^{3}.

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.

The concept of acoustic black holes, based on the kinematical analogue between the motion of sound wave in a convergent fluid flow and the motion of a scalar field in the background of Schwarzschild spacetime, had been suggested initially in 1981 [

As stated above, the acoustic black hole is based on only the kinematical analogue, irrelevant to the dynamics. But for Schwarzschild black holes, once the radiation starts, it will get hotter and hotter by losing energy, which is evident from the relation that the temperature is inversely proportional to the mass. This description is beyond kinematics and is mainly based on thermodynamics that is definitely dependent on the dynamics [

In the earlier discussion about thermodynamics of acoustic black holes, the dynamics is fixed in advance. In this paper, we want to study whether the dynamics of fluid can support the description of thermodynamics spontaneously only if the acoustic horizon has formed. This reminds us of a method which is well-known in gravity; that is, Einstein’s equation can be derived from the thermodynamics plus the knowledge from the black hole physics [

The structure of the paper is as follows. We will first revisit the concept of the acoustic black hole with the model used in the initial paper of Unruh and explain the Hawking-like radiation with the perturbed action. In Section

Consider an irrotational, barotropic fluid that was also considered in the seminal paper [

The variation of

The propagation of sound wave on the background fields can be obtained by a wave equation,

Furthermore, if the scalar field

At first, let us review how the Einstein equation is identified with the thermodynamic relation [

For the metric (

Before applying the thermodynamic identification to acoustic black holes, we have to be sure whether there is the dynamic connection between the fluid and gravitational models only by the kinematical analogue. Now we give some direct kinematical relations between 2D black holes and acoustic black holes, which were once given in [

Generally, 4D Einstein gravity can decay into 2D by the method of spherical reduction if the line element can be written as

Comparing the metric (

When we endow the thermodynamic-like description to the acoustic black hole, the role played by relations (

As well known, the acoustic metric is obtained through the linear perturbation, so in this subsection we will give a brief proof that, at the perturbative level, the two cases are still equivalent. For the acoustic fluid, the corrected action is given by (

For 2D dilaton gravity, the corrected action can be given by the Polyakov-Liouville action (see [

Within the Schwarzschild gauge,

As far as we know, the method of thermodynamic identification has not been applied for 2D gravity. Here we give a brief implementation for this. At the horizon of black hole (

Now we begin to discuss the thermodynamics for an acoustic black hole. Taking the derivative with regard to

It is evident that the thermodynamics of acoustic black holes cannot be Schwarzschild-like, since the Schwarzschild analogue usually takes

As a consistent check, we will see that the kinematical relation (

In order to present the universal property of thermodynamic identification (

With the temperature of the acoustic black hole known in advance (note that the speed of sound here is not constant), (

From the discussion above, we have known the expressions of thermodynamic quantities of acoustic black holes which conform to the first law of thermodynamics. Actually, more importantly, we want to discuss how to use these quantities to describe the evolution of acoustic physical systems, that is, the change caused by emission of thermal radiation from the acoustic horizon. The calculation of back-reaction provided a fundamental method to answer this question, but here we want to estimate it via specific heat of an acoustic black hole, which will give the information about the change of temperature during the radiation. Generally, the temperature in black hole theory is a geometric quantity related closely to the spacetime background, so the change of temperature will indicate the change of spacetime background that is also called back-reaction if the change is not so violent.

According to our results in (

The specific heat of 2D dilaton black holes is also easy to be gotten:

From the analysis above, it is seen that the derivative of the parameter

In this paper, we have reinvestigated the concept of acoustic black holes and discussed background fluid equations. Even though the fluid equations cannot give the complete expressions for each field without any extra knowledge besides the equations of motion, they still include a wealth of information. Via fluid’s equation of motion as a thermodynamic identity, we have read off the mass and the entropy for acoustic black holes with the temperature known in advance. This identification is guaranteed by the kinematical connection which can lead to the dynamic connection between the fluid and gravitational models in two dimensions. In particular, through the analysis for the kinematical relations, we have found that the thermodynamics of acoustic black holes reproduces the thermodynamics of two-dimensional dilaton black holes exactly, and so the fluid can be regarded as a natural analogue of two-dimensional dilaton gravity, which is significant for many ongoing related experimental observations. Novelly, we have also found that the entropy for acoustic black holes is model-independent and has an interpretation similar to the entropic gravity. Moreover, it is found that the derivative of velocity

With the mass and the temperature identified by thermodynamics, we have proceeded to get the specific heat for the acoustic black hole and found the sign of the specific heat is model-dependent, which means that some extra knowledge besides the equations of motion must be given to estimate the thermodynamic stability. However, when we want to model some kind of black holes, that is, Schwarzschild black holes, the relation about the parameters in the fluid can be known only by the fluid equation and the thermodynamic correspondence, as made in (

The author declares that they have no competing interests.

The author acknowledges the support by Grant no. 11374330 from the National Natural Science Foundation of China and from Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics and from the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (no. CUG150630).