (2) is indepen- between pρ and t can be obtained. Setting t ¼ 0, λ ¼ 1, dent of t and ϕ, the quantities, thus 2πT ¼ 1, and r near the horizon, we obtain pρ ∼ 0.9997t dt e as shown in Fig. 4. Notice that t only shifts the ϵ ¼ð1 − 2Þ ð Þ v χ ; 54 pρ − t curve horizontally, it does not affect the exponential d relationship between pρ − t. After careful verifications, we 2 2 dϕ observe that as r approaches the horizon λ, ln pρ approaches l ¼ r sin θ ; ð55Þ t dχ 2πT. This shows that the acoustic black hole still satisfies the chaos bound of the momentum growth, which may are conserved along sound wave orbits. If the normalization of χ is chosen so that u coincides with the momentum p of 1For the discussions on redshift of sound wave propagation, a beam of sound wave moving along the null geodesic, then one may refer to the Appendix. ϵ and l are the sound wave’s energy and angular momentum

104009-8 GEOMETRY OUTSIDE OF ACOUSTIC BLACK HOLES IN … PHYS. REV. D 102, 104009 (2020)

u u ¼ dxα dxβ ¼ 0 dϕ dϕ dr d at infinity. Since · gαβ χ χ , considering the ¼ ¼ ð Þ d d 2 : 63 equatorial plane condition θ ¼ π=2,wehave dt dr dt r

2 2 2 −1 2 2 ϕ 2 Therefore, −ð1 − v ÞðutÞ þð1 − v Þ ðurÞ þ r ðu Þ ¼ 0: ð56Þ ¼ ð Þ Utilizing Eqs. (54) and (55),wehave b d: 64   2 2 This shows that the constant b is a parameter of sound 2 −1 2 2 −1 dr l −ð1 − v Þ ϵ þð1 − v Þ þ ¼ 0: ð57Þ waves ray reaching infinity and is defined to be positive. dχ r2 Figure 6 shows Weff as a function of r on the left. Itp goesffiffiffi 2 2 ¼ 2λ Multiplied by ð1 − v Þ=l , it can be written as to zero at large r, and it has a maximum at r . ¼ 2λ   If b , thep circularffiffiffi orbit of the sound wave at the 1 1 dr 2 maximum r ¼ 2λ. The maximum is ¼ þ W ; ð58Þ b2 l2 dχ eff pffiffiffi 1 W ð 2λÞ¼ : ð65Þ therein eff 4λ2

2 2 b2 ¼ l2=ϵ2; ð59Þ If b ¼ 4λ , it can be seen from formula (58) that the circularpffiffiffi orbit of the sound wave is possible at radius and r ¼ 2λ in the first pair of Fig. 6. However, the circular orbit is unstable. A small perturbation in b will result in 1 2 orbit far away from the maximum. A stable sound wave W ¼ ð1 − v Þ: ð60Þ eff r2 circular orbit is impossible around the two-dimensional acoustic black hole in our model. The qualitative character- In the equatorial plane, istics of other sound wave orbits depend on whether 1=b2 is greater than the maximum value of W , as shown in ¼ ϕ ¼ ϕ ð Þ eff x r cos ;yr sin ; 61 Fig. 6. First of all, consider that orbits start at infinity, if 1=b2 < 1=4λ2, the orbit will have a turning point and suppose a beam of sound wave moves parallel to the x-axis, returns to infinity again,also as shown in the first line of and the distance from the x-axis is d, as shown in Fig. 5. Fig. 6. We will discuss the size of the deflection angle in At a distance from the source of an acoustic black hole, 2 2 detail in a minute. If 1=b > 1=4λ , the sound wave will fall the sound wave moves in a straight line. For r ≫ λ, the all the way to the origin and be captured, as in the second quantity b is line of Fig. 6. The angle of interest is the deflection angle δϕ , showed in Fig. 7. This angle shows a property of the l r2 dϕ dϕ def ¼ ¼ ≈ 2 ð Þ shape of the sound wave orbit. This shape of a beam of b ϵ 1 − 2 r : 62 v dt dt sound wave can be calculated in the same way as the shape ϕ χ ϕ ≈ ≈−1 of a vortex orbit. Solve (55) for d =d , solve (58) for For very large r there are d=r, and dr=dt ,it dr=dχ, divide the second into the first, and then predigest becomes using (59) and (60) to find   −1 dϕ 1 1 2 ¼ − W : ð66Þ dr r2 b2 eff

The plus or minus sign indicates the direction of the orbit. When the sound wave comes in from infinity and comes back again, the total sweep angle Δϕ is twice the sweep angle from the turning point r ¼ r1 to infinity. Therefore, Z   2 −1 ∞ 1 1 − 2 Δϕ ¼ 2 dr − v ð Þ 2 2 2 : 67 r1 r b r

2 2 2 The radius r1 satisfies 1=b ¼ð1 − v Þ=r1; i.e., the radius where the bracket in the preceding expression vanishes. FIG. 5. The orbit of the sound wave comes from infinity. Set a new variable w defined by

104009-9 QING-BING WANG and XIAN-HUI GE PHYS. REV. D 102, 104009 (2020)

0.30 4 0.25

0.20 2

0.15 0

0.10 2

0.05 4 0.00 0 2 4 6 8 10 4 2 0 2 4

0.30 4 0.25

0.20 2

0.15 0

0.10 2

0.05 4 0.00 0 2 4 6 8 10 4 2 0 2 4

1 2 FIG. 6. The picture shows three tracks corresponding to different b values. The relationship between potential Weff and =b is shown in the left column. The effective potential of the acoustic black hole and the behavior of the sound wave passing through the acoustic black hole: when the beam of sound wave energy is equal to the maximum value of the effective potential of the acoustic black hole, it is a critical state; when the beam of sound wave energy is less than the height of the barrier, the sound wave does not fall into the acoustic black hole; when the beam of sound wave energy is greater than the barrier height, the sound wave will fall into the black hole.

b where w1 ¼ b=r1 is the value of w at the turning point. For r ¼ ; ð68Þ w v2 ¼ λ2=r2 ¼ λ2w2=b2,wehave Z   2 −1 the expression of Δϕ turns out to be w1 λ 2 Δϕ ¼ 2 1 − 2 1 − dw w 2 Z 0 r w1 2 2 −1 Z   Δϕ ¼ 2 ½1 − ð1 − Þ 2 ð Þ 2 2 −1 dw w v ; 69 w1 λ w 2 0 ¼ 2 1 − 2 1 − ð Þ dw w 2 : 70 0 b

When the value of v is much less than 1, i.e., r is much larger than λ and λw=b is far less than 1, we can recast Eq. (70) as Z     2 2 −1 2 2 −1 −1 w1 λ 2 λ 2 Δϕ ¼ 2 1 − w 1 − w − 2 dw 2 2 w : 0 b b ð71Þ

Expand the exponential term of ð1 − λ2w2=b2Þ as a power series, FIG. 7. In this figure, b is the linear distance from the black hole Z 2 2 2 to the incident direction of the sound wave. A beam of sound w1 1þ2λ π Δϕ ¼ 2 w =b ¼ ð Þ wave coming from infinite distances pass through the vicinity of dw 2 2 2 2 1 2 2 3 ; 72 δϕ 0 ð1þλ w =b −w Þ2 ð1−λ =b Þ2 acoustic black holes and deflect in a direction of def .

104009-10 GEOMETRY OUTSIDE OF ACOUSTIC BLACK HOLES IN … PHYS. REV. D 102, 104009 (2020) where w1 is a root of the denominator in Eq. (72). For small increase. According to Eq. (54) for dt=dχ and Eq. (58) for λ=b, it becomes dr=dχ, we can obtain   2     3 λ 2 −1 −1 Δϕ ¼ π 1 þ ð Þ dt 1 λ 1 2 2 2 : 73 ¼ 1 − − W ; ð76Þ b dr b r2 b2 eff Therefore, the deflection angle is where the positive sign is suitable for increasing radius, and 3 λ2 the negative sign is suitable for decreasing radius. From the δϕ ¼ π: ð74Þ def 2 b2 left of Fig. 8, we can see that the total time for sound wave travel forth and back between observer and reflector is Reinserted the factor of cs (sound velocity), the above formula can be written as (b has dimensions of length) Δ ¼ 2 ð Þþ2 ð Þ ð Þ ttotal t r1;r2 t r1;r3 ; 77 3 2λ2 δϕ ¼ cs π ð Þ def 2 ; 75 where the time required for a beam of sound wave moving 2 b from a radius of r1 to a radius of r is for small c λ=b. This shows that the deflection angle of s Z     2 −1 −1 the sound wave is directly proportional to the square of r 1 λ 1 2 ð Þ¼ 1 − − ð Þ the csλ=b. t r1;r dr 2 2 Weff ; 78 r1 b r b B. Time delay of sound waves among them, Since the sound wave trajectory bends near the acoustic     black hole, the path of sound wave transmission at the λ2 −1 1 λ2 2 ¼ 2 1 − ¼ 1 − ð Þ same two points becomes longer, as shown in Fig. 8. The b r1 2 ;Weff 2 2 : 79 quantities r1, r2, and r3 are radii of the orbits of the closest r1 r r point, reflector, and observer to the acoustic black hole, respectively. Thus, the time for the sound wave to travel We consider λ=r as a small quantity, simplify and approxi- back and forth between the observer and the reflector will mate the above formula, then

FIG. 8. Sound waves bend by trajectories near acoustic black holes: the left figure shows the situation when the trajectory bends, while the right figure corresponds to the trajectory without bending. Obviously, when the trajectory bends, the trajectory becomes longer and the reflected signal is relatively delayed.

104009-11 QING-BING WANG and XIAN-HUI GE PHYS. REV. D 102, 104009 (2020) Z     2 −3 2 −1 r λ 2 2 vortex. It is a little bit surprising that it saturates the chaos ð Þ¼ 1 − 1 − r1 t r1;r dr 2 2 bound since that acoustic black holes can only simulate r1 r r Z    2 2 −1 the kinetic aspects of real black holes, but it does not satisfy r 3λ r 2 ≈ dr 1 þ 1 − 1 : ð Þ the Einstein equation. It would be interesting to verify this 2 2 2 80 r1 r r point experimentally. Taking vortices as relativistic particles outside acous- The integral can be evaluated as tic black holes, we calculated in detail the effective qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   potential and found that there is no stable circular 3λ2 π 2 2 r1 orbit for vortices. As the radial falling orbit is further tðr1;rÞ¼ r − r þ − arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð81Þ 1 2 2 2 2 r1 r − r1 considered, there is no orbital precession. Sound wave deflection and sound wave time delay are further Thus, investigated. The results show that the deflection angle qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   of sound waves is inversely proportional to the square of 3λ2 π the distance to the acoustic black hole. The time delay of Δ ¼ 2 − 2 þ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffir1 ð Þ ttotal r r1 arctan 2 2 : 82 sound waves passing through the vicinity of the acoustic 2r1 2 r − r 1 black hole has the greatest correlation with the distance Then the radar echo delay is r2 when the sound wave trajectory is closest to the acoustic black hole. It is inversely proportional to the qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi initial distance r1 and the final distance r3 and r1.Thatis Δ ¼ Δ − 2 2 − 2 − 2 2 − 2 ð Þ texcess ttotal r2 r1 r3 r1: 83 to say, the greater the r is, the less the effect on the time delay will be. Considering r1=r2 ≪ 1;r1=r3 ≪ 1 and sound velocity cs, we can recast (83) as ACKNOWLEDGMENTS     3 2λ2 π 1 1 We would like to thank Mikio Nakahara, Sang-Jin Δ ≈ cs π − r1 − r1 ¼ 3 2λ2 − − texcess cs : Sin, Yu-Qi Lei, and Hai-Ming Yuan for helpful r1 r2 r3 r1 r2 r3 discussions. This work is partly supported by NSFC, ð84Þ China (Grant No. 11875184).

We can see that the time delay is mostly related to λ. The APPENDIX: ACOUSTIC REDSHIFT larger the λ is, the more the time delay is. The parameter λ corresponds to the flow velocity around the acoustic black Let us consider an observer who emits a sound signal hole, which is equivalent to the mass of the black hole. from a fixed radius r near the acoustic black hole. The ω Note that λ also corresponds to the radius of the horizon. original signal has frequency as measured by this Therefore, the time delay is closely related to the nature of stationary observer. The sound signal is measured by the central acoustic black hole. Then, since π=r1 is much another stationary observer (see Fig. 9), where its larger than 1=r2 and 1=r3, this reflects that r1 is the closest point to the acoustic black hole, which has a greater impact on the time delay effect. This can be understood as the closer to the acoustic black hole is, the more obvious the bending of the sound trajectory is, and the greater time delay effect is. emitting VII. DISCUSSION AND CONCLUSION observer In summary, the geometry outside a (2 þ 1)-dimensional acoustic black hole encodes interesting physics. For vor- tices falling into the acoustic black holes along the radial receiving sound axis, the growth of the radial momentum near the horizon observer 2πTt wave has the relation pr ∼ e , signalizing the chaos behavior Λ ¼ 2π with the Lyapunov exponent Lyapunov T. In the case of the AdS-Schwarzschild background, the increase of the momentum near the horizon of acoustic black holes is 2πTt e . These are in agreement with the chaos bound FIG. 9. A sound wave traveling from r to infinity, and the proposed in [16]. The result is further confirmed from sound frequency measured by the receiver is lower than that of the numerical calculation of the radial infalling orbit of the the sender.

104009-12 GEOMETRY OUTSIDE OF ACOUSTIC BLACK HOLES IN … PHYS. REV. D 102, 104009 (2020)

α frequency ω∞ received by an observer at infinity is less where ξ is the Killing vector ξ ¼ð1; 0; 0Þ corresponding to than ω. The sound wave energy related to the viewer’s the fact that the metric is time-independent. Therefore, for a “ ” u three-velocity obs is [32] stationary observer at radius r, one has

2 −1 ¼ −p u ð Þ u ð Þ¼ð1 − Þ 2ξ ð Þ E · obs: A1 obs r v : A6 ℏω ¼ −p u Since the energy of a beam of sound wave is E ¼ ℏω, Using · obs in Eq. (A6), the frequencies of ℏω ¼ −p u · obs given the relationship between the observ- the sound wave measured by the stationary observer at ’ “ ” u er s measured frequency and the three-velocity obs.For radius r and infinite radius r∞ are j a stationary observer, the spatial component uobs of the 2 −1 ℏω ¼ð1 − Þ 2ð−ξ pÞ ð Þ “three-velocity” is zero (j ¼ r; ϕ). The time component v · r ; A7 ut ðrÞ at radius r for a stationary observer depend on the obs 2 −1 ℏω ¼ð1 − Þ 2ð−ξ pÞ ð Þ normalization condition, ∞ v∞ · r∞ : A8

α β ξ p ’ u ðrÞ · u ðrÞ¼gαβu ðrÞu ðrÞ¼−1: ðA2Þ The quantity · is conserved along the sound wave s obs obs obs obs ð−ξ pÞ ¼ð−ξ pÞ geodesic. That is, · r · r∞ . Therefore, the ui ð Þ¼0 relationship between the frequencies is For static black hole, obs r , and this means    2    2 1 λ 1 2 1 t 2 1 − 2 1 − 2 2 λ 2 g ðrÞ½u ðrÞ ¼ −1: ðA3Þ v r tt obs ω∞ ¼ ω ¼ ω ¼ ω 1 − : 1 − 2 λ2 2 v∞ 1 − 2 r r∞ In other words, ðA9Þ 2 −1 t ð Þ¼ð1 − Þ 2 ð Þ uobs r v : A4 In the r → ∞ limit, the frequency is less than the frequency 2 2 1=2 at r by a factor ð1 − λ =rÞ . The sound wave should be “ ” Thus, the time-component of the three-velocity is influenced by the “gravitational” redshift of the acoustic given by metric. If the position r of the signal sender approaches the 2 −1 2 −1 α acoustic horizon infinitely, the frequency observed at t ð Þ¼½ð1 − Þ 2 0 0¼ð1 − Þ 2ξ ð Þ uobs r v ; ; v ; A5 infinity will be infinitely low.

[1] J. R. Nova, K. Golubkov, V. I. Kolobov, and J. Steinhauer, [7] X. H. Ge, S. F. Wu, Y. Wang, G. H. Yang, and Y. G. Shen, Observation of thermal Hawking radiation and its temper- Acoustic black holes from supercurrent tunneling, Int. J. ature in an analogue black hole, Nature (London) 569, 688 Mod. Phys. D 21, 1250038 (2012). (2019). [8] X. H. Ge and S. J. Sin, Acoustic black holes for relativistic [2] J. Drori, Y. Rosenberg, D. Bermudez, Y. Silberberg, and U. fluids, J. High Energy Phys. 06 (2010) 087. Leonhardt, Observation of Stimulated Hawking Radiation [9] X. H. Ge, J. R. Sun, Y. L. Zhang, X. N. Wu, and Y. Tian, in an Optical Analogue, Phys. Rev. Lett. 122, 010404 Holographic interpretation of acoustic black holes, Phys. (2019). Rev. D 92, 084052 (2015). [3] W. G. Unruh, Experimental Black-Hole Evaporation, Phys. [10] C. Barceló, S. Liberati, and M. Visser, Towards the Rev. Lett. 46, 1351 (1981). observation of Hawking radiation in Bose-Einstein con- [4] M. Visser, Acoustic black holes: Horizons, , and densates, Int. J. Mod. Phys. A 18, 3735 (2003). Hawking radiation, Classical 15, 1767 [11] C. Yu and J. R. Sun, Note on acoustic black holes (1998). from black D3-brane, Int. J. Mod. Phys. D 28, 1950095 [5] K. G. Zloshchastiev, Acoustic phase lenses in superfluid (2019). helium as models of composite spacetimes in general [12] R. Balbinot, A. Fabbri, R. A. Dudley, and P. R. Anderson, relativity: Classical and quantum features, Acta Phys. Particle production in the interiors of acoustic black holes, Pol. B 30, 897 (1998), arXiv:gr-qc/9802060. Phys. Rev. D 100, 105021 (2019). [6] X. H. Ge, M. Nakahara, S. J. Sin, Y. Tian, and S. F. Wu, [13] E. Bittencourt, V. A. De Lorenci, R. Klippert, and L. S. Acoustic black holes in curved spacetime and the emergence Ruiz, Effective acoustic geometry for relativistic viscous of analogue Minkowski spacetime, Phys. Rev. D 99, 104047 fluids, Phys. Rev. D 98, 064042 (2018). (2019). [14] L. Susskind, Why do Things Fall?, arXiv:1802.01198.

104009-13 QING-BING WANG and XIAN-HUI GE PHYS. REV. D 102, 104009 (2020)

[15] L. Susskind and Y. Zhao, Complexity and momentum, [27] J. M. Duan, Mass of a vortex line in superfluid 4He: Effects arXiv:2006.03019. of gauge-symmetry breaking, Phys. Rev. B 49, 12381 [16] J. Maldacena, S. H. Shenker, and D. Stanford, A bound on (1994). chaos, J. High Energy Phys. 08 (2016) 106. [28] Z. Zhou and J. P. Wu, Particle motion and chaos, Adv. High [17] M. Visser and C. Molina-París, Acoustic geometry Energy Phys. 2020, 1670362 (2020). for general relativistic barotropic irrotational fluid flow, [29] A. R. Brown, H. Gharibyan, A. Streicher, L. Susskind, L. New J. Phys. 12, 095014 (2010). Thorlacius, and Y. Zhao, Falling toward charged black [18] N. Bilić, Relativistic acoustic geometry, Classical Quantum holes, Phys. Rev. D 98, 126016 (2018). Gravity 16, 3953 (1999). [30] S. P. Kim, Hawking radiation as quantum tunneling in [19] V. Moncrief, Stability of stationary, spherical accretion Rindler coordinate, J. High Energy Phys. 11 (2007) 048. onto a Schwarzschild black hole, Astrophys. J. 235, 1038 [31] S. M. Carroll, Spacetime and Geometry (Addison-Wesley, (1980). San Francisco, 2004), pp. 205–222. [20] J. R. Sun and Y. Sun, On the emergence of gravitational [32] J. B. Hartle, Gravity: An Introduction to Einstein’s dynamics from tensor networks, arXiv:1912.02070. General Relativity (Addison-Wesley, Reading, MA, 2003), [21] B. Rezník, Origin of the thermal radiation in a solid-state pp. 186–217. analog of a black hole, Phys. Rev. D 62, 044044 (2000). [33] B. Carter, Global structure of the Kerr family of gravita- [22] S. A. Hartnoll, Lectures on holographic methods for con- tional fields, Phys. Rev. 174, 1559 (1968). densed matter physics, Classical Quantum Gravity 26, [34] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Rotating 224002 (2009). black hole: Locally nonrotating frames, energy extraction, [23] N. J. Cornish and J. Levin, Lyapunov timescales and black and scalar synchrotron radiation, Astrophys. J. 178, 347 hole binaries, Classical Quantum Gravity 20, 1649 (2003). (1972). [24] G. E. Volovik, The Universe in a Helium Droplet (Clarendon [35] F. de Felice, Angular momentum and separation constant in Press and Oxford University Press, Oxford, 2003). the , J. Phys. A 13, 1701 (1980). [25] P. M. Zhang, L. M. Cao, Y. S. Duan, and C. K. Zhong, [36] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics Transverse force on a moving vortex with the acoustic (Oxford University Press, Oxford, 2013), pp. 52–58. geometry, Phys. Lett. A 326, 375 (2004). [37] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation [26] V. N. Popov, Quantum vortices and phase transitions in (Princeton University Press, Princeton, NJ, 2017), Bose systems, Sov. Phys. JETP 37, 341 (1973), http://jetp.ac pp. 901–917. .ru/cgi-bin/dn/e_037_02_0341.pdf.

104009-14