Dirac quasinormal modes of Born-Infeld black hole spacetimes

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Hong Ma and Jin Li. Dirac quasinormal modes of Born-Infeld black hole spacetimes[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/9/095102
Hong Ma and Jin Li. Dirac quasinormal modes of Born-Infeld black hole spacetimes[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/9/095102 shu
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Dirac quasinormal modes of Born-Infeld black hole spacetimes

    Corresponding author: Hong Ma, michaelmahong@126.com
  • College of Physics and Electronic Information Engineering, Qinghai Normal University, Xining 810000, China and Joint Research Center for Physics, Lanzhou University and Qinghai Normal University, Xining, 810000, China
  • Department of Physics, Chongqing University, Chongqing 401331, China

Abstract: Quasinormal modes (QNMs) for massless and massive Dirac perturbations of Born-Infeld black holes (BHs) in higher dimensions are investigated. Solving the corresponding master equation in accordance hypergeometric functions and the QNMs are evaluated. We pay more attention to discuss the relationships between QNM frequencies and spacetime dimension. Meanwhile, we discuss the stability of the Born-Infeld BH by calculating the temporal evolution of the perturbation field. Both the perturbation frequencies and the decay rate increase with the enhance of the dimension of spacetime n. This shows that the Born-Infeld BHs become more and more unstable in higher dimensions. Furthermore, the traditional finite difference method is improved, so that it can be used to calculate massive Dirac field. And we elucidate the dynamical evolution of Born-Infeld BHs in massive Dirac field. Because the number of extra dimensions is related to the string scale, there is a relationship between spacetime dimension n and the properties of Born-Infeld BHs which might be advantageous to the development of extra-dimensional brane worlds and string theory.


    1.   Introduction
    • The first gravitational waves (GWs) are directly detected from the coalescence of binary black holes (BBHs) [1-8] by Advanced LIGO [9] and Virgo [10]. The recent experimental precision satisfies the requirement of general relativity theory, but the observations precision of GWs also leaves the window for alternative theories of gravity open [11]. In addition, the Event Horizon Telescope (EHT) collaboration recently released the first shadow image of a supermassive BH M87 [12, 13] in the galaxy. These have proved the existence of BHs in the universe, thus giving birth to a new era of the astronomy of astrophysical compact objects [14]. The interaction between two BHs can be conditionally divided into three phases: the inspiral phase [15], the merger phase [16], the ringdown stage [17]. The portion of the GWs signal associated with the single BH oscillations is referred to as the ringdown phase, as the perturbed BH rings down akin to a struck bell. The ringdown phase is the brief oscillation stage before the newly formed black hole reaches its final stable state. Damping times and frequencies associated with a given BH are known as quasinormal modes (QNMs) [18]. Therefore, QNMs play a central role in the final stage, ringdown phase, from the coalescence of BBHs [19-23].

      In addition, BHs are nonlinear solutions to a highly nonlinear theory. It is always a difficult task to study their dynamics challenge. Perturbation method is an effective tool to study the interaction between BH and basic test field. The study of black hole perturbation started with Regge and Wheeler's [24] analysis of perturbation of axisymmetric gravitational field. Subsequently, Zerilli [25-27] studied polar symmetric perturbation. The BH perturbations were then systematically summarized in Chandrasekhar’s monograph [28]. The disturbed BH can be considered as a dissipative system, and the perturbation has a discrete spectrum. In consequence, QNM frequencies are complex numbers, giving also damping of the oscillations. The imaginary part of the frequencies represents the decay of the amplitude and the real part corresponds to the oscillations of the perturbations. The decay time scale and oscillation frequency only depend on the spacetime background of the BH, independent of the initial disturbance. Since specific black holes have specific QNMs, which can well reflect the properties of spacetime of BHs. It is graphically known as the characteristic sound of a BH, which become a powerful tool to reveal the intrinsic properties of BHs. A good review for QNM theory can be found in [17, 18, 29, 30].

      Therefore, the mass, angular momentum and charge of the BH can be determined by detecting the quasinormal frequencies and damping rates. Even QNMs can also be used to test the no-hair theorem and quantization of black holes [31, 32]. In addition, in gauge/gravity duality theory, there is a connection between QNMs and the poles of propagators in dual field theory, which is why physicists use it as a tool to study strongly coupled gauge theory (or holography) [33, 34]. Therefore, there has been great attention to modified theories of gravity.

      In this paper, the charged BHs in the Born-Infeld gravity are investigated. In 1930, born and infield proposed the nonlinear theory of electrodynamics and obtained the self-energy of a finite point charge in a nonlinear system [35]. The main motivation is to observe it occurring in D-branes and open superstrings. The low energy efficiency of open superstrings leads to the Born-Infeld type action [36]. Along with the development of superstring theory, the dynamics of some super-gravity soliton solutions D-branes [37] are controlled by the interaction of born-field action. Whereafter, Garcia et al. obtained the Born-Infeld black hole solution [38]. And it has extended to nonlinear charged black hole in general relativity, characterized by spacetime dimension n, mass M, charge Q and nonlinear parameter $ \beta $. Major goal of our research in this article is to explore physical characteristics of Born-Infeld BHs, these spacetimes are perturbed and the QNMs generated are probed by the perturbation. Most of the research on the QNMs of the Born-Infeld BHs are focused on scalar fields, electromagnetic fields and gravitational field perturbations (that is, fields with integer spin). Fernando [57] calculated the gravitational perturbation of charged black hole under the Born-Infeld gravity. Liu et al. [40] studied the QNMs of the scalar field interacting with the electromagnetic field of the Born-Infeld AdS BH. To make the study more completes, based on the frame in high dimensional spherically symmetric BHs [41, 42], we study the spacetime structure and the QNMs for massive and massless Dirac fields of Born-Infeld BHs. Using the WKB method, the effect of the dimension of spacetime n, charge Q and the multipole numbers $ |k| $ on the QNMs of Born-Infeld BHs are studied. Specially, the dynamical evolution of the Dirac perturbation fields in Born-Infeld spacetime is investigated by the finite difference method. The results show that quasinormal behavior of Born-Infeld BHs depends on the dimension of spacetime n.

      The structure of this article is as follows: Section II explains the research background of the paper, which is mainly based on the QNMs under Dirac field perturbation. We study the stability of the Born-Infeld BH with the finite difference method in Section III. In addition, we utilize the WKB method to calculate the QNMs numerically in Section IV, including two parts: Part A, QNMs for massless fields are calculated. Part B, the QNMs of Born-Infeld BHs in the massive case are analyzed. In the last section, the important consequences and expectations are showed.

    2.   Spherically symmetric static born-infeld bh solutions
    • The general action describing Born-Infeld interaction in a $ (n+1) $-dimensional ($ n \geqslant 3 $) background without cosmological constant $ \Lambda $ has the form [41, 44]

      $ S = \int d^{n+1}x\sqrt{-g}\left(\frac{R}{16\pi G}+L(F)\right), $


      here R is scalar curvature, Born-Infeld $ L(F) $ part of action is decomposed as

      $ L(F) = 4\beta^{2}\left(1-\sqrt{1+\frac{F^{\mu\nu}F_{\mu\nu}}{2\beta^{2}}}\right ). $


      Where $ L(F) $ is a function of the electrodynamic field strength $ F_{\mu \nu} $ of Born-Infeld. $ \beta $ is BI parameters with dimensions $ length^{-(n+1)/2} $. For simplicity, let's assume $ 16\pi G = 1 $. In the limit $ \beta\rightarrow\infty $, Born-Infeld $ L(F) $ tends to be Maxwell's electrodynamics with $ -F^2 $, and $ L(F) $ is

      $ L(F) = -F^{\mu\nu}F_{\mu\nu}+{\cal O}(F^{4}). $


      By changing the action of gauge field $ A_{\mu} $ and metric field $ g_{\mu\nu} $, the equations of motion and Einstein equations of electromagnetic field can be derived

      $ \partial_{\mu}\left (\frac{\sqrt{-g}F^{\mu\nu}}{\sqrt{1+\dfrac{F^{\mu\nu}F_{\mu\nu}}{2\beta^{2}}}}\right ) = 0, $


      $ {R}_{\mu\nu}-\frac{1}{2}{R}g_{\mu\nu} = \frac{1}{2}g_{\mu\nu}L(F)+ \frac{2F_{\mu \alpha}F^{\; \alpha}_{\nu}}{\sqrt{1+\dfrac{F^{\mu\nu}F_{\mu\nu}}{2\beta^{2}}}}. $


      Assuming metric ansatz is in this form

      $ ds^{2} = -f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+ R^{2}(r)h_{ij}dx^{i}dx^{j}, $


      $ h_{ij} $ is a function of coordinates $ x^{i} $. It spans a hypersurface whose curvature in $ (n-1) $-dimensional is scalar curvature $ (n-1)(n-2)k $. $ f(r) $ and $ R(r) $ are two functions of r. For the metric (6), non-vanishing components of Ricci tensor are obtained as [45],

      $ {R}^{t}_{t} = -\frac{f''}{2} - (n-1)\frac{f'R'}{2R}, $


      $ {R}^{r}_{r} = -\frac{f''}{2}-(n-1)\frac{f'R'}{2R}-(n-1)\frac{fR''}{R}, $


      $ {R}^{i}_{j} = \left( \frac{n-2}{R^2}k -\frac{1}{(n-1)R^{n-1}}[f(R^{n-1})']'\right)\delta^{i}_{j}. $


      This is the derivative of a prime in regard to the coordinate r.

      Except for $ F^{rt} $, the set of $ F^{\mu\nu} $ equals $ 0 $ satisfying the class of solutions to the eq.(4), which yields

      $ F^{rt} = \frac{\sqrt{(n-1)(n-2)}\beta q}{\sqrt{2\beta^{2}R^{2n-2}+(n-1)(n-2)q^{2}}}, $


      here q is an integral constant associated with the electrodynamic charge. The electric charge defined by $ Q = \dfrac{1}{4\pi} \int \ ^*F d\Omega $, we have

      $ Q = \frac{\sqrt{(n-1)(n-2)}\omega_{n-1}} {4\pi\sqrt{2}}q, $


      here $ \omega_{n-1} $ is the volume of the hypersurface with curvature defined as $ h_{ij}dx^idx^j $. There is $ F^{rt} $ in the $ \beta $ limit as $ F^{rt} \sim \dfrac{q}{r^{n-1}} $. Electric field is finite at $ r = 0 $. The cosmological constant is redefined as $ \Lambda = 0 $, solving the equation (6) yields

      $ \begin{split} f(r) =& k -\frac{m}{r^{n-2}} +\left( \frac{4\beta^2r^2}{n(n-1)} \right) \\ &-\frac{2\sqrt{2} \beta}{(n-1)r^{n-2}}\int \sqrt{2\beta^2r^{2n-2} +(n-1)(n-2)q^2 }dr. \end{split}$


      For $ k = 1 $, the integral in (12) can be represented in terms of hypergeometric functions,

      $ \begin{split} f(r) =& 1-\frac{m}{r^{n-2}}+\left(\frac{4\beta^{2} r^{2}}{n(n-1)}\right) \\ &-\frac{2\sqrt{2}\beta}{n(n-1)r^{n-3}}\sqrt{2\beta^{2}r^{2n-2}+(n-1)(n-2)q^{2}} \\ &+\frac{2(n-1)q^{2}}{nr^{2n-4} } \ _2F_{1}[\frac{n-2}{2(n-1)},\frac{1}{2}, \frac{3n-4}{2(n-1)}, -\frac{(n-1)(n-2)q^{2}}{2\beta^{2}r^{2n-2}}]. \end{split}$


      Here m is integral constant that is depended on the configured ADM mass M. This is given by

      $ M = (n-1)\omega_{n-1}m. $


      The analytic function within $ |z| <1 $ can be parsed to the full z plane. Therefore, $ _2F_1(a,b,c,z) $ is a single-valued analytic function which expands along the real number line in the z plane [45]. For $ |z| <1 $, $ _2F_1(a,b,c,z) $ has a convergent series expansion. This gives

      $ f(r) = 1 - {m\over{r^{n-2}}} + {q^2\over{r^{2n-4}}} - {(n-1)(n-2)^2 q^4\over{8 \beta^2 (3n -4)r^{4n-6}}}, $


      when $ \beta \rightarrow \infty $ and $ n = 3 $ the function $ f(r) $ describes Maxwell's electrodynamics Reissner-Norström(RN) BH [46, 47]. These black holes are described in Table 1. Meanwhile, the spacetime structures of Born-Infeld BHs are exhibited in Fig. 1(a) and Fig. 1(b) ($ |z| <1 $). For the asymptotically flat spacetime, the spacetime of Born-Infeld goes flat at inflnity faster with increasing spacetime dimensions of n.

      ${\rm{EBI}}\;{\rm{BH}}$$f(r)= 1-\frac{m}{r^{n-2}}+\frac{4{\beta}^2r^2}{n(n-1)} -{{\frac{2{\sqrt2}\beta}{(n-1)r^{n-2}}{\int{\sqrt{2{\beta}^2 r^{2n-2}+(n-1)(n-2)q^2}}dr}}}$
      $|z<1|$$f(r) = 1 - {m\over{r^{n-2}}} + {q^2\over{r^{2n-4}}} - {(n-1)(n-2)^2 q^4\over{8 \beta^2 (3n -4)r^{4n-6}}}$
      $|z<1|$, $\beta \rightarrow \infty$$f(r) = 1 - {m\over{r^{n-2}}} + {q^2\over{r^{2n-4}}}$
      $|z<1|$, $\beta \rightarrow \infty$, $n=3$$f(r)_{RN} = 1 -\frac{m}{r}+ {q^2\over{r^{2}}}$

      Table 1.  It shows a summary of Born-Infeld BHs. More details can be found in Refs [41, 44-49].

      Figure 1.  The structures of the Born-Infeld metric function $f(r)$ with different spacetimes dimension n: (from top to bottom: $n=3, 4, 5, 6, 7, 8, 9, 10$) (a)Born-Infeld metric spacetimes with $M=1$, $Q=0.1$, $b=0.1$ and $r_+=1$. (b)Born-Infeld metric spacetimes with $|z|<1$, $M=1$, $Q=0.1$, $b=0.1$ and $r_+=1$.

    3.   Dirac perturbation in the einstein-born-infeld spacetime
    • The general equation for a massive Dirac spinor field in the high-dimensional spacetime can be expressed as [50, 51]

      $ [\gamma^{\mu}D_{\mu}+\frac{m}{\hbar}]\Psi = 0, \qquad \mu = t, r, \theta, \phi,x^{\eta},\cdots, {x}^{n+1}, $


      where $ {x}^{n+1} $ is extra-dimensional coordinates,

      $ D_{\mu} = \partial_{\mu}+\frac{i}{2}{\Gamma^{\alpha}\,_{\mu}}^{\beta}\Pi_{\alpha\beta}, $


      here $ \Pi_{\alpha\beta} = \dfrac{i}{4}[\gamma^{a},\gamma^{b}] $ and $ \Gamma_{\mu} = \dfrac{1}{8}[\gamma^{a},\gamma^{b}]e_{a}^{\nu}e_{b\nu;\mu} $ is spin connection, the gamma matrices satisfy the condition that,

      $ \{\gamma^{\mu}, \gamma^{\nu}\} = 2g^{\mu\nu}I, $


      the spior wave function $ \Psi $ in Eq.(16) can be written as

      $ \Psi = \left[ \begin{array}{c} A_{(m/2) \times 1}(t,r,\theta,\phi,x^{\eta},\cdots x^{n+1}) \\ B_{(m/2) \times 1}(t,r,\theta,\phi,x^{\eta},\cdots x^{n+1}) \\ \\ \end{array}\right] e^{(-i\omega t)S(t,r,\theta,\phi,x^{\eta},\cdots x^{n+1})} , $


      the solution form of a statically spherically symmetric BH is given by Eq.(6)

      $ e_{\nu}^{a} = {\rm diag}(f(r)^{1/2},f(r)^{-1/2},r,r\sin \theta, x^{\eta}, \cdots, x^{n+1}). $


      Where $ f(r) $ represents the Born-Infeld spacetime function listed in Table 1. The Dirac spinor can be written as,

      $ \Psi = f(r)^{-1/4}\Phi, $


      higher-dimensional spacetime, then the Dirac equation simplifies to

      $ \left[\frac{1}{f}\left(\frac{\partial S}{\partial t}\right)+f\left(\frac{\partial S}{\partial r}\right)+g^{\theta\theta}\left(\frac{\partial S}{\partial \theta}\right) +g^{\phi\phi}\left(\frac{\partial S}{\partial \phi}\right)+g^{\eta\eta}\left(\frac{\partial S}{\partial x^{\eta}}\right)+ \cdots+g^{x^{n+1}x^{n+1}}\left(\frac{\partial S}{\partial x^{n+1}}\right)+m\right]\Phi = 0. $


      Wave functions must be defined separately as follows [52, 53].

      $ \begin{split}&\Phi(t,r,\theta,\phi, x^{\eta}, \cdots, x^{n+1}) = \\&\quad\left( \begin{array}{c} \dfrac{iG^{\pm}(r)}{r}\varphi^{\pm}_{jm}(\theta,\phi) H(x^{\eta}, \cdots, x^{n+1})\\ \dfrac{F^{\pm}(r)}{r}\varphi^{\mp}_{jm}(\theta,\phi)H (x^{\eta}, \cdots, x^{n+1})\\ \end{array}\right) e^{-i\omega t}, \end{split}$



      $ \varphi^{+}_{jm} = \left( \begin{array}{c} \sqrt{\dfrac{l+1/2+m}{2l+1}}Y^{m-1/2}_{l} \\ \sqrt{\dfrac{l+1/2-m}{2l+1}}Y^{m+1/2}_{l} \\ \end{array} \right) \; \; \; \left({\rm{for}}\; j = l+\frac{1}{2}\right), $


      $ \varphi^{-}_{jm} = \left( \begin{array}{c} \sqrt{\dfrac{l+1/2-m}{2l+1}}Y^{m-1/2}_{l} \\ -\sqrt{\dfrac{l+1/2+m}{2l+1}}Y^{m+1/2}_{l} \\ \end{array} \right) \; \; \; \left({\rm{for}}\; j = l-\frac{1}{2}\right). $


      Radial functions ($ G^{\pm} $ and $ F^{\pm} $) are considered. Eq.(22) is rewritten as

      $\begin{split}& \frac{d}{{d{r_*}}}\left( {\begin{array}{*{20}{c}} {{F^ \pm }}\\ {{G^ \pm }} \end{array}} \right) - \sqrt {f(r)} \left( {\begin{array}{*{20}{c}} {{k_ \pm }/r}&m\\ m&{ - {k_ \pm }/r} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{F^ \pm }}\\ {{G^ \pm }} \end{array}} \right) =\\&\quad \left( {\begin{array}{*{20}{c}} 0&{ - \omega }\\ \omega &0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{F^ \pm }}\\ {{G^ \pm }} \end{array}} \right),\end{split}$


      where $ d/dr_{*} = f(r)d/dr $. Let's do some transformation of $ F^{\pm} $ and $ G^{\pm} $ [50],

      $ \left( {\begin{array}{*{20}{c}} {{{\hat F}^ \pm }}\\ {{{\hat G}^ \pm }} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\sin \dfrac{\theta }{2}}&{\cos \dfrac{\theta }{2}}\\ {\cos \dfrac{\theta }{2}}&{ - \sin \dfrac{\theta }{2}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{F^ \pm }}\\ {{G^ \pm }} \end{array}} \right), $


      where $ \theta = \tan^{-1}(mr/|k|) $. Introducing $ \hat{r}_{*} = r_{*}+\tan^{-1}(mr/|k|)/2\omega $ into Eq.(26), then

      $ \frac{d}{{d{{\hat r}_*}}}\left( {\begin{array}{*{20}{c}} {{{\hat F}^ \pm }}\\ {{{\hat G}^ \pm }} \end{array}} \right) + {W_ \pm }\left( {\begin{array}{*{20}{c}} { - {{\hat F}^ \pm }}\\ {{{\hat G}^ \pm }} \end{array}} \right) = \omega \left( {\begin{array}{*{20}{c}} {{{\hat G}^ \pm }}\\ { - {{\hat F}^ \pm }} \end{array}} \right),$



      $ W_{\pm} = \frac{\sqrt{f(r)\left(\frac{k_{\pm}^{2}}{r^{2}}+m^{2}\right)}}{1+\dfrac{f(r)m|k_{\pm}|}{2\omega(k_{\pm}^{2}+m^{2}r^{2})}}, $



      $ \frac{d}{d\hat{r}_{*}} = \frac{f(r)}{1+\dfrac{m|k_{\pm}|f(r)}{2\omega(k^{2}+m^{2}r^{2})}}\frac{d}{dr}. $


      The following derivation radial functions written as

      $ \left(-\frac{d^{2}}{d\hat{r}^{2}_{*}}+\bar{V}_{\pm}\right)\hat{F}^{\pm} = \omega^{2}\hat{F}^{\pm}, $


      $ \left(-\frac{d^{2}}{d\hat{r}^{2}_{*}}+\tilde{V}_{\pm}\right)\hat{G}^{\pm} = \omega^{2}\hat{G}^{\pm}, $



      $ \bar{V}_{\pm} = \frac{dW_{\pm}}{d\hat{r}_{*}}+W^{2}_{\pm}, $



      $ \tilde{V}_{\pm} = -\frac{dW_{\pm}}{d\hat{r}_{*}}+W^{2}_{\pm}. $


      Here $ k_\pm $ is the number associated with the angular momentum quantum number l, $ k_{-} = -l $ for $ j = l-1/2 $ and $ k_{+} = l+1 $ for $ j = l+1/2 $. In the following context, we only consider $ j = l-1/2 $. Moreover, in spherically symmetric EBI spacetime, Dirac particles and antiparticles have the same QNMs. Therefore the radial function $ \hat{F}^- $ can represent all the physics of Dirac field evolution.

    4.   Qnms of massless dirac perturbations in born-infeld spacetime

      4.1.   The QNMs frequency calculated by WKB

    • In this section the QNM frequency is calculated, the properties of the effective potential need to be determined. In the massless case, $ \hat{F}^- $ was chosen as an example, thus

      $ \left(-\frac{d^{2}}{d\hat{r}^{2}_{*}}+V\right)\hat{F}^{-} = \omega^{2}\hat{F}^{-}, $



      $ V = \tilde{V}_{-}|_{m = 0} = -\frac{dW_{-}}{d\hat{r}_{*}}+W^{2}_{-}, $



      $ W_{-} = \sqrt{f(r)}\frac{|k_{-}|}{r}. $


      By simplifying the symbols, the '$ - $' subscripts and superscripts of the rest are removed. $ r_+ = 1 $ is used.

      In order to calculate the QNMs frequency of the BH, the properties of the potential function $ V(r) $ are first considered. The Dirac effective potential of Born-Infeld BH is in the form of a barrier, $ V(r) $ depends on $ |k| $, spacetime dimensions n, the Born-Infeld parameter b and charge Q. Fig. 2(a) shows the variation of potential function $ V(r) $ with space-time dimension n of Born-Infeld BHs under massless Dirac field perturbation. As we seen Fig. 2(a) shows the peak value of the barrier grow when the spacetime dimensions n increases. Meanwhile, as shown in Fig. 2(b), as $ |k| $ increases, the peak of the barrier becomes higher and higher. From Fig. 2(b), we can find the position of the peak respectively. Similarly, The Born-Infeld BHs ($ |z|<1 $) potential function has the same property, even slight differences are mainly caused by the hyperfunction.

      Figure 2.  Behavior of Born-Infeld BH potential under massless condition: (a) different values of n (from top to bottom: $n=3, 4, 5, 6, 7, 8, 9, 10$). (b) different values of $|k|$ (from top to bottom: $|k|=1, 2, 3, 4, 5$).

      $ r_{max}(|k|\longrightarrow \infty )\longrightarrow 1.34, $


      and Born-Infeld BHs $ |z|<1 $

      $ r_{max}(|k|\longrightarrow \infty )\longrightarrow 1.26. $


      In order to numerically calculate the quasinormal mode frequencies of Born-Infeld BHs, we adopt the WKB approximation developed by Schutz, Will, and Iyer [54-56]. The values for different spacetime dimensions n are listed in Table 2, Table 3 and Table 4, here M is the mass of Born-Infeld BH, Q is electric charge, N is the overtone number, $ |k| $ represents the related to angular quantum number values and b is Born-Infeld parameter. Here spacetime dimensions n in the fundamental mode ($ N = 0 $) is focused. When $ \beta \rightarrow \infty $ and $ n = 3 $, the Born-Infeld BHs returns to RN BHs, and we compare the results in BI black holes with RN black holes. The results indicates that $ {\rm{Re}}(\omega) $ and $ |{\rm{Im}}(\omega)| $ both enhances as the spacetime dimensions n increases. It shows that the QNMs with higher spacetime dimensions the rate of decay is accelerating than the low dimension ones. For another, $ {\rm{Re}}(\omega) $ of the frequencies enhances as angular momentum number $ |k| $ enhances with same spacetime dimensions n. But the magitude of the imaginary part $ |{\rm{Im}}(\omega)| $ is the opposite. Furthermore, $ {\rm{Re}}(\omega) $ and $ |{\rm{Im}}(\omega)| $ both increase significantly with increasing charge Q while $ |k| $ is constant. The results show that the damping of QNMs is affected by the amount of charge Q. In addition, gravitational perturbations of Born-Infeld BHs are studied in Refs. [57]. The QNM for the gravitational perturbations are computed also using the WKB method. It is interesting to note that although we chose a different range of parameters to calculate, we came to a consistent conclusion: for the Born-Infeld BHs when the charge increases, the imaginary part of the QNM continue to increase.

      $ n$$|k|=3$$|k|=4$$|k|=5$

      Table 2.  Fundamental modes ($N=0$, $r_{+}=1$, $b=0.1$, $Q=0.1$) of Dirac perturbations calculated by WKB method


      Table 3.  Fundamental modes ($N=0$, $r_{+}=1$, $b=0.1$, $Q=0.2$) of Dirac perturbations calculated by WKB method


      Table 4.  Fundamental modes ($N=0$, $r_{+}=1$, $b=0.1$, $Q=0.3$) of Dirac perturbations calculated by WKB method

    • 4.2.   The QNMs frequency calculated by the finite difference method

    • Here, we use the finite difference method [58, 59] to illustrate the dynamic evolution of Born-Infeld BHs. We study the ringing of BHs spacetimes, which can directly reflect the (in)stability of the Born-Infeld BHs in the temporal evolution images with all the frequencies. Hence, using a numerical integration scheme [60], Eq.(28) can be expressed in light-cone coordinates:

      $ \omega G^{\pm} = -\frac{f(r)dF^{\pm}}{dr}+\sqrt{f(r)}\frac{k}{r}F^{\pm}+m\sqrt{f(r)}G^{\pm}, $


      $ \omega F^{\pm} = \frac{f(r)dG^{\pm}}{dr}+\sqrt{f(r)}\frac{k}{r}G^{\pm}-m\sqrt{f(r)}F^{\pm}. $


      Multiply both sides by $ \omega $, Eq.(40) and Eq.(41) are rewritten as

      $ \omega^{2} G^{\pm} \!=\! -\frac{f(r)d(\omega F^{\pm})}{dr}\!+\!\sqrt{f(r)}\frac{k}{r}(\omega F^{\pm})\!+\!m\sqrt{f(r)}(\omega G^{\pm}), $


      $ \omega^{2} F^{\pm} \!=\! \frac{f(r)d(\omega G^{\pm})}{dr}\!+\!\sqrt{f(r)}\frac{k}{r}(\omega G^{\pm})\!-\!m\sqrt{f(r)}(\omega F^{\pm}). $


      Putting Eq.(40) and (41) into Eq.(42) and (43),

      $ \begin{split} \frac{m\sqrt{f}}{2}f'F^{\pm} =& f^{2}G''^{\pm}+ff'G'^{\pm}\\&+\left[\frac{kf'}{2r}\sqrt{f}-\frac{k^{2}}{r^{2}}f- \frac{k}{r^{2}}f^{3/2}+\omega^{2}-m^{2}f\right]G^{\pm}; \\ \frac{m\sqrt{f}}{2}f'G^{\pm} =& f^{2}F''^{\pm}+ff'F'^{\pm}\\&+\left[-\frac{kf'}{2r}\sqrt{f}-\frac{k^{2}}{r^{2}}f+ \frac{k}{r^{2}}f^{3/2}+\omega^{2}-m^{2}f\right]F^{\pm}. \end{split}$


      Here $ ' $ means $ \partial/\partial r $, $ \omega^{2} = -\partial^{2}/\partial^{2} t $. $ (t,r) \rightarrow (\mu,\nu) $ is applied, where $ \mu = t-r_{*},\nu = t+r_{*} $, yielding

      $ \frac{\partial}{\partial r} = \frac{1}{f}\left(\frac{\partial}{\partial \nu}-\frac{\partial}{\partial \mu}\right),\; \; \; \; \; \; \frac{\partial}{\partial t} = \frac{\partial}{\partial \nu}+\frac{\partial}{\partial \mu}. $


      $ F^{\pm}(\mu,\nu) $ and $ G^{\pm}(\mu,\nu) $ are deduced by Eq.(44), which can be integrated by the finite difference method [61-63]. In Fig. 3 and Fig. 4 the plus '$ + $' and minus '$ - $' signs of $ F^{\pm}(\mu,\nu) $ are unified to be F.

      Figure 3.  The dynamical evolution of Born-Infeld BHs with varying spacetime dimensions n (from top to bottom: $n=3, 4, 5, 6, 7, 8, 9, 10$): (a) show Born-Infeld BHs with $M=1$, $Q=0.1$, $b=0.1$, $|k|=3$ and $r_+=1$. (b) show Born-Infeld BHs with $|z|<1$, $M=1$, $Q=0.1$, $b=0.1$, $|k|=1$ and $r_+=1$.

      Figure 4.  The dynamical evolution of Born-Infeld BHs with various $|k|$.

      Fig. 3 illuminates the evolution of Dirac field in Born-Infeld black holes with $ |k| = 3 $, $ Q = 0.1 $ and $ n = 3, 4, 5, 6, 7, 8, 9, 10 $. The effects of different spacetime dimensions of n on the Born-Infeld BHs are concentrated. The absolute value of imaginary parts of quasinormal frequencies enhance as spacetime dimensions of BH increase to 10. It shows that it takes less time for QNMs to completely decay outside the Born-Infeld BHs. In other words, it takes less time to restore equilibrium under perturbation. It indicates that it will affect the QNMs of Born-Infeld BHs with the increase of spacetime dimensions n. Furthermore, Fig. 4 describes that given n, $ {\rm{Re}}(\omega) $ of the frequency increases but $ |{\rm{Im}}(\omega)| $ of the frequency decreases with the increase of $ |k| $. Note that the oscillation frequency of QNMs is faster but the decay is slower.

    5.   Qnms of massive dirac perturbations in born-infeld spacetime
    • Next we exactly calculate the QNMs of Born-Infeld BHs in massive Dirac field. In the massive case, potential function $ V(r) $ depends not only on the mass m of the Dirac field, but also on $ \omega $. That makes it more complicated to utilize the WKB method to calculate the quasinormal mode frequencies. Fig. 5 shows the dependence of $ V(r) $ on m for $ \omega = 1 $. The effective potential functions are still in the form of a barrier. Fig. 5 shows peak of the potential function $ V(r) $ increase with the enhances of m, $ V(r) $ exhibits the following

      Figure 5.  Behavior of Born-Infeld BH potential under massive condition.

      $ V(r\rightarrow\infty) = m^{2}. $


      As m increases, the peak potential is slowly increasing. Eventually, when $ r\rightarrow \infty $ the summit of the peak value is less than the asymptotic value $ m^{2} $. It is important to note that $ \omega $ is unknown at the beginning of the calculation (it is considered a typical value of $ \omega = 1 $) and must be self-consistently determined. Consequently, Fig. 5 can only represent the general behavior of the potential $ V(r) $.

      The QNM frequency of massive Dirac field is calculated by WKB method. Table 5, Table 6 and Table 7 are listed the values of the QNMs frequencies of Born-Infeld BHs for the parameter range $ b = 0.1 $, $ Q = 0.1 $. As it is shown in the table, when the spacetime dimension n increases, the real and imaginary parts of the quasinormal frequency increase. In addition, the rate of oscillation $ {\rm{Re}}(\omega) $ increases, but the rate of decay $ |{\rm{Im}}(\omega)| $ decreases as $ |k| $ increases. As mass m enhances, the real part of $ {\rm{Re}}(\omega) $ decreases and the imaginary part of $ |{\rm{Im}}(\omega)| $ increases. The results show that the frequency oscillation of QNMs is slower and the decay is faster. One possible explanation is that when a disturbance occurs, the massive of particles are absorbed by the black hole, so the disturbance in the form of a black hole carries away the energy of the gravitational wave.


      Table 5.  $\omega$ in Born-Infeld BH ($r_{+}=1$, $Q=0.1$, $m=0.1$)


      Table 6.  $\omega$ in Born-Infeld BH ($r_{+}=1$, $Q=0.1$, $m=0.2$)


      Table 7.  $\omega$ in Born-Infeld BH ($r_{+}=1$, $Q=0.1$, $m=0.3$)

    6.   Conclusion
    • In this work, we have calculated the QNMs of the perturbation of the massless and massive Dirac field in the backgrounds of the Born-Infeld BH. Here the quasinormal mode frequencies are found out and tabulated using the WKB method, and the dynamic evolution of the Born-Infeld BHs is described by the finite difference method, which varies the multipole number $ |k| $ and spacetime dimension n.

      The findings of this article are as follows: For massless Dirac perturbations, when given Q, the oscillation frequency $ {\rm{Re}}(\omega) $ enhances but decay rate $ |{\rm{Im}}(\omega)| $ decreases slowly with the increase of $ |k| $. Meanwhile, the real part of the quasinormal mode frequencies $ {\rm{Re}}(\omega) $ and the imaginary part $ |{\rm{Im}}(\omega)| $ became larger with higher dimension.

      For massive Dirac perturbations of Born-Infeld BH, the potential function $ V(r) $ depends on the mass m and spacetime dimension n. The peak value of the potential function enhances with the increase of mass m. This implies that for the massive particle, the slower it oscillates, the faster it decays. The faster Born-Infeld BH oscillates in higher-dimensional spacetime, the faster it decays. The massive field particles themselves have energy, so they are absorbed by the black hole when disturbed. Although the disturbance will cause the black hole to take away the energy. The energy will be replenished, so the massive of the field particle, the slower the oscillation, but the faster the decay. In higher dimensions, the more energy the perturbation of the black hole will propagate outward, which will oscillate faster and decay faster. This is a description of the excited states of fermions (such as neutrinos) near the Born-Infeld black hole.

      In conclusion, the following conclusions can be obtained: The asymptotically late time oscillation doesn't depend too much on the field spin and the Born-Infeld parameter. They decay is according to the oscillation, which depends more on the number of extra dimensions and the multipole number.

      A natural extension of this work is to research the BHs in Anti-de Sitter spacetime. The development of string theory on AdS/CFT duality, and the relationship between Born-Infeld theory and string theory is the driving force behind this phenomenon. So we think it's worth understanding the various properties of BH solutions in this theory. In addition, we should broaden our study of the high-frequency part to fully understand its physical nature. These interesting ideas can be used as follow-up research in the near future.

Reference (63)



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