Superradiance in a ghost-free scalar theory
SSuperradiance in a ghost-free scalar theory
Valeri P. Frolov ∗ and Andrei Zelnikov † Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2E1
We study superradiance effect in the ghost-free theory. We consider a scattering of a ghost-freescalar massless field on a rotating cylinder. We assume that cylinder is thin and empty inside, sothat its interaction with the field is described by a delta-like potential. This potential besides thereal factor, describing its height, contains also imaginary part, responsible for the absorption of thefield. By calculating the scattering amplitude we obtained the amplification coefficient both in thelocal and non-local (ghost-free) models and demonstrated that in the both cases it is greater than 1when the standard superradiance condition is satisfied. We also demonstrated that dependence ofthe amplification coefficient on the frequency of the scalar field wave may be essentially modified inthe non-local case.
I. INTRODUCTION
In this paper we discuss the superradiance effect ina ghost-free theoriy. In general, the superradiance is aphenomenon where radiation is enhanced. This effectis known in different areas of physics, such as quantumoptics, electromagnetism, fluid dynamics and quantummechanics. The famous example of the superradiance isan amplification of an electromagnetic wave by a rapidlyrotating conducting body [1–3]. If the angular velocityof the rotating body is Ω, then an infalling wave with thepositive frequency ω and an azimuthal angular momen-tum m is amplified when the superradiance condition ismet 0 < ω < m Ω . (1.1)Suppose that a rotating body is a cylinder of radius R ,then the condition (1.1) implies that the linear veloc-ity R Ω of the surface of the cylinder is faster than thephase velocity of the wave. Zel’dovich realized that inthe quantum physics there should exist a similar effectof spontaneous emission from vacuum of quanta satisfy-ing the superradiance condition (1.1). He also suggestedthat the superradiance may exist in rotating black holes.The effect of amplification of the waves by rotating blackholes was studied in [4–6], while Unruh [7] demonstratedthe existence of the quantum spontaneous superradianceemission by direct calculations.The superradiance effect is quite similar to a wideclass of phenomena connected with a so-called anomalousDoppler effect [8]. For example, an electrically neutralbody with internal degrees of freedom, moving uniformlythrough the media may emit electromagnetic waves evenif it starts off in its ground state. It happens when thevelocity of the body is higher than the speed of light inthe media [9–11].Recently, the existence of classical effect of the super-radiance was demonstrated by studying the behavior of ∗ [email protected] † [email protected] sound and surface waves [12, 13]. More recent discussionof the superradiance effect and its interesting astrophys-ical applications can be found e.g in [14–16]. See alsoreferences therein.As we already mentioned, our purpose is to study thesuperradiance effect in the framework of the ghost-freetheory. In such a theory the field equations are modifiedby introducing a non-local form factor. The latter is cho-sen so that it does not introduce ghosts and the numberof degrees of freedom of the modified theory is the sameas for a local one. In a flat spacetime the correspond-ing form-factor is Lorentz invariant and usually it hasthe form ∼ exp(( − (cid:96) (cid:3) ) N ), where N is a positive integernumber and (cid:96) is the characteristic scale where non-localmodification becomes important. Ghost-free theories ofthis type are usually called GF N theories [17]. Nice re-cent reviews of ghost-free theories can be found in [18–20]. One of interesting applications of ghost-free theoriesis study of ghost-free modifications of the Einstein grav-ity [21–24]. It was demonstrated that such modificationsmay help to resolve problem of singularities in cosmologyand black holes [25–29]. In the present paper we restrictourselves by studying of the effect of superradiance forthe case of scalar fields. However the obtained resultscan be easily generalized to the scattering of ghost-freefields with non-zero spins, including linearized ghost-freegravity.For study of the superradiance in the ghost-free theorywe consider a simple model. Namely we consider a scat-tering of the ghost-free massless scalar field ϕ on a rotat-ing infinitely long thin cylinder of radius R . This interac-tion is described by δ -like potential, which besides a pa-rameter, characterising its height, contains also some ab-sorption coefficient . We describe this model in SectionII. We calculate the amplification coefficient in the localtheory by two methods giving the same results: by us-ing jump conditions and by solving Lippmann-Schwingerequation (Sections II and III). In Section IV we demon-strate that a similar problem is exactly solvable in the A similar model in the local quantum mechanics was briefly dis-cussed in [16] a r X i v : . [ h e p - t h ] S e p ghost-free theory of the scalar massless field. In the Sec-tions V-VI we study the properties of the amplificationcoefficients for the ghost-free case. Section VII containsbrief discussion of the obtained results. II. SUPERRADIANCE IN A LOCAL SCALARTHEORY
Let us remind a derivation of a classical superradi-ance effect in a local scalar theory. In the presence ofan absorbing medium the massless scalar field satisfiesthe equation (see Appendix A) (cid:3) ϕ − V ϕ = 0 . (2.1)We consider a special case when the operator V has theform V = δ ( ρ − R ) R (cid:0) β + αR u µ ∂ µ (cid:17) . (2.2)Such a potential describes the matter distribution local-ized on the surface of a cylinder of radius R . Here u µ is the four-velocity vector of an element of an absorbingmedium. The term proportional to β characterizes the”height” of a semi-transparent cylindrical barrier. Theother term, which is proportional to α , describes an in-teraction with absorbing medium, which is located onthe surface of the rotating cylinder. We chose normaliza-tion of the height β of the potential and the absorptioncoefficient α so that they are dimensionless quantities.The metric in cylindrical coordinates reads ds = − dt + dz + dρ + ρ dφ . (2.3)The linear velocity of the surface of the rotating cylinderis u µ = 1 √ − Ω R (cid:2) , , , Ω (cid:3) , (2.4)where Ω is its angular velocity. This linear velocity tendsto the speed of light in the limit R Ω → ϕ = (cid:88) ω,k,m e − iωt + ikz + imφ ϕ ωkm ( ρ ) . (2.5)Here (cid:88) ω,k,m ≡ π ∞ (cid:88) m = −∞ (cid:90) ∞−∞ d k π (cid:90) ∞−∞ d ω π . (2.6)For a real field ϕ the radial harmonics obey the property ϕ − ω − k − m ( ρ ) = ϕ ∗ ωkm ( ρ ) . (2.7) Equations of this type naturally appear in consideration of su-perradiance in stars [1, 30] and many other applications [31, 32].
Every mode ψ ( ρ ) ≡ ϕ ωkm ( ρ ) satisfies a one-dimensional equation( ˆ F − V ωm ) ψ = 0 . (2.8)Here ˆ F = 1 ρ ∂ ρ ( ρ ∂ ρ ) + (cid:104) ω − k − m ρ ] (2.9)and the δ -like complex potential V ωm is V ωm = Λ δ ( ρ − R ) R ,
Λ = β + iγ,γ = α ( m Ω − ω ) R √ − Ω R . (2.10) V ωm is the Fourier transform of V . Note that parameters α, β, γ, Λ are dimensionless in the units when the lightspeed c = 1. The absorption condition on the cylindercorresponds to α > ρ → ∞ ) thereis a condition ω − k >
0. The solution of (2.8) is of theform ψ = (cid:40) C J m ( (cid:36)ρ ); ρ < R,C J m ( (cid:36)ρ ) + C Y m ( (cid:36)ρ ); ρ > R, (2.11) (cid:36) = (cid:112) ω − k . (2.12)The continuity and the jump conditions lead to theequations for the complex constants C , , ( C − C ) J m − C Y m = 0 , (2.13) (cid:36) (cid:2) ( C − C ) J m +1 − C Y m +1 (cid:3) = Λ R C J m . (2.14)Here and later, when the argument of the Bessel functionsis (cid:36)R , we omit it and denote J m = J m ( (cid:36)R ), Y m = Y m ( (cid:36)R ), H (1) m = H (1) m ( (cid:36)R ), H (2) m = H (2) m ( (cid:36)R ).The solution of these equations reads C C = 1 + Λ (cid:36)R Y m ∆ , C C = − Λ (cid:36)R J m ∆ , (2.15)∆ = J m Y m +1 − J m +1 Y m J m = − π(cid:36)R J m . (2.16)In the last equality we used the following property of theWronskian W { J m ( z ) , Y m ( z ) } = J m ( z ) Y (cid:48) m ( z ) − Y m ( z ) J (cid:48) m ( z )= J m +1 ( z ) Y m ( z ) − Y m +1 ( z ) J m ( z ) = 2 πz . (2.17) Discussion of the properties of a δ -like complex potentials and itsapplication to description of absorbtion dynamics can be foundin [33]. Thus we have C C = 1 − π J m Y m , C C = π J m ) . (2.18)At large ρ (cid:29) (cid:36) − , ρ > R the asymptotic behavior ofthe solution is ψ (cid:39) √ π(cid:36)ρ (cid:104) ( C − iC ) e + i (cid:2) (cid:36)ρ − π (cid:0) m + (cid:1)(cid:3) +( C + iC ) e − i (cid:2) (cid:36)ρ − π (cid:0) m + (cid:1)(cid:3)(cid:105) . (2.19)The outgoing wave corresponds to ψ ∼ (cid:40) e + i(cid:36)ρ ω > ,e − i(cid:36)ρ ω < . (2.20)Correspondingly one can write for ω > ψ (cid:39) ( C + iC ) √ π(cid:36)ρ (cid:104) A e + i (cid:2) (cid:36)ρ − π (cid:0) m + (cid:1)(cid:3) + e − i (cid:2) (cid:36)ρ − π (cid:0) m + (cid:1)(cid:3)(cid:105) , (2.21)and for ω < ψ (cid:39) ( C − iC ) √ π(cid:36)ρ (cid:104) e + i (cid:2) (cid:36)ρ − π (cid:0) m + (cid:1)(cid:3) + (cid:101) A e − i (cid:2) (cid:36)ρ − π (cid:0) m + (cid:1)(cid:3)(cid:105) , (2.22)Here the complex relative amplitudes are A ≡ C − iC C + iC = 1 − i π Λ J m H (2) m i π Λ J m H (1) m = πJ m + (cid:0) γJ m − βY m (cid:1) − i (cid:0) βJ m + γY m (cid:1) πJ m − (cid:0) γJ m + βY m (cid:1) + i (cid:0) βJ m − γY m (cid:1) (2.23)and (cid:101) A ≡ C + iC C − iC = 1 A . (2.24)The amplification factor Z is defined as Z = (cid:40) | A | − , ω > , | (cid:101) A | − , ω < . (2.25)Thus we have | A | = (cid:2) πJ m + (cid:0) γJ m − βY m (cid:1)(cid:3) + (cid:0) βJ m + γY m (cid:1) (cid:2) πJ m − (cid:0) γJ m + βY m (cid:1)(cid:3) + (cid:0) βJ m − γY m (cid:1) . (2.26)The amplitude | (cid:101) A | can be obtained from | A | by substi-tution γ → − γ or | (cid:101) A | = | A | − . (2.27) Hence, for positive frequencies we get Z = 8 γπ (cid:0) πJ m − γJ m − βY m (cid:1) + (cid:0) βJ m − γY m (cid:1) . (2.28)Depending on the sign of the parameter γ this amplifi-cation factor can be either positive or negative. One cansee that superradiance amplification occurs when Z > γ >
0, or0 < ω < m Ω . (2.29)This condition means that the angular phase velocity ofthe mode Ω Phase = dφdt = ωm (2.30)obeys the inequality 0 < Ω Phase < Ω . (2.31)In this derivation we assumed ω >
0. Because the veloc-ity of the surface of a cylinder can not exceed the speed oflight we also have a restriction 0 < R Ω <
1. It means, inparticular, that the argument (cid:36)R of the Bessel functionsentering (2.28) lies in the interval0 ≤ (cid:36)R ≤ m, m > . (2.32)In this range of the argument one has J m > Y m <
0. Note that the first zeros j m, and y m, of J m and Y m ,correspondingly, obey the inequality (see Eq.(10.21.3) inthe book [34]) j m, > y m, > m. (2.33)Therefore for the case β ≥ (cid:101) Z = − γπ (cid:0) πJ m + γJ m − βY m (cid:1) + (cid:0) βJ m + γY m (cid:1) . (2.34)The amplification factor of the complex conjugated mode ϕ ∗ ωkm ( ρ ) can be obtained from (2.34) by substitution m → − m and γ → − γ . As a result it gives exactly thesame expression (2.28) as for the ϕ ωkm ( ρ ) mode. Thus,every component of a real wave, which is the sum e − iωt + ikz + imφ ϕ ωkm ( ρ ) + e iωt − ikz − imφ ϕ ∗ ωkm ( ρ ) , (2.35)has the same amplification factor (2.28). III. LIPPMANN–SCHWINGER METHOD
Now let us recalculate the amplification factor usingthe Lippmann–Schwinger approach. The solution of (2.8)can be written in terms of the Green function G of theoperator ˆ Fψ ( ρ ) = ψ ( ρ ) − (cid:90) d ρ (cid:48) ρ (cid:48) G ( ρ, ρ (cid:48) ) V ( ρ (cid:48) ) ψ ( ρ (cid:48) ) , (3.1)where ψ is a solution of the free equationˆ F ψ = 0 . (3.2)The Green function G ( ρ, ρ (cid:48) ) satisfies an inhomogeneousequation ˆ F G ( ρ, ρ (cid:48) ) = − δ ( ρ − ρ (cid:48) ) ρ (cid:48) . (3.3)and outgoing asymptotic condition G ( ρ, ρ (cid:48) ) G ( ρ, ρ (cid:48) ) ∼ √ ρ e i(cid:36)ρ , ρ (cid:29) ρ (cid:48) , (3.4)for ρ → ∞ . G ( ρ, ρ (cid:48) ) is the Fourier transform of theretarded Green function. G ( ρ, ρ (cid:48) ) = (cid:90) d t d z d φ e iωt − ikz − imφ G R ( x, x (cid:48) ) , (3.5)where (cid:3) G R ( x, x (cid:48) ) = − δ ( x, x (cid:48) ) . (3.6)Substitution of (2.10) to (3.1) leads to an algebraic equa-tion ψ ( ρ ) = ψ ( ρ ) − Λ G ( ρ, R ) ψ ( R ) . (3.7)If the complex expression1 + Λ G ( R, R ) (cid:54) = 0 , (3.8)we obtain a solution ψ ( ρ ) = ψ ( ρ ) − Λ G ( ρ, R )1 + Λ G ( R, R ) ψ ( R ) . (3.9)The Green function G ( ρ, ρ (cid:48) ) can be expressed in termsof the Bessel functions G ( ρ, ρ (cid:48) ) = i π J m ( (cid:36)ρ < ) × (cid:40) H (1) m ( (cid:36)ρ > ) , ω > , − H (2) m ( (cid:36)ρ > ) , ω < , (3.10)where ρ < = min( ρ, ρ (cid:48) ) , ρ > = max( ρ, ρ (cid:48) ) . (3.11)For ρ (cid:29) ρ (cid:48) the asymptotic reads G ( ρ, ρ (cid:48) ) (cid:12)(cid:12)(cid:12) ρ →∞ = i (cid:114) π (cid:36)ρ J m ( (cid:36)ρ (cid:48) ) × (cid:40) + e + i(cid:36)ρ − i π m − i π , ω > , − e − i(cid:36)ρ + i π m + i π , ω < . (3.12) Now we choose ψ describing a regular at the centerwave of some arbitrary amplitude. It consists of an in-coming and outgoing waves of the same amplitude ψ = 2 C J m ( (cid:36)ρ ) = C [ H (2) m ( (cid:36)ρ ) + H (1) m ( (cid:36)ρ )] . (3.13)Then at large ρ the solution with the potential (3.9) con-sists of an incoming and outgoing waves (3.13) ψ = ψ in + ψ out , (3.14) ψ in = C (cid:40) H (2) m ( (cid:36)ρ ) , ω > ,H (1) m ( (cid:36)ρ ) , ω < . (3.15)Using (3.9) in the case of ω > ψ out = CH (1) m ( (cid:36)ρ ) − C Λ G ( ρ, R )1 + Λ G ( R, R ) J m ,G ( ρ, R ) = i π J m H (1) m ( (cid:36)ρ ) ,G ( R, R ) = i π J m H (1) m ,ψ out = CH (1) m ( (cid:36)ρ ) 1 − i π Λ J m H (2) m i π Λ J m H (1) m . (3.16)In the case of ω < ψ out = CH (2) m ( (cid:36)ρ ) − C Λ G ( ρ, R )1 + Λ G ( R, R ) J m ,G ( ρ, R ) = − i π J m H (2) m ( (cid:36)ρ ) ,G ( R, R ) = − i π J m H (2) m ,ψ out = CH (2) m ( (cid:36)ρ ) 1 + i π Λ J m H (1) m − i π Λ J m H (2) m . (3.17)The complex relative amplitudes A and (cid:101) A are definedas the ratio of amplitudes of the incoming and outgoingwaves evaluated at infinity. Therefore one gets A = 1 − i π Λ J m H (2) m i π Λ J m H (1) m , (cid:101) A = A − . (3.18)This result exactly reproduces previously calculated val-ues (2.23)-(2.24). IV. SUPERRADIANCE IN THE GHOST-FREESCALAR THEORY
The standard method, as it is described in section II, isnot applicable to the case of the non-local ghost-free the-ory. However, the Lippmann–Schwinger approach workspretty well [35].The ghost-free scalar field satisfies the equation a ( (cid:3) ) (cid:3) ϕ − V φ = 0 . (4.1)The function a ( z ) is an analytical function having theform of an exponent of an entire function and is chosensuch that a (0) = 1. The following choice of the form-factor a is often used in the literature a ( (cid:3) ) = exp[( − (cid:96) (cid:3) ) N ] . (4.2)We refer for a ghost-free theory of this type as to GF N theory [17].Expanding in modes (2.5) one gets the equation for themodes [ a ( ˆ F ) ˆ F − V ωm ] ψ = 0 , (4.3)where the operator ˆ F and the potential V ωm are given by(2.9) and (2.10). Let ψ be a solution of the homogeneousequation ˆ F ψ = 0 . (4.4)Evidently, it satisfies also the nonlocal equation withouta potential a ( ˆ F ) ˆ F ψ = 0 . (4.5)Because of the properties of the ghost-free theories theform-factor a ( ˆ F ) does not lead to any extra freely prop-agating modes.Let us also introduce the Green function G GF ( ρ, ρ (cid:48) ) asa solution of the equation a ( ˆ F ) ˆ F G GF ( ρ, ρ (cid:48) ) = − δ ( ρ − ρ (cid:48) ) ρ (cid:48) , (4.6)which is regular at ρ = 0 and obeys the outgoing bound-ary conditions.The solution for (4.3) can be found using the Lipp-mann–Schwinger approach when applied to the non-localoperator a ( ˆ F ) ˆ F . This solution is similar to (3.9). Theonly difference is that G GF is substituted for G ψ ( ρ ) = ψ ( ρ ) − Λ G GF ( ρ, R )1 + Λ G GF ( R, R ) ψ ( R ) . (4.7)It should be emphasized that, though the solutions ψ of the homogeneous equations in the local and non-localcases coincide, the Green functions G GF and G differ,because they satisfy the inhomogeneous equations with δ -like source.The Green function G GF can be computed using themomentum representation of the retarded Green functioncorresponding to the operator a ( (cid:3) ) (cid:3) a ( (cid:3) ) (cid:3) G GF R ( x, x (cid:48) ) = − δ ( x, x (cid:48) ) . (4.8)In the non-local ghost-free theory the standard time or-dering does not work well in the vicinity of the null coneand the retarded Green function is defined as the solutionof (4.8) which satisfies the same asymptotic conditionsas the local G R ( x, x (cid:48) ) when | x − x (cid:48) | → ∞ , that is far away from the null cone. This requirement, in particular,means that G GF ( ρ, R ) (cid:12)(cid:12) ( ρ − R ) →∞ → G ( ρ, R ) . (4.9)Similarly to the local case the amplitude of the reflectedwave can be defined using (4.7) in the asymptotic domainat large ρ . One can see that in this asymptotic the onlydifference between local and ghost-free cases is the valueof G GF ( R, R ) instead of G ( R, R ) in the denominator ofthe second term in (4.7).Using momentum representation in the Cartesian co-ordinates X = ( t, x ) = ( t, x, y, z ), where x = ρ sin φ and y = ρ cos φ , the retarded Green functions in the local andghost-free theories are G R ( X − X (cid:48) ) = − (cid:90) dωd k (2 π ) e − iω ( t − t (cid:48) )+ i k ( x − x (cid:48) ) × ν + i sgn( ω ) (cid:15) , (4.10) G GF R ( X − X (cid:48) ) = − (cid:90) dωd k (2 π ) e − iω ( t − t (cid:48) )+ i k ( x − x (cid:48) ) × a − ( ν ) ν + i sgn( ω ) (cid:15) . (4.11)Here ν = ω − k = (cid:36) − k x − k y , k = k x + k y + k z , (4.12)and (cid:15) is an infinitesimal positive constant. Because ofanalytical properties of the function a ( ν ) the retardedGreen function, as well as the advanced Green functionand the Feynman propagator, can be written in the form G GF R,A,F ( X − X (cid:48) ) = G R,A,F ( X − X (cid:48) ) + ∆ G ( X − X (cid:48) ) . (4.13)Here ∆ G is a universal non-local correction to the localpropagators∆ G ( X − X (cid:48) ) = (cid:90) dωd k (2 π ) e − iω ( t − t (cid:48) )+ i k ( x − x (cid:48) ) × − a − ( ν ) ν . (4.14)It is this universal correction ∆ G , which is responsiblefor the violation of local causality in the vicinity of thenull cone [18]. The integrand in this expression does nothave poles and, hence, the integral is well defined and un-ambiguous. Note that in order to compute this integralone does not need to analytically continue this expressionto the complex plane. Moreover, analytical continuationwould be a bad idea, because a − ( ν ) may diverge insome complex directions and, hence, the contour inte-gration is not well behaved at infinities. V. SUPERRADIANCE IN THE GF SCALARTHEORY
Let us consider the GF scalar theory [17] which cor-responds to the choice a ( (cid:3) ) = e − (cid:96) (cid:3) . (5.1)Here (cid:96) is the characteristic length scale of non-locality.Typically it is assumed to be larger or of the order of thePlanck scale. Substituting (5.1) to (4.14) and using anintegral representation1 − e (cid:96) ν ν = − (cid:90) (cid:96) ds e sν , (5.2)one can write∆ G ( X − X (cid:48) ) = − (cid:90) (cid:96) ds (cid:90) dωd k (2 π ) × e − iω ( t − t (cid:48) )+ i k ( x − x (cid:48) ) e s ( ω − k ) . (5.3)Performing the Fourier transform in t, z coordinates firstwe obtain∆ G ωk ( x − x (cid:48) , y − y (cid:48) ) = − (cid:90) (cid:96) ds e s(cid:36) (cid:90) ∞−∞ dk x dk y (2 π ) e ik x ( x − x (cid:48) )+ ik y ( y − y (cid:48) ) e − s ( k x + k y ) , (5.4)where k = k z , (cid:36) = (cid:112) ω − k . (5.5)Using the coordinate transformation x = ρ sin φ , y = ρ cos φ we express it in polar coordinates and computethe angular Fourier component∆ G ωkm ( ρ, ρ (cid:48) ) = − (cid:90) (cid:96) ds e s(cid:36) (cid:90) π − π dφ e im ( φ − φ (cid:48) ) (cid:90) ∞−∞ dk x dk y (2 π ) e − s ( k x + k y ) e ik x [ ρ sin φ − ρ (cid:48) sin φ (cid:48) ] e ik y [ ρ cos φ − ρ (cid:48) cos φ (cid:48) ] = − (cid:90) (cid:96) ds πs e s(cid:36) (cid:90) π − π dφ e im ( φ − φ (cid:48) ) e − ρ ρ (cid:48) − ρρ (cid:48) cos( φ − φ (cid:48) )4 s . (5.6)The result of integration over the angle variable can beexpressed in terms of the modified Bessel function I m ∆ G ωkm ( ρ, ρ (cid:48) ) = − π (cid:90) (cid:96) dss e s(cid:36) − ρ ρ (cid:48) s I m (cid:0) ρρ (cid:48) s (cid:1) . (5.7)One can see that this non-local correction to the propa-gators is finite, real, and negative for all values of its realarguments. The asymptotics of the Bessel function are I m ( x ) = 1 √ πx e x (cid:104) O (cid:16) x (cid:17)(cid:105) , x → ∞ ,I m ( x ) = 12 m m ! x m + O ( x m +2 ) , x → . (5.8)It means that the asymptotic of the ∆ G ωkm , when ρρ (cid:48) (cid:29) (cid:96) , reads∆ G ωkm ( ρ, ρ (cid:48) ) ≈ − √ π √ ρρ (cid:48) (cid:90) (cid:96) ds √ s e s(cid:36) − ( ρ − ρ (cid:48) )24 s . (5.9)One can write is in an explicit form∆ G ωkm ( ρ, ρ (cid:48) ) ≈ iπ(cid:36) √ ρρ (cid:48) × (cid:110) e i(cid:36) ( ρ − ρ (cid:48) ) (cid:104) erf (cid:16) ρ − ρ (cid:48) (cid:96) + i(cid:36)(cid:96) (cid:17) − (cid:105)(cid:111) − e − i(cid:36) ( ρ − ρ (cid:48) ) (cid:104) erf (cid:16) ρ − ρ (cid:48) (cid:96) − i(cid:36)(cid:96) (cid:17) − (cid:105)(cid:111) . (5.10)For a fixed ρ (cid:48) = R and large ρ (cid:29) R the correction tothe Green function is exponentially small.∆ G ωkm ( ρ, R ) ∼ e − ρ (cid:96) . (5.11)Therefore, it does not affect the asymptotic behavior atlarge radii.Computation of the amplification coefficient requirescomputation of the value g ≡ ∆ G ωkm ( R, R )= − π (cid:90) (cid:96) dss e s(cid:36) − R s I m (cid:0) R s (cid:1) . (5.12)This function is real and negative. If the radius of therotating cylinder R (cid:29) (cid:96) , then we obtain g ≈ − π(cid:36)R erfi( (cid:36)(cid:96) ) . (5.13)In general, g is the function of the parameters m, (cid:36), R, (cid:96) ,but in the limit when R (cid:29) (cid:96) , it does not depend on theangular momentum m .In the other approximation, when (cid:36)(cid:96) (cid:28) R arbi-trary, the integral (5.7) can be evaluated as follows g ≈ − πm (cid:104) − ξ m F (cid:0) m, m + ; m + 1 , m + 1; − ξ (cid:1) m m ! (cid:105) − π(cid:36) R m − (cid:104) m ( m + 1) − ξ m − F (cid:0) m − , m + ; m, m + 1; − ξ (cid:1) m m ! (cid:105) , (5.14)where ξ = R (cid:96) . (5.15)Similarly to the local case we have ψ = ψ in + ψ out . (5.16)For ω > ψ in = CH (2) m ( (cid:36)ρ ) ,ψ out = CH (1) m ( (cid:36)ρ ) − C Λ G GF ( ρ, R )1 + Λ G GF ( R, R ) J m , (5.17)where G GF ( R, R ) = i π J m H (1) m + g. (5.18)The asymptotic of G GF ( ρ, R ) at ρ → ∞ reads G GF ( ρ, R ) (cid:39) i π J m H (1) m ( (cid:36)ρ ) . (5.19)If the radius of the cylinder is much larger that the scaleof nonlocality R (cid:29) (cid:96) , then the function g can be approx-imated by (5.13). For small frequencies ω (cid:28) (cid:96) − it alsoleads to a simple asymptotic g → − √ π (cid:96)R (cid:104) (cid:36) (cid:96) + O ( (cid:36) (cid:96) ) (cid:105) . (5.20)Eventually we obtain the asymptotic of ψ out at large ρψ out (cid:39) CH (1) m ( (cid:36)ρ ) (cid:104) − i π J m i π Λ J m H (1) m + Λ g (cid:105) . (5.21)The complex relative amplitude A is defined as the ratioof amplitudes of the incoming and outgoing waves eval-uated at infinity. Thus we obtain A = 1 − i π Λ J m H (2) m + Λ g i π Λ J m H (1) m + Λ g . (5.22)For ω <
0, similarly, one gets ψ in = CH (1) m ( (cid:36)ρ ) ,ψ out = CH (2) m ( (cid:36)ρ ) − C Λ G GF ( ρ, R )1 + Λ G GF ( R, R ) J m . (5.23)Here G GF ( R, R ) = − i π J m H (2) m + g. (5.24)At large ρ we have an asymptotic G GF ( ρ, R ) (cid:39) − i π J m H (2) m ( (cid:36)ρ ) , (5.25)and, therefore, ψ out (cid:39) CH (2) m ( (cid:36)ρ ) (cid:104) i π J m − i π Λ J m H (2) m + Λ g (cid:105) . (5.26) The complex relative amplitude (cid:101) A is defined as the ratioof amplitudes of the incoming and outgoing waves eval-uated at infinity. Thus one gets (cid:101) A = 1 + i π Λ J m H (1) m + Λ g − i π Λ J m H (2) m + Λ g . (5.27)It satisfies the usual relation (cid:101) A = A − . (5.28)Taking into account that Λ = β + iγ one can compute theamplification factor Z = | A | − Z = 8 γπ (cid:104)(cid:0) γJ m + βY m − βg ) πJ m (cid:1) + (cid:0) βJ m − γY m + γgπJ m (cid:1) (cid:105) . (5.29)Let us fix the parameters (cid:96), R, (cid:36) and find the value of γ at which the amplification is maximal. It is easy tofind that is happens when γ max = NM , (5.30)where L = πJ m Y m − g,M = (cid:112) π ( J m ) + L ,N = (cid:112) (2 − βL ) + β π ( J m ) . (5.31)Both M and N are positive functions. The maximalvalue of Z is then given by Z max = 4 π ( J m ) M N − π ( J m ) . (5.32)The value of β which maximizes Z is β max = 2 LM . (5.33)Remarkably it does not depend on the value of γ .In this paper we restrict our consideration with onlypositive value of the parameter β , which corresponds toa semitransparent potential barrier, rather than well. Inthe local theory g = 0 and β max is always negative for m >
0, 0 ≤ R ≤ Ω − and 0 ≤ (cid:36)R ≤ m . The lastrestriction on (cid:36) comes from requirement of positivityof γ , what is equivalent to considering only amplifiedmodes. In fact in the local theory the amplification factormonotonically decreases with growth of the strength β of δ -potential. As a result, the maximum amplificationoccurs when β = 0. On the other hand, in the non-localtheory β max may be either negative or positive. In thelatter case the maximum of the amplification factor islocated not at β = 0 but at β max > Z diverges when β = β max (5.34)and γ = γ max ( β max ) = 2 π ( J m ) M . (5.35)These values correspond exactly to the complex condition1 + Λ G GF ( R, R ) = 0 . (5.36)This expression naturally enters the Lipp-mann–Schwinger equation (3.7) and is assumed notto be vanishing in the derivation of its solution (3.9). VI. PROPERTIES OF SUPERRADIANCEAMPLIFICATION
First of all let us consider superradiance in the localtheory, i.e., when (cid:96) = 0 and hence g = 0. Let us in-troduce dimensionless frequency of rotation and modemomentum P = Ω R, p = (cid:112) ω − k R = (cid:36)R, (6.1)and substitute γ = α mP − p √ − P (6.2)to the (5.29). The mode is amplified when γ > p is in the interval0 < p < mP. (6.3)When p = mP the factor γ vanishes and, therefore, Z vanishes as well. When the angular velocity of thecylinder reaches its maximum P = 1 and the factor γ di-verges. In this limit amplification factor Z vanishes bothin local and non-local theories. The strongest amplifica-tion is achieved typically, but not always, for the highangular velocities of the cylinder. Now consider depen-dence of Z on the momentum parameter p . At small p the Bessel function J m ( p ) ≈ p m − m /m !. Thus for m ≥ p = 0.Let us discuss now how these results, obtained forthe local theory, are modified when we include the non-locality effects. To simplify the presentation, in whatfollows we put m = 1 and k = 0. . In Fig. 1 we depictedtypical frequency dependence of the amplification factor Z ( p ). In order to demonstrate how nonlocality affects the In the case of a vanishing of damping factor α = 0 = γ , thecondition (5.36) is closely related to the condition of existence ofa bound state in the δ -potential and corresponding quasinormalmodes [35]. The theory without absorption is invariant under the boostsalong z axis. In this case one can put k = 0 without loss ofgenerality. However, in the case of a non-vanishing absorptionthis symmetry is broken and there appears dependence on k viathe combination √ ω − k . superradiance we present plots with a few ratios of theradius R of a rotating cylinder to the non-locality scale (cid:96) . In Fig. 2 one can see that for (cid:96)/R = (0 . , . , .
4) theeffect of nonlocality is quite modest, the superradiance isof the same order of magnitude. But when (cid:96)/R ≈ . (cid:96)/R ≈ . times stronger than inthe local theory. There is a range of parameters, wherea resonant non-local amplification of superradiance oc-curs. Usually it happens in a quite narrow corridor in theparameter space. A example of a non-locally enhancedsuperradiance for quite low rotations Ω = 0 . R − and atmoderate frequencies ω ∼ . R − is depicted in Fig. 4. VII. DISCUSSION
Let us summarize the results. We considered scatteringof a ghost-free scalar massless field by a rotating (with theangular velocity Ω) cylinder, and demonstrated that inthe presence of absorption of the wave by the cylinder, itsamplitude can be amplified. It happens for the same su-perradiance condition as for the local case 0 < ω < m
Ω.However, the dependence of the amplification coefficienton the frequency of the wave, as well as the parame-ters characterizing the cylinder (its height, β , and itsabsorption factor, α ), might considerably differ from thelocal case. We demonstrated that for a chosen model ofthin and empty inside cylinder the scattering problemin both (local and non-local) cases is exactly solvable.The required solution was found by using the Lippmann-Schwinger equation. The solution contains cylindricalharmonics of the free retarded Green function of the cor-responding problem in the absence of the potential. Themodification of the amplification coefficient in the ghost-free case is related with a non-local contribution into theretarded Green function. We demonstrated the depen-dence of this coefficient on the parameter of the non-locality (cid:96) . In particular, we found that the superradi-ance effect can be greatly amplified for special relationbetween the frequency of the radiation, parameters ofthe potential and of the non-locality. A similar effect wasobserved earlier in the scattering of a ghost-free wave onthe delta-like potential barrier [35]. We studied super-radiance in the simplest case of GF theory. It has anadvantage that all computations can be performed ana-lytically and the result can presented in an explicit form.Disadvantage of GF theory is that it suffers instabilitiesat very high frequencies ω (cid:29) (cid:96) − (see [17]). However,for consideration of superradiance this drawback is notof our concern, because superradiance is important atlow frequencies. GF N theories with even N do not sufferwith this instability and at the same time one can expectthat superradiance qualitatively behaves in a very similarmanner.Zel’dovich [1] used the analogy between a scattering ofthe waves on a rotating absorbing cylinder and the scat- Z p Z p FIG. 1. The amplification factor Z ( p ) in the local scalar theory for Ω = 0 . R − , m = 1, β = 0 (on the left) and β = 1(on theright). p R l ln | | / | 0 | A A l R l R l
1, 0 p R l ln | | / | 0 | A A l R l R l
1, 1
FIG. 2. Enhancement ln[(1 + Z ) / (1 + Z | (cid:96) =0 )] = ln[ | A | / | A | (cid:96) =0 ] in the non-local scalar theory for Ω = 0 . R − , m = 1, β = 0(on the left) and β = 1(on the right) and a few values of non-locality scale (cid:96) . Local theory would corresponds to (cid:96) = 0. p
1, 0 ln | | / | 0 |
A A l R l p R l ln | | / | 0 | A A l
1, 1
FIG. 3. Enhancement ln[(1 + Z ) / (1 + Z | (cid:96) =0 )] = ln[ | A | / | A | (cid:96) =0 ] in the non-local scalar theory for Ω = 0 . R − , m = 1, β = 0(on the left) and β = 1(on the right). One can see that at (cid:96)/R ≈ . β = 1 there is a huge amplification of thesuperradiance, about ∼ . · times stronger, as compared to the local theory. This happens at p ≈ . α , β , and (cid:96)/R . tering of similar waves on a rotating black hole in orderto predict the effect of the black hole induced superra-diance. An interesting question is: Is this analogy alsovalid for the ghost-free field. This point requires explana-tions. In the case of a rotating absorbing cylinder, which we considered in this paper, the potential describing thecylinder modifies the free propagation of the field. Inother words, this is a so-called off-shell problem and thecorresponding retarded Green function, which is used todescribe this effect, is a solution of the inhomogeneous0 p R l ln | | / | 0 | A A l R l R l FIG. 4. Enhancement ln[(1+ Z ) / (1+ Z | (cid:96) =0 )] = ln[ | A | / | A | (cid:96) =0 ] in the non-local scalar theory for moderate values of Ω = 0 . R − , m = 1, α = 0 .
853 and β = 6. The strongest resonance amplification is at (cid:96)/R ≈ . (cid:96)/R significantly changes both the amplitude of the amplification and the characteristic frequency(see dotted and dashed lines). equation with the delta-function in its right-hand side.Such Green functions are different for the local and non-local cases (see e.g. [36]). In the case of a rotating blackhole, the background metric enters the form-factor of thenon-locality, so that, at least at first glance, for the de-scription of the wave scattering by the black hole it issufficient to use only on-shell quantities, which are iden-tical for the local and ghost-free non-local cases. Thiswould imply, that the superradiance is also valid for theghost-free field, but the effects of the non-locality do notmanifest themselves in this case. Anyway, it would bevery interesting to perform accurate calculations of thesuper-radiant scattering of the ghost-free field on a ro-tating black hole and to check whether the Zel’dovich’sanalogy is still valid and complete. ACKNOWLEDGMENTS
The authors thank the Natural Sciences and Engineer-ing Research Council of Canada and the Killam Trust fortheir financial support.
Appendix A: Absorption on δ -potential Let us consider a complex massless scalar field ϕ in acurved spacetime interacting with an absorbing mediumconcentrated on a surface Σ. It obeys the equation (cid:3) ϕ − V ϕ = 0 , (A1)where V = δ (Σ)[ b + au µ ∇ µ ] . (A2)Σ is a timelike surface and u µ is a unit future-directedtimelike vector tangent to Σ. The real coefficients b and a characterize the “height” of the δ -potential and its “ab-sorption capacity”, respectively. Let us denote j µ = − i [ ϕ ∗ ∇ µ ϕ − ϕ µ ∇ ϕ ∗ ] . (A3) Equation (A1) and its conjugated imply ∇ µ j µ − au µ j µ δ (Σ) = 0 . (A4)Let σ and σ be two spacelike surfaces. Let us assumethat current j µ vanishes fast enough at spatial infinity,then integrating over the 4-volume V between σ and σ and using the Stokes theorem one obtains Q [ σ ] − Q [ σ ] = − a (cid:90) dvu µ j µ δ (Σ) . (A5)Here Q [ σ ] = − (cid:90) σ n µ j µ √ hd y . (A6) h is the induced metric on σ and n µ is a future-directedunit normal to σ vector.The equation (A5) has a simple meaning. Namely, thetotal charge Q , carried by the field ϕ changes with timeas a result of its absorption by the potential V . For a realquantized scalar field in the flat spacetime one can relate ϕ and ϕ ∗ with its positive and negative frequency parts.In this case a relation similar to (A5) would describe theabsorption of the scalar quanta by the potential V .Let us adapt the obtained relations to a simple caseof the 4-dimensional flat spacetime. Let its Cartesiancoordinates be ( T, X, Y, Z ) and the surface Σ is a 3-plane Z = 0. One has u µ ∂ µ = ∂ T . (A7)We choose σ as a surface T =const, so that n µ ∂ µ = ∂ T and Q [ T ] = (cid:90) dXdY dZ j T , (A8)and one obtains dQdT = − a (cid:90) j T (cid:12)(cid:12) Z dXdY . (A9)1This relation shows that the rate of the change of “num-ber of particles” is proportional to the “particle density” in the vicinity of the barrier, and the coefficient a char-acterizes how fast this process is. [1] Y. B. Zel’dovich, “Generation of waves by a rotatingbody,” Sov. Phys. JETP Letters , 180 (1971).[2] Y. B. Zel’dovich, “Amplification of cylindrical electro-magnetic waves reflected from a rotating body,” Sov.Phys. JETP , 1085 (1972).[3] B. M. Bolotovskii and S. N. Stolyarov, “Current statusof the electrodynamics of moving media (infinite media),” Phys. Usp. , 875 (1975).[4] A. A. Starobinsky, “Amplification of waves reflected froma rotating ”black hole”.” Sov. Phys. JETP , 28 (1973),[Zh. Eksp. Teor. Fiz.64,48(1973)].[5] A. A. Starobinskii and S. M. Churilov, “Amplificationof electromagnetic and gravitational waves scattered by arotating ”black hole”,” Sov. Phys. JETP , 1 (1974).[6] R. A. Breuer, P. L. Chrzanowksi, H. G. Hughes, andC. W. Misner, “Geodesic synchrotron radiation,” Phys.Rev. D8 , 4309 (1973).[7] W. G. Unruh, “Second quantization in the Kerr metric,” Phys. Rev.
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