Acoustic analogies with covariant electromagnetism, general relativity, and quantum field theory
aa r X i v : . [ g r- q c ] J u l Masovic, Sarradj
Sound waves – weak disturbances of acoustic spacetime
Drasko Masovic a) and Ennes Sarradj Department of Engineering Acoustics, Technical University of Berlin,Einsteinufer 25, 10587 Berlin, Germany (Dated: 8 July 2019)
Acoustic spacetime is a four-dimensional manifold analogue to the relativistic space-time with the reference speed of sound replacing the speed of light. In spite of theformal similarities between the two, the analogy between acoustics in fluids and gen-eral relativity is considered to be limited by different governing equations only towave propagation in various background flows. In this paper, the linearized Einsteinfield equations are used for describing both sound generation and propagation byinterpreting sound waves as a weak disturbance of the background metric and treat-ing them relativistically. The quadrupole source of jet noise is found to correspondclosely to the quadrupole source of gravitational waves and Lighthill’s 8 th -power lawis obtained. Exact solutions for the acoustic monopole (pulsating sphere) and dipole(oscillating sphere) are also reproduced in the relativistic framework. The extendedanalogy between general relativity and acoustics in fluids can have far-reaching im-plications on theoretical and experimental studies in both fields. a) [email protected] I. INTRODUCTION
The analogy between acoustics and general relativity is based on the formal similaritybetween certain relativistic phenomena and acoustics in subsonic flows of fluids when thereference speed of light (here denoted with c ) is replaced by the speed of sound. It shouldbe distinguished from the relativistic acoustics which is relevant when the speed of acousticwaves is actually comparable to the speed of light. The first observations on the similaritybetween special relativity and sound propagation date back to W. Gordon . However, theanalogy is commonly attributed to W. G. Unruh , who used it for studying Hawking ra-diation by means of much better understood acoustic effects in transonic flows. Althoughcommonly associated with quantum phenomena, Hawking radiation is almost fully describ-able with classical mechanics .After its establishment, the analogy was further developed and promoted by M. Visserand C. Barcel´o in their studies of analogue models of gravity . The authors have shownthat sound propagation in moving fluids can be described with the differential geometryof a curved spacetime. Similarly as Unruh, they used the analogy to achieve new insightsinto the gravitational phenomena with the aid of simpler Newtonian physics. For example,gravitational ergo-surfaces are observed in the analogue models as surfaces in fluids whereMach number of the background flow equals one . The aim of this work is to extend theanalogy beyond propagation to wave generation and show that the Einstein field equationscan be used universally as governing equations for acoustics in fluids.2ound waves – weak disturbances of acoustic spacetimeIn contrast to the appreciable interest in the analogy for studying gravitation, the rel-ativistic approach has been used very sporadically for illuminating acoustic phenomena.The recent works of Gregory et al. are rare examples in this direction, although Lorentztransformations, which are typical for special relativity, were used much earlier for soundpropagation in uniform mean flows . Similarly as Unruh and others, they treat acousticproblems of sound propagation in uniform and non-uniform background flows with the aidof geometric algebra of flat and curved four-dimensional spacetime.In spite of the successful implementations, the analogy between acoustics in fluids andgeneral relativity appears to be limited by different governing equations (the Einstein fieldequations and conservation laws of fluid dynamics) only to four-dimensional geometricalinterpretations of sound propagation. In this work we consider the full analogy, includingboth sound generation and propagation, based on the hypothesis that, analogously to gravi-tational waves, sound waves are weak disturbances of acoustic spacetime. First we introducethe concept of acoustic spacetime as already established in literature and show how it cancharacterize sound propagation in background flows. Then we give the essentials of the(linearized) theory of gravitation in section III, which provides a framework for treatmentof waves as perturbations of the background spacetime. The distinction is made betweentransverse gravitational waves and longitudinal sound waves in fluids and their representa-tions with the second-order tensor of weak metric perturbation. The analogy based on thefour-dimensional acoustic spacetime is further extended to sound generation in section IV.Three main types of the sources of sound in fluids – monopole, dipole, and quadrupole –are analysed using the relativistic approach. The 8 th -power law for the quadrupole source of3ound waves – weak disturbances of acoustic spacetimejet noise is reproduced, as well as the exact solutions for compact pulsating and oscillatingspheres. In the last section the main conclusions which follow from the extended analogyare summarized and the potentials for further investigations are discussed. II. ACOUSTIC SPACETIME ANALOGY
In order to demonstrate the analogy in the form already present in the literature, weobserve a symmetric second-order metric tensor g , which is in a specific frame given by itscomponents g αβ . By definition, pseudo-Riemannian spaces, which are considered here, aredifferentiable manifolds supplied at each point with a metric describing their shape. Weuse bold symbols for second-order tensors and arrow above a symbol (for example, ~v ) forvectors. Greek letters are used for the four-dimensional components ( α, β, ... = 0 ...
3, where0 denotes the temporal coordinate) and Latin letters for the three spatial components only(for example, i = 1 ... x = c t , where c is constant reference speed of soundor light and x to x are usual spatial coordinates, for example, x = x , x = y , and x = z .For relativistic quantities we use the notation very similar to Ref. 12.In general curved spacetime, d’Alembertian of a scalar φ equals ✷ φ = ( g αβ φ ,β ) ; α = 1 √− g ( √− gg αβ φ ,β ) ,α . (1)Comma denotes usual derivative with respect to the coordinate which follows it (for example, φ ,α = ∂φ/∂x α ), while semicolon is used for covariant derivative in curved manifolds ( V β ; α = V β,α + V µ Γ βµα , where Γ βµα are Christoffel symbols). Determinant of the matrix ( g αβ ) is g = det( g αβ ). We also use Einstein’s convention, which implies summation over each letter4ound waves – weak disturbances of acoustic spacetimewhich appears in an expression once as a subscript and once as a superscript. The twopositions of the letters refer to the covariant and contravariant vector bases. It is interestingto note that the d’Alembertian in Eq. (1) is also the differential operator of the masslessKlein-Gordon equation, which is the simplest Lorentz-invariant equation of motion whicha scalar field can satisfy . Therefore, it is natural that the same operator appears in therelativistic framework for sound fields in fluids, which are described with a single scalar.Adopting the mixed signature [ − + ++], the simplest flat (Minkowski) spacetime is givenwith the metric g αβ = η αβ = − . (2)In this particular case, g = −
1, the spacetime is flat, covariant derivative becomes simplederivative, and the d’Alembertian becomes the classical wave operator: ✷ φ = η αβ φ ,αβ = φ ,α,α = (cid:18) − c ∂ ∂t + ∇ (cid:19) φ, (3)since , = ∂/∂x = (1 /c ) ∂/∂t and ∇ = ∂ /∂ ( x ) + ∂ /∂ ( x ) + ∂ /∂ ( x ) . Multiplicationwith η αβ raises the index α (or β ). Similarly, multiplication with η αβ lowers α or β .5ound waves – weak disturbances of acoustic spacetimeAnother example of a flat spacetime is given with the metric g αβ = − − M − M − M , (4)where M < ✷ φ = g αβ φ ,αβ = (cid:18) − c D Dt + ∇ (cid:19) φ. (5)Here we recognize the operator D/Dt = ∂/∂t + M c ∂/∂x of the convected wave equation,which describes sound propagation in a uniform subsonic background flow with Mach number M , which is in this case directed along the x -axis. Since we can orient the axes arbitrarily, itfollows that the d’Alembertian in Eq. (1) can describe sound propagation in both quiescentfluid and uniform mean flow, when used with appropriate metric tensors. The metrics suchas the one in Eq. (4), which typically occur in acoustic problems, are sometimes also calledacoustic metrics .Unruh and others have further shown that sound propagation in non-uniform back-ground flows can be described with metrics of curved spacetimes. In particular, one canobtain the wave operator of the Pierce equation , which describes sound propagation ininhomogeneous and unsteady low Mach number flows, the characteristic length and timescales of which are larger than those of the acoustic perturbations. Similar metrics are used6ound waves – weak disturbances of acoustic spacetimeto describe a horizon of a black hole and even more general forms are discussed by Bergliaffaet al. .Different metrics of acoustic spacetime are thus proven to capture the effects of back-ground flows on sound propagation, such as convection and refraction. This has beennoted by several authors who used the analogy for both acoustics and relativity. However,they restrict it to sound propagation only, due to apparently different physics (governingequations) , nature of the corresponding waves, and mechanisms of their generation. Inthe following, we expand the analogy to sound generation and complete it by hypothesizingthat sound waves in fluids are, like gravitational waves, small disturbance (curvature) ofthe background acoustic spacetime. In doing so, we will apply usual linearized theory ofgravitation. For simplicity, we will suppose that the background spacetime is flat Minkowskispacetime. In other words, we assume a quiescent fluid with constant density ρ and speed ofsound c in which the sound waves propagate. After the basic description of sound waves inacoustic spacetime, we will consider the mechanism of their aeroacoustic generation withinthe same relativistic framework. III. SOUND WAVES IN ACOUSTIC SPACETIME
Governing equations of general relativity are the Einstein field equations . They relatecurvature of spacetime, represented by the metric tensor g , with its source, the stress-energytensor T : G + Λ g = kGc T , (6)7ound waves – weak disturbances of acoustic spacetimewhere G is Einstein tensor (which depends only on the metric tensor up to its second-orderderivatives) and Λ (in m − ), dimensionless k (not to be confused with wave number), and G (in m /(kg s )) are constants. We use bold symbols for second-order tensors. Equation (6) iswritten in a frame-independent form. Local conservation of energy and momentum providethe additional conditions, which expressed in terms of the components in a given frame read G αβ ; β = T αβ ; β = 0 . (7)If we are not interested in steady (or slowly varying) solutions, we can adopt usual Λ = 0.In fact, we will suppose that the only disturbance of otherwise flat spacetime is due to thewaves. In an appropriate frame, the metric tensor can be written as a sum of η αβ and aweak component h αβ : g αβ = η αβ + h αβ , (8)with | h αβ | ≪ that there always exists¯ h αβ ( | ¯ h αβ | ≪ h αβ = h αβ − η αβ h ν ν , (9)such that ¯ h αβ ,β = 0 (10)and, consequently, G αβ = − ✷ ¯ h αβ . (11)The term h νν is the trace of h αβ and Eq. (10) is the Lorenz gauge condition. For example,if the Lorenz gauge condition is not satisfied directly by ¯ h αβ in Eq. (9) in a certain frame,8ound waves – weak disturbances of acoustic spacetimeone can introduce a small change of coordinates (gauging) x α → x α + ξ α , (12)which transforms the metric as h αβ → h αβ − ξ α,β − ξ β,α , (13)such that ✷ ξ α = ξ α,β,β = (cid:18) h αβ − η αβ h νν (cid:19) ,β = 0 (14)and therefore ¯ h αβ = h αβ − η αβ h νν − ξ α,β − ξ β,α + η αβ ξ ν,ν (15)does satisfy it. Note that¯ h αα = η αβ ¯ h αβ = η αβ (cid:18) h αβ − η αβ h νν (cid:19) = h αα − η αα h νν = h αα − h αα = − h αα , (16)so h αβ and ¯ h αβ are mutually trace reverse and Eq. (9) can be inverted to h αβ = ¯ h αβ − η αβ ¯ h ν ν , (17)which will be used later. Inserting Eq. (11) into Eq. (6) with Λ = 0 gives the linearizedEinstein field equations: ✷ ¯ h αβ = − kGc T αβ . (18)First we will exclude the source term and consider only wave propagation. Since thebackground spacetime is flat, Eq. (18) is classical wave equation with the operator fromEq. (3), that is ✷ ¯ h αβ = (cid:18) − c ∂ ∂t + ∇ (cid:19) ¯ h αβ = 0 . (19)9ound waves – weak disturbances of acoustic spacetimeThe simplest solution has the form of a plane wave, the real part of¯ h αβ = A αβ e jk ν x ν , (20)where the components of the polarization tensor, A αβ , are complex constants and the four-vector k α is null vector in the flat Minkowski spacetime: k α k α = η αβ k β k α = 0. Forexample, if we suppose that the plane wave propagates in the direction of the x -axis, k α = [ ω/c , , , ω/c ], k α = η αβ k β = [ − ω/c , , , ω/c ], and we can obtain the usual expo-nent − jω ( t − z/c ), after replacing x with c t and x with z .The gauge condition in Eq. (10) gives additional constraint k β A αβ = 0 , (21)which follows from the equality ¯ h αβ ,ν = jk ν ¯ h αβ = jk ν A αβ e jk µ x µ . Since it involves fourequations, the condition decreases the number of the unknown components A αβ from 10 to6. This number can be decreased further. We note that Eq. (14) suggests that a small changeof coordinates, achieved by adding a vector ζ µ to ξ µ from Eq. (12), leaves equations (8) and(10) satisfied if ✷ ζ µ = 0 . (22)This gauge invariance allows further freedom in selecting a specific gauge within the classof Lorenz gauges and the introduction of additional constraints.A typical treatment of plane gravitational waves takes the solution of Eq. (22) (with thelowered index), ζ α = B α e jk µ x µ , (23)10ound waves – weak disturbances of acoustic spacetimein which B α = 1 U ν k ν (cid:18) − k α U β B β − jU β A αβ + j A ββ U α (cid:19) (24)and U β B β = − j U ν k ν (cid:18) U β U α A βα + 12 A µµ (cid:19) . (25)Here, ~U denotes dimensionless four-velocity vector. In a particle’s momentarily comovingreference frame, it is a constant timelike unit vector (time basis vector): U α = δ α , where δ αβ is Kronecker delta. Hence, in this frame ~U · ~U = −
1. For a slowly moving (non-relativistic)particle, three-dimensional velocity satisfies | ~v | ≪ c and the spatial components of the four-velocity vector are v j /c , which justifies naming it velocity. However, it should be notedthat even when the background space is not flat, we can always make a background Lorentztransformation such that U β = δ β in the specific frame. We can also point the spatial axesin this frame such that k α = [ ω/c , , , ω/c ], so the treatment here is completely general.After the additional gauging with ζ α , Eq. (15) gives¯ h T Tαβ = ¯ h αβ − ζ α,β − ζ β,α + η αβ ζ ν,ν . (26)This specifies the so-called transverse-traceless (TT) gauge in which A T T αα = 0 (zero trace)and A T Tαβ U β = A T Tαβ δ β = 0. From the second equality it follows that A T Tα = A T T α = 0 andfrom Eq. (21) k A T T α = ωA T T α /c = 0 for k α = [ ω/c , , , ω/c ], which explains why the11ound waves – weak disturbances of acoustic spacetimegauge is transverse. In the transverse-traceless gauge, the polarisation tensor equals A T Tαβ = A T T A T T A T T − A T T
00 0 0 0 . (27)We can also note that gauging in equations (15) and (26) does not change the order ofmagnitude of the metric perturbation: | h αβ | ∼ | ¯ h αβ | ∼ | ¯ h T Tαβ | ≪ . (28)It is the maximum order of magnitude and the solution in transverse-traceless gauge isrelevant for the gravitational waves, which are transverse waves with two polarizations rep-resented by the two components A T T and A T T . The two components are analogous to theelectric and magnetic components of transverse electromagnetic waves.Since it suppresses all longitudinal components, transverse-traceless gauge is suitablefor the transverse gravitational waves. It may also be relevant for acoustics in solids, iftransverse waves dominate over the longitudinal waves. However, much more suitable forlongitudinal acoustic waves in fluids is Newtonian gauge, which is used for calculations ofthe corrections of classical Newtonian gravitational potential. To show how it can describesound waves in acoustic spacetime, we first note that each free particle in curved spacetimeobeys the geodesic equation 1 c d ~Udτ = 0 , (29)12ound waves – weak disturbances of acoustic spacetimewhere τ is proper time ( c dτ = − ds , where ds is interval between two infinitesimally closeevents in spacetime; for example, in a flat spacetime ds = − c dt + dx + dy + dz ). Thegeodesic equation says that a free particle follows its world line. In a curved spacetime itcan also be written as U α ; β U β = 1 c dU α dτ + Γ αµν U µ U ν = 0 . (30)Christoffel symbols Γ αµν are related to the metric by the equalityΓ αµν = 12 g αβ ( g βµ,ν + g βν,µ − g µν,β ) . (31)If the particle is moving slowly (compared to c ) in an essentially flat spacetime, its four-acceleration due to a small metric perturbation h αβ is in the first order approximation ofEq. (30) dU α dτ = − c Γ α = − c η αβ ( h β , + h β, − h ,β ) , (32)with ~U = d~x/d ( c τ ). The condition for a non-relativistically moving particle is satisfied byweak acoustic waves as well, since the particle velocity is much smaller than the speed ofsound. For such a particle, τ ≈ t and therefore the three-dimensional acceleration equals d x k dt = − c η kl ( h l , + h l, − h ,l ) . (33)Finally, in the Newtonian form | h l | = | h l | ≪ | h | and consequently d x k dt = c h ,k . (34)Motion of the particle due to the wave is expressed with a single scalar h , as we expectfrom the compressible acoustic waves in fluids. In this way we can obtain a measurablequantity (acceleration) from the relativistic analogy.13ound waves – weak disturbances of acoustic spacetimeAs a summary, the physical sound wave is naturally described by the metric component h in the Newtonian gauge. Lorenz gauge condition discussed above can be used first inorder to solve the linearized Einstein equations with ¯ h αβ , but eventually we have to switchto the Newtonian gauge (rather than transverse-traceless gauge) in order to obtain classicalacoustic quantities. From Eq. (34) we can determine other relevant quantities. Since acousticvelocity and potential φ are related by the equality v k = dx k dt = φ ,k , (35)we can associate h with φ : dφdt = c h . (36)If the fluid is quiescent, acoustic pressure and potential are related over the equality p = − ρ dφdt = − ρ c h . (37)Finally, for a plane sound wave, component of the particle velocity in the direction of wavepropagation (say, x -axis) equals v = pρ c = − c h (38)and, indeed, | v | ≪ c for | h | ≪ h in the Newtonian gauge does not have to be ofthe same order of magnitude as ¯ h T Tαβ (or ¯ h αβ or h αβ from which it is derived) if the transversemetric perturbations dominate over the longitudinal components. In fact, we will see thatfor compact sources h is smaller than ¯ h T Tαβ by factor ( ωL/c ) , where L is characteristiclength scale of the source and ( ωL/c ) → h is generated by different sources of sound.14ound waves – weak disturbances of acoustic spacetime IV. GENERATION OF SOUND WAVES
In the previous section we showed how metric perturbation can describe a sound wave inacoustic spacetime by purely geometric means. In this section we inspect several mechanismsof wave generation. As in general relativity at small (non-relativistic) velocities, it is thematter satisfying the conservation laws which causes the curvature of spacetime around it.
A. Aeroacoustic sound generation
In order to study sound generation, we should refer back to the linearized Eq. (18) withthe source term. The source in acoustic spacetime has to satisfy the conservation laws,Eq. (7): T αβ ; β = 0 . (39)In general relativity, stress-energy tensor of a perfect fluid equals T = ( ρc + p ) ~U ⊗ ~U + p g − , (40)where ρ is energy density, p is pressure, ~U is four-velocity of particles ( ~U ⊗ ~U = U α U β ), and g is metric tensor. If all particles of the fluid move with small three-dimensional velocity, | ~v | ≪ c , then the approximation ~U ≈ (1 , ~v/c ) = [1 , v /c , v /c , v /c ] holds and the15ound waves – weak disturbances of acoustic spacetimecomponents of the stress-energy tensor in nearly flat spacetime (Eq. (2)) are T αβ = ρc P c v P c v P c v P c v P v v + p P v v P v v P c v P v v P v v + p P v v P c v P v v P v v P v v + p , (41)where P = ρ + p/c .We shall notice the similarity between the spatial part of the stress-energy tensor ( T jk )and Lighthill’s tensor , which is a pure aeroacoustic source of sound in fluids in free space(without boundaries). Apart from the different value of c (speed of sound instead of speed oflight), the main differences are that in classical fluid dynamics ρ denotes usual density of thematter, rather than energy density, and the appearance of pressure in the sum P = ρ + p/c in the momentum and stress terms. Usually, however, p/c ≪ ρ when | ~v | ≪ c , whichis in the context of aeroacoustics satisfied in subsonic flows with low Mach number value.Pressure p should not be confused with acoustic pressure. Moreover, the mass energy usuallydominates over the energy of massless particles in general relativity, as well, especially atnon-relativistic speeds, which makes the analogy even deeper. The exception are highlynon-relativistic fluids (for example, the early universe is considered to be a soup of masslessparticles). It is interesting to note that such a fluid still posses compressibility and thespeed of sound can be defined . The maximum speed of sound is then of the same order ofmagnitude as the speed of light, smaller only by the factor √
3. This justifies the relativistictreatment of acoustics even in fundamental theories.16ound waves – weak disturbances of acoustic spacetimeAnother important difference is that Lighthill’s tensor is derived from the conservationlaws after the weak acoustic part (represented by the terms with acoustic pressure or density)is shifted to the left-hand side of the wave equation. Therefore, it alone cannot satisfy theconservation laws. In contrast to this, the full T αβ which we consider as the source satisfiesthe conservation laws in Eq. (39). It is the source of the perturbations of spacetime itself,so we do not need to split it into the source and propagation parts in terms of the dynamicquantities.In the further treatment of aeroacoustic sound generation by the stress-energy tensor,we follow Misner et al. and consider a single isolated source of waves, far from which thespacetime is asymptotically flat towards the infinity, with the background metric given inEq. (2). Small metric perturbation is defined by Eq. (8) everywhere (including the sourceregion). Linearized Eq. (18) holds under the condition in Eq. (10) even if the source isrelativistic, with high subsonic velocities in the source region . Furthermore, we expectthat only small part of the stress-energy tensor is responsible for the radiation of waves. Inorder to emphasize this, we can formally split it into the dominant effective stress-energytensor, T eff αβ , and the small component t αβ : T αβ = T eff αβ + t αβ . Thus, we have ✷ ¯ h αβ = − kGc ( T eff αβ + t αβ ) . (42)Such a split may resemble the one in Lighthill’s aeroacoustic analogy, but no part of thestress energy tensor, including the radiation-related part, is shifted to the left-hand side ofEq. (42). The general solution for both ingoing ( ǫ = −
1) and outgoing ( ǫ = +1) wave is¯ h αβ = kG πc Z [ T eff αβ + t αβ ] ( t − ǫR/c ) R d ~y, (43)17ound waves – weak disturbances of acoustic spacetimewhere the integral is over the entire three-dimensional space and R = | x i − y i | . The valuesof T eff αβ and t αβ are to be evaluated at the retarded time t − ǫR/c .Next, we will assume that the source is compact, so that its characteristic length scalesatisfies L ≪ c /ω , where ω is characteristic angular frequency of the oscillations. Thisactually comes down to the condition for non-relativistic motion, | ~v | ≪ c , since | ~v | ∼ Lω .For this reason, the condition is also referred to as slow-motion condition. We will alsoconsider far geometric field ( R ≫ L ). Thus, we can approximate¯ h αβ = kG πrc Z [ T eff αβ + t αβ ] ( t − ǫr/c ) d ~y, (44)where r is radial coordinate of the spherical coordinate system with the source in its origin.From the conservation laws, Eq. (39), one can deduce the identity c d dt Z ( T eff00 + t ) x j x k d ~x = 2 Z ( T eff jk + t jk ) d ~x. (45)It relates the spatial components of the stress-energy tensor with the (scalar) component T , which ultimately removes the need for a split as in Lighthill’s analogy. In fact, thisis allowed by the fact that the stress-energy tensor satisfies the conservation laws, whichdoes not hold for Lighthill’s tensor. The second-order time derivative on the left-hand sideof Eq. (45) corresponds to the second-order derivatives of the source terms in Lighthill’sanalogy. They naturally appear when the second-order tensor should be contracted to ascalar. The integral on the left-hand side of Eq. (45) multiplied with 1 /c represents thesecond moment of the mass distribution and it is called quadrupole moment tensor of themass distribution. It is usually denoted with I jk , which is, thus, by definition I jk = 1 c Z ( T eff00 + t ) x j x k d ~x. (46)18ound waves – weak disturbances of acoustic spacetimeThe quantity which appears to be more convenient for mathematical description of wavegeneration is reduced quadrupole moment, defined as I jk = I jk − δ jk I ll . (47)From equations (44)-(46), ¯ h jk = kG πrc d dt I jk ( t − ǫr/c ) . (48)We will now limit ourselves to the derivation of Lighthill’s scaling law for the acousticpower of the quadrupole source of jet noise. For this, it is sufficient to estimate reaction ofthe source to the far-field radiation, which can actually be done in the acoustic near field.Expanding ¯ h jk in powers of r for ωr/c ≪ ǫ , which correspond to the wave radiation, gives (after replacing ǫ = 1 for an outgoingwave) ¯ h react jk = − kG πc d dt I jk ( t ) − kG πc r d dt I jk ( t ) + ..., (49)where ... denotes the omitted higher-order terms. Using Eq. (10), we can also find¯ h react0 j = − kG πc x k d dt I jk ( t ) − kG πc r x k d dt I jk ( t ) + ... (50)and ¯ h react00 = − kG πc d dt I jj ( t ) − kG πc ( r δ jk + 2 x j x k ) d dt I jk ( t ) + ... (51)These components represent reaction potentials in Lorenz gauge (reaction to the radiation).In terms of gravitation, the omitted terms which do not contribute to the radiation arecorrections of the Newtonian potential producing the effects such as perihelion shift. In the19ound waves – weak disturbances of acoustic spacetimeacoustic analogy, they describe incompressible fluctuations which do not propagate into thefar field.In order to arrive at a Newtonian form which describes the acoustic radiation, we firstswitch back to h αβ = ¯ h αβ − ¯ h ν ν η αβ / x µ + ξ µ ,with ξ = − kG πc d dt I ll ( t ) + kG πc x j x k d dt I jk ( t ) − kG πc r d dt I ll ( t ) (52)and ξ j = − kG πc x k d dt I jk ( t ) + kG πc x j d dt I ll ( t ) . (53)In this gauge, h react00 = − kG πc x j x k d dt I jk ( t ) (54)to the lowest order, while the components h react0 j ∼ ( ωL/c ) h react00 are of higher order. There-fore, we have derived the metric component h in the Newtonian form and it should describethe acoustic longitudinal perturbation of acoustic spacetime in the near field of the isentropicquadrupole source ρ~v~v in Lighthills analogy. The geodesic equation (34) gives the accelera-tion of a non-relativistic particle affected by the perturbation: d x l dt = c h react ,l = − kG πc (cid:18) x j x k d dt I jk ( t ) (cid:19) ,l . (55)We are now ready to derive Lighthill’s scaling law for the source power. We suppose that T jk scales as ρ | ~v | , where ρ is density of essentially incompressible fluid. From equations(45)-(47), | I jk | ∼ ρ L , so the acoustic particle velocity from Eq. (55) scales as | ~v ac | ∼ kGρ L c ( ωL ) . (56)20ound waves – weak disturbances of acoustic spacetimeThe intensity (energy flux) scales as | ~I | ∼ ρ c | ~v ac | ∼ k G ρ L c (cid:18) | ~v | c (cid:19) , (57)where we also replaced ωL ∼ | ~v | . For L ∼ r in the near field, Lighthill’s 8 th -power law givesthe scaling : | ~I | ∼ ρ c ( | ~v | /c ) . Neglecting the multiplication constant which depends onthe dimensionless constant k , the two solutions become equivalent if we set G = c L M ∼ c ρ L , (58)where M is total mass of the source. In this way, we can identify the length scale L asacoustic Schwarzschild radius (characteristic radius of a source of gravitational waves): L = 2 GMc . (59)In cosmology, Schwarzschild radius determines the length scale of a black hole and its eventhorizon. It makes sense that a similar concept appears to determine the length scale of anaeroacoustic source in acoustic spacetime. This points to the similar behaviour of vorticity,as the source of aeroacoustic sound, and merging black holes, as the quadrupole sources ofgravitational waves.If T ∼ ρc dominates aeroacoustic sound generation, as in jets with combustion orsignificant vapour condensation , the acoustic power scales as ( ωL ) ∼ | ~v | , which is en-tirely because the waves in question are longitudinal (compare with squared Eq. (76) insection IV B). In contrast to this, transverse gravitational waves are characterized with asecond-order tensor, which gives the order of magnitude higher by the factor ( ωL/c ) − .The dependence of amplitude on the square of frequency appears in Lighthill’s analogy as a21ound waves – weak disturbances of acoustic spacetimeconsequence of the double differentiation of the source terms. However, the representationby means of acoustic spacetime seems to be more natural, since the full stress-energy tensoris used as the source of waves, as already discussed. There is no need for splitting it intothe source and propagation terms, or selecting an appropriate dynamic quantity for theanalogy (acoustic pressure or density), since the waves are purely geometric perturbationsof spacetime. Transition from the second-order tensor to the scalar is done by expressingthe components of the obtained metric in the Newtonian gauge, rather than by double di-vergence of the stress-energy tensor on the right-hand side. Possible effects of the mean flowon sound propagation outside the source region, such as convection and refraction, have tobe taken into account with appropriate background metric replacing the metric η αβ fromEq. (8). This was discussed in section II and in more details in and . B. Pulsating sphere
Acoustic monopole has no counterpart in the theory of gravitation or electromagnetismand pulsating spherical objects cannot produce gravitational or electromagnetic waves. Thereason is that the sound waves in fluids are compressible and described with scalar poten-tial. Therefore, it is worthwhile to study the appearance of a monopole source in acousticspacetime.As an example of an ideal monopole source, we will consider radiation of a compactpulsating sphere, for which the exact solution exists. We will solve Eq. (42), ✷ ¯ h αβ = − kGc T αβ = − kGc ( T eff αβ + t αβ ) , (60)22ound waves – weak disturbances of acoustic spacetimesimilarly as in . We suppose simple oscillations of t αβ with frequency ω and amplitude S αβ : t αβ = S αβ e − jωt . (61)Since the source is compact, ωL/c ≪
1, where L is radius of the sphere. Outgoing wavesolution of Eq. (60) in the far field has the form of a spherical wave,¯ h αβ = A αβ r e − jω ( t − r/c ) , (62)with r denoting distance from the source and A αβ complex constants. We neglected anyterms of order 1 /r − (the far-field approximation). After cancelling the time dependence onboth sides of Eq. (60), we obtain a form of the Helmholtz equation:[( ω/c ) + ∇ ] (cid:18) A αβ r e jωr/c (cid:19) = − kGc S αβ . (63)Integrating the left-hand side of the equation over the source region gives the followingterms: Z V ω c A αβ r e jωr/c d ~y = 4 πL ω c A αβ e jωL/c = 4 π (cid:18) ωLc (cid:19) A αβ , (64)where A αβ is taken to be constant within the compact sphere, and Z V ∇ (cid:18) A αβ r e jωr/c (cid:19) d ~y = I S ~n · ∇ (cid:18) A αβ r e jωr/c (cid:19) d ~y = 4 πL ddr (cid:18) A αβ r e jωr/c (cid:19) r = L = 4 πL (cid:18) − A αβ r e jωr/c + jωc A αβ r e jωr/c (cid:19) r = L = − πA αβ + j π (cid:18) ωLc (cid:19) A αβ , (65)23ound waves – weak disturbances of acoustic spacetimewith ~n unit vector normal to the surface of the sphere pointing outwards. We also approxi-mated e jωL/c ≈ ωL/c . Although much stronger, the latterterms describe transverse waves only. In fact, the first of the two, which is the strongest,is associated with gravitational waves (described by a second-order tensor). Longitudinal(scalar) component of the metric perturbation is removed in the context of gravitation byutilizing the gauge invariance in Eq. (22). The weaker acoustic waves are thus left out asthe higher order gauge terms. As a consequence, a source oscillating spherically symmetric,such as a pulsating sphere, generates only longitudinal waves and cannot radiate transversegravitational waves.The leading term from Eq. (65) and Eq. (63) give the following expression for the polar-ization tensor of gravitational waves: A αβ = kG πc Z V S αβ d ~y. (66)After comparing the term from Eq. (64) with the leading term from Eq. (65), we can concludethat the acoustic waves are weaker by the order ( ωL/c ) (only a small fraction of t αβ actuallygenerates the longitudinal waves). For the calculation of acoustic A αβ , the additional factorof -1/3 should be multiplied with 2 in order to include fraction of the ∇ term in Eq. (63)which is of the same order as the ω /c term and also provided by the source S αβ . Thus, we24ound waves – weak disturbances of acoustic spacetimeobtain the polarization tensor A αβ = − kG πc (cid:18) ωLc (cid:19) Z V S αβ d ~y (67)and from Eq. (62): ¯ h αβ = − kG πc (cid:18) ωLc (cid:19) e jωr/c r Z V t αβ d ~y. (68)This metric perturbation in Lorenz gauge captures the longitudinal sound wave in acousticspacetime due to the compact pulsating sphere.Physically, acoustic monopole radiation can originate from a source with unsteady massor varying Schwarzschild radius given in Eq. (59). The latter is the case with pulsatingsphere, although mass injection comes similarly down to the effect of displacement of avolume fraction of the fluid surrounding it . The only non-zero component ¯ h equals¯ h = − kG πc (cid:18) ωLc (cid:19) e jωr/c r Z V t d ~y. (69)This solution also follows from the Schwarzschild metric, which is the only spherically sym-metric and asymptotically flat (for r → ∞ ) solution of the Einstein field equations invacuum. For linearized Eq. (60), the solution in Lorenz gauge for large r (in the far field)is ¯ h = kM G/ (2 πrc ), ¯ h j = ¯ h jk = 0. The perturbation of mass m = M − M eff , whichfor a pulsating sphere replaces the mass part of the volume integral of t = ( ρ − ρ eff ) c and which is of interest for acoustic radiation is scaled with the factor − ωL/c ) / Z V t d ~y = ρ c
43 ( L + ¯ le − jωt ) π ≈ ρ c L π ¯ le − jωt , (70)25ound waves – weak disturbances of acoustic spacetimewhere ¯ l ≪ L is amplitude of the oscillations around L and obtain¯ h = − kGρ L c (cid:18) ωLc (cid:19) ¯ l e − jω ( t − r/c ) r . (71)From Eq. (58) we have G = 3 c ρ L π (72)and therefore ¯ h = − k π (cid:18) ωLc (cid:19) ¯ l e − jω ( t − r/c ) r . (73)Switching from ¯ h αβ to h αβ , Eq. (17) gives h = ¯ h − η ¯ h νν = ¯ h + 12 η µν ¯ h µν = ¯ h + 12 η ¯ h = 12 ¯ h (74)and h j = 0. The metric perturbation is already in the Newtonian gauge and from Eq. (34)component of the particle velocity in radial direction due to the acoustic wave equals v r ac = k π c (cid:18) ωLc (cid:19) ¯ l e − jω ( t − r/c ) r . (75)We again neglected the component ∼ /r . The metric does not produce any transversewaves and is indeed purely acoustic.The classical solution for a compact ( ωL/c ≪
1) pulsating sphere reads v r ac = − c (cid:18) ωLc (cid:19) ¯ l e − jω ( t − r/c ) r . (76)Hence, we can adopt k = − π to match the two results. In the context of gravitation, thedimensionless constant is k = 8 π . The opposite sign followed from the different signs inequations (66) and (67). It implies that mass acts as an attracting source of gravity and (itsunsteady component) as a repelling source in acoustics.26ound waves – weak disturbances of acoustic spacetime C. Oscillating sphere
In this section we consider sound radiation from a sphere with radius L which oscillateswith velocity ~v ∼ e − jωt and magnitude | ~v | around the origin of the coordinate system. Thederivation is very similar as in section IV B, except that we replace the source integral inEq. (70) with ( r , φ , and z are now cylindrical coordinates) Z V t j d ~y = ρ c v j Z L rdr Z π dφ Z L dz = ρ c L πv j . (77)One half of the translating sphere is acting on the surrounding fluid with the scatteringcross-sectional area L π . Equation (68) then gives the only non-zero components of themetric perturbation in Lorenz gauge:¯ h j = − kGρ L c (cid:18) ωLc (cid:19) e jωr/c r v j . (78)The Lorenz gauge condition is ¯ h j ,j = 0, which is satisfied to the lowest order of r becausethe flow around the compact sphere is incompressible ( v j ,j = 0). However, we are interestedin the compressible longitudinal component radiated to the far field, so we introduce thechange of coordinates: ξ = − kGρ L c (cid:18) ωLc (cid:19) e jωr/c r v j x j (79)and ξ j = 0 . (80)In this gauge, equations (13) and (17) with ¯ h ν ν = 0 give h αβ = ¯ h αβ − ξ α,β − ξ β,α (81)27ound waves – weak disturbances of acoustic spacetimeand so h = ¯ h − ξ , − ξ , = − c ∂ξ ∂t = − j k πc (cid:18) ωLc (cid:19) e jωr/c r v j x j (82)and h j , h jk = 0 to the lowest order of ωL/c and 1 /r . We have also replaced the constant G from Eq. (58): G = c L/ M = 3 c ρ L π , (83)with the Schwarzschild radius L/
2, since only one half of the sphere effectively pushes thefluid.The metric perturbation is now in the Newtonian gauge and we can use Eq. (34) to obtainat the lowest order v r ac = kc ωπ (cid:18) ωLc (cid:19) | ~v | cos( θ ) e − jω ( t − r/c ) r , (84)where we also used the equality ( v j x j ) ,r = ( | ~v | cos( θ ) e − jωt x r ) ,r = | ~v | cos( θ ) e − jωt in which θ denotes the angle between ~v and the position vector ~r . This matches the classical solution for k = − π and thus confirms the value of k . V. CONCLUSION
In the preceding sections we showed that sound waves in fluids can be treated as weakperturbations (curvatures) of acoustic spacetime which obey the governing equations ofgeneral relativity with c the reference speed of sound. The quadrupole source of jet noisein fluids, which is classically described by Lighthill’s aeroacoustic analogy, corresponds tothe quadrupole source of gravitational waves. In the suitable Newtonian gauge, the 8 th -power law was obtained. For compact sources, this is much weaker radiation than in the28ound waves – weak disturbances of acoustic spacetimecase of transverse gravitational waves, described with a second-order tensor. In fact, thelongitudinal component is then completely removed in the transverse-traceless gauge. Theexact solutions were also obtained for ideal monopole (pulsating compact sphere) and dipole(oscillating compact sphere). This lead to equal value of the dimensionless constant k as ingeneral relativity, only with opposite sign.Although only the problems involving small perturbations of the background spacetimewere treated using the linearized theory of relativity, the fact that Eq. (39) is satisfied influids by the conservation laws suggests that the full Einstein field equations, Eq. (6) with k = − π , may be applicable for general acoustic problems in subsonic flows. Nevertheless,the applicability for non-linear acoustics and high Mach number flows should be furtherinspected. The linearized formulation for small perturbations of otherwise flat acousticspacetime is given in Eq. (18) under the Lorenz gauge condition in Eq. (10). Physicallyrelevant and measurable particle motion due to an acoustic wave can be obtained fromthe component h of the metric perturbation in the Newtonian gauge (Newtonian form),according to the geodesic equation (34).The proposed complete analogy between the two apparently remote physics can havefar-reaching consequences. In a didactic sense, it places the scalar acoustic fields next tothe vector electromagnetic and second-order tensor gravitational fields in the relativistictheories of fields, allowing equivalent treatment. More specifically, quadrupole sources ofgravitational waves, such as merging stars and black holes, can be compared with acousticallycompact vorticity, which constitutes the aeroacoustic quadrupole source of sound in fluids.The analogy reveals new possibilities for indirect studies of gravitational waves with the29ound waves – weak disturbances of acoustic spacetimeaid of acoustic experiments and calculations. It also opens important questions for furtherfundamental work in both areas with regard to the nature of the two types of spacetimes,as well as possible effects of boundaries in such a framework. REFERENCES W. Gordon, “Zur Lichtfortpflanzung nach der Relativit¨atstheorie,” (“Towards propagationof light based on the theory of relativity”) Ann. Phys. , 421–456 (1923). W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett. (21), 1351–1353(1981). W. G. Unruh, “Has Hawking radiation been measured?,” Found. Phys. , 532–545 (2014). C. Barcel´o, S. Liberati, and M. Visser, “Analogue gravity,” Living Rev. Relativ. (3),1–151 (2011). C. Barcel´o, S. Liberati, S. Sonego, and M. Visser, “Causal structure of analogue space-times,” New J. Phys. (186), 1–48 (2004). M. Visser, “Survey of analogue spacetimes,” in
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