Acoustic Black Holes and De Laval Nozzle

Alma Mater Studiorum · Università di Bologna Scuola di Scienze Dipartimento di Fisica e Astronomia Corso di Laurea in Fisica Acoustic black holes and de Laval nozzle Relatore: Presentata da: Prof. Roberto Balbinot Marco De Vito Anno Accademico 2017/2018 Che cosa pensa allora il fanciullo mentre fa queste scoperte? Innanzi tutto si meraviglia. Questo sentimento di stupore è la sorgente e la fonte inestinguibile della sua sete di conoscere. Max Planck, Significato e limiti della scienza esatta (saggio datato 1941 e 1947) Sommario I buchi neri acustici sono l’analogo sonoro dei buchi neri gravitazionali. La loro base teorica è l’analogia formale tra due tipi di equazione d’onda: quella per la perturbazione del potenziale della velocità in un fluido non viscoso e inomo- geneo e quella per un campo scalare in uno spazio-tempo curvo. Nel presente lavoro, in primo luogo vengono introdotti gli elementi fondamentali della Rela- tività Generale. In secondo luogo, vengono mostrate la metrica di Schwarzschild e la sua rappresentazione nelle coordinate di Painlevé-Gullstrand. Successiva- mente, la parte principale di questa tesi è dedicata alla derivazione della metrica acustica e allo studio di buchi neri acustici a simmetria sferica o unidimensionali. Per quanto riguarda questi ultimi, viene illustrato l’ugello di Laval, essendo un possibile modello per realizzare sperimentalmente buchi neri acustici. Abstract Acoustic black holes are the sonic analogue of gravitational black holes. Their theoretical basis is the formal analogy between two types of wave equation: the one for the velocity potential perturbation in a non-homogeneous inviscid fluid and the one for a scalar field in a curved space-time. In the present work, firstly the basics of General Relativity are introduced. Secondly, the Schwarzschild metric and its representation in the Painlevé-Gullstrand coordinates are shown. Then, the main part of this thesis is dedicated to the derivation of the acoustic metric and to the study of spherically-symmetric or one-dimensional acoustic black holes. Concerning these last ones, the de Laval nozzle is depicted, being a possible model to realize acoustic black holes experimentally. Contents Introduction i 1 Introduction to General Relativity 1 1.1 Equivalence Principle and geodesics equation . .1 1.2 Principle of General Relativity and Principle of General Covariance6 1.3 Space-time curvature and Einstein field equations . .9 2 Schwarzschild metric and Painlevé-Gullstrand coordinates 13 2.1 Schwarzschild metric . 13 2.2 Painlevé-Gullstrand coordinate system . 19 3 Black holes hydrodynamics 23 3.1 Fundamental equations of fluid dynamics . 24 3.2 Perturbative approach . 25 3.3 The acoustic metric . 29 3.4 The acoustic metric in Painlevé-Gullstrand coordinates . 31 3.5 De Laval nozzle . 33 Conclusions 41 A Wave equation in curved space-time 43 Introduction The aim of this bachelor thesis is to describe the fundamental aspects of black holes hydrodynamics. This branch has its foundations on the formal analogy between the behaviour of acoustical disturbances in a non-homogeneous inviscid fluid and the wave propagation in a curved space-time. First of all, this requires an overview of the basic elements of Einstein’s General Relativity, to which Chapter 1 is dedicated. It begins with a discussion about the Equivalence Principle, whose one of the main consequences is the geodesics equation. We then deal with two postulates of General Relativity, that are the Principles of General Relativity and General Covariance, which both lead to the necessity of general tensorial physical equations. The first chapter ends with further geo- metrical considerations about space-time curvature and with the derivation of Einstein field equations. In Chapter 2 we shall concentrate on a specific solution to Einstein field equations, that is the Schwarzschild metric for a spherically-symmetric gravitational source. After its derivation, we will study the physical phenomena occuring at the Schwarzschild radius, which results to be a coordinate singularity. For this reason, in the following section we will introduce the Painlevé-Gullstrand coordinate system, which permits to write the Schwarzschild metric in such a way to eliminate this coordinate singularity. Chapter 1 and 2 will provide a solid background for the last chapter. Chapter 3 is the core of this thesis. We will start by recalling the fundamental equations of fluid dynamics, that are the continuity equation and Euler’s equation. In the second section, the first order perturbative expansion of these equations will be described. This leads to their linearization, which will result in the wave equation for the velocity potential perturbation in a non-homogeneous inviscid fluid. This equation will end up determining the propagation of sound in such fluid and presenting the same formal structure as the wave equation in curved space-time. In the following, this correspondence will be deepened up to obtain the so-called acoustic metric. Theoretically, this represents the only metric to which sound waves couple, thus showing an analogous behaviour to a scalar field propagating in a Lorentzian space-time manifold. Furthermore, by i means of the Painlevé-Gullstrand coordinates, we will observe how the acoustic metric conformally achieves reproducing the Schwarzschild space-time for a spherically-symmetric massive source. The last section of Chapter 3 plays a significant experimental role. We will provide an example of one-dimensional acoustic black hole realization, known as de Laval nozzle. It is a converging-diverging nozzle in which sound waves behave analogously to electromagnetic waves when subject to an hypothetical one-dimensional gravitational black hole. This confirms the theoretical framework depicted in the previous sections. Finally, this thesis concludes with an appendix where a mathematical proof of the wave equation in curved space-time is given. ii Chapter 1 Introduction to General Relativity In this chapter we shall provide an overview of the basic concepts of Einstein’s General Theory of Relativity. This will give us a sufficient base to introduce the Schwarzschild solution to Einstein field equations and its representation in Painlevé-Gullstrand coordinates in the next chapter. 1.1 Equivalence Principle and geodesics equation Einstein’s General Theory of Relativity was invented as an attempt to extend Newton’s theory of gravity in the framework of Special Relativity. However, this theoretical project appeared impossible quite immediately. Indeed, Newtonian gravity is described by the Poisson’s law r2φ = 4πGρ, (1.1.1) where φ denotes the gravitational potential, ρ the matter density and G the Newtonian constant of gravitation. Being a non-tensorial equation, this equation can be shown not to be covariant under Lorentz transformation. Furthermore, let us consider the relativistic tensorial form of Maxwell’s equations in terms of the 4-potential Aµ and the 4-current J µ in the Lorentz gauge 4π 2Aµ = J µ: (1.1.2) c Unlike this, Poisson’s equation for gravity (1.1.1) contains the Laplace operator r2, which is only a spatial differential operator. Therefore, this implies an instant interaction, thus not leading to an interaction propagating at a finite speed, as required by Special Relativity. Such problems motivated Einstein to develop a completely new field theory for gravity, no longer inside the framework 1 CHAPTER 1. INTRODUCTION TO GENERAL RELATIVITY x P* g S' g S Figure 1.1.1: A system P* of free-falling particles in the Earth gravitational field. S denotes a reference frame comoving with the Earth and can be assumed to be inertial. On the other hand, S0 stands for a free-falling reference frame. S0 is not inertial in Newtonian gravity, while it is considered inertial in General Relativity. of Special Relativity. Finally, he came up with an innovative and more general description of space-time itself. A core element of General Relativity is the Equivalence Principle. Its base is the experimental equality between inertial mass m(i) and gravitational mass m(g). Let us consider the Earth gravitational field as homogeneous and a system of free- falling particles (Figure 1.1.1) . The reference frame comoving with the Earth can be assumed to be inertial, neglecting rotational effects. Then, Newton’s equation of motion for the N-th particle is d2~x m(i) N = m(g) ~g: (1.1.3) N dt2 N We shall now perform the non-galileian coordinate transformation ( ~x0 = ~x − 1 ~gt2 2 : (1.1.4) t0 = t 2 1.1. EQUIVALENCE PRINCIPLE AND GEODESICS EQUATION Then, the equation of motion becomes d2~x0 ¨ ¨¨ m(i) N + m¨(i) ~g = m¨(g) ~g; (1.1.5) N dt2 ¨ N ¨ N (i) (g) being mN = mN . This means that gravity cannot be detected in a free-falling reference frame, since everything in the frame is subject to the same homogeneous gravitational force. Let us now consider a parallel situation. We assume no gravity and an inertial reference frame K where some particles are at rest. We now define a reference frame K0 through the same non-galileian coordinate transformation (1.1.4) (Fig- ure 1.1.2) ( ~x = ~x0 + 1 ~gt2 K ! K0 ) 2 ; t = t0 where now ~g is the acceleration owed by K0 with respect to K. x g P* K' K Figure 1.1.2: A system P* of particles in the absence of gravity. K denotes an inertial reference frame in which the particles are at rest. On the other hand, K0 stands for a reference frame moving upward with acceleration ~g with respect to K. K0 is not inertial in Newtonian gravity but it conveys the same physical description as the inertial reference frame S in Figure 1.1.1. 3 CHAPTER 1. INTRODUCTION TO GENERAL RELATIVITY 2 d ~xN 0 Since for the N-th particle dt2 = 0 holds (it is at rest in K), in K we have 2 0 d ~xN dt2 = −~g.

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