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Thermodynamics of Acoustic Black Holes in Two Dimensions

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Thermodynamics of Acoustic Black Holes in Two Dimensions Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 5710625, 8 pages http://dx.doi.org/10.1155/2016/5710625 Research Article Thermodynamics of Acoustic Black Holes in Two Dimensions Baocheng Zhang School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China Correspondence should be addressed to Baocheng Zhang; [email protected] Received 12 May 2016; Accepted 11 July 2016 Academic Editor: Emmanuel N. Saridakis Copyright © 2016 Baocheng Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. It is well-known that the thermal Hawking-like radiation can be emitted from the acoustic horizon, but the thermodynamic-like understanding for acoustic black holes was rarely made. In this paper, we will show that the kinematic connection can lead to the dynamic connection at the horizon between the fluid and gravitational models in two dimensions, which implies that there exists the thermodynamic-like description for acoustic black holes. Then, we discuss the first law of thermodynamics for the acoustic black hole via an intriguing connection between the gravitational-like dynamics of the acoustic horizon and thermodynamics. We obtain a universal form for the entropy of acoustic black holes, which has an interpretation similar to the entropic gravity. We also discuss the specific heat and find that the derivative of the velocity of background fluid can be regarded as a novel acoustic analogue of the two-dimensional dilaton potential, which interprets why the two-dimensional fluid dynamics can be connected to the gravitational dynamics but it is difficult for four-dimensional case. In particular, when a constraint is added for the fluid, the analogue of a Schwarzschild black hole can be realized. 1. Introduction Asstatedabove,theacousticblackholeisbasedononly the kinematical analogue, irrelevant to the dynamics. But for The concept of acoustic black holes, based on the kinematical Schwarzschild black holes, once the radiation starts, it will analogue between the motion of sound wave in a convergent get hotter and hotter by losing energy, which is evident from fluid flow and the motion of a scalar field in the background of the relation that the temperature is inversely proportional to Schwarzschild spacetime, had been suggested initially in 1981 the mass. This description is beyond kinematics and is mainly [1],andthenmanyphysicalsystemshadbeenusedtopresent based on thermodynamics that is definitely dependent on the a similar concept, such as Bose-Einstein condensates (BEC) dynamics [16, 17]. Recently, it was found that when the back- [2], superfluid He [3], slow light [4], electromagnetic waveg- reaction [18, 19] was considered (some different views for this uide [5], and light in a nonlinear liquid [6]; see review [7]. were seen in [20–22], but how to discuss the back-reaction Recently, the observation of the acoustic horizon had been exactly is still open now [7]), the acoustic black hole will not made in different physical systems [8–11]. But the Hawking- present similar thermodynamic behaviors of Schwarzschild like radiation [12, 13] is still difficult to be observed due to black holes while it looks more like a near-extremal Reissner- the small radiation temperature, while the recent report [14] Nordstrom¨ black hole. This is an important step for the on the observation of Hawking-like radiation was related to a possible thermodynamics of an acoustic black hole. Next, self-amplifying mechanism called black hole laser suggested as for thermodynamics of a gravitational black hole, the byCorleyandJacobson[15].Thus,theanalogueofblackholes description usually has to be made with the help of proper provides a prospective avenue to observe the Hawking radia- expressions for the mass and the entropy of the acoustic black tion experimentally, since the direct observation from astro- hole. Some recent developments had shown that the entropy physical black holes is nearly impossible due to the tempera- could be endowed to an acoustic black hole by understanding ture that is several orders of magnitude lower than the cosmic the microscopical modes in some situations [23–26]. But the microwave background. mass is definitely relevant to the classical dynamics, and thus 2 Advances in High Energy Physics its definition requires an analogue for gravity’s equation. An where is the mass density, is the velocity potential, that is, interesting work [27] had been attempted for this, in which V⃗ = ∇⃗ ,and() is the internal energy density. In this paper, the mass and the entropy were defined by using an analogue we only involve the linear perturbation for the derivation of between two-dimensional (2D) dilaton black holes and the the acoustic metric, so action (1) is enough for our discussion. acoustic black holes but at the same time the fluid dynamics The variation of will give the equations of motion, and hadbeenfixedbythetwo-dimensionaldilatongravity[28]. one of them is the Bernoulli equation: Thus, by the analogue for gravitational dynamics, Cadoni [27] 1 2 realized the thermodynamics for an acoustic black hole in two +̇ V⃗ +() =0, (2) dimensions. 2 In the earlier discussion about thermodynamics of acous- where () = /, and the other one is the continuity ticblackholes,thedynamicsisfixedinadvance.Inthispaper, equation: we want to study whether the dynamics of fluid can support ⃗ the description of thermodynamics spontaneously only if the +̇ ∇⋅(V⃗)=0. (3) acoustic horizon has formed. This reminds us of a method We linearize the density and the velocity potential by replac- which is well-known in gravity; that is, Einstein’s equation →+ →+ can be derived from the thermodynamics plus the knowledge ing and ,where and are related to the background fluid determined by the equations of motion, from the black hole physics [29, 30]. Thus, Einstein’s equation could be regarded as a thermodynamic identity; see the and and are small perturbations. When we put the new review paper [31]. That means that, at the horizon, Einstein’s velocity potential and mass density after perturbations into equation is equivalent to the thermodynamic first law, and the the initial action (1), a new added term, up to quadratic order, thermodynamicquantitiessuchasthemassandtheentropy appears as can be read off directly from Einstein’s equation. Then, one 1 2 2 =− ∫ 4[(∇)⃗ − (+̇ V⃗ ⋅ ∇)⃗ ], might ask whether this can be extended to the fluid model, or 2 2 (4) whether it is permitted to read off the mass and the entropy of an acoustic black hole from the fluid equation. It is feasible 2 where the speed of sound is defined as = (/) and if the kinematical analogue can give the connection between +(/2)(+̇ V⃗⋅∇)⃗ = 0 two kinds of dynamics. Moreover, the application of the the equation of motion for , ,isused. method should be able to give consistent results with those It is noted that, for some cases such as presented in Unruh’s obtainedbyothermethods[18,19,27].Inthispaper,wewill original model [1], a completely appropriate approximation try to explore these from the equation of motion for the fluid, can be taken for the speed of sound as a position-independent but our discussion is limited in two dimensions. Once the constant, but there are also some other cases such as BEC [2, 32]whichdidnottakethespeedofsoundasaconstant.Inthis mass and the entropy of the acoustic black hole are given, we paper, we take the speed of sound as constant, in addition to can estimate the thermodynamic evolution using the specific the special case where the nonconstant speed of sound will be heat, as used usually for the Schwarzschild black hole. By the pointed out. estimation,weareabletofindhowtheelementsinafluid The propagation of sound wave on the background model influence the evolution of the acoustic black hole. 2 fields can be obtained by a wave equation, ∇ = (1/ The structure of the paper is as follows. We will first revisit ] √−)(√− ]),where] canbereadofffromtheso- the concept of the acoustic black hole with the model used called acoustic metric: in the initial paper of Unruh and explain the Hawking-like radiation with the perturbed action. In Section 3, we will 2 = [−22 + −⃗ V⃗ 2]. ( ) (5) analyze the background fluid equations and construct the kinematical connection between acoustic black holes and 2D This metric, if considered as a metric of acoustic black hole, black holes, which leads to the dynamic connection between can be understood from a model called the river model of them. Then, we identify the corresponding thermodynamic black holes [33, 34]: the fluid as the background is flowing quantities by a similar method for Einstein’sequation. Wealso along a direction to the region beyond the Newtonian escape discuss the same identification from the equations for BEC as velocity (i.e., the local velocity of sound), and the point where an acoustic analogue of black holes. Section 4 contributes to the velocity of background fluid equals the sound velocity the thermodynamic stability of the acoustic black hole with represents the horizon of the black hole. From this descrip- theaidofthespecificheat.Finally,wediscussandsummarize tion, it is noted that the essence of the acoustic black hole, in our results in Section 5. particular, Hawking radiation which propagates against the fluid, can be understood with a two-dimensional model. So 2. Acoustic Black Hole in the paper we will consider only the case of two dimensions, for which we take the direction of fluid flowing as the - Consider an irrotational, barotropic fluid that was also con- axis and thus the components of the speed V⃗ along the other sidered in the seminal paper [1] of Unruh with the following two directions will be suppressed naturally. Moreover, one action [19]: can also reduce the dimensions by the spherical symmetry, 1 2 in which this analogue will reproduce the results for the so- =−∫ 4[+̇ (∇)⃗ + ()] , 2 (1) called “dirty black holes” [35].
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