Near-horizon quasinormal modes of charged scalar around a general spherically symmetric black hole
aa r X i v : . [ h e p - t h ] D ec Near-horizon quasinormal modes of charged scalar around ageneral spherically symmetric black hole
Takol Tangphati, ∗ Supakchai Ponglertsakul, † and Piyabut Burikham ‡ High Energy Physics Theory Group, Department of Physics,Faculty of Science, Chulalongkorn University,Phyathai Rd., Bangkok 10330, Thailand (Dated: December 27, 2018)
Abstract
We study the quasinormal modes (QNMs) of charged scalar in the static spherically symmetricblack hole background near the event and cosmological horizon. Starting with numerical analysis ofthe QNMs of black hole in the dRGT massive gravity, the mathematical tool called the AsymptoticIteration Method (AIM) is used to calculate the quasinormal frequencies. The parameters such asthe mass and charge of the black hole, the cosmological constant, the coefficient of the linear termfrom massive gravity γ , and the mass of the scalar are varied to study the behavior of the QNMs.We found the tower pattern of the near-horizon quasinormal frequencies from the numerical resultsby AIM where the real parts depend only on product of the charge of the black hole and the scalarfield and the imaginary parts depend only on the surface gravity. To confirm the numerical finding,we analytically determine the exact QNMs of the charged scalar near the horizons of any staticspherically symmetric black hole background in the simple universal forms; ω = qQr h + iκ h n and ω = qQr c + i | κ c | n where n is a non-positive integer and κ h,c is the surface gravity, for the event andcosmological horizon respectively. Extending our analysis, we also compute the four towers of thenear-horizon QNMs that can reach the far region. The four kinds of QNMs converge to certainasymptotic values with equally spacing imaginary parts and the real parts proportional to qQ/r h,c .These modes do not match with the all-region (WKB) modes of the real background since theyare originated from the linearly approximated metric.Keywords: Quasinormal modes, dRGT black hole, Schwarzschild black hole, Reissner-Nordstr¨omblack hole ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Black hole has a unique property as its boundary is defined by the event horizon fromwithin which nothing even light can escape. Dynamics of the spacetime around the blackhole is governed by the Einstein field equations, the event horion can be perturbed and thefluctuations of the spacetime can propagate and carry energy away from the black hole in theform of gravitational waves. Oscillation modes of horizon usually have small amplitudes dueto the high scale of the Planck mass which results in extreme stiffness of spacetime. However,the frequencies of the oscillating horizon as well as any fields around it are determined by theblack hole physical parameters such as mass, charge and spin. In 2015, LIGO remarkablydetected the gravitational waves from the merging of two black holes [1]. The ringdownfrequency profile of the black hole after binary merging is characterized by the quasinormalmodes (QNMs) of the black hole.QNM of a black hole is analogous to the damped oscillation of an oscillator. The real partof QNM represents the energy or physical oscillating frequencies whilst the imaginary partgives the decaying time or relaxation time, it represents dissipation of the energy. QNMs ofa black hole are thus signature of each black hole that provides information of the intrinsicphysical parameters, i.e., mass, charge and spin according to the General Relativity (GR).A vast number of studies of black hole QNMs were explored in many contexts. We referinteresting readers to nice reviews on the subject in Ref. [2–4].GR is the theory describing gravitational interaction mediated by a massless spin-2 parti-cle called graviton (in its quantized extension which is still problematic). Many phenomenahad been predicted by GR and were experimentally confirmed e.g. gravitational time dila-tion [5, 6], precession of Mercury’s orbit [7] and gravitational waves [1]. GR works incrediblywell in many scenarios, however it fails to give a satisfying explanation to an acceleratedexpansion of the universe [8, 9] in the extragalactic scale and the asymptotically flat rota-tion curves of the galaxies [10]. This failure indicates that either there are overwhelminglyunknown form of dark matter and dark energy in the Universe or GR itself may need tobe modified at such scales. In contrast to GR, one can consider the theory where gravityis mediated by a massive graviton. This type of alternative theory of gravity is formallyknown as massive gravity.The theory of massive gravity was originated long before the discovery of accelerated2xpansion of the universe. By adding an appropriated mass term (of graviton) to the lin-earised Einstein’s general relativity, Fierz and Pauli firstly constructed a linear theory ofmassive gravity in 1939 [11]. However, the mass term in Fierz-Pauli (FP) massive gravityleads to van Dam-Veltman-Zakharov (vDVZ) discontinuity [12, 13] where the prediction oflinear theory of massive gravity in massless limit does not agree with those of GR. It wasrealized later that the vDVZ discontinuity was in fact caused by the extra degree of freedom[14]. Vainshtein suggested [14] that the vDVZ discontinuity can be eliminated by consider-ing the massive theory in the non-linear regime. However the generic non-linear theory ofmassive gravity usually suffers from the Boulware-Deser (BD) ghost [15]. Nevertheless, deRham, Gabadadze, and Tolley (dRGT) have successfully constructed the non-linear theoryof massive gravity without the BD ghost in 2010 [16]. Since then the dRGT has becomeuseful in cosmological study. Its cosmological solution can be used to explain an acceleratedexpansion of the universe in the sense that the existence of graviton mass naturally providesthe effect of cosmological constant [17, 18]. Moreover, the dRGT model can also explain avast varieties of the galactic rotation curves [19].On the other hand, the physics of compact object in dRGT massive gravity is also inter-esting. A static black hole solution in ( n + 2) dimensional massive gravity is found in [20].Static spherically symmetric black holes with and without electric charge in dRGT massivegravity are constructed and investigated in [21]. Similar to cosmological solution, the effec-tive cosmological constant appears naturally. Therefore one can consider these black holes asgeneralized version of Schwarzschild/Reissner-Nordstr¨om black holes with positive and neg-ative cosmological constant. In this work, we shall focus particularly on the QNMs of blackholes in dRGT massive gravity [2–4] as the generalized form of the black hole spacetime.In [22], numerical study of the QNMs of charged dRGT black hole reveals some intriguingbehaviour about the real parts and imaginary parts of the QNMs of charged scalar near theevent and cosmological horizon. The real parts can be explicitly shown to be proportionalto the product of the charge of scalar field and the black hole. While the imaginary partsexhibit some relation with the surface gravity at each horizon. Here in this article we tryto fill the gap by providing an analytical explanation to the observed behaviour for thosenear horizon modes found in [22] and make corrections to the numerical procedure andresults of Ref. [22]. This paper is organized as follow. In Sec. II, we setup the equationof motion for the charged scalar field in the curved black hole background and review the3symptotic iteration method (AIM) [23–26] which shall be employed to the numerical studyof the QNMs in dRGT massive gravity. Then we highlight some intriguing behaviour of ournumerical results in Sec. III. In Sec. IV, we provide analytical calculations of the QNMs fornear event horizon and near cosmic horizon modes for any static and spherically symmetricblack hole background. Section V devotes to discussions and conclusions. In this article,natural units c = 1 , ~ = 1 , and G = 1 are used throughout. II. THE SETUP AND NUMERICAL ANALYSIS
We consider perturbation of a charged scalar particle around the black hole spacetime.The dynamic of the field is described by the generalized Klein-Gordon equation with thepresence of the gauge field,( ∇ α − iqA α )( ∇ α − iqA α ) φ − m s Φ = 0 , (1)where Φ is the scalar field, ∇ α is the covariant derivative, q is the charge of the scalar field, A µ is the gauge field, and m s is the scalar field’s mass. The static spherically symmetricblack hole metric can be expressed as ds = − f ( r ) dt + dr f ( r ) + r d Ω . (2)First we will use the iterative method called the Asymptotic Iteration Method (AIM) inorder to find the numerical results of quasinormal frequencies of the field in the generic blackhole spacetime with mass, charge, linear term in radial coordinate r and the cosmologicalconstant term. A number of exotic matters and modified gravity models can give suchbackground, a well-known example is the dRGT massive gravity [19, 21]. The process ofAIM is split into two steps. The first one is to change the coordinates of the Klein-Gordonequation and to scale out the divergences at the boundary conditions. The second step isto apply the iterative algorithm of AIM to the equation of motion. We also modify AIMto eliminate the derivatives in each iteration which improves the computational time andprecision of the numerical results [22, 25]. 4 . Setting the form of differential equation By the separation method, the scalar field can be decomposed intoΦ( t, r, θ, ϕ ) = e − iωt φ ( r ) r Y ( θ, Φ ) , (3)where Y ( θ, ϕ ) is the spherical harmonic and the gauge field A µ = {− Q/r, , , } . The radialscalar field then obeys [22] d dr ∗ φ ( r ) + ( ω − qQ/r ) φ ( r ) − f ( r ) (cid:18) l ( l + 1) r + f ′ ( r ) r + m s (cid:19) φ ( r ) = 0 , (4)where dr ∗ = dr/f ( r ). The quasinormal frequencies satisfy the boundary conditions (asymp-totic behaviours), φ ∼ e − i ˜ ωr ∗ as r → r h e i ˆ ωr ∗ as r → r c , (5)where ˜ ω = ω − qQr h , ˆ ω = ω − qQr c , r h is the event horizon and r c is the cosmological horizon. Ifthere is no cosmological constant, r c → ∞ . Performing the radial coordinate transformation r = 1 /ξ [22, 25–28], we obtain φ ′′ ( ξ ) + p ′ ( ξ ) p ( ξ ) φ ′ ( ξ ) + ( ω − qQξ ) p ( ξ ) φ ( ξ ) − p ( ξ ) (cid:18) l ( l + 1) − ξ df (1 /ξ ) dξ + m s ξ (cid:19) φ ( ξ ) = 0 , (6)where p ( ξ ) ≡ ξ f (1 /ξ ) . (7)Assume there are n singularities ( ξ , ξ , ...ξ n ) corresponding to f (1 /ξ i ) = 0. We have toscale out the divergences of the singularities from the radial part of the scalar field. Firstwe apply the boundary condition near the cosmological horizon from Eq. (5) [24, 26, 28] φ ( ξ ) = e i ˆ ωr ∗ u ( ξ ) , (8)where e i ˆ ωr ∗ ≡ n Y i =1 ( ξ − ξ i ) i ˆ ω κi , (9)and κ i is the surface gravity at ξ i . The radial part of the scalar field equation Eq. (6) reads u ′′ + ( p ′ − i ˆ ω ) p u ′ + (( ω − qQξ ) − ˆ ω ) p u − p " l ( l + 1) − ξ df (1 /ξ ) dξ + m s ξ u = 0 . (10)5ext we apply the boundary condition near the event horizon [24, 26, 28] to eliminate thecorresponding divergence, u ( ξ ) = ( ξ − ξ h ) − i ˜ ω κh ( ξ − ξ h ) − i ˆ ω κh χ ( ξ ) . (11)Note the difference in the scaling factors in (8) and (11) from Ref. [22], the correct ˆ ω (˜ ω ) mustbe used for elimination of the divergence at the cosmological (event) horizon respectively.Remarkably, the all-region (WKB) modes are unchanged and consistent with the valuescalculated from other methods such as the WKB and analytic formula regardless of thescaling factors at the horizons. The all-region QNMs are insensitive to the divergencesaround the horizon for the analysis performed by AIM.Finally, Eq. (10) can be written as [22, 25, 26], χ ′′ ( ξ ) = λ ( ξ ) χ ′ ( ξ ) + s ( ξ ) χ ( ξ ) , (12)where λ and s are the coefficients of the equation. With this form, we are ready to usethe algorithm of AIM in the next step. B. The algorithm of AIM
From the homogeneous linear second-order differential equation in Eq. (12), AIM usesthe asymptotic aspect by taking higher-order differentiation in order to find the a generalsolution and its eigenvalues. First, taking differentiation of Eq. (12), we find that [25, 29] χ ′′′ ( ξ ) = λ ( ξ ) χ ′ ( ξ ) + s ( ξ ) χ ( ξ ) , (13)where λ ( ξ ) = λ ′ ( ξ ) + s ( ξ ) + λ ( ξ ) (14)and s ( ξ ) = s ′ ( ξ ) + s ( ξ ) λ ( ξ ) . (15)Taking the n -time derivation of Eq. (12) to obtain χ ( n +2) ( ξ ) = λ n ( ξ ) χ ′ ( ξ ) + s n ( ξ ) χ ( ξ ) , (16)where λ n ( ξ ) = λ ′ n − ( ξ ) + s n − ( ξ ) + λ ( ξ ) λ n − ( ξ ) (17)6nd s n ( ξ ) = s ′ n − ( ξ ) + s ( ξ ) λ n − ( ξ ) . (18)With large enough n , the asymptotic of AIM claims that [22, 25, 29] s n ( ξ ) λ n ( ξ ) = s n − ( ξ ) λ n − ( ξ ) ≡ β ( ξ ) . (19)Finally, we get the quantization condition for this method to solve the quasinormal modes, s n ( ξ ) λ n − ( ξ ) = s n − ( ξ ) λ n ( ξ ) . (20)One can use this iterative algorithm to find the eigenvalues ω of Eq. (4). On the other hand,there is a giant drawback in this algorithm because, in each step, we have to differentiate s and λ from the previous step. This causes not just only a lot of time but the divergence ofthe numerical results. We improve AIM by using Taylor series expansion of s and λ aroundthe point ¯ ξ as follows [22, 25], λ n ( ξ ) = ∞ X i =0 c in ( ξ − ¯ ξ ) i , (21) s n ( ξ ) = ∞ X i =0 d in ( ξ − ¯ ξ ) i , (22)where c in and d in are the i th Taylor coefficient’s of λ n ( ξ ) and s n ( ξ ) respectively. Substitutingthese Taylor expansions, Eq. (21) and Eq. (22), into Eq. (17) and Eq. (18) leads to c in = ( i + 1) c i +1 n − + d in − + i X k =0 c k c i − kn − , (23) d in = ( i + 1) d i +1 n − + i X k =0 d k c i − kn − . (24)The quantization condition can be written in terms of these coefficients, d n c n − − d n − c n = 0 . (25)With this improvement of AIM, there is no derivative operator in the iteration process. Wecan apply the recursive method to find the quasinormal modes.7 II. NUMERICAL RESULTS
In this section, we numerically calculate the quasinormal frequencies of the charged scalarin the generalized black hole background by AIM. Generically, there are three kinds ofQNMs that can be found by AIM; near event horizon, near cosmological horizon and all-region (WKB) modes [22]. We consider the near horizon QNMs in 4 cases; the QNMs ofthe near extremal Schwarzschild de-Sitter dRGT black hole in subsection III A, the QNMsof the near extremal Reissner-Nordstr¨om de-Sitter dRGT black hole in subsection III B, theQNMs of the non-extremal Schwarzschild de-Sitter dRGT black hole in subsection III C, andthe QNMs of the non-extremal Reissner-Nordstr¨om de-Sitter dRGT black hole in subsectionIII D.The results of QNMs in this section are calculated at two ¯ ξ expansion points. The firstpoint is near the event horizon, ¯ ξ = 0 . ξ h = 0 . /r h . The second point is near thecosmological horizon, ¯ ξ = 1 . ξ c = 1 . /r c . A. The Near-Extremal Schwarzschild de-Sitter dRGT cases
In this case, the equation of motion of the scalar field is (Eq. (4) with zero charge) [22] d dr ∗ φ ( r ) + ω φ ( r ) − f ( r ) (cid:18) l ( l + 1) r + f ′ ( r ) r + m s (cid:19) φ ( r ) = 0 , (26)which f ( r ) = 1 − Mr − Λ r γr + ζ , (27)where M is the mass of the black hole, Λ is the cosmological constant, γ and ζ are thecoefficients from the dRGT effects. The condition for the near extremal black hole is r c − r h r h ≪ . (28)To consider QNMs of the near-extremal black hole, we set parameters, corresponding to thecondition Eq. (28), as M = 1 , Λ = 5 . , ζ = 2 . , l = 0 , γ = 0 . , and m s = 0 . r h = 0 . r c = 0 . ω = iκ h . 8 QNMs( ω n ) by AIM ∆ ω n by AIM0 1.41 × − i -1 -0.01216 i -0.01216 i2 -0.02433 i -0.01216 iTABLE I. The quasinormal modes of uncharged near-extremal dRGT black hole with 100 iterationsof AIM for the parameters, M = 1 , Λ = 5 . , ζ = 2 . , l = 0 , γ = 0 . , and m s = 0 .
00 at ¯ ξ . κ h = 0 . ∼ − κ c . B. The Near-Extremal Reissner-Nordstr¨om de-Sitter dRGT cases
The equation of motion for the zero-charge scalar field in the charged black hole spacetimeis Eq. (26) with the metric [22] f ( r ) = 1 − Mr + Q r − Λ r γr + ζ , (29)where Q is the black hole charge.We set the spacetime parameters to obtain two near-extremal cases; the event horizon isvery close to the cosmological horizon (small universe), and the event horizon is very close tothe Cauchy horizon. The resulting QNMs are shown in Table II and Table III respectively. n QNMs( ω n ) by AIM Im(∆ ω n ) by AIM0 4.27 × − i -1 -0.0006779 i -0.0006779 i2 -0.001356 i -0.0006779 iTABLE II. The quasinormal modes of charged near-extremal dRGT black hole with 100 itera-tions of AIM for the parameters, M = 1 , Q = 0 . , Λ = 0 . , q = 0 . , ζ = 0 . , l = 0 , γ = − . , and m s = 0 .
01 at the point ¯ ξ , κ h = 0 . Table II shows the near-horizon QNMs by AIM for the 3 lowest frequencies. The param-eters are chosen such that the extremal condition Eq. (28) is satisfied and the point we solveusing AIM is near the event horizon ¯ ξ = 0 . ξ h . The numerical results of the QNMs arestill a tower pattern where the gap in the imaginary parts is equal to the surface gravity atthe horizon. 9 QNMs( ω n ) by AIM Im(∆ ω n ) by AIM0 3.022 × − i -1 -0.009264 i -0.009264 i2 -0.01852 i -0.009264 iTABLE III. The quasinormal modes of charged near-extremal dRGT black hole with 200 itera-tions of AIM for the parameters, M = 1 , Q = 0 . , Λ = 0 . , q = 0 . , ζ = 0 . , l = 2 , γ =0 . , and m s = 0 .
01 at the point ¯ ξ , κ = 0 . We consider another case of the near extremal black hole; when the event horizon isclose to the Cauchy horizon or r − ∼ r h . The quasinormal frequencies generated by AIMare presented in Table III. Again, the gaps between frequencies of the near-horizon QNMsare precisely equal to the value of the surface gravity at the horizon. This is similar tothe behaviour of the all-region WKB modes of the near-extremal black hole (see also e.g.Ref. [27] for the black string version with almost identical formula and derivation to theblack hole case). C. The Non-Extremal Schwarzschild de-Sitter dRGT cases
In this section, we numerically calculate the near-horizon QNMs of the scalar in thenon-extremal Schwarzschild de-Sitter background for both event and cosmological horizonmodes. Again, we choose to observe at the points ¯ ξ = 0 . ξ h = 0 . /r h (near the eventhorizon) and ¯ ξ = 1 . ξ c = 1 . /r c (near the cosmological horizon).The dynamic of the scalar field follows the Eq. (26) with the same function f ( r ) inEq. (27). However, the set of parameter does not satisfy the condition in Eq. (28). To inves-tigate the pattern of the tower in QNMs near the horizons, we vary all physical parametersin Eq. (26).Variation of cosmological constant Λ has effect on the gaps between the QNMs as shownin Table IV. The patterns of tower remain at the points near both the event and cosmologicalhorizons.Next, the numerical results from the variation of the coefficient in the linear term γ atpoint ¯ ξ and ¯ ξ are listed in Table V. In this case, we set the mass of the scalar field andthe angular momentum to be nonzero ( m s = 0 . , l = 2). The numerical values of QNMs in10 n ω (at ¯ ξ ) ω (at ¯ ξ ) Im(∆ ω n (at ¯ ξ )) Im(∆ ω n (at ¯ ξ )) κ h κ c × − i 6.73 × − i - -1 -1.265 i -0.4002 i -1.265 i -0.4002 i 1.265 -0.40022 -2.529 i -0.8004 i -1.265 i -0.4002 i3 -3.794 i -1.2006 i -1.265 i -0.4002 i0.005 0 3.74 × − i -2.13 × − i - -1 -1.262 i -0.4021 i -1.262 i -0.4021 i 1.262 -0.40212 -2.524 i -0.8041 i -1.262 i -0.4021 i3 -3.786 i -1.2062 i -1.262 i -0.4021 i0.05 0 -5.56 × − i -6.20 × − i - -1 -1.235 i -0.4191 i -1.235 i -0.4191 i 1.235 -0.41912 -2.469 i -0.8382 i -1.235 i -0.4191 i3 -3.704 i -1.2573 i -1.235 i -0.4191 i0.5 0 -6.76 × − i -3.41 × − i - -1 -0.9343 i -0.4826 i -0.9343 i -0.4826 i 0.9343 -0.48262 -1.8687 i -0.9653 i -0.9343 i -0.4826 i3 -2.8030 i -1.4479 i -0.9343 i -0.4826 iTABLE IV. The quasinormal modes of non-extremal uncharged dRGT black hole with 100 itera-tions of AIM for the parameters, M = 1 , γ = 0 . , ζ = 0 , l = 0 , and m s = 0 .
00 around the point ¯ ξ and ¯ ξ . these tables have vanishing real parts. The imaginary parts have constant gaps which canbe well approximated by the values of the corresponding surface gravity.The scalar mass has no effect on the function f ( r ), thus the surface gravity near the eventand cosmological horizons κ h , κ c do not change under the the scalar mass variation. Fromthe results in the Table VI, there is again a tower pattern in the QNMs near the event andcosmological horizon. 11 n ω (at ¯ ξ ) ω (at ¯ ξ ) Im(∆ ω n (at ¯ ξ )) Im(∆ ω n (at ¯ ξ )) κ h κ c -0.1 0 -7.88 × − i -1.47 × − i - -1 -0.07847 i -0.03173 i -0.07847 i -0.03173 i 0.07847 -0.031732 -0.1569 i -0.06345 i -0.07847 i -0.03173 i3 -0.2354 i -0.09518 i -0.07847 i -0.03173 i0.0 0 -5.90 × − i -1.20 × − i - -1 -0.2487 i -0.01757 i -0.2487 i -0.01757 i 0.2487 -0.017572 -0.4973 i -0.03514 i -0.2487 i -0.01757 i3 -0.7460 i -0.05270 i -0.2487 i -0.01757 i0.1 0 -3.61 × − i 4.61 × − i - -1 -0.3916 i -0.0532 i -0.3916 i -0.0532 i 0.3916 -0.05322 -0.7832 i -0.1064 i -0.3916 i -0.0532 i3 -1.1749 i -0.1596 i -0.3916 i -0.0532 iTABLE V. The quasinormal modes of uncharged non-extremal dRGT black hole with 100 iterationsof AIM for the parameters, M = 1 , Λ = 0 . , ζ = 0 , l = 2 , and m s = 0 . ξ and¯ ξ . D. The Non-Extremal Reissner-Nordstr ¨o m de-Sitter dRGT cases The equation of motion for the charged scalar field in the non-extremal Reissner-Nordstr¨om de-Sitter dRGT black hole satisfies Eq. (4) with f ( r ) given by Eq. (29).The structures of tower have been found in the near-horizon QNMs for this case inRef. [22] (with errors in the real parts by a factor of 1 / qQ/r h and qQ/r c for the event and cosomological horizon respectively. Wevary parameters such as cosmological constant Λ, the linear term γ , the charge of blackhole Q , and the mass of scalar field m s . The results are presented in Table VII-IX. Ineach case, we report the numerical results of the first four quasinormal frequencies, thedifference between overtones in the imaginary parts, and the values of the surface gravities.We choose ¯ ξ = 0 . ξ h , and ¯ ξ = 1 . ξ c to be points on spacetime for our observation ofthe quasinormal frequencies. 12 s n ω (at ¯ ξ ) ω (at ¯ ξ ) Im(∆ ω n (at ¯ ξ )) Im(∆ ω n (at ¯ ξ )) κ h κ c × − i -6.32 × − i - -1 -0.5238 i -0.1048 i -0.5238 i -0.1048 i 0.5238 -0.10482 -1.0476 i -0.2096 i -0.5238 i -0.1048 i3 -1.5713 i -0.3144 i -0.5238 i -0.1048 i0.01 0 2.44 × − i -6.32 × − i - -1 -0.5238 i -0.1048 i -0.5238 i -0.1048 i 0.5238 -0.10482 -1.0476 i -0.2096 i -0.5238 i -0.1048 i3 -1.5713 i -0.3144 i -0.5238 i -0.1048 i0.1 0 2.44 × − i -6.32 × − i - -1 -0.5238 i -0.1048 i -0.5238 i -0.1048 i 0.5238 -0.10482 -1.0476 i -0.2096 i -0.5238 i -0.1048 i3 -1.5713 i -0.3144 i -0.5238 i -0.1048 iTABLE VI. The quasinormal modes of uncharged non-extremal dRGT black hole with 100 itera-tions of AIM for the parameters, M = 1 , Λ = 0 . , ζ = 0 , l = 1 , and γ = 0 . ξ and ¯ ξ . In Table VII, we present the QNMs near the event and cosmological horizons by thevariation of the cosmological constant from 0 .
001 to 0 .
1. The magnitudes of the real partequal qQ/r i where r i is either the event horizon or the cosmological horizon. Moreover, thegaps between overtones of QNMs converge to the surface gravities.In Table VII, we consider the quasinormal frequencies at the points ¯ ξ = 0 . ξ h and¯ ξ = 1 . ξ c and vary the coefficient of the linear term from the effect of dRGT. Thestructures of tower appear in the numerical result of QNMs where the real parts shift withthe value qQ/r i and the gaps between overtones are of the surface gravities.The sign of the real parts of the quasinormal frequencies depends on the sign of the blackhole charge. As shown in Table IX, the real parts of the QNMs are indeed given by qQ/r i .Furthermore, the gaps of the imaginary parts are again equal to the surface gravities. Thestructure of tower is still found in the numerical results.With variations of the 4 parameters for general cases displayed in Table VII-IX, we donot only get the structures of tower for QNMs but also learn that the gaps in imaginary13 n ω (at ¯ ξ ) ω (at ¯ ξ ) qQ/r h qQ/r c κ h κ c × − i 5.90 × − - 3.0 × − i1 0.0541 - 0.3216 i 5.90 × − - 0.0308 i 0.0541 5.90 × − × − - 0.0617 i3 0.0541 - 0.9648 i 5.90 × − - 0.0926 i0.01 0 0.05365 + 5.6 × − i 0.00385 + 1.2 × − i1 0.05365 - 0.3109 i 0.00385 - 0.0593 i 0.05365 0.00385 0.3109 -0.05932 0.05365 - 0.6219 i 0.00385 - 0.1186 i3 0.05365 - 0.9328 i 0.00385 - 0.1779 i0.1 0 0.04763 + 2.6 × − i 0.01948 + 1.1 × − i1 0.04763 - 0.1861 i 0.01948 - 0.1058 i 0.04763 0.01948 0.1861 -0.10582 0.04763 - 0.3722 i 0.01948 - 0.2115 i3 0.04763 - 0.5582 i 0.01948 - 0.3173 iTABLE VII. The quasinormal modes of charged non-extremal dRGT black hole with 100 iterationsof AIM for the parameters, M = 1 , Q = 0 . , q = 0 . , ζ = 0 , l = 0 , m s = 0 . γ = 0 .
05 aroundthe point ¯ ξ and ¯ ξ . parts of the overtones are equal the corresponding suface gravities. This is not surprisingsince perturbations of fields near the horizons should be governed entirely by the physicalparameters of the horizons and fields ( q but not m s , l ). IV. THE EXACT SOLUTIONS OF NEAR-HORIZON QUASINORMAL FRE-QUENCIES
According to the notable numerical results of the QNMs of the black holes near theevent and cosmological horizons in the previous section, it is expected that the quasinormalfrequencies are generically forming the tower patterns in the vicinity of the horizons. Inthis section, we analytically calculate the exact solutions of quasinormal frequencies nearthe event and cosmological horizons in general static spherically symmetric spacetime. Thisis an extension of the analysis in Ref. [30] to include the effect of gauge field on the chargedscalar. The results are exact and universal, the Coulomb potential shifts the real parts of14 n ω (at ¯ ξ ) ω (at ¯ ξ ) qQ/r h qQ/r c κ h κ c -0.05 0 0.04302 + 6.8 × − i 0.01026 - 7.4 × − i1 0.04302 - 0.1554 i 0.01026 - 0.04623 i 0.04302 0.01026 0.1554 -0.046432 0.04302 - 0.3108 i 0.01026 - 0.09290 i3 0.04302 - 0.4661 i 0.01026 - 0.1393 i0.00 0 0.04894 + 8.5 × − i 0.006104 - 1.8 × − i1 0.04894 - 0.2364 i 0.006104 - 0.05020 i 0.04894 0.006104 0.2364 -0.050252 0.04894 - 0.4730 i 0.006104 - 0.1005 i3 0.04894 - 0.7095 i 0.006104 - 0.1508 i0.05 0 0.05365 - 5.6 × − i 0.003845 + 1.3 × − i1 0.05365 - 0.3109 i 0.003845 - 0.05930 i 0.05365 0.003845 0.3109 -0.059312 0.05365 - 0.6219 i 0.003845 - 0.1186 i3 0.05365 - 0.9328 i 0.003845 - 0.1779 iTABLE VIII. The quasinormal modes of charged non-extremal dRGT black hole with 100 iterationsof AIM for the parameters, M = 1 , Q = 0 . , Λ = 0 . , q = 0 . , ζ = 0 , l = 2 and m s = 1 .
00 aroundthe point ¯ ξ and ¯ ξ . the QNMs by qQ/r a for r a = ( r h , r c ) of the event and cosmological horizon respectively. A. Exact solution of the QNMs near the event and cosmological horizons in thestatic spherically symmetric spacetime
The evolution of charged scalar field on the spherically symmetric spacetime can bedescribed by Eq. (4). It is convenient to redefine the radial function such that φ ( r ) = rR ( r ).Then Eq. (4) takes the form d R ( r ) dr + (cid:18) f ′ ( r ) f ( r ) + 2 r (cid:19) dR ( r ) dr + (cid:20) f ( r ) (cid:18) ω − qQr (cid:19) − l ( l + 1) f ( r ) r − m s f ( r ) (cid:21) R ( r ) = 0 , (30)where in this case f ( r ) is a generic function of r with at least two singular points i.e. f ( r h ) = f ( r c ) = 0. We now define a new coordinate z = 1 − r a r , (31)15 n ω (at ¯ ξ ) ω (at ¯ ξ ) qQ/r h qQ/r c κ h κ c -0.2 0 -0.05462 - 1.6 × − i -0.00388 + 1.1 × − i1 -0.05462 - 0.3107 i -0.00388 - 0.0593 i -0.05462 -0.00388 0.3107 -0.05932 -0.05462 - 0.6213 i -0.00388 - 0.1186 i3 -0.05462 - 0.9320 i -0.00388 - 0.1779 i0.5 0 0.1451 + 2.4 × − i 0.009708 + 1.4 × − i1 0.1451 - 0.3071 i 0.009708 - 0.0593 i 0.1451 0.009708 0.3071 -0.05932 0.1451 - 0.6142 i 0.009708 - 0.1187 i3 0.1451 - 0.9212 i 0.009708 - 0.1780 iTABLE IX. The quasinormal modes of uncharged non-extremal dRGT black hole with 100 itera-tions of AIM for the parameters, M = 1 , Λ = 0 . , γ = 0 . , q = 0 . , ζ = 0 . , l = 1 and m s = 0 . ξ and ¯ ξ . where a = { h, c } associates with each horizon. Therefore in this coordinate system, thehorizons are located at z = 0. Near each horizon, the metric function can be approximatedas f ≈ f ′ z where f ′ is dfdz (cid:12)(cid:12)(cid:12) z =0 . Hence the scalar field equation (30) can be rewritten in thenear horizon limit as d R ( z ) dz + 1 z dR ( z ) dz + r a (1 − z ) (cid:16) ω − qQ (1 − z ) r a (cid:17) f ′ z − m s f ′ z − l ( l + 1) (1 − z ) f ′ zr a R ( z ) = 0 . (32)We can simplify this equation by introducing the following radial field function R ( z ) = z α (1 − z ) β F ( z ) . (33)By substituting this into Eq. (32), we find that F ( z ) satisfies the standard hypergeometricdifferential equation z (1 − z ) d F ( z ) dz + [ c − ( a + b + 1) z ] dF ( z ) dz − abF ( z ) = 0 , (34)where we have taken z to be small and ignored the higher order power of z . The powerfactor α and β are constrained to be α = ± ( ωr a − qQ ) i r a κ a , (35)16 = 12 ± κ a r a rh r a (cid:16) κ a (2 l ( l + 1) + r a ( κ a + 6 µ r a )) − r a ω (cid:17) − q Q + 6 qQr a ω i , (36)where the surface gravity is defined as κ a = f ′ / r a . The hypergeometric parameters can bedefined as the following a = α + β − δ, (37) b = α + β + δ, (38) c = 1 + 2 α, (39)where δ = p r a (2 qQω + 4 κ a m s r a − r a ω )2 κ a r a . (40)Thus for z →
0, the general solution of Eq.(32) has the form R ( z ) = A z α (1 − z ) β F ( a, b, c, z ) + A z − c + α (1 − z ) β F (1 + a − c, b − c, − c, z ) , (41)where F ( a, b, c, z ) is the hypergeometric function and A and A are arbitrary constant fornon-integer c .However if c is an integer n , this gives an exact formula for the discrete frequencies as ω = qQr a ± iκ a ( n − , (42)the plus sign corresponds to choosing negative value of α . The solutions have different formsfor positive and negative integer cases. If c is a positive integer, the general solution around z = 0 becomes R ( z ) = z α (1 − z ) β [ A F ( a, b, c, z ) + A H ( a, b, c, z )] . (43)For c is zero and negative integer, the general solution is given by R ( z ) = z − α (1 − z ) β [ A F (1 + a − c, b − c, − c, z ) + A H (1 + a − c, b − c, − c, z )] . (44)The second linearly independent solution H is defined such that H ( a, b, c, z ) = F ( a, b, c, z ) ln z − c − X k =1 ( c − k − c − k − − a ) k (1 − b ) k ( − z ) − k (45)17 ∞ X k =0 ( a ) k ( b ) k ( c ) k k ! z k [ ψ ( a + k ) + ψ ( b + k ) − ψ (1 + k ) − ψ ( c + k )] , (46)where ψ ( z ) is the digamma function. In this case, the QNMs for minus sign choice (plussign for cosmic horizon) of α are given by ω = qQr a + i | κ a | ( n − , n = 0 , − , − , ... (47)This is exactly the modes found numerically by AIM. The mode with zero imaginary partis also obtained when c = 1 in the positive integer case. B. Exact solution of the QNMs in all-region modes of the linearly approximatedmetric
In general, the quasinormal frequencies can be categorized into three kinds which are thenear event horizon, the near cosmological horizon, and the all-region modes [22]. In thissection, we derive analytic formulae in terms of the roots of quartic equation (though theywill not be presented due to the lengthy expressions) for the quasinormal frequencies of theall-region modes in the near-horizon coordinates. These are the modes that are derived inthe near-horizon approximation and yet can reach the asymptotically far region. Since themetric is linearly approximated, the modes naturally are not equal to the all-region QNMsfound by the WKB method or AIM in the usual coordinates without approximation.As mentioned earlier in section IV A, Eq. (34) can be regarded as a hypergeometricdifferential equation in the z → R ( z ) = A z α (1 − z ) β F ( a, b, c, z ) + A z − c + α (1 − z ) β F (1 + a − c, b − c, − c, z ) , (48)where A and A are arbitrary constants. a , b , c are defined as in Eq. (37–39), and we choose α = i ( qQ − r a ω )2 κ a r a β = 12 + p κ a r a (3 r a m s + l + l ) + 6 ωr a ( qQ − ωr a ) + κ a r a − q Q κ a r a . (49)Thus near the event horizon, we must choose A = 0. Then the solution satisfying theboundary condition at the horizon becomes R ( z ) = A z α (1 − z ) β F ( a, b, c, z ) . (50)18n order to satisfy the outgoing-wave boundary condition at large r i.e. z →
1, we make useof the following transformation [31], F ( a, b, c, z ) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) F ( a, b, a + b − c + 1 , − z )+(1 − z ) c − a − b Γ( c )Γ( a + b − c )Γ( a )Γ( b ) F ( c − a, c − b, c − a − b + 1 , − z ) . (51)At large r , only outgoing wave is allowed. Thus we must have the second term on the RHSvanishes a = − n, or b = − n, n = 0 , , , ... (52)Eq. (52) is a quartic equation of ω and there are four roots of the QNMs. These four rootsform four towers of the QNMs as we will numerically demonstrate subsequently.Set the value of parameters M = 1 , Q = 0 . , Λ = 0 . , q = 0 . , ζ = 0 , l = 5 , γ =0 . , and m s = 0 .
01 and display the frequencies of the QNMs in Figure 1 and 2. The fourtowers of the QNMs are found (the separated green dot closed to the real axis is the n = 0mode lying at ω = 0 . . i of the green tower).In Figure 1, we show the QNMs ( n = 0 , , , , ..., and 50000) which are sep-arated into 4 towers. The first tower is represented in the blue dots and it is the unstablemodes since the imaginary parts are positive. The remaining 3 towers have negative imag-inary parts, they are the decaying quasinormal frequencies represented in the green, blackand red towers.In Figure 2 and Table X, we show that, as the mode number increases, the real parts ofeach tower of modes converge to some constant values. Moreover, the gaps in the imaginaryparts between modes are also converging to constant values as presented in Table X. Forquasinormal modes (represented in green, black, and red towers), the magnitude of the gapin the imaginary parts increases as the real part of QNMs increases.Finally, we compare these 4 towers of QNMs with the values obtained by AIM and WKBin the usual coordinates, e.g. as in Ref. [22], and found that they are not equal. They arethe near-horizon modes that can reach the far region of the linearly approximated metricbackground. 19 IG. 1. The quasinormal modes ( n = 0 , , , , ..., and 50000) of charged non-extremaldRGT black hole for the parameters, M = 1 , Q = 0 . , Λ = 0 . , q = 0 . , ζ = 0 , l = 5 , γ =0 . , and m s = 0 .
01 for the all-region QNMs.Modes n → ∞ Real parts The gap in imaginary partsThe first tower (unstable) 0.01449 0.1862The second tower 0.03108 -0.1143The third tower 0.05512 -0.3450The fourth tower 0.07172 -2.097TABLE X. The near-horizon quasinormal modes that can reach far region for M = 1 , Q = 0 . , Λ =0 . , q = 0 . , ζ = 0 , l = 5 , γ = 0 . , and m s = 0 .
01 in the large n limit. C. Exact solution of the QNMs near the event and cosmological horizons in theRindler coordinate
From the metric in Eq. (2), we use the approximation f ( r ) = f ( r a ) + f ′ ( r a )( r − r a ) + O ( r − r a ) to rewrite the metric as ds = − κ a ( r − r a ) dt + 12 κ a ( r − r a ) dr + r a d Ω . (53)20 IG. 2. The first 100 quasinormal modes (zoom in) of charged non-extremal dRGT black hole forthe parameters, M = 1 , Q = 0 . , Λ = 0 . , q = 0 . , ζ = 0 , l = 5 , γ = 0 . , and m s = 0 .
01 for theall-region QNMs.
With the definition of the Rindler coordinate dx = dr κ a ( r − r a ) , the metric in the Rindlerspacetime is ds = − κ a x dt + dx + r a d Ω . (54)The Klein-Gordon equation in the Rindler coordinate takes the form x ddx (cid:18) x dR ( x ) dx (cid:19) + (cid:18) κ a (cid:18) ω − qQr a (cid:19) − (cid:18) l ( l + 1) r a + m s (cid:19) x (cid:19) R ( x ) = 0 . (55)The general form of solutions can be written as R ( x ) = C J ν − ix p m r a + l ( l + 1) r a ! + C Y ν − ix p m r a + l ( l + 1) r a ! , (56)where ν = − i ( r a ω − qQ ) κ a r a (57)21nd J ν , Y ν are the Bessel function of the first and second kind respectively. The QNMs forthe case when ν is nonzero integer and half-integer (half odd-integer) are ω = qQr a + i | κ a | n − , for n = 0 , − , − , ... (58)The QNMs with negative imaginary parts are the tower of decaying modes. Comparing toEq. (47), the modes with imaginary parts equal to half-integer of the surface gravity, e.g. iκ a / , iκ a / , ... emerge in the Rindler coordinate. Numerical results by the modified AIMconfirms emergence of these new modes. V. DISCUSSIONS AND CONCLUSIONS
The near-horizon quasinormal frequencies of the charged scalar field in the black holespacetime are investigated numerically and analytically. Using AIM, the tower patterns inthe QNMs for any generalized black hole parameters are numerically found. For the near-extremal Schwarzschild and Reissner-Nordstr¨om dRGT cases, the QNMs near the eventhorizon of the (uncharged) scalar field are purely imaginary with the gaps between overtonesequal to the surface gravity at the horizon.For the non-extremal Schwarzshild de-Sitter dRGT cases, the quasinormal modes are alsopurely imaginary. The value of the gaps between each overtone equals to the correspondingsurface gravity depending on the observing point. If we observe QNMs near the event (cos-mological) horizon, the steps then equal to the surface gravity at the event (cosmological)horizon respectively. For the charged black holes interacting with the charged scalar, thereal parts of the QNMs are universally given by qQ/r h (near the event horizon) and qQ/r c (near the cosmological horizon). The gaps between overtones are equal to the correspondingsurface gravity.Finally, we find the analytic solution of the QNMs near the horizons of static sphericallysymmetric black holes. The tower pattern of quasinormal frequencies near the horizonis ω = qQr a + i | κ a | n , where r a is the corresponding horizon, κ a is the surface gravity at r a , and n is a non-positive integer. Extending the analysis to the far region of the linearlyapproximated metric with outgoing-wave boundary condition, four kinds of all-region QNMsare computed as the four roots of a quartic equation. Each of the four kinds of the QNMsconverge to asymptotically constant real parts and constant-spacing imaginary parts. One22ower of the QNMs is unstable while the other three towers are the damping modes.Ringdown frequency profile of black hole after its formation could contain both the near(event) horizon modes and the all-region (WKB) mode. The near-horizon modes usuallyhas vanishing real parts since most black holes do not carry charge but their imaginary partscontain direct information of the surface gravity of the black hole itself. By more precisedata to be collected in the years to come from LIGO/VIRGO and other collaborations, itbecomes possible to analyze the damping characteristic of the black hole and detect suchmodes, gaining very precise information of the black hole from its surface gravity. ACKNOWLEDGMENTS
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