Absorption of massless scalar field by furry black holes in de Rham-Gabadadze-Tolley theory
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Absorption of massless scalar fieldby furry black holes in deRham-Gabadadze-Tolley theory
Ping Li, a Yang Huang, b Xin-Zhou Li a a Division of Mathematical and Theoretical Physics, Shanghai Normal University, 100 GuilinRoad, Shanghai 200234, China b School of Physics and Technology, University of Jinan, 336 West Road of Nan Xinzhuang,Jinan, Shandong 250022, ChinaE-mail: [email protected], [email protected], [email protected]
Abstract.
We study the absorption of massless scalar field by two types furry charged blackholes in de Rham-Gabadadze-Tolley (dRGT) theory. The absorption cross section is cal-culated in high frequency limit σ hf and low frequency limit σ lf . We show that the highfrequency limit σ hf is the area of shadow and the low frequency limit σ lf is the area of hori-zon. The ratio R f = σ hf σ lf is used to measure the impact of charges on the absorption spectraof furry black hole. If the black hole possess an extra charge except mass, the interval value ofabsorption ratio R f is different: [1 , ] for electric charge, [0 . , ] for positive charge and [ , . for negative charge. We also use a numerical method to compute the absorptioncross section in the finite frequency domain. A series of numerical results are presented. a r X i v : . [ g r- q c ] F e b ontents R f = σ hf σ lf General relativity (GR) is a uniquely theory of massless spin-2 in four dimensions. Meanwhile,dRGT theory [1] may be the theory of massive spin-2. A comprehensive review of massivegravity can be found in [2].Similar to GR, the spherically symmetric vacuum solution plays a crucial role in massivegravity. However, there is no conventional Schwarzschild solution in the unitary gauge [3].In order to restore the diffeomorphism invariance, the Stückelberg fields φ a are introduced indRGT theory. Compared with the process to seek the spherically symmetric vacuum solutionin GR, there existed an additional invariant I ab = g µν ∂ µ φ a ∂ ν φ b in massive gravity [3]. In theunitary gauge φ a = x µ δ aµ , any inverse metric g µν that has divergence including the coordinatesingularity in GR would exhibit a singularity in the invariant I ab . Then, we use a nonunitarygauge to seek healthy spherically symmetric vacuum solutions. In the Ref.[4], we obtainseven solutions including the Schwarzschild solution, Reissner-Nordstr ¨ o m (RN) solution, andfurry black holes in dRGT theory. All these solutions have no coordinate singularity in I ab .Therefore, these solutions may become candidates for black holes in dRGT.The spherically symmetric black holes in GR depend on two parameters: mass M andelectric charge Q . There are numerous papers [5]-[10] and books [11] have studied the scatter-ing problem in these black holes. For recently work, Dolan et al. studied Fermi field scattering– 1 –y Schwarzschild black hole in Ref.[12]. Crispino et al. have studied the electromagnetic ab-sorption of RN black hole [13]. The absorption of massive scalar field by RN black hole isstudied detailed in Ref.[14].Furry black holes possess an event horizon that depends on the Schwarzschild radius r s , electric charge Q and a new parameter S . The new parameter S introduce a new degreeof freedom in event horizon of black hole, which is not the Mass M and Electric Charge Q .Thus, we call this furry black hole. This may be a new classification of black hole beyondKerr-Newman family.The furry black holes have some features different from black hole in Kerr-Newmanfamily. In Ref.[15], we have shown the supperradiant instabilities for furry black hole in acavity depend on the black hole’s parameters. As a comparison, the supperradiant instabilitiesfor RN black hole in a cavity is free of M and Q , which is shown by Degollado et al. [16].In this paper, we study the absorption of neutral massless scalar field by two types furryblack holes in detail. We also show a clear difference between the absorption spectra ofblack holes in GR and the absorption spectra of furry black holes in massive gravity. Toillustrate this difference, we define the ratio R f = σ hf σ lf , whose value can reflect the degreeof a curved background by black hole. The typical value R f for the absorption of masslessfield by Schwarzschild black hole is . In the case of RN black hole, the absorption ratio R f is the range [1 , ] . In the case of type (I) furry black hole, the ratio R f has a maximum in Schwarzschild case, the electric charge Q can reduce this value to 1 at most; while thepositive charge k + can reduce this value to 0.7675 at most. In the case of type (II) furry blackhole, for a fixed Q , the absorption ratio R f has a maximum when k − is the minimum and theabsorption ratio R f has a minimum when k − is the maximum. At intermediate frequencies,the numerical method is used to compute the absorption cross section. In the end, the fullspectrum description of absorption is obtained together with the different methods.This paper is organized as follows. In Sec.II, we derive the expression of the absorptioncross section by two types furry black holes. In Sec.III and Sec.IV, we calculate the absorptioncross section in high and low frequency limit. In Sec.V, we numerical calculate the absorptionratio influenced by different charges. In Sec.VI, the numerical method is used to obtain thefull spectrum of the absorption cross section. A series of results are presented together withthe absorption cross section in high and low frequency limit. Finally, Sec.VII is a conclusionabout our result. The action of dRGT theory is given by S = M pl (cid:90) d x √− g [ R + m U ( g, φ a )] , (2.1)where R is the Ricci scalar of physical metric g µν , φ a is the St ¨ u ckelberg field and U is apotential for the graviton. The potential is composed of three parts, U ( g, φ a ) = U + α U + α U , (2.2)where α and α are dimensionless parameters, and U = [ K ] − [ K ] , (2.3) U = [ K ] − K ][ K ] + 2[ K ] , (2.4) U = [ K ] − K ] [ K ] + 8[ K ][ K ] + 3[ K ] − K ] . (2.5)– 2 –ere the square brackets denote the trace, i.e. [ K ] = K µµ and K µν = δ µν − (cid:112) g µα ∂ α φ a ∂ ν φ b η ab ≡ δ µν − √ Σ µν , (2.6)where the matrix square root is √ Σ µα √ Σ αν = Σ µν . Variation the action with respect to themetric leads to the modified Einstein equations G µν + m T ( K ) µν = T ( m ) µν , (2.7)where T ( K ) µν = √− g δ ( √− gU ) δg µν . (2.8)In order to obtain the static spherically symmetric black hole, we showed that a self-consistent ans ¨ a tz should be written as [4] ds = − b ( r ) dt + a ( r ) dr + r d Ω , (2.9) φ = t + ˜ h ( r ) , (2.10) φ i = βx i , (2.11)where a ( r ) = b − ( r ) = f ( r ) and β = 1 or α + 1 . Actually, we obtain seven solutionsincluding the Schwarzschild solution, Reissner-Nordstr ¨ o m (RN) solution, and furry black holesolutions in Ref.[4]. Moreover, all of them can avoid the singularity in the invariant I ab fromthe divergence of g µν in the unitary gauge. Among the above solutions, the furry black holeexcite our interests. For furry black hole with electric charge Q , the function f ( r ) can bewritten as f ( r ) = 1 − Mr + 2 + λ − λ Q r − Sr λ , (2.12)where the cosmological constant term is neglected and λ > . As a classic theory, we expect λ is an integer. The term r − λ , which dubbed to St ¨ u ckelberg hair, is introduced by St ¨ u ckelbergfield φ a in dRGT model. To make the effect of r − λ most obvious in large scale r >> M , wechoose λ = 3 . It also has an general meaning for the integer of λ > .We define H ( r ) ≡ f ( r ) r as the horizon function, where the horizon r = h are thenull points of H ( r ) at r > . Based on rigorous mathematical analysis of function H ( r ) , weactually found two different horizons described by metric (2.12) in the case λ = 3 . When S isin the range [0 , + ∞ ) , there existed only one null point H ( r ) at r > . This case correspondsto black hole with only one horizon similar to Schwarzschild. When S ∈ ( − M g + , , where g ± ≡ Q M ± (4 + 15 Q M ) , there are two null points of H ( r ) at r > . This casecorresponds to black hole with two horizons similar to RN [15]. When S < − M g + , thereare no null points of H ( r ) at r > , which corresponds to a naked singularity.As we have known, for black holes in Kerr family, their horizon(s) are decided by conser-vation charge. For a set of conservation charge of black holes, it decides one and only one typeof horizon. If a set of conservation charge have more than one type horizon, then conservationcharge cannot decide all properties of black hole. It means that in order to distinguish whichone is practical, a nonphysical method is needed. We don’t think this is physical. Thus, forblack holes beyond Kerr family, we can also expected that a set of conservation charge decidesone and only one type of horizon.According to the horizon(s) of H ( r ) , we can define two different types of conservationcharge as follow: – 3 –. Positive Charge k + : The furry black hole has only one horizon h which is the zeropoint of function H ( r ) in the range r > . Thus, function H ( r ) can be factorized to H ( r ) = ( r − h )( r + ar + b ) . In order to recover (2.12), we can relate parameter S asthe set of conservation charge ( M, Q, k + ) S = ( M + (cid:113) M + 5 Q + k ) k ≡ hk , (2.13)where h is the horizon of furry black hole. Then the metric function f ( r ) can befactorized to f I ( r ) = ( r − h )( r + ( h − M ) r + k ) r . (2.14)This actually happens for S > and the range of k + is [0 , ∞ ) . We call this type (I)black hole.2. Negative Charge k − : The furry black hole have two horizons h and h similar to RNblack hole. And function H ( r ) can be factorized to H ( r ) = ( r − h )( r − h )( r + a ) .Similar to type (I) black hole, we can define parameter S as the set of conservationcharge ( M, Q, k − ) S = k − h h , (2.15)where h = 12 (2 M − k − − (cid:113) M + 20 Q + 4 M k − − k − ) , (2.16) h = 12 (2 M − k − + (cid:113) M + 20 Q + 4 M k − − k − ) (2.17)are two horizons of furry black hole. The metric function f ( r ) now can be factorized to f II ( r ) = ( r − h )( r − h )( r − k − ) r . (2.18)This actually happens for S ∈ ( − M g + , and we call this type (II) black hole. Inorder to satisfy S ∈ ( − M g + , , the range of k − is ( ( M − (cid:112) Q + 4 M ) , M − (cid:112) M + 5 Q ) . Notice that k − is always negative, thus we call k − negative charge.The strength of St ¨ u ckelberg hair is now described by k ± which can affect the size of horizon h . Actually, the configuration of Stückelberg field led to new feature of black hole in dRGTtheory. Thus, both Positive Charge k + and Negative Charge k − are known as St ¨ u ckelbergcharge.In the following part, we will calculate the absorption of neutral massless scalar fieldby two types black holes respectively. The Klein-Gordon equation for scalar fields in such acurved background can be written as ∇ µ ∇ µ Φ = 0 . (2.19)Assume that the incident wave is along the z -axis, thus the complex scalar field can beseparated as Φ ωl = ψ ωl ( r ) r P l (cos θ ) e − iωt , (2.20)– 4 –here P l (cos θ ) is a Legendre polynomial. Plugging the metric (2.9) into evolution equation(2.19), we obtain the radial equation f ψ (cid:48)(cid:48) ωl + f f (cid:48) ψ (cid:48) ωl + (cid:2) ω − f ( f (cid:48) r + l ( l + 1) r ) (cid:3) ψ ωl = 0 . (2.21)The tortoise coordinate r ∗ is defined as dr ∗ = drf ( r ) . In both types (2.14) and (2.18), we obtain r ∗ I = r + β arctan (cid:2) r + h − √ k − ( h − (cid:3)(cid:113) k − ( h − + β ln[ r − h ] + β ln[ r + ( h − r + k ] , (2.22) r ∗ II = r + h ln( r − h )( h − h )( h − k − ) + h ln( r − h )( h − h )( h − k − ) − k ln( r − k − )( h − k − )( h − k − ) , (2.23)where β = h ( h − + 2(2 + h − h ) k − k k + 2 h ( h − , (2.24) β = h k + 2 h ( h − , (2.25) β = − h ( h − − k k + 2 h ( h − , (2.26)and we have chosen M = 1 . By using the tortoise coordinate r ∗ , the radial equation (2.21)reduce to a schr ¨ o dinger-like form d dr ∗ ψ ωl ( r ) + (cid:2) ω − V ( r ) (cid:3) ψ ωl ( r ) = 0 , (2.27)where V ( r ) = f ( r ) (cid:0) f (cid:48) ( r ) r + l ( l + 1) r (cid:1) , (2.28)is the effective potential.Taking the asymptotic limits of Eq.(2.27), we obtain the solutions approaching theboundary ψ ωl ( r ) ≈ (cid:26) T ωl e − iωr ∗ , for r → h , e − iωr + R ωl e iωr , for r → ∞ , (2.29)where | R ωl | and | T ωl | are interpreted as the reflection and transmission coefficients respec-tively. The conservation of flux indicate a relationship between them | R ωl | + | T ωl | = 1 . (2.30)The absorption cross section is the ratio of the flux in Φ passing into the black hole tothe current in the incident wave. The total absorption cross section σ is a sum of partial crosssections σ l σ = ∞ (cid:88) l =0 σ l , (2.31)where σ l is defined by the transmission/reflection coefficients σ l = πω (2 l + 1)(1 − | R ωl | ) = π (2 l + 1) ω | T ωl | . (2.32)– 5 – igure 1 . The critical radius r c and critical impact parameter b c as the function of Q for type (I)black hole. As the frequency getting higher and higher, the particle properties become more and moreapparent. When the wavelength of the field becomes very small in comparison to the size ofhorizon, the wavefront propagates along geodesics of background. Thus, the absorption crosssection is going to approaching the geodesic capture cross section. Without loss of generality,we consider the plane motion for a free particle in θ = π . Through Killing vector K µ , we candefine two conserved quantity: energy E and angular momentum LE = f ( r ) dtdλ , (3.1) L = r dφdλ , (3.2)where λ is the affine parameter. Along the geodesic, the mass of particle is also a conservedquantity µ = g µν dx µ dλ dx ν dλ . (3.3)Using the expression for E and L , we obtain − E + (cid:0) drdλ (cid:1) + f ( r ) L r = 0 . (3.4)The effective kinetic energy along geodesic is described by T ( r ) ≡ (cid:0) drdλ (cid:1) .For a free massless particle incoming from infinity toward central black hole, its finialstate of motion have three different cases. Case (a) the particle will reach the perihelion r p ,and then be reflected to infinity. Case (c) the particle will fall into black hole. (Or saying,the particle is captured by black hole.) There is a critical case between them – Case (b)the particle will approach an unstable bound r c , but never reach it. The geodesic capturecross section is defined in this critical case. Case (b) must satisfy two conditions at the sametime: (i) ddr T ( r ) | r = r p = 0 and (ii) T ( r ) | r = r p = 0 . Condition (i) indicates that there is anperihelion r p along geodesic. While condition (ii) indicates that the free particle never reachthe perihelion r p . (If the particle has a positive kinetic energy T > when reaching theperihelion r p , it would keep moving and be reflect to infinity.) In this case, the critical bound r p ≡ r c is also known as photon sphere. – 6 – igure 2 . The critical radius r c and critical impact parameter b c as the function of k − for type (II)black hole. Two conditions (i) and (ii) decide two variables: one is the critical bound r c , the otheris the critical impact parameter b c . The impact parameter is defined by b ≡ LE . To make therelationship obviously, we introduce the function T ( r, b ) ≡ T ( r ) L = 1 b − f ( r ) r . (3.5)Since L is constant, the two conditions turn into (i) T ( r c , b c ) = 0 and (ii) ∂ r T ( r c , b c ) = 0 .For the solution ( r c , b c ) of condition (i) and (ii), all geodesics have no perihelion in the range r < r c ; meanwhile, all geodesics with parameter b < b c are captured by black hole. Thus, thegeodesic capture cross section is defined by σ hf = πb c . In fact, the critical impact parameteris also the radius of shadow. Thus, the high frequency limit is also the area of shadow froma far view. We seek the solutions of conditions (i) and (ii) by Wolfram Mathematica soft. In the actualcalculation, there may be multiple sets of solutions. A rigorous mathematical analysis showsthat there existed one and only one solutions ( r c , b c ) in the range r > h . Actually, we onlyneed the set in which both of element is largest. As a common sense, the mass of black holeis chosen to be 1. When all parameters are fixed, the equations of conditions (i) and (ii) areactually polynomial equations. Thus, we use the NSolve method to seek the solutions. Toget rid of unwanted solutions, we add the conditions r > h and reals when doing numericalcalculation.Both the electric charge Q and positive charge k + can influence the high frequency limit σ hf . For a fixed Q , the high frequency limit σ hf increases as the positive charge k + growth. Asimilar behavior would happen when the parameter k + is fixed and the parameter Q increases.Fig.1 shows that the critical radius r c and critical impact parameter b c as the function of Q for various values of k + . In the Schwarzschild limit ( Q = k + = 0) , the critical radius r c = 3 and critical impact parameter b c = 27 , which is exactly shown in Fig.1. In the case of type (II) black hole, the most different thing is the limited impact of k − onthe horizon h . The range of negative charge k − also depends on electric charge Q . We plotcritical radius r c and critical impact parameter b c as the function of k − in Fig.2, where thecritical radius r c = 3 and critical impact parameter b c = 27 when parameters approachingSchwarzschild limit ( Q = k − = 0) . – 7 – Low-Frequency Regime
In this section, we are interested in the solutions when wavelength is larger than the horizon ofblack hole ω << h . Following Ref.[17] and Ref.[14], we use a asymptotic expansion techniqueto solve the equation (2.27) in various regions. In order to obtain the analytical solution, allterms of higher than (1 /r ) are neglected. As a result, the St ¨ u ckelberg charge k ± appeared interm r − is also neglected. However, since k ± appeared in h , the final result still contain theeffect of k ± . The most important thing we must emphasize that, in the case of low frequency,the l = 0 mode dominant all other l terms [14]. Therefore, in the following calculation weonly concern l = 0 mode.We will consider the solution in three different regions: the region near the horizon(Region I), the intermediate region (Region II) and the region far form the black hole (RegionIII). At last, we match these solutions in different types of black hole.1. Region I: The Eq.(2.27) is easy to approximate (cid:0) d dr ∗ + ω (cid:1) ψ = 0 , (4.1)where r ∗ is the tortoise coordinate.2. Region II: The term of ω is much smaller than all other terms: (cid:0) ψr (cid:1) (cid:48)(cid:48) + ( 2 r + f (cid:48) f ) (cid:0) ψr (cid:1) (cid:48) = 0 . (4.2)3. Region III: All terms of order higher than (1 /r ) are neglected in Eq.(2.27) ( f ψ ) (cid:48)(cid:48) + (cid:0) ω + 4 M ωr (cid:1) f ψ = 0 . (4.3) For type (I) black hole, it would be more easier when we neglect r − , the tortoise coordinate r ∗ become r ∗ I,kmin = r + h h − M ) ln( r − h ) − ( h − M ) h − M ) ln( r + h − M ) . (4.4)The solution of three different regions are given by ψ = A tra e − iωr ∗ I,kmin , for Region I; (cid:18) ζ ln (cid:0) r − hr + h − M (cid:1) + τ (cid:19) r, for Region II; aF ( η, ωr ) + bG ( η, ωr ) , for Region III. (4.5)where F l ( η, ωr ) and G l ( η, ωr ) are the regular and irregular spherical wave functions respec-tively and η = − M ω . There are five constants A tra , ζ, τ, a, b we need to determine in thefollowing matching process. – 8 –et us consider the solution in the overlap of Region I and II. Near the horizon h , weonly consider the dominant term of solution in Region I ψ I = A tra ( r − h ) − iωα , (4.6)where α = h h − M ) . This term can also be expanded as the series of ω as ψ I ≈ A tra (1 − iωα ln( r − h )) , (4.7)where we neglect the terms higher than order ω . Meanwhile, taking the limit r → h ofsolution in Region II, we obtain ψ II ∼ h ( ζ ln( r − h ) − ζ ln(2 h − M ) + τ ) . (4.8)Matching these two solutions, we have ζ = − iωαh A tra , τ = (1 − iωβ ) h A tra , (4.9)where β = α ln(2 h − M ) .The overlap between Region II and Region III is defined in r >> h but >> ωr . For l = 0 , the Coulomb wave functions have following form F ( η, x ) = ρx, G ( η, x ) = 1 ρ , (4.10)where ρ = ηe η − . (4.11)Thus, the solution in the low frequency limit in Region III reduce to ψ III = aρωr + bρ . (4.12)In the asymptotic limit, the solution in Region II can expressed as ψ II ≈ τ r − ζ · h − M ) . (4.13)Thus, matching these solutions, we obtain a = 1 − iωβρω A tra , b = ih ωρA tra . (4.14)The non-normalized incidence coefficient A inc and reflection coefficient A ref are relatedto a and b by A inc = − a + ib i , A ref = a + ib i . (4.15)Therefore, the reflection coefficient is given by | R ω | = (cid:12)(cid:12)(cid:12)(cid:12) A ref A inc (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − h ω ρ − iωβ h ω ρ − iωβ (cid:12)(cid:12)(cid:12)(cid:12) . (4.16)Further considering the low frequency limit ω ≈ and expand all term as ω , we obtain thefirst term σ Ilf = A , (4.17)where A = 4 πh is the area of black hole. – 9 – igure 3 . The absorption ratio R f as the function of k + . Following a similar calculation process, we can obtain the total absorption cross section inthe low frequency limit for type (II) black hole σ IIlf = 4 πh . (4.18)It is easy to check our result. The conclusion is universal that the total absorption crosssection of black hole in low frequency limit is the area of event horizon. The total absorptioncross section of Schwarzschild in low frequency limit was firstly given by Unruh [17] σ Schlf = A Sch , (4.19)where A Sch = 4 πr s is the area of Schwarzschild black hole. R f = σ hf σ lf No matter the black hole with hair or without hair, the behavior of the frequency limit for theabsorption of massless field are very similar. However, an significant difference is still existedbetween them. In order to discuss this difference, we introduce the absorption ratio R f = σ hf σ lf . (5.1)Different types of conserved charges have different effects on the frequency limit of the ab-sorption cross section. We have shown as the Stückelberg charge k ± increasing in the case offixed Q , both high frequency limit σ hf and low frequency limit σ lf increase. While, the lowfrequency limit σ lf increases faster. Therefore, the absorption ratio R f is no longer the samefor black hole with different charges. We use a numerical method to calculate R f , which is plotted in Fig.3. The lower limit of R f is only determined by the positive charge k + . Whatever the value of electric charge Q is, thelower limit R fmin = 0 . is achieved when the positive charge k + approaching to infinity.The upper limit of R f is achieved with no positive charge k + . The maximum of upper limitis R f = in the Schwarzschild case. If the electric charge Q approaching to infinity, theminimum of upper limit is 1, which is exactly the lower limit R fmin = 1 decreased by electriccharge in the case of RN black hole. – 10 – igure 4 . The absorption ratio R f as the function of Q , where kM in = (1 − (cid:112) Q + 4) and kM ax = 1 − (cid:112) Q + 1 . Figure 5 . The partial and total absorption cross sections of type (I) black hole, where the parametersare chosen to be Q = 0 . , k = 0 . . In the case of absorption massless field by type (II) black hole, things would be very different.Since the negative charge has a range k − ∈ ( (1 − (cid:112) Q + 4) , − (cid:112) Q + 1) , the lower limitof R f is achieved when k − = kM ax = 1 − (cid:112) Q + 1 and the upper limit of R f is achievedwhen k − = kM in = (1 − (cid:112) Q + 4) . The negative charge k − influenced the absorptionratio R f depend on electric charge Q . In the case of no electric charge Q = 0 , the absorptionratio is in the range R f ∈ [ , . . The electric charge Q can also reduce the absorptionratio R f to 1 in the limit Q → ∞ and k − = kM ax . In this section, we use the numerical method to solve the radial Eq.(2.21). We expand thesolution (2.29) near the horizon and approaching infinity. Using Taylor series of solution nearthe horizon, we integrate the radial Eq.(2.21) numerically. The integration end at a large r . By matching the numerical solution and Taylor series of solution approaching infinity, weobtain the reflection and transmission coefficient. We present the numerical results for type (I) black hole in this subsection. We calculate partialcross sections σ l and sum them to obtain the total absorption cross section, see Fig.5. Fig.6shows the total absorption cross sections σ as the function of frequency ω . They fitted wellin the high frequency limit and Low frequency limit respectively. Fig.7 shows the absorption– 11 – igure 6 . Total absorption cross sections of type (I) black hole in various k + . The electric chargeis taken to be Q = 0 . . Figure 7 . Total absorption cross sections of type (I) black hole, where low frequency σ lf limit canexceed high frequency limit σ hf . The electric charge is also taken to be Q = 0 . . Figure 8 . The partial and total absorption cross sections of type (II) black hole, where theparameters are chosen to be Q = 0 . , k − = − . . ratio R f can be smaller than 1. These figures also demonstrate the total absorption crosssection σ increases as the positive charge k + growth. For type (II) black holes, there are similar numerical results presented in this subsection. InFig.8, we show the partial and total absorption cross sections for Q = 0 . and k − = − . .Fig.9 shows the total absorption cross sections σ for different values of k − . For fixed Q , asthe negative charge k − decreasing, the total absorption cross section σ decreases.– 12 – igure 9 . Total absorption cross sections of type (II) black hole in various k − . The electric chargeis taken to be Q = 0 . . In this paper, we obtain full spectrum description of absorption spectra of massless field bytwo types furry black holes. In the high frequency limit, the absorption cross section is thearea of shadow. In the low frequency limit, the absorption cross section is the area of horizon.As a general conclusion, the bigger the black hole is, the more it absorbs the field,which is also shown by the numerical results. The charge k ± , as a new parameter which canaffect the horizon, is introduced by the Stückelberg field φ a in dRGT theory. Thus, the newproperties appearing on the absorption cross section is also caused by the configuration ofStückelberg field φ a .Furthermore, we have shown different charges have a different influence on the absorptionratio R f . If a black hole only has mass M , the absorption ratio is a constant R f = . If italso has a electric charge Q , as the electric charge Q increasing, the absorption ratio decreases.In the extremely case, the electric charge Q can reduce the absorption ratio to R f = 1 . TheStückelberg hair affects the absorption ratio with similar behavior but totally different value.In the extremely case, the positive charge k + can reduce the absorption ratio to R f = 0 . .While, the negative charge k − can rise the absorption ratio to R f = 3 . at most. References [1] C. de Rham, G. Gabadadze and A. J. Tolley,
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