Black hole acoustics in the minimal geometric deformation of a de Laval nozzle
BBlack hole acoustics in the minimal geometric deformation of a de Laval nozzle
Rold˜ao da Rocha
Centro de Matem´atica, Computa¸c˜ao e Cogni¸c˜ao,Universidade Federal do ABC - UFABC09210-580, Santo Andr´e, Brazil. ∗ The correspondence between sound waves, in a de Laval propelling nozzle, and quasinormal modesemitted by brane-world black holes deformed by a 5D bulk Weyl fluid are here explored and scruti-nised. The analysis of sound waves patterns in a de Laval nozzle at a laboratory, reciprocally, is hereshown to provide relevant data about the 5D bulk Weyl fluid and its on-brane projection, comprisedby the minimal geometrically deformed compact stellar distribution on the brane. Acoustic pertur-bations of the gas fluid flow in the de Laval nozzle are proved to coincide to the quasinormal modesof black holes solutions deformed by the 5D Weyl fluid, in the geometric deformation procedure.Hence, in a phenomenological E¨otv¨os-Friedmann fluid brane-world model, the realistic shape of ade Laval nozzle is derived and its consequences studied.
PACS numbers: 04.50.Gh, 04.70.Bw, 11.25.-w
I. INTRODUCTION
General Relativity (GR) is a well-succeeded theory,widely tested by experiments and observations, howevera limited one to comprise some recent questions, like thedark energy/dark matter. GR can be recovered frommodels involving higher dimensions as a very restrictedcase [1, 2]. In brane-world models, the brane self-energydensity manifest as the brane tension ( σ ), which is as-sumed to be infinite in the GR limit. Nevertheless, afinite value for the brane tension, in codimension onemodels, is an ubiquitous feature of the brane that alsocomprises the warped five-dimensional (5D) geometry,besides the brane self-gravity.Although an enormous brane tension value recoversGR at low energy regimes, phenomenological evidenceindicates a variable brane tension [3–6]. At an immenselyhot universe, the tension of the brane had attained anugatory value. Afterwards, the brane tension increasedand the universe cooled down [3–5]. Fluid membranesplay a central role in modelling this scenario, whereina temperature-dependent tension is ruled by the E¨otv¨osprinciple [6], that governs a Friedmann brane with a scalefactor that drives the expansion of the universe [7].4D gravity can be effectively formulated on a brane-world by two complementary methods. The first onecomprises the Shiromizu-Maeda-Sasaki implementationof the Gauss-Codazzi on-brane projection routine [8].Nevertheless, this method does not represent a consis-tent system of equations, since the 5D bulk Weyl ten-sor can not be determined from data on the brane. Infact, there is no action, whose projected Euler-Lagrangeequations onto the brane can be derived [2]. A comple-mentary technique does involve an action that, at lowenergy regimes, derives the effective 4D theory describedby the respective Euler-Lagrange equations [9]. Hence, ∗ [email protected] these two complementary procedures can be employed inthe construction of an effective 4D theory [5].Among successful efforts to formulate theories beyondGR, the procedure consisting in accomplishing a mini-mal geometric deformation of the Schwarzschild solutionin a brane-world was derived [10–12]. It comprises ex-act solutions of the Einstein’s equations for the 4D ef-fective theory on the brane [10–12]. The minimal geo-metric deformation procedure incorporates high-energyimprovements to GR, when the (brane) vacuum state ispercolated by a 5D Weyl fluid in the bulk [10, 13, 14].These analytical solutions – of the 4D brane effectiveEinstein’s field equations – encode compact distributionssupporting stellar systems that can even exhibit solidcrusts [13, 14], driven by the 5D bulk Weyl fluid, andalso peculiar generalizations [15]. The deformation itselfcomprises the brane tension as the managing parameterof high energy regimes. This setup has GR correspond-ing to an ideally rigid brane ( σ → ∞ ), at low energies.A more refined setup can be implemented by consider-ing a variable tension fluid brane [3, 16]. This approachhas been comprehensively and successfully constrainedby experimental and observational bounds, provided bythe perihelion precession of Mercury, the deflection oflight by the Sun, the gravitational redshift and the radarecho delay, recently obtained in Ref. [17]. Besides, ob-servational lensing effects by the minimal geometricallydeformed black holes have a typical signature that maybe soon probed by the European Space Agency satellitemission [18]. Despite of this comprehensive list of exper-imental and observational possible signatures, regardingthe brane-world black hole that is geometrically deformedby a 5D Weyl fluid, any study of the quasinormal modesproduced by this kind of black holes lacks, still.On the other hand, acoustic perturbations of a gasflow, in the so-called de Laval nozzle, have been shownto correspond to the general form of perturbations ofSchwarzschild black holes [19–21]. It introduces the fea-sibility to produce and observe their quasinormal reso-nances in a laboratory. The de Laval nozzle is an ex- a r X i v : . [ h e p - t h ] M a r ample among propelling nozzles, that are widely studieddevices that turn a fluid (gas) turbine into a jet engine.A de Laval nozzle is constituted by an hourglass-shapedtube, strained in the middle, utilized to accelerate a hotpressurised gas to a high supersonic speed into the thrustdirection. The fluid thermal energy is commuted into ki-netic energy, and the fluid velocity increases. The energyto accelerate the gas stream induces the gas to adiabati-cally expand with high efficiency to a final – transonic, su-personic, or even hypersonic speed – propelling jet. Thede Laval nozzle is constructed upon the theory of quasi-1D flows, where a fluid moves at the magnitude of thespeed of sound. In this regime, the changes in the fluiddensity turns significant and compress the flow. The deLaval nozzle is based upon the Venturi effect.Fluid flows have been studied, in this context, in a deLaval nozzle, aiming to observe acoustic black holes inthe Schwarzschild setup [19–21]. The acoustic black holesurface gravity was experimentally derived in a labora-tory, in Ref. [22]. As argued in Refs. [20, 21], sonic re-gions in a fluid can induce a surface for the sound waves,known as the acoustic horizon, that emulates a black holeevent horizon. Perturbations of sound waves have beenshown to be analogue to the quasinormal modes, corre-sponding to black holes gravitational excitations [23, 24],being moreover successfully explored in different contexts[25–27].Our point here is to derive and analyse the correctionto a de Laval nozzle trend, using also its analogy to abrane-world black hole in the minimal geometric defor-mation setup, regarding a variable brane tension. Be-sides, another goal here is to study the analogy betweenwaves in a de Laval nozzle in a laboratory and quasi-normal modes of minimal geometrically deformed brane-world black holes. Hence, to scrutinise sonic waves ina de Laval nozzle can circumvent the indeterminacy ofthe 5D Weyl tensor on the bulk [2, 28] that encrypts thebulk geometry. Since the minimal geometric deformationis generated by a 5D Weyl fluid on the bulk, experimentsregarding a de Laval nozzle in a laboratory may recip-rocally provide relevant data about the 5D bulk Weylfluid.This paper is organized as follows: Sect. II is devotedto a brief review, regarding minimal geometrically de-formed compact systems, further refined by phenomeno-logical E¨otv¨os-Friedmann fluid branes. In Sect. III, a gasflow is perturbed in a de Laval nozzle, whose wave equa-tion is analogue to the wave equation regarding spin- s perturbations of minimal geometrically deformed brane-world black holes. Hence, the current bound for the vari-able brane tension provides corrections to the expressionfor the trend of de Laval nozzles. Moreover, quasinormalmodes from these black holes are here proposed to bestudied in a laboratory, when the wave equation in a deLaval nozzle equals the wave equation of spin- s pertur-bations of brane-world black holes undergoing a minimalgeometric deformation. Sect. IV is dedicated to discussour results, to summarise the conclusions and to providerelevant perspectives. II. THE MINIMAL GEOMETRICDEFORMATION SETUP AND FLUID BRANES
Acoustic analogues of brane-world black holes havebeen reported [29], in the context of both Randall-Sundrum and Dvali-Gabadadze-Porrati [30, 31] setups.However, no approach has regarded realistic data onthe brane yet, comprising the variable brane tensionparadigm on fluid branes. Employing the minimal ge-ometric deformation technique incorporates high-energyrefinements to general relativity, by permeating the branevacuum outer of a compact distribution with a Weyl fluidin the 5D bulk [10, 13, 14]. Brane-world models encom-passing a variable brane tension are best implemented byE¨otv¨os-Friedmann fluid branes, where the brane temper-ature drives the brane tension, according to the E¨otv¨os’rule across the universe expansion [3, 6].The most stringent brane tension bound σ (cid:39) . × − GeV has been provided in the context of the min-imal geometric deformation of black holes, formed asBose-Einstein condensates of gravitons that weakly inter-act among themselves [32]. The associated entropic infor-mation content can also predict the Chandrasekhar crit-ical density of compact objects in this paradigm [32, 33].The effective 4D Einstein’s equations can be derivedby the Gauss-Codazzi on-brane projection method, fromthe 5D bulk equations. [2, 8]. In natural units, the 4DEinstein’s effective equations were obtained in Ref. [8](hereon Greek indexes run in the set of Minkowski space-time indexes): G µν + Λ g µν − T µν = 0 , (1)where G µν denotes the Einstein’s tensor and the cosmo-logical parameter on the brane is denoted by Λ. The ef-fective stress-energy tensor T µν = T µν + σ − S µν + E µν + P µν + L µν encrypts the matter stress-energy tensor on thebrane ( T µν ), the 5D bulk Weyl tensor electric projectionon the brane ( E µν ) – that comprise data (constituting aWeyl fluid) about the gravitational field out of the brane– and S µν is the traceless irreducible component, propor-tional to the brane extrinsic curvature [8] – regards 5Deffects onto the brane; the tensor components L µν en-crypt the asymmetric embedding of the brane, into thebulk, and P µν stands for the pullback onto the brane ofthe stress-energy tensor, designating eventual 5D non-standard model fields, comprising radiation of quantumorigin, dilatonic, and even moduli fields [2, 7, 8]. De-viations from the usual Einstein’s standard equations inGR can be also generated by excitations of 5D gravitons,whose effects are encompassed in the P µν tensor. Even-tually, some of the terms constituting T µν can equal zero.Exact solutions of the 4D effective Einstein’s equationsare rare, due to the intricacy of the system of equationsand to initial data out of the brane as well. Compactdistributions, modelling stellar structures are sphericallysymmetric, static, solutions of Eq. (1), of type ds = − A ( r ) dt + ( B ( r )) − dr + r d Ω , (2)for d Ω representing the 2-sphere surface element. Thedeformation on the radial component in Eq. (2) is causedby the bulk constituents, encrypting not only anisotropiceffects originated from the bulk gravity but also the ef-fects of a 5D Weyl fluid in the bulk, whose brand perme-ates the brane vacuum.The minimal geometric deformation is implemented byfixing the temporal component in (2) and deforming theouter radial component [12, 13], B ( r ) = 1 − Mr + ς e Θ( r ) , (3)where Θ( r ) = (cid:90) r R f ( A (a)) f ( A (a)) d a , (4)for [12] f ( A ) = ln( A ) (cid:48) r (cid:18) ln( A ) (cid:48) + 2 r (cid:19) + AA (cid:48)(cid:48) A (cid:48) − f ( A ) = (cid:18) r + 12 ln( A ) (cid:48) (cid:19) − . (5b)The prime denotes the derivative with respect to the ra-dial coordinate, and R denotes the compact distributioneffective radius [10]. The ς parameter in (3) regards theWeyl fluid in the bulk and its induced deformation of thebrane 4D vacuum [17]. The region inner to the stellardistribution is regular at the origin. The inner and outerregion to a star have a shared boundary constituted of asolid incrustation. The (variable) brane tension and thestellar effective radius are parameters that determine thestar crust width [13, 14]. The outer region r > R [10]then promotes the deformed metric [12] A ( r ) = 1 − Mr , (6a) B ( r ) = (cid:34) ς l r (cid:0) − M r (cid:1) (cid:35) (cid:18) − Mr (cid:19) , (6b)where [12] l ≡ (cid:0) − M R (cid:1) − (cid:0) − M (cid:1) R . Refs. [10, 16]show that the metric radial component (6a) can be splitas – hereon in this section the subindex “0” refers to theGR limit σ → ∞ (or, equivalently, ς = 0): B ( r ) = B ( r ) + B ς ( r ) , (7)where B ( r ) = lim ς → B ( r ) = 1 − Mr , (8a) B ς ( r ) = − (cid:0) − M r (cid:1) r − M r ( l | M ) ς , (8b)where B ς ( r ) evinces a high energy correction to theSchwarzschild solution to order O (1 /σ ), for M = M + O (1 /σ ).The parameter ς is proportional to the stellar distribu-tion compactness and drives the geometric deformation of the Schwarzschild solution, having the explicit expres-sion in terms of the brane tension [10, 12]: ς ≈ α τ (R)98 π σ ( l | M )R (cid:20) α +390 α − α + 13595263 α (cid:21) ∝∼ σ − R ( l | M ) ≡ − d σ − R , (9)for α ≡ d R and τ ( r ) ≡ (cid:18) α (cid:16) r R (cid:17) (cid:19) − (cid:18) α (cid:16) r R (cid:17) (cid:19) − . (10)Typically d (cid:117) . . The GR σ → ∞ limit yields ς = 0in Eqs. (6a, 6b), leading to the standard Schwarzschildmetric solution of Einstein’s equations.The current experimental and observational data wasshown to enforce the strongest bound | ς | (cid:46) . × − (obtained by the perihelion precession classical test ofGR) and the weakest bound | ς | (cid:46) . × − (derivedfrom the radar echo delay classical test of GR) on the adi-mensional deformation parameter, in Ref. [17]. Besides,the most recent brane tension bound σ (cid:39) . × − GeV has been obtained by the informational entropy of theminimal geometrically deformed Bose-Einstein conden-sate of gravitons [32]. Eq. (9) implies a negative value forthe parameter ς . Hence, the gravitational field strengthis mitigated by the finite value of the brane tensionand by the 5D encompassing scenario [17], which at-tains a maximum at the stellar surface r = R. Denotingthe stellar distribution density by ρ star , the bound σ (cid:39) . × − GeV complies with the condition ρ star σ (cid:28) T . The E¨otv¨os law asserts that σ ≈ T − τ ,[3, 34], for τ a crucial constant parameter that drives σ into positive values, subsequently to the Big Bang [3, 6].The scale factor constant value a fixes the beginning ofthe universe at a τ temperature [3, 34]. Ref. [3] com-puted the temperature dependence upon the scale factoras T ( t ) ≈ a ( t ) [3], then yielding a variable brane tensionthat is time dependent [3, 4] σ ( t ) σ = κ κ = 1 − a a ( t ) , (11)where κ denotes the 4D coupling “constant” and κ =(8 πG ) − is the late-time coupling constant, for G denot-ing the Newton’s constant.At early times, until the radiation density had equatedthe brane matter density, the brane tension could betaken as having a slight value. However, the brane ten-sion and the 4D coupling parameter likewise magnified asthe universe expanded. The time-dependent brane ten-sion expression yields Λ = Λ − a a ( t ) (cid:16) − a a ( t ) (cid:17) κ σ [3, 4]. Black string and black brane solutions in a variablebrane-world context were studied in Refs. [16, 35, 36]. III. THE MINIMAL GEOMETRICDEFORMATION OF A DE LAVAL NOZZLE
A de Laval nozzle is deeply based on the Venturi ef-fect. When the flow of a (gas) fluid passes through a con-stricted part of a tube with variable cross section A ( x ), itoriginates the reduction in the fluid pressure, whereas thefluid velocity increases. Modelling de Laval nozzles con-siders quasi-1D flows, which are isentropic, adiabatic, andfrictionless ones, which shall be considered hereon. Theregarded fluid can consist of an ideal gas and expressedby the equation of state p = ρRT , where p denotes thefluid pressure, T is the absolute temperature, and R isthe universal gas constant. An ideal gas is known tohave a constant heat capacity, at constant pressure andconstant volume – respectively denoted by C p and C V .Hence R = C p − C V and the specific heat ratio reads γ = C p /C V . For instance, the heat capacity ratio forHelium is γ = 1 .
66, whereas Nitrogen has γ = 1 .
4. Here-upon diatomic gas molecules shall be regarded.An isentropic gas flow, from an initial to a final statehas the property [37] p = ρ γ = T γγ − , (12)where, hereon, these quantities shall be normalised bythe initial state. Prominent properties of isentropic flowscomprise the uniform expansion of the gas, composingthen a shock-free, continuous, flow. A relevant parameterof a compressible flow is the Mach number, M ( x ) = v ( x ) c s ( x ) , where c s = dpdρ = γRT is the local speed of sound and x denotes the transversal nozzle coordinate, namely, thecoordinate along the de Laval nozzle, and v is the localflow velocity. The Mach number is employed to cate-gorise the distinct regimes of flow . Besides, the massflow rate dmdt is the flux per unit throat area ρAv , mean-ing the mass of gas that passes through a cross section ofthe tube per unit time, also known as the fluid discharge[37]. A quasi-1D fluid flow is ruled by the Euler-Lagrangeequations and the continuity relation in fluid mechanics,given by [37] ∂∂t ( ρA ) + ∂∂x ( ρAv ) = 0 , (13a) ∂∂t ( ρAv ) + ∂∂x [( ρv + p ) A ] = 0 , (13b) ∂∂t (cid:18) ρv − p − γ A (cid:19) + ∂∂x (cid:20)(cid:18) ρv − γ − γ (cid:19) Av (cid:21) = 0 . (13c)Instead of Eq. (13b), one can use the Euler’s equation ρ (cid:18) ∂v∂t + v ∂v∂x (cid:19) + ∂p∂x = 0 , (14) Those regimes include hypersonic, supersonic, transonic, sonic,and subsonic flows. which can be led to the Bernoulli’s equation12 (cid:18) ∂ Φ ∂x (cid:19) + (cid:90) ρ − dp = − ∂ Φ ∂t , (15)where the last term in Eq. (15) represents the heatfunction of a barotropic fluid, identified to the enthalpy,and Φ = (cid:82) v dx denoting the velocity potential. FromEq. (15), the linearized equation for sound waves canbe then obtained, considering perturbations φ and δρ ,respectively around Φ and ρ [19, 20].A quasi-1D fluid flow in a de Laval nozzle has a stag-nation state, which is a state attained by the fluid if itis conveyed to rest into an isentropic state and withoutwork. The stagnation speed of sound is denoted by c s .Now, the acoustic analogue of the tortoise coordinate, x (cid:63) ,is defined by x (cid:63) = c s (cid:90) (cid:2) c s ( x )(1 − M ( x ) ) (cid:3) − dx. (16)Perturbing the system of equations (13a – 13c) in a nozzleyields [19]: (cid:20) d dx (cid:63) + ω c s − V ( x (cid:63) ) (cid:21) φ ( ω, x (cid:63) ) = 0 , (17)where the associated potential, representing the soundwaves curvature scattering on the acoustic black hole,reads V ( x (cid:63) ) = 12g (cid:32) g d g dx (cid:63) − (cid:18) d g √ dx (cid:63) (cid:19) (cid:33) , (18)for [19, 20]g( x ) ≡ ρ ( x ) A ( x ) c s ( x ) ∝∼ A ( x )2 ρ ( γ − / , (19) φ ( ω, x (cid:63) ) = (cid:90) (cid:112) g( x (cid:63) ) φ ( t, x (cid:63) ) e iω [ t − f ( x (cid:63) )] dt, (20)where f ( x ) in Eq. (20) is a function defined by df ( x ) dx = | v | c s − v .A de Laval nozzle trend is constructed upon the nozzlethroat cross-sectional area. Dimensionless quantities for ρ ( x ) and A ( x ) are obtained by measuring them in unitsof the stagnation gas density ρ and of the throat nozzlecross-sectional area, respectively. Moreover, [20, 21] A ∝∼ (cid:16) − ρ ( γ − (cid:17) / ρ, (21)which by Eq. (19) yields g = ρ − γ ρ − γ − / , following that ρ − γ = 2g − (cid:112) g − γ − M ≥ , (22)yielding M = 2 γ − − (cid:112) g − − . (23)The Mach number equals the unit at the horizon, whereinthus g have to be finite,g horizon = 1 + γ √ γ − √ ≥ . (24)Replacing Eq. (22) into (19) implies the cross-sectionalnozzle area expressed in terms of g [19], A = 2 γ γ − g − γ γ − (cid:16) g − (cid:112) g − (cid:17) − γγ − , (25)= 1 M (cid:20)(cid:18) γ − M (cid:19) γ + 1 (cid:21) γγ − . (26)On the other hand, the analogy between fluid flows in a de Laval nozzle and the brane-world black hole – un-dergoing a minimal geometric deformation – can be im-plemented. In fact, scalar field perturbations in theminimally-geometric deformed brane-world black holebackground are known to yield the wave-like equation[25] (cid:18) d dr ∗ + ω − V ( r ∗ ) (cid:19) Ψ( r ∗ ) = 0 , (27)where dr ∗ = drB ( r ) and the effective potential for the quasi-normal ringing of the brane-world black hole under theminimal geometric deformation reads V ( r ) = (cid:32) ς l r (cid:0) − M r (cid:1) (cid:33) (cid:18) − Mr (cid:19) (cid:40) (cid:96) ( (cid:96) +1) r +(1 − s ) (cid:34) M (cid:32) ς l r (cid:0) − M r (cid:1) (cid:33) − ς l r (cid:0) − M r (cid:1) (cid:18) − Mr (cid:19)(cid:35) r (cid:41) . (28)Eq. (27) is analogue to Eq. (17). In fact, to find a scalarfunction g that produces the same potential, the tortoisecoordinate of the black hole solution is identified to thede Laval nozzle, dr ∗ = dx (cid:63) , yielding dx (cid:63) = ρ − γ (1 − M ) dx = 2g − (cid:112) g − − (cid:104) − γ − (cid:16) − (cid:112) g − − (cid:17) − (cid:105) dx . (29)The differential equation for g ( r ) then reads[ B ( r )g (cid:48) ( r )] (cid:48) − B ( r ) B (cid:48) ( r )g (cid:48) ( r ) − ( B ( r )g (cid:48) ( r )) r ) = V ( r )g( r ) . (30)One can elect an unit event horizon radius, yielding thenozzle coordinate to be written with respect to the eventhorizon.In the limit ς →
0, Eq. (7) is reduced to Eq. (30) inRef. [20], which has the solution in Eq. (31) in that ref-erence. Substituting Eq. (7) into (30) yields an intricateequation that can not be analytically solved. However,by splitting the solution of (30) into a sum of a purelyGR component (g ( r ) ≡ lim σ →∞ g( r )) and a componentthat is induced by the 5D Weyl fluid,g( r ) = g ( r ) + g ς ( r ) , (31)we can substitute the solution of (30) for ς = 0,g ( r ) = 1 + γ √ γ − (cid:96) (cid:88) j = s (cid:18) ( (cid:96) + j )!( j − s )!( s + j )!( (cid:96) − j )! r j +1 (cid:19) , obtained in Ref. [20], to find the g ς ( r ) function, itera-tively solving Eq. (30). The solution of Eq. (30) has two integration constants, determined by Eq. (24). Eq. (29)and Eq. (6b) then provide the nozzle coordinate x withrespect to r , x = r (cid:90) r (cid:104) (¯ r ) − r ) (cid:112) g (¯ r ) − − (cid:105) − − γ (1 − γ ) B (¯ r ) (cid:16) (¯ r ) − r ) (cid:112) g (¯ r ) − − (cid:17) / d ¯ r. (32)The integral lower limit can be made consistent with thefact that the coordinate x is null at the sonic point, byimposing r = 1.Hence, the de Laval nozzle cross-section A ( x ) can befinally derived, modelling the nozzle shape. In fact, Eq.(32) can be input into Eq. (30), whose numerical solu-tions for g( r ) derive the corrections (due to the 5D Weylfluid) g ς ( r ), in Eq. (31). Subsequently, we rewrite thenumerical solution g( r ) in Eq. (31) with respect to thetransversal nozzle coordinate x in Eq. (32), substitutingit into the expression for the nozzle cross section A ( x ) inEq. (26).In what follows, the solid gray areas in Figs. 1 - 3 indi-cate the cross section A ( x ) of the de Laval nozzle and itsshape, in the σ → ∞ GR limit (gray area limited by thecontinuous gray line) and its minimal geometric deforma-tion due to a 5D bulk Weyl fluid (gray area limited bythe dotted gray line). The black strips respectively rep-resent the effective potential for those both cases. Thebrane tension bound adopted σ ≈ . × − GeV wasderived in Ref. [32] throughout the numerical computa-tion of the minimal geometric deformation case, as wellas the weakest experimental bound | ς | (cid:46) . × − onthe minimal geometric deformation parameter [17]. - x - - Nozzle shape ; V ( x ) FIG. 1. The nozzle profile for s = (cid:96) = 0. The gray filledarea, limited by the continuous [dashed] line denotes the deLaval nozzle in the Schwarzschild, GR σ → ∞ , limit [in theminimal geometric deformation of the de Laval nozzle]. Theblack lines represent V ( x ) for the GR limit (continuous line)and its minimal geometric deformation (dashed line). - - - x - - - Nozzle shape ; V ( x ) FIG. 2. The nozzle profile for s = (cid:96) = 1. The gray filledarea, limited by the continuous [dashed] line denotes the deLaval nozzle in the Schwarzschild, GR σ → ∞ , limit [in theminimal geometric deformation of the de Laval nozzle]. Theblack lines represent V ( x ) for the GR limit (continuous line)and its minimal geometric deformation (dashed line). - - - x - - - ������ ����� ; V ( x ) FIG. 3. The nozzle profile for s = (cid:96) = 2. The gray filledarea, limited by the continuous [dashed] line denotes the deLaval nozzle in the Schwarzschild, GR σ → ∞ , limit [in theminimal geometric deformation of the de Laval nozzle]. Theblack lines represent V ( x ) for the GR limit (continuous line)and its minimal geometric deformation (dashed line). The minimal geometric deformation constricts the noz-zle throat cross-sectional area. In addition, by comparingthe Schwarzschild versus minimal geometric deformation,the nozzle corrections due to the influence of a 5D bulkWeyl fluid that permeates the vacuum on the brane arenotorious. Hence, the quasinormal modes of black holessolutions deformed by the 5D bulk Weyl fluid can beprobed by their analogy with acoustic perturbations ofa diatomic gas fluid flow in a de Laval nozzle, using thegeometric deformation technique. Reciprocally, the sig-nature and the lacking data about the Weyl tensor in thebulk can be probed in a laboratory, by analysing sonic waves throughout a de minimal geometrically deformedde Laval nozzle. The next section is devoted to furtherexplore and analyse the consequences of our results.
IV. CONCLUDING REMARKS AND OUTLOOK
The minimal geometric deformation of a de Laval noz-zle can have double-handed applications. The de Lavalnozzle associated with black hole analogues produced inthe laboratory can present their trend slightly modifiedby a 5D bulk Weyl fluid effects. On the other hand, 5Deffects can be also probed in the laboratory, due to theanalogy heretofore presented.Using a phenomenological E¨otv¨os-Friedmann fluidbrane setup, describing an inflationary brane-world uni-verse, the perturbation of a fluid flow in a de Laval noz-zle was considered, providing a wave equation that issimilar to the wave equation regarding perturbations ofminimal geometrically deformed brane-world black holes.The precise bounds for the variable brane tension valueprovided corrections to the shape of de Laval nozzles inthis context. Figs. 1 - 3 plot the de Laval nozzle profileand also the analysis of the nozzle deformation with re-spect to the Schwarzschild solution. Such a deformationis generated by a 5D Weyl fluid permeating a compactdistribution described by the Schwarzschild metric solu-tion of the 4D brane effective Einstein’s equations.Moreover, quasinormal modes of brane-world blackholes undergoing a minimal geometric deformation canbe, then, produced and observed in a laboratory, byanalysing the sonic waves throughout the associated de-formed Laval nozzle. Hence, the solution for the inversetechnique, consisting of the correspondence between theshape of the de Laval nozzle and the general trend ofperturbations in brane-world black holes deformed by aminimal geometric deformation of Schwarzschild blackholes, has been here implemented. The corrections tothe Schwarzschild solution on the brane, permeated by a5D bulk Weyl fluid, affect how the pressure is dispersedacross the deformed de Laval nozzle. The finite branetension, then, specifies a protocol to the analysis consist-ing of whether the de Laval nozzle highest thrust can beachieved and the search for best flow properties that arebeing attained, for the derived de Laval nozzle shape.Using the sonic analogue to black holes [38], the ther-mal spectrum of sound waves was given out from thesonic horizon in transsonic fluid flows, also in the con-text of analogue gravity [39–41]. These approaches canbe further explored, using the methods here introduced,together with more fluid analogies phenomena regardingblack holes in the laboratory [42, 43]. Still, further typesof black holes can be studied [27, 44]. Finally, the ex-tended MGD approach [45] can also be used to derivefurther corrections to the de Laval nozzle profile.
Acknowledgements
RdR is grateful to CNPq (Grant No. 303293/2015-2),to FAPESP (Grant No. 2015/10270-0), for partial finan- cial support, and to Dr. A. Zhidenko for worth discus-sions. [1] Antoniadis I, Arkani-Hamed N, Dimopoulos S andDvali G R 1998
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