Quasi-normal modes and absorption probabilities of spin-3/2 fields in D-dimensional Reissner-Nordström black hole spacetimes

Quasi-normal modes and absorption probabilities of spin-3/2 fields in -dimensional Reissner-Nordström black hole spacetimes

C.-H. Chen chunhungchen928@gmail.com Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan    H. T. Cho htcho@mail.tku.edu.tw Department of Physics, Tamkang University, Tamsui District, New Taipei City, Taiwan 25137    A. S. Cornell alan.cornell@wits.ac.za National Institute for Theoretical Physics; School of Physics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa    G. Harmsen gerhard.harmsen5@gmail.com National Institute for Theoretical Physics; School of Physics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa    X. Ngcobo 540630@students.wits.ac.za National Institute for Theoretical Physics; School of Physics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
July 16, 2019
Abstract

In this paper we consider spin-3/2 fields in a -dimensional Reissner-Nordström black hole spacetime. As these spacetimes are not Ricci-flat, it is necessary to modify the covariant derivative to the supercovariant derivative, by including terms related to the background electromagnetic fields, so as to maintain the gauge symmetry. Using this supercovariant derivative we arrive at the corresponding Rarita-Schwinger equation in a charged black hole background. As in our previous works, we exploit the spherically symmetry of the spacetime and use the eigenspinor-vectors on an -sphere to derive the radial equations for both non-transverse-traceless (non-TT) modes and TT modes. We then determine the quasi-normal mode and absorption probabilities of the associated gauge-invariant variables using the WKB approximation and the asymptotic iteration method. We then concentrate on how these quantities change with the charge of the black hole, especially when they reach the extremal limits.

pacs:
04.62+v, 04.65+e, 04.70.Dy

I Introduction

In supergravity theories dasfre (); gripen () the gravitino is described by a spin-3/2 field. The equations of motion of these spin-3/2 fields are given by the Rarita-Schwinger equation:

(1.1)

where

(1.2)

is antisymmetric product of Dirac gamma matrices, is the covariant derivative, and is the spin-3/2 field. In four dimensional black hole spacetimes the Rarita-Schwinger equations are usually analyzed in the Newman-Penrose formalism. However, this formalism cannot be extended to higher dimensions in a straightforward way. In our previous works Chen2015 (); Chen2016 () we have tried an alternative approach to deal with spherically symmetric black hole cases. Using a complete set of eigenspinor-vectors on -spheres, we were able to separate the radial and angular parts of the Rarita-Schwinger equation. In this paper we would like to extend our considerations to charged black hole spacetimes.

The Rarita-Schwinger equation is invariant under the gauge transformation

(1.3)

where is a gauge spinor, provided that the background spacetime is Ricci-flat Chen2015 (); Chen2016 (). This is not the case for charged black holes, nor for black holes in de Sitter or anti-de Sitter spaces. To maintain the gauge symmetry in those cases it is necessary to modify the covariant derivative into the so-called the “supercovariant derivative”. This is done by adding terms related to the cosmological constant and the electromagnetic field of the black hole. Here we shall concentrate on charged Reissner-Nordström black holes in asymptotically flat spacetimes, where in the following section we shall show in detail how the supercovariant derivative is constructed in this case.

Using the supercovariant derivative we are able to obtain the Rarita-Schwinger equation for spin-3/2 fields in Reissner-Nordström black hole spacetimes. Since the spacetime is still spherically symmetric, it is possible, as in our previous works, to derive the radial equations for each component of the spin-3/2 field using eigenspinor-vectors on the -sphere. However, the component fields are not gauge invariant, while the physical fields should be. Hence we shall, as in Ref. Chen2016 (), construct a combination of the component fields, which is gauge invariant. That is, we shall use the same gauge invariant variables and work out the corresponding radial equations.

As the aim in this paper is to study spin-3/2 fields near a Reissner-Nordström black hole, we will focus on how the charge Q of the black hole affects the behavior of the fields. This is done by studying the quasi-normal models (QNMs) associated to our fields, where QNMs are characterized by their complex frequencies. The real parts of the frequencies represent the frequencies of oscillations, while the imaginary parts the decay constants of damping. These QNMs are uniquely determined by the parameters of the black hole konzen (), where in order to determine these QNMs we will use the WKB and improved Asymptotic Iterative Method (AIM), where these methods and how to implement them are given in Refs. kon (); chocor1 (); Chen2016 (). Finally, using the WKB method we are able to obtain the absorption probabilities associated to our spin-3/2 fields, which can give us an insight into the grey-body factors and cross-sections of the black hole.

As such, this paper is set out as follows: In the next section we give a brief motivation for the form of our supercovariant derivative, and in Sec. III we use this supercovariant derivative to determine our equations of motion, and the potential functions for both the “non-TT” and the “TT eigenmodes” of our fields. In Sec. IV, we present the QNMs for our spin-3/2 fields near the Reissner-Nordström black hole. The corresponding absorption probabilities are laid out in Sec. V. Finally, in Sec. VI, we give concluding remarks on our results.

Ii Supercovariant derivative

In order to ensure that our Rarita-Schwinger equation, , remains true, we must require that our spinor-vectors, , are invariant under the transformation in Eq. (1.3). This is guaranteed if

(2.4)

which is satisfied when the metric is a Ricci flat spacetime Chen2016 (). However, in the case of the Reissner-Nordström metric this is not necessarily true, and so we must first determine the supercovariant derivative, . We make the assumption that the derivative has the form:

(2.5)

which needs to satisfy:

(2.6)

where , is the electromagnetic field, and , are unknown constants. Plugging Eq. (2.5) into Eq. (2.6) we have:

(2.7)

Setting equal to zero we have , and together with Eq. (2.7) we have:

(2.8)

In order to remove the terms we require that

(2.9)

where

(2.10)

and with the term equal to zero only when . Next we consider the and terms. In the four dimensional case, and , which is the Maxwell equation, and Eq. (2.8) is automatically satisfied. In the five dimensional case, is proportional to the identity matrix, such that we have to set the equations of motion for the electromagnetic field (where ) as:

(2.11)

where is the Levi-Civita tensor. In higher dimensional cases we can take the equations of motion for electromagnetic field to be , and the term vanishes if the condition is fulfilled. Finally the supercovariant derivative for the spin-3/2 field in a general dimensional Reissner-Nordström black hole spacetime can be written as:

(2.12)

Note that this is consistent with the results of Ref. Liu2014 (), though we must emphasize that in our construction one cannot find an appropriate supercovariant derivative for a “charged” spin-3/2 field in the Reissner-Nordström black hole spacetime.

Iii Potential function

In this section we will determine both the radial equations and the potential functions for our spin-3/2 fields near the Reissner-Nordström black hole, using the same approach as we have done for the dimensional Schwarzschild black hole. For completeness we reproduce some of the results from the Schwarzschild case in this paper, where a full explanation of this method can be found in Ref.Chen2016 ().

iii.1 Rarita-Schwinger field near D dimensional Reissner-Nordström black holes

Our line element is given as Liu2014 ():

(3.13)

where and . The term denotes the metric of the sphere , where we will use over-bars to represent terms from this metric. Next, the electromagnetic field takes the Coulomb form which is given as KodaIshi2004 ():

(3.14)

The relation of and is

(3.15)

where is a constant defined by the Einstein field equation,

(3.16)

In order to be consistent with the supercovariant derivative above, one should take as in Eq. (2.10). Since we represent the wave functions of our fields as spinor-vectors, which can be constructed from the “non TT eigemodes” and the “TT eigenmodes” on , we will use the massless form of the Rarita-Schwinger equation Chen2016 ():

(3.17)

where is the supercovariant derivative in Eq. (2.12).

iii.2 Non-TT eigenfunctions

The radial and temporal wave functions can be written as:

(3.18)

where is an eigenspinor on , with eigenvalue . The eigenvalues are given by , where Chen2016 (). Our angular wave function is written as:

(3.19)

where , are functions of and which behave like 2-spinors. We begin by using the Weyl gauge, , and then introduce a gauge invariant variable, which we also use to determine the equations of motion. Looking at , and equations in Eq. (3.17) separately, we determine the 4 appropriate equations of motion. Note that our choice of the gamma tensors and the spin connections can be found in Ref. Chen2016 ().

iii.2.1 Equations of motion

Firstly consider the case of in Eq. (3.17):

(3.20)

Using the definitions of our wave functions we determine our first equation of motion to be:

(3.21)

Next we consider , and get the second equation of motion as:

(3.22)

Finally for the case of we obtain two more equations:

(3.23)
(3.24)

It can be shown that these four equations of motion are not independent. One of them can be obtained from the other three. Hence we shall work with only Eqs. (3.21), (3.22) and (3.24) in the following.

iii.2.2 Effective potential

The functions , and are not gauge invariant, and as such we apply a gauge invariant variable to our equations of motion. Using the same arguments as we have used in Ref.Chen2016 () we obtain the following gauge invariant variable

(3.25)

Plugging this into Eq. (3.21), Eq. (3.22) and Eq. (3.24) we obtain the equation of motion for the gauge invariant variable ,

(3.26)

Component-wise, can be written as:

(3.27)

where and are purely radially dependent terms. Furthermore we set

(3.28)

where

(3.29)

Applying Eq. (3.27) and Eq. (3.28) to Eq. (3.26) we get the following set of coupled equations:

(3.30)

where

(3.31)

Decoupling these two equations we obtain the following radial equations

(3.32)

where is the tortoise coordinate with the definition , and

Setting in Eqs. (3.32) we recover the Schwarzschild potential as given in Ref. Chen2016 ().

iii.3 TT eigenfunctions

iii.3.1 Equations of motion

Setting the and to be the same as in the “non-TT eigenfunctions” case given in Eq. (3.18), the angular part is now given as:

(3.33)

where is the TT mode eigenspinor-vector which includes the “TT mode I” and “TT mode II”, as described in Ref. Chen2016 (), and behaves like a 2-spinor. We again initially use the Weyl gauge, and in this case apply the TT conditions on a sphere, giving us Chen2016 (). Our only non-zero equation of motion is then determined to be

(3.34)

where in this case is already gauge invariant. We can therefore use this equation to determine our radial equation.

iii.3.2 Effective potential

We can rewrite as

(3.35)

and substituting Eq.(3.35) into Eq. (3.34) we get the following set of coupled equations:

(3.36)

Setting

(3.37)

we can simplify the equations in Eq. (3.36), and get the following

(3.38)

where

(3.39)

We now decouple the equations in Eq. (3.38) and obtain the radial equations

(3.40)

where

and our eigenvalue is given as with .

As noted in Ref. Chen2016 (), the Schwarzschild case of this potential is the same as for Dirac particles in a general dimensional Schwarzschild black hole cho2007split (). This, however, is not true for the Reissner-Nordström case. For the spin-3/2 field one needs to use the supercovariant derivative in the charged black hole spacetime, whereas for the Dirac field one would still use the ordinary covariant derivative. The extra terms in the supercovariant derivative would render the effective potential of the spin-3/2 field in the TT mode to be different from that of the Dirac field in the same spacetime.

Iv QNMs

In order to obtain the QNMs we have chosen to use the WKB method, to 3rd and 6th order, and AIM. We would like to investigate how the quasi-normal frequencies change with the charge of the black hole, where a particularly interesting case of the Reissner-Nordström black hole would be the extremal case . In this section we present the QNMs in the cases , , and for both “non-TT eigenfunction related” and “TT eigenfunction related” potentials from to .

iv.1 non-TT eigenfunctions related

In this subsection we consider the QNMs of the radial equations in Eq. (3.32). Since and are isospectral, we can choose either one to work with. Here we shall concentrate on the first equation, that is, the potential . In the case of the WKB methods for the QNMs associated to the “non-TT eigenfunctions”, the full explanation of how to determine the QNMs Refs. iyewil (); kon (). For the AIM, a detailed discussion can be found in Refs. chocor1 (); Chen2015 (); Chen2016 ().

For the case of the Reissner-Nordström background, we shall start with the definition of the tortoise coordinate

(4.41)

where . The relation between and can be obtained by

(4.42)
(4.43)

In the AIM we first single out the asymptotic behavior of , which is due to the QNM boundary conditions,

(4.44)

Note that we use “” to signify that for convenience we choose the asymptotic function to include the leading term and some of the subleading terms of , but not necessarily the whole exponential. Next, a necessary coordinate transformation in the AIM will be

(4.45)

The coefficients for the lowest order can be obtained as:

(4.46)

We next find the higher order and by the relation

(4.47)

and the corresponding by the equation

(4.48)

Note that

(4.49)

Iterating this method for a sufficiently large number of iteration, becomes stable, indicating that this is the QNM we are looking for. For example, for the first mode of the extremal case in 6 dimensions, the relation between iteration number and the QNM frequency is plotted in Fig 1 (with ). This mode is one of the modes that the 3rd and the 6th order WKB methods do not give a reasonable result for, but the AIM does.

Figure 1: The relation between iteration number and the value of the real part and the imaginary part of the QNM frequency for , , and .

In Tabs. 1-6 we present the QNMs for the Reissner-Nordström black hole for dimensions to . The results are given in units of , that is, we have set . We note that as the value of increases, for fixed values of () and , the real part of the QNM decreases and the magnitude of the imaginary part increases, this being the same behavior as we have seen for the Schwarzschild black hole. This result suggests that the lower modes are easier to detect compared to the higher less energetic modes. Furthermore, they also decay the slowest. We also note that an increase in the number of dimensions results in the QNM being emitted more energetically. This can be understood by considering the change in the potentials as the dimension is increased as shown in Fig. 2. From to the maximum value of the potential increases as is increased. Hence, the real part of the QNM frequency would also increase. Lastly, when the charge is increased, the real part of the frequency for the same mode increases, while the magnitude of the imaginary part also increases. This is consistent with the change of the effective potentials as is increased, as shown in Fig. 3. As is increased from 0 to 1 (in units of ), the maximum value of the potential increases, hence the real part of the QNM frequency increases. On the other hand, the potential tends to sharpen as is increased, this implies that the field can decay more easily, giving a large decay constant, or a large absolute value of the imaginary part of the frequency.

Figure 2: Potential function with and for to .

Note that in the tables there are several blank entries. The reason for leaving these entries out is that we think the numbers we have obtained are not reliable. For the WKB approximation we found that higher order terms dominate over the lower order terms. This is unreasonable as the WKB method is generated from a series expansion. As for the AIM, the results do not converge when the number of iterations is increased. As such, in these cases we have left these entries as blank.

We also find that there is a strong disagreement for the WKB methods in the cases of , for dimensions higher than . The reason for this disagreement is two-fold. The first is again the problem with the WKB series expansion, as mentioned above. The second one is due to the peculiar behavior of the effective potential. As shown in Fig. 2, it is clear that for the potentials in the cases , a second local maximum will develop. This happens not just for the cases but also for potentals with other values. For larger values of , the dimension at which the potential will have this behavior is higher. The presence of a second maximum renders the WKB approximation and the AIM unreliable, so we have only listed the results up to in these instances.

Figure 3: The non-TT effective potential with and for to .
4 Dimensions 5 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 0.3161-0.0909i 0.3185-0.0910i 0.3185-0.0909i 0 0 0.6360-0.2147i 0.6484-0.2191i 0.6484-0.2191i
1 0 0.5370-0.0942i 0.5375-0.0942i 0.5374-0.0942i 1 0 1.0773-0.2343i 1.0808-0.2343i 1.0807-0.2343i
1 1 0.5180-0.2871i 0.5191-0.2867i 0.5191-0.2867i 1 1 0.9917-0.7241i 1.0008-0.7209i 1.0008-0.7209i
2 0 0.7423-0.0952i 0.7425-0.0952i 0.7424-0.0952i 2 0 1.4730-0.2411i 1.4742-0.2411i 1.4742-0.2411i
2 1 0.7285-0.2880i 0.7289-0.2879i 0.7289-0.2878i 2 1 1.4100-0.7347i 1.4126-0.7342i 1.4126-0.7342i
2 2 0.7041-0.4860i 0.7035-0.4871i 0.7034-0.4870i 2 2 1.2998-1.2522i 1.2922-1.2636i 1.2922-1.2636i
3 0 0.9423-0.0957i 0.9424-0.0957i 0.9424-0.0956i 3 0 1.8520-0.2443i 1.8527-0.2443i 1.8527-0.2443i
3 1 0.9315-0.2884i 0.9316-0.2884i 0.9316-0.2883i 3 1 1.8018-0.7401i 1.8040-0.7395i 1.8040-0.7395i
3 2 0.9114-0.4848i 0.9109-0.4853i 0.9109-0.4852i 3 2 1.7104-1.2530i 1.7101-1.2546i 1.7101-1.2546i
3 3 0.8845-0.6853i 0.8819-0.6887i 0.8819-0.6887i 3 3 1.5882-1.7838i 1.5787-1.8012i 1.5787-1.8012i
4 0 1.1398-0.0959i 1.1399-0.0959i 1.1398-0.0958i 4 0 2.2228-0.2461i 2.2232-0.2461i 2.2232-0.2461i
4 1 1.1309-0.2886i 1.1310-0.2886i 1.1309-0.2886i 4 1 2.1809-0.7432i 2.1824-0.7429i 2.1824-0.7429i
4 2 1.1139-0.4840i 1.1136-0.4842i 1.1135-0.4842i 4 2 2.1027-1.2530i 2.1031-1.2534i 2.1031-1.2534i
4 3 1.0905-0.6827i 1.0886-0.6845i 1.0886-0.6845i 4 3 1.9958-1.7778i 1.9911-1.7852i 1.9911-1.7852i
4 4 1.0620-0.8848i 1.0575-0.8910i 1.0575-0.8909i 4 4 1.8659-2.3161i 1.8545-2.3415i 1.8545-2.3415i
5 0 1.3360-0.0960i 1.3360-0.0960i 1.3360-0.0960i 5 0 2.5889-0.2471i 2.5891-0.2471i 2.5891-0.2471i
5 1 1.3283-0.2887i 1.3284-0.2887i 1.3283-0.2887i 5 1 2.5529-0.7451i 2.5536-0.7450i 2.5536-0.7450i
5 2 1.3136-0.4834i 1.3134-0.4836i 1.3133-0.4835i 5 2 2.4846-1.2528i 2.4838-1.2536i 2.4838-1.2536i
5 3 1.2929-0.6809i 1.2916-0.6819i 1.2916-0.6819i 5 3 2.3895-1.7731i 2.3822-1.7802i 2.3822-1.7802i
5 4 1.2674-0.8812i 1.2640-0.8849i 1.2640-0.8849i 5 4 2.2727-2.3056i 2.2533-2.3315i 2.2533-2.3315i
5 5 1.2378-1.0842i 1.2315-1.0935i 1.2315-1.0935i 5 5 2.1371-2.8485i 2.1024-2.9129i 2.1024-2.9129i
Table 1: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 4,5 and Q=0.1M.
6 Dimensions 7 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 0.9364-0.3422i 0.9408-0.3027i 0.9408-0.3027i 0 0 1.2845-0.4857i 1.2111-0.5136i 1.2110-0.5136i
1 0 1.5164-0.3578i 1.5292-0.3540i 1.5292-0.3540i 1 0 1.9288-0.4776i 1.9388-0.4621i 1.9388-0.4621i
1 1 1.3214-1.1165i 1.3680-1.0786i 1.3680-1.0786i 1 1 1.5920-1.4970i 1.6160-1.3961i 1.6160-1.3961i
2 0 2.0379-0.3705i 2.0418-0.3699i 2.0418-0.3699i 2 0 2.5348-0.4880i 2.5430-0.4836i 2.5430-0.4836i
2 1 1.8967-1.1346i 1.9117-1.1279i 1.9117-1.1279i 2 1 2.2870-1.4974i 2.3243-1.4620i 2.3243-1.4620i
2 2 1.6451-1.9511i 1.6488-1.9502i 1.6488-1.9502i 2 2 1.8288-2.5991i 1.8401-2.5026i 1.8401-2.5026i
3 0 2.5341-0.3774i 2.5360-0.3772i 2.5360-0.3772i 3 0 3.1136-0.4968i 3.1174-0.4954i 3.1174-0.4954i
3 1 2.4214-1.1470i 2.4285-1.1444i 2.4285-1.1444i 3 1 2.9167-1.5115i 2.9352-1.5003i 2.9352-1.5003i
3 2 2.2133-1.9532i 2.2124-1.9547i 2.2124-1.9547i 3 2 2.5422-2.5869i 2.5506-2.5600i 2.5506-2.5600i
3 3 1.9314-2.8004i 1.8910-2.8487i 1.8910-2.8487i 3 3 2.0267-3.7434i 1.9395-3.7554i 1.9395-3.7554i
4 0 3.0173-0.3816i 3.0183-0.3814i 3.0183-0.3814i 4 0 3.6764-0.5028i 3.6784-0.5022i 3.6784-0.5022i
4 1 2.9227-1.1550i 2.9267-1.1537i 2.9267-1.1537i 4 1 3.5114-1.5233i 3.5215-1.5186i 3.5215-1.5186i
4 2 2.7444-1.9555i 2.7428-1.9567i 2.7428-1.9567i 4 2 3.1931-2.5872i 3.1959-2.5773i 3.1959-2.5773i
4 3 2.4979-2.7896i 2.4678-2.8176i 2.4678-2.8176i 4 3 2.7445-3.7136i 2.6843-3.7312i 2.6843-3.7312i
4 4 2.1951-3.6553i 2.1087-3.7702i 2.1087-3.7702i 4 4 2.1908-4.9064i 1.9854-5.0614i 1.9854-5.0614i
5 0 3.4926-0.3842i 3.4932-0.3841i 3.4932-0.3841i 5 0 4.2293-0.5070i 4.2305-0.5067i 4.2305-0.5067i
5 1 3.4109-1.1603i 3.4133-1.1596i 3.4133-1.1596i 5 1 4.0865-1.5321i 4.0927-1.5296i 4.0927-1.5296i
5 2 3.2547-1.9572i 3.2531-1.9581i 3.2531-1.9581i 5 2 3.8085-2.5899i 3.8092-2.5851i 3.8092-2.5851i
5 3 3.0353-2.7818i 3.0130-2.7991i 3.0130-2.7991i 5 3 3.4108-3.6971i 3.3672-3.7097i 3.3672-3.7097i
5 4 2.7630-3.6346i 2.6964-3.7065i 2.6964-3.7065i 5 4 2.9125-4.8608i 2.7597-4.9595i 2.7597-4.9595i
5 5 2.4444-4.5127i 2.3116-4.7073i 2.3116-4.7073i 5 5 2.3296-6.0794i 1.9987-6.4037i 1.9987-6.4037i
Table 2: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 6,7 and Q=0.1M
4 Dimensions 5 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 0.3616-0.0953i 0.3634-0.0952i 0.3621-0.0881i 0 0 0.7048-0.2213i 0.8005-0.1499i 0.7065-0.2259i
1 0 0.5905-0.0971i 0.5908-0.0971i 0.4050-0.1854i 1 0 1.1471-0.2356i 1.1504-0.2347i 1.1504-0.2346i
1 1 0.5736-0.2950i 0.5744-0.2948i 0.3674-0.3656i 1 1 1.0756-0.7253i 1.0842-0.7160i 1.0842-0.7159i
2 0 0.8044-0.0976i 0.8046-0.0976i 0.8045-0.0975i 2 0 1.5506-0.2420i 1.5519-0.2420i 1.5519-0.2420i
2 1 0.7920-0.2948i 0.7923-0.2947i 0.7922-0.2947i 2 1 1.4946-0.7364i 1.4995-0.7345i 1.4995-0.7345i
2 2 0.7698-0.4967i 0.7692-0.4977i 0.7691-0.4976i 2 2 1.3974-1.2527i 1.4031-1.2505i 1.4031-1.2505i
3 0 1.0131-0.0977i 1.0132-0.0977i 1.0131-0.0977i 3 0 1.9368-0.2448i 1.9374-0.2448i 1.9374-0.2448i
3 1 1.0032-0.2945i 1.0033-0.2945i 1.0033-0.2944i 3 1 1.8913-0.7411i 1.8931-0.7408i 1.8931-0.7408i
3 2 0.9849-0.4946i 0.9844-0.4950i 0.9844-0.4950i 3 2 1.8088-1.2532i 1.8074-1.2559i 1.8074-1.2559i
3 3 0.9602-0.6986i 0.9579-0.7015i 0.9579-0.7015i 3 3 1.6992-1.7820i 1.6869-1.8027i 1.6869-1.8027i
4 0 1.2193-0.0978i 1.2193-0.0978i 1.2193-0.0978i 4 0 2.3142-0.2462i 2.3145-0.2462i 2.3145-0.2462i
4 1 1.2110-0.2943i 1.2111-0.2943i 1.2111-0.2942i 4 1 2.2759-0.7433i 2.2770-0.7432i 2.2770-0.7432i
4 2 1.1954-0.4932i 1.1951-0.4934i 1.1951-0.4934i 4 2 2.2047-1.2523i 2.2036-1.2538i 2.2036-1.2538i
4 3 1.1738-0.6952i 1.1722-0.6968i 1.1722-0.6967i 4 3 2.1077-1.7753i 2.0983-1.7871i 2.0983-1.7871i
4 4 1.1476-0.9004i 1.1437-0.9057i 1.1437-0.9057i 4 4 1.9902-2.3108i 1.9669-2.3518i 1.9669-2.3518i
5 0 1.4240-0.0978i 1.4241-0.0978i 1.4241-0.0978i 5 0 2.6865-0.2470i 2.6868-0.2470i 2.6868-0.2470i
5 1 1.4170-0.2941i 1.4170-0.2941i 1.4170-0.2941i 5 1 2.6535-0.7445i 2.6542-0.7444i 2.6542-0.7444i
5 2 1.4034-0.4922i 1.4032-0.4923i 1.4032-0.4923i 5 2 2.5910-1.2510i 2.5901-1.2519i 2.5901-1.2519i
5 3 1.3843-0.6928i 1.3831-0.6937i 1.3831-0.6937i 5 3 2.5041-1.7694i 2.4970-1.7766i 2.4970-1.7766i
5 4 1.3606-0.8962i 1.3576-0.8994i 1.3576-0.8994i 5 4 2.3977-2.2993i 2.3787-2.3250i 2.3787-2.3250i
5 5 1.3333-1.1022i 1.3277-1.1101i 1.3277-1.1101i 5 5 2.2747-2.8389i 2.2403-2.9032i 2.2403-2.9032i
Table 3: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 4,5 and Q=0.5M.
6 Dimensions 7 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 0.9546-0.1667i 0.9545-0.1666i 0 0 1.1968-0.3978i 1.1967-0.3978i
1 0 1.5887-0.3502i 1.5411-0.3743i 1.5411-0.3743i 1 0 1.9675-0.4197i
1 1 1.4709-1.0935i 1.2200-1.3800i 1.2200-1.3800i 1 1 1.6232-1.2937i
2 0 2.1178-0.3667i 2.1190-0.3673i 2.1190-0.3673i 2 0 2.6056-0.4703i 2.5998-0.4897i 2.5998-0.4897i
2 1 2.0024-1.1232i 2.0024-1.1224i 2.0024-1.1224i 2 1 2.4023-1.4443i 2.3449-1.6313i 2.3449-1.6313i
2 2 1.8050-1.9277i 1.7720-1.9392i 1.7720-1.9392i 2 2 2.0392-2.4977i 1.9354-3.3791i 1.9354-3.3791i
3 0 2.6220-0.3750i 2.6235-0.3751i 2.6235-0.3751i 3 0 3.1972-0.4877i 3.1984-0.4909i 3.1984-0.4909i
3 1 2.5225-1.1395i 2.5271-1.1384i 2.5271-1.1384i 3 1 3.0274-1.4858i 3.0248-1.5096i 3.0248-1.5096i
3 2 2.3417-1.9388i 2.3351-1.9436i 2.3351-1.9436i 3 2 2.7137-2.5414i 2.6649-2.6679i 2.6649-2.6679i
3 3 2.1010-2.7748i 2.0523-2.8252i 2.0523-2.8252i 3 3 2.2912-3.6638i 2.1445-4.1160i 2.1445-4.1160i
4 0 3.1119-0.3795i 3.1129-0.3796i 3.1129-0.3796i 4 0 3.7688-0.4968i 3.7706-0.4975i 3.7706-0.4975i
4 1 3.0261-1.1488i 3.0295-1.1482i 3.0295-1.1482i 4 1 3.6211-1.5062i 3.6269-1.5094i 3.6269-1.5094i
4 2 2.8655-1.9440i 2.8632-1.9468i 2.8632-1.9468i 4 2 3.3407-2.5582i 3.3302-2.5804i 3.3302-2.5804i
4 3 2.6457-2.7704i 2.6165-2.8004i 2.6165-2.8004i 4 3 2.9521-3.6661i 2.8719-3.7744i 2.8719-3.7744i
4 4 2.3779-3.6250i 2.2969-3.7378i 2.2969-3.7378i 4 4 2.4763-4.8284i 2.2632-5.1748i 2.2632-5.1748i
5 0 3.5931-0.3823i 3.5937-0.3823i 3.5937-0.3823i 5 0 4.3286-0.5023i 4.3299-0.5025i 4.3299-0.5025i
5 1 3.5180-1.1543i 3.5204-1.1540i 3.5204-1.1540i 5 1 4.1979-1.5186i 4.2034-1.5187i 4.2034-1.5187i
5 2 3.3751-1.9464i 3.3738-1.9481i 3.3738-1.9481i 5 2 3.9462-2.5673i 3.9440-2.5725i 3.9440-2.5725i
5 3 3.1756-2.7646i 3.1555-2.7829i 3.1555-2.7829i 5 3 3.5902-3.6620i 3.5428-3.7021i 3.5428-3.7021i
5 4 2.9296-3.6087i 2.8698-3.6796i 2.8698-3.6796i 5 4 3.1475-4.8060i 2.9968-4.9618i 2.9968-4.9618i
5 5 2.6429-4.4752i 2.5247-4.6616i 2.5247-4.6616i 5 5 2.6308-5.9957i 2.3192-6.4154i 2.3192-6.4154i
Table 4: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 6,7 and Q=0.5M.
4 Dimensions 5 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 0.5410-0.0867i 0.5414-0.0865i 0.5414-0.0864i 0 0 0.8519-0.1875i
1 0 0.8173-0.0874i 0.8174-0.0874i 0.8174-0.0873i 1 0 1.3394-0.2077i 1.3414-0.2098i 1.3413-0.2097i
1 1 0.8027-0.2638i 0.8032-0.2636i 0.8032-0.2636i 1 1 1.2704-0.6320i 1.2809-0.6422i 1.2809-0.6422i
2 0 1.0810-0.0878i 1.0811-0.0878i 1.0811-0.0877i 2 0 1.7723-0.2144i 1.7735-0.2147i 1.7735-0.2147i
2 1 1.0701-0.2643i 1.0704-0.2643i 1.0703-0.2642i 2 1 1.7224-0.6490i 1.7266-0.6494i 1.7266-0.6494i
2 2 1.0491-0.4435i 1.0491-0.4435i 1.0490-0.4435i 2 2 1.6288-1.0985i 1.6337-1.1014i 1.6337-1.1014i
3 0 1.3395-0.0880i 1.3396-0.0880i 1.3395-0.0879i 3 0 2.1865-0.2174i 2.1872-0.2175i 2.1872-0.2175i
3 1 1.3307-0.2646i 1.3309-0.2646i 1.3308-0.2645i 3 1 2.1461-0.6559i 2.1486-0.6558i 2.1486-0.6558i
3 2 1.3136-0.4430i 1.3136-0.4430i 1.3135-0.4430i 3 2 2.0689-1.1048i 2.0716-1.1049i 2.0716-1.1049i
3 3 1.2889-0.6239i 1.2880-0.6245i 1.2880-0.6245i 3 3 1.9604-1.5675i 1.9574-1.5732i 1.9574-1.5732i
4 0 1.5953-0.0881i 1.5953-0.0881i 1.5953-0.0881i 4 0 2.5913-0.2189i 2.5917-0.2190i 2.5917-0.2190i
4 1 1.5879-0.2648i 1.5880-0.2648i 1.5880-0.2648i 4 1 2.5572-0.6595i 2.5587-0.6594i 2.5587-0.6594i
4 2 1.5734-0.4427i 1.5734-0.4427i 1.5734-0.4427i 4 2 2.4912-1.1076i 2.4929-1.1075i 2.4929-1.1075i
4 3 1.5523-0.6225i 1.5517-0.6228i 1.5517-0.6228i 4 3 2.3969-1.5663i 2.3948-1.5691i 2.3948-1.5691i
4 4 1.5252-0.8047i 1.5232-0.8060i 1.5232-0.8060i 4 4 2.2784-2.0370i 2.2655-2.0507i 2.2655-2.0507i
5 0 1.8494-0.0882i 1.8494-0.0882i 1.8494-0.0882i 5 0 2.9905-0.2199i 2.9908-0.2199i 2.9908-0.2199i
5 1 1.8430-0.2649i 1.8431-0.2649i 1.8431-0.2649i 5 1 2.9610-0.6616i 2.9620-0.6616i 2.9620-0.6616i
5 2 1.8305-0.4425i 1.8305-0.4425i 1.8305-0.4425i 5 2 2.9034-1.1090i 2.9045-1.1090i 2.9045-1.1090i
5 3 1.8121-0.6216i 1.8117-0.6218i 1.8117-0.6218i 5 3 2.8201-1.5649i 2.8186-1.5665i 2.8186-1.5665i
5 4 1.7883-0.8026i 1.7869-0.8034i 1.7869-0.8034i 5 4 2.7143-2.0307i 2.7049-2.0388i 2.7049-2.0388i
5 5 1.7596-0.9857i 1.7563-0.9879i 1.7563-0.9879i 5 5 2.5887-2.5066i 2.5646-2.5311i 2.5646-2.5311i
Table 5: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 4,5 and Q=M.
6 Dimensions 7 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 0.9646 - 0.3481i 0 0
1 0 1.7521-0.3082i 1.7516 - 0.3053i 1 0
1 1 1.6678-0.9419i 1.6970 - 0.9918i 1 1
2 0 2.3058-0.3275i 2.3115-0.3240i 2.3081 - 0.3401i 2 0 2.7749-0.4333i 2.8918-0.3419i 2.7749-0.4333i
2 1 2.2126-0.9992i 2.2297-0.9708i 2.2375 - 1.0523i 2 1 2.7194-1.3659i 2.7194-1.3659i
2 2 2.0483-1.7097i 2.0302-1.6426i 2.1478 - 1.8272i 2 2 2.7162-2.4111i 2.7162-2.4111i
3 0 2.8322-0.3371i 2.8345-0.3368i 2.8345-0.3368i 3 0 3.3789-0.4424i 3.3883-0.4341i 3.3883-0.4341i
3 1 2.7461-1.0212i 2.7554-1.0156i 2.7554-1.0156i 3 1 3.2568-1.3524i 3.3296-1.2462i 3.3296-1.2462i
3 2 2.5841-1.7318i 2.5948-1.7106i 2.5948-1.7106i 3 2 3.0467-2.3221i 3.3425-1.7230i
3 3 2.3621-2.4758i 2.3449-2.4367i 2.3449-2.4367i 3 3 2.7896-3.3500i 3.9401-1.1831i
4 0 3.3424-0.3426i 3.3438-0.3426i 3.3438-0.3426i 4 0 3.9683-0.4525i 3.9713-0.4516i 3.9713-0.4516i
4 1 3.2663-1.0344i 3.2720-1.0332i 3.2720-1.0332i 4 1 3.8448-1.3713i 3.8600-1.3555i 3.8600-1.3555i
4 2 3.1198-1.7450i 3.1269-1.7402i 3.1269-1.7402i 4 2 3.6112-2.3277i 3.6410-2.2421i 3.6410-2.2421i
4 3 2.9124-2.4820i 2.9058-2.4774i 2.9058-2.4774i 4 3 3.2891-3.3340i 3.3018-3.0389i 3.3018-3.0389i
4 4 2.6545-3.2475i 2.6054-3.2637i 2.6054-3.2637i 4 4 2.8991-4.3888i 2.7814-3.5851i 2.7814-3.5851i
5 0 3.8425-0.3459i 3.8435-0.3460i 3.8435-0.3460i 5 0 4.5446-0.4595i 4.5466-0.4595i 4.5466-0.4595i
5 1 3.7753-1.0426i 3.7792-1.0422i 3.7792-1.0422i 5 1 4.4302-1.3874i 4.4389-1.3841i 4.4389-1.3841i
5 2 3.6441-1.7533i 3.6494-1.7517i 3.6494-1.7517i 5 2 4.2074-2.3414i 4.2195-2.3230i 4.2195-2.3230i
5 3 3.4554-2.4847i 3.4524-2.4854i 3.4524-2.4854i 5 3 3.8883-3.3348i 3.8773-3.2798i 3.8773-3.2798i
5 4 3.2166-3.2404i 3.1862-3.2573i 3.1862-3.2573i 5 4 3.4880-4.3739i 3.3887-4.2548i 3.3887-4.2548i
5 5 2.9346-4.0206i 2.8505-4.0854i 2.8505-4.0854i 5 5 3.0202-5.4581i 2.7165-5.2505i 2.7165-5.2505i
Table 6: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 6,7 and Q=M.

iv.2 TT eigenfunctions related

For the “TT eigenfunction related” cases, both the WKB and AIM present reasonable results. We have to note that there is no “TT eigenfunction related” case in the -dimensional Reissner-Nordström spacetime because of the absence of the TT eigenmodes on the -sphere. In Tabs. 7-12 we present the TT QNMs for , , and from to . The change in QNM frequencies is similar to that for non-TT cases when either or is changed. To see how the frequencies are affected by the change in the charge , we plot in Fig. 4 the first three modes in more detail for to 1. The red, blue and green colors represent , , and dimensions. The result indicates that when becomes larger, the real part decreases and the absolute value of imaginary part also decreases. This is consistent with the change of the TT potential with , which is plotted in Fig. 5. We can see that when is increased, the maximum value of the potential decreases. This implies that the real part of the QNM frequency decreases accordingly. In addition to this the potential broadens when is increased, so the mode decays slower, which implies that the absolute value of the imaginary part of the frequency becomes smaller. Note that this trend is the exact opposite to that for the non-TT cases for .

Figure 4: The change of the first three QNM frequencies when increases from 0 to 1.
Figure 5: The TT effective potential of and for to .
5 Dimensions
l n WKB 3rd order WKB 6th order AIM
0 0 0.8443-0.2482i 0.8538-0.2493i 0.7755-0.1916i
1 0 1.2093-0.2487i 1.2124-0.2490i 1.2355-0.2633i
1 1 1.1297-0.7659i 1.1384-0.7640i 0.8044-0.6729i
2 0 1.5677-0.2490i 1.5690-0.2490i 1.5690-0.2490i
2 1 1.5069-0.7580i 1.5108-0.7572i 1.5108-0.7572i
2 2 1.4010-1.2904i 1.4002-1.2962i 1.4002-1.2962i
3 0 1.9239-0.2492i 1.9245-0.2492i 1.9245-0.2492i
3 1 1.8747-0.7545i 1.8767-0.7541i 1.8767-0.7541i
3 2 1.7854-1.2767i 1.7840-1.2795i 1.7840-1.2795i
3 3 1.6663-1.8167i 1.6530-1.8392i 1.6530-1.8392i
4 0 2.2791-0.2493i 2.2795-0.2493i 2.2795-0.2493i
4 1 2.2379-0.7527i 2.2390-0.7525i 2.2390-0.7525i
4 2 2.1608-1.2687i 2.1596-1.2703i 2.1596-1.2703i
4 3 2.0557-1.7997i 2.0451-1.8126i 2.0451-1.8126i
4 4 1.9280-2.3440i 1.9017-2.3890i 1.9017-2.3890i
5 0 2.6340-0.2493i 2.6342-0.2493i 2.6342-0.2493i
5 1 2.5983-0.7517i 2.5990-0.7516i 2.5990-0.7516i
5 2 2.5307-1.2637i 2.5297-1.2647i 2.5297-1.2647i
5 3 2.4367-1.7883i 2.4286-1.7961i 2.4286-1.7961i
5 4 2.3212-2.3250i 2.2998-2.3533i 2.2998-2.3533i
5 5 2.1873-2.8720i 2.1482-2.9430i 2.1482-2.9430i
Table 7: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 5 and Q=0.1M.
6 Dimensions 7 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 1.2853-0.3861i 1.3064-0.3961i 1.3063-0.3961i 0 0 1.7137-0.5075i 1.7430-0.5408i 1.7430-0.5408i
1 0 1.7613-0.3889i 1.7692-0.3917i 1.7692-0.3917i 1 0 2.2717-0.5149i 2.2844-0.5247i 2.2844-0.5247i
1 1 1.5809-1.2099i 1.6097-1.2106i 1.6097-1.2106i 1 1 1.9511-1.6139i 2.0126-1.6368i 2.0126-1.6368i
2 0 2.2243-0.3901i 2.2281-0.3906i 2.2282-0.3904i 2 0 2.8099-0.5180i 2.8176-0.5196i 2.8176-0.5196i
2 1 2.0867-1.1939i 2.1008-1.1922i 2.1008-1.1922i 2 1 2.5668-1.5899i 2.6000-1.5911i 2.6000-1.5911i
2 2 1.8433-2.0498i 1.8483-2.0618i 1.8483-2.0618i 2 2 2.1240-2.7539i 2.1468-2.7766i 2.1468-2.7766i
3 0 2.6826-0.3906i 2.6846-0.3906i 2.6846-0.3906i 3 0 3.3403-0.5191i 3.3445-0.5191i 3.3445-0.5191i
3 1 2.5712-1.1868i 2.5787-1.1853i 2.5787-1.1853i 3 1 3.1444-1.5797i 3.1633-1.5766i 3.1633-1.5766i
3 2 2.3663-2.0196i 2.3672-2.0240i 2.3672-2.0240i 3 2 2.7746-2.7018i 2.7858-2.7031i 2.7858-2.7031i
3 3 2.0899-2.8930i 2.0549-2.9462i 2.0549-2.9462i 3 3 2.2680-3.9031i 2.1975-3.9815i 2.1975-3.9815i
4 0 3.1389-0.3909i 3.1400-0.3908i 3.1400-0.3908i 4 0 3.8673-0.5196i 3.8696-0.5194i 3.8696-0.5194i
4 1 3.0451-1.1830i 3.0493-1.1821i 3.0493-1.1821i 4 1 3.7024-1.5743i 3.7135-1.5716i 3.7135-1.5716i
4 2 2.8685-2.0023i 2.8678-2.0044i 2.8678-2.0044i 4 2 3.3855-2.6730i 3.3908-2.6705i 3.3908-2.6705i
4 3 2.6248-2.8550i 2.5975-2.8847i 2.5975-2.8847i 4 3 2.9404-3.8338i 2.8874-3.8700i 2.8874-3.8700i
4 4 2.3261-3.7390i 2.2457-3.8554i 2.2457-3.8554i 4 4 2.3917-5.0594i 2.2039-5.2494i 2.2039-5.2494i
5 0 3.5942-0.3910i 3.5948-0.3910i 3.5948-0.3910i 5 0 4.3926-0.5198i 4.3940-0.5197i 4.3940-0.5197i
5 1 3.5129-1.1808i 3.5156-1.1802i 3.5156-1.1802i 5 1 4.2497-1.5710i 4.2566-1.5691i 4.2566-1.5691i
5 2 3.3578-1.9914i 3.3567-1.9926i 3.3567-1.9926i 5 2 3.9722-2.6553i 3.9746-2.6529i 3.9746-2.6529i
5 3 3.1401-2.8297i 3.1191-2.8476i 3.1191-2.8476i 5 3 3.5760-3.7890i 3.5363-3.8079i 3.5363-3.8079i
5 4 2.8703-3.6960i 2.8067-3.7686i 2.8067-3.7686i 5 4 3.0802-4.9785i 2.9360-5.0898i 2.9360-5.0898i
5 5 2.5549-4.5872i 2.4278-4.7818i 2.4278-4.7818i 5 5 2.5001-6.2217i 2.1857-6.5663i 2.1857-6.5663i
Table 8: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 6,7 and Q=0.1M.
5 Dimensions
l n WKB 3rd order WKB 6th order AIM
0 0 0.8029-0.2388i 0.8131-0.2411i 0.3915-0.2511i
1 0 1.1737-0.2414i 1.1769-0.2418i 1.1768-0.2418i
1 1 1.0949-0.7434i 1.1046-0.7419i 1.1045-0.7419i
2 0 1.5374-0.2427i 1.5387-0.2428i 1.5387-0.2428i
2 1 1.4783-0.7388i 1.4825-0.7381i 1.4825-0.7381i
2 2 1.3751-1.2575i 1.3754-1.2628i 1.3754-1.2628i
3 0 1.8990-0.2436i 1.8996-0.2436i 1.8996-0.2436i
3 1 1.8516-0.7375i 1.8537-0.7372i 1.8537-0.7372i
3 2 1.7654-1.2475i 1.7647-1.2502i 1.7647-1.2502i
3 3 1.6504-1.7747i 1.6389-1.7957i 1.6389-1.7957i
4 0 2.2597-0.2442i 2.2601-0.2442i 2.2601-0.2442i
4 1 2.2202-0.7372i 2.2213-0.7370i 2.2213-0.7370i
4 2 2.1463-1.2422i 2.1454-1.2437i 2.1454-1.2437i
4 3 2.0453-1.7616i 2.0361-1.7736i 2.0361-1.7736i
4 4 1.9228-2.2938i 1.8992-2.3356i 1.8992-2.3356i
5 0 2.6201-0.2446i 2.6203-0.2446i 2.6203-0.2446i
5 1 2.5860-0.7373i 2.5867-0.7372i 2.5867-0.7372i
5 2 2.5214-1.2392i 2.5207-1.2400i 2.5207-1.2400i
5 3 2.4316-1.7530i 2.4245-1.7603i 2.4245-1.7603i
5 4 2.3212-2.2786i 2.3020-2.3048i 2.3020-2.3048i
5 5 2.1933-2.8141i 2.1580-2.8799i 2.1580-2.8799i
Table 9: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 5 and Q=0.5M.
6 Dimensions 7 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 1.2225-0.3706i 1.2484-0.3817i 1.2484-0.3816i 0 0 1.6352-0.4888i 1.6769-0.5175i 1.6769-0.5175i
1 0 1.7034-0.3764i 1.7124-0.3790i 1.7130-0.3777i 1 0 2.1983-0.4993i 2.2147-0.5067i 2.2147-0.5067i
1 1 1.5222-1.1710i 1.5568-1.1692i 1.5568-1.1692i 1 1 1.8724-1.5650i 1.9497-1.5713i 1.9497-1.5713i
2 0 2.1698-0.3789i 2.1739-0.3794i 2.1739-0.3794i 2 0 2.7387-0.5037i 2.7474-0.5046i 2.7471-0.5046i
2 1 2.0344-1.1593i 2.0504-1.1572i 2.0504-1.1572i 2 1 2.4972-1.5452i 2.5363-1.5413i 2.5363-1.5413i
2 2 1.7938-1.9903i 1.8052-1.9976i 1.8052-1.9976i 2 2 2.0540-2.6772i 2.0923-2.6761i 2.0923-2.6761i
3 0 2.6314-0.3803i 2.6335-0.3804i 2.6335-0.3804i 3 0 3.2711-0.5057i 3.2757-0.5056i 3.2757-0.5056i
3 1 2.5230-1.1553i 2.5313-1.1538i 2.5313-1.1538i 3 1 3.0786-1.5383i 3.1001-1.5339i 3.1001-1.5339i
3 2 2.3230-1.9653i 2.3273-1.9684i 2.3273-1.9684i 3 2 2.7136-2.6303i 2.7337-2.6236i 2.7337-2.6236i
3 3 2.0525-2.8150i 2.0259-2.8605i 2.0259-2.8605i 3 3 2.2117-3.8007i 2.1600-3.8499i 2.1600-3.8499i
4 0 3.0911-0.3813i 3.0923-0.3813i 3.0923-0.3813i 4 0 3.8003-0.5069i 3.8028-0.5067i 3.8028-0.5067i
4 1 3.0003-1.1538i 3.0051-1.1529i 3.0051-1.1529i 4 1 3.6393-1.5354i 3.6518-1.5324i 3.6518-1.5324i
4 2 2.8293-1.9521i 2.8307-1.9537i 2.8307-1.9537i 4 2 3.3291-2.6061i 3.3398-2.6008i 3.3398-2.6008i
4 3 2.5928-2.7828i 2.5712-2.8088i 2.5712-2.8088i 4 3 2.8920-3.7372i 2.8530-3.7610i 2.8530-3.7610i
4 4 2.3025-3.6437i 2.2336-3.7479i 2.2336-3.7479i 4 4 2.3523-4.9324i 2.1898-5.0856i 2.1898-5.0856i
5 0 3.5499-0.3819i 3.5506-0.3819i 3.5506-0.3819i 5 0 4.3278-0.5077i 4.3294-0.5076i 4.3294-0.5076i
5 1 3.4716-1.1532i 3.4746-1.1527i 3.4746-1.1527i 5 1 4.1890-1.5341i 4.1967-1.5323i 4.1967-1.5323i
5 2 3.3221-1.9444i 3.3224-1.9454i 3.3224-1.9454i 5 2 3.9188-2.5922i 3.9248-2.5888i 3.9248-2.5888i
5 3 3.1120-2.7621i 3.0951-2.7781i 3.0951-2.7781i 5 3 3.5322-3.6979i 3.5026-3.7110i 3.5026-3.7110i
5 4 2.8514-3.6067i 2.7965-3.6724i 2.7965-3.6724i 5 4 3.0476-4.8579i 2.9241-4.9499i 2.9241-4.9499i
5 5 2.5466-4.4755i 2.4344-4.6523i 2.4344-4.6523i 5 5 2.4801-6.0708i 2.1989-6.3684i 2.1989-6.3684i
Table 10: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 6,7 and Q=0.5M.
5 Dimensions
l n WKB 3rd order WKB 6th order AIM
0 0 0.7636-0.2260i 0.7748-0.2267i 0.7748-0.2266i
1 0 1.1542-0.2239i 1.1579-0.2242i 1.1578-0.2242i
1 1 1.0713-0.6900i 1.0828-0.6866i 1.0827-0.6866i
2 0 1.5404-0.2232i 1.5420-0.2233i 1.5420-0.2233i
2 1 1.4797-0.6787i 1.4853-0.6776i 1.4853-0.6776i
2 2 1.3701-1.1560i 1.3734-1.1563i 1.3734-1.1563i
3 0 1.9256-0.2229i 1.9264-0.2229i 1.9264-0.2229i
3 1 1.8779-0.6740i 1.8809-0.6735i 1.8809-0.6735i
3 2 1.7885-1.1394i 1.7906-1.1393i 1.7906-1.1393i
3 3 1.6656-1.6221i 1.6574-1.6323i 1.6574-1.6323i
4 0 2.3105-0.2227i 2.3110-0.2227i 2.3110-0.2227i
4 1 2.2713-0.6717i 2.2731-0.6714i 2.2731-0.6714i
4 2 2.1961-1.1305i 2.1975-1.1304i 2.1975-1.1304i
4 3 2.0903-1.6027i 2.0852-1.6076i 2.0852-1.6076i
4 4 1.9591-2.0886i 1.9383-2.1120i 1.9383-2.1120i
5 0 2.6953-0.2226i 2.6957-0.2226i 2.6957-0.2226i
5 1 2.6620-0.6703i 2.6631-0.6701i 2.6631-0.6701i
5 2 2.5973-1.1252i 2.5982-1.1252i 2.5982-1.1252i
5 3 2.5048-1.5905i 2.5014-1.5932i 2.5014-1.5932i
5 4 2.3884-2.0674i 2.3738-2.0806i 2.3738-2.0806i
5 5 2.2513-2.5556i 2.2176-2.5942i 2.2176-2.5942i
Table 11: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 5 and Q=M.
6 Dimensions 7 Dimensions
l n WKB 3rd order WKB 6th order AIM l n WKB 3rd order WKB 6th order AIM
0 0 1.1597-0.3598i 1.1816-0.3675i 1.1816-0.3674i 0 0 1.5619-0.4825i 1.5886-0.5104i 1.5886-0.5104i
1 0 1.6468-0.3575i 1.6557-0.3593i 1.6557-0.3593i 1 0 2.1226-0.4825i 2.1369-0.4882i 2.1369-0.4882i
1 1 1.4621-1.1141i 1.4941-1.1081i 1.4941-1.1081i 1 1 1.7977-1.5148i 1.8602-1.5197i 1.8602-1.5197i
2 0 2.1237-0.3565i 2.1282-0.3568i 2.1282-0.3568i 2 0 2.6667-0.4819i 2.6752-0.4824i 2.6752-0.4824i
2 1 1.9865-1.0905i 2.0037-1.0867i 2.0037-1.0867i 2 1 2.4248-1.4788i 2.4613-1.4728i 2.4613-1.4728i
2 2 1.7385-1.8755i 1.7499-1.8733i 1.7499-1.8733i 2 2 1.9778-2.5675i 2.0051-2.5593i 2.0051-2.5593i
3 0 2.5979-0.3559i 2.6003-0.3559i 2.6003-0.3559i 3 0 3.2055-0.4811i 3.2103-0.4809i 3.2103-0.4809i
3 1 2.4891-1.0802i 2.4990-1.0780i 2.4990-1.0780i 3 1 3.0132-1.4630i 3.0353-1.4574i 3.0353-1.4574i
3 2 2.2845-1.8379i 2.2927-1.8346i 2.2927-1.8346i 3 2 2.6446-2.5031i 2.6659-2.4884i 2.6659-2.4884i
3 3 2.0038-2.6373i 1.9791-2.6610i 1.9791-2.6610i 3 3 2.1348-3.6254i 2.0774-3.6483i 2.0774-3.6483i
4 0 3.0710-0.3555i 3.0725-0.3555i 3.0725-0.3555i 4 0 3.7422-0.4805i 3.7452-0.4803i 3.7452-0.4803i
4 1 2.9808-1.0748i 2.9869-1.0736i 2.9869-1.0736i 4 1 3.5823-1.4546i 3.5963-1.4508i 3.5963-1.4508i
4 2 2.8077-1.8174i 2.8134-1.8150i 2.8134-1.8150i 4 2 3.2705-2.4682i 3.2860-2.4568i 3.2860-2.4568i
4 3 2.5640-2.5920i 2.5492-2.6021i 2.5492-2.6021i 4 3 2.8265-3.5433i 2.7943-3.5431i 2.7943-3.5431i
4 4 2.2621-3.4002i 2.1950-3.4647i 2.1950-3.4647i 4 4 2.2767-4.6878i 2.1118-4.7819i 2.1118-4.7819i
5 0 3.5438-0.3552i 3.5448-0.3552i 3.5448-0.3552i 5 0 4.2782-0.4800i 4.2801-0.4799i 4.2801-0.4799i
5 1 3.4667-1.0716i 3.4707-1.0709i 3.4707-1.0709i 5 1 4.1410-1.4495i 4.1502-1.4471i 4.1502-1.4471i
5 2 3.3169-1.8051i 3.3209-1.8036i 3.3209-1.8036i 5 2 3.8711-2.4475i 3.8824-2.4397i 3.8824-2.4397i
5 3 3.1025-2.5636i 3.0932-2.5684i 3.0932-2.5684i 5 3 3.4802-3.4920i 3.4614-3.4866i 3.4614-3.4866i
5 4 2.8329-3.3502i 2.7864-3.3853i 2.7864-3.3853i 5 4 2.9864-4.5939i 2.8744-4.6353i 2.8744-4.6353i
5 5 2.5157-4.1645i 2.4036-4.2794i 2.4036-4.2794i 5 5 2.4079-5.7543i 2.1250-5.9489i 2.1250-5.9489i
Table 12: Low-lying (, with ) spin-3/2 field quasi-normal frequencies using the WKB and the AIM with D = 6,7 and Q=M.

V Absorption probabilities

In this section we present the absorption probabilities associated with our spin-3/2 fields near a Reissner-Nordström black hole. We use the same approach as in Ref.Chen2016 (), and as such only present the analysis of the results here, and refer the reader to Ref. Chen2016 () for the implementation of the method.

v.1 Non-TT eigenmodes related

In Fig. 6 we see that for a specific and , the behavior of the absorption probability shifts from lower energy to higher energy as increases, and this trend is similar to the Schwarzschild case. For fixed and , we can compare the scale of each subplot and realize a lower energy to higher energy shift as increase. For a fixed and , the absorption probability also shifts from left to right as increases. This is due to the maximum value of the corresponding potential increasing as increases, as shown in Fig. 3 for the case of , . An exception is in , case, where the curve shifts to the left instead. This is because the maximum value of the potential decreases instead of increasing as