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Analytical solutions in a cosmic string Born–Infeld-dilaton black hole geometry: quasinormal modes and quantization

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Abstract

Chargeless massive scalar fields are studied in the spacetime of Born–Infeld dilaton black holes (BIDBHs). We first separate the massive covariant Klein–Gordon equation into radial and angular parts and obtain the exact solution of the radial equation in terms of the confluent Heun functions. Using the obtained radial solution, we show how one gets the exact quasinormal modes for some particular cases. We also solve the Klein–Gordon equation solution in the spacetime of a BIDBHs with a cosmic string in which we point out the effect of the conical deficit on the BIDBHs. The analytical solutions of the radial and angular parts are obtained in terms of the confluent Heun functions. Finally, we study the quantization of the BIDBH. While doing this, we also discuss the Hawking radiation in terms of the effective temperature.

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Acknowledgements

We wish to thank the Editor and anonymous Referee for their valuable comments and suggestions. I. S. is grateful to Dr. Edgardo Cheb-Terrab (Waterloo-Canada) for his valuable comments on the confluent Heun functions. A. Ö. acknowledges financial support provided under the Chilean FONDECYT Grant No. 3170035. A. Ö. is grateful to Prof. Robert Mann for hosting him as a research visitor at Waterloo University.

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Authors and Affiliations

  1. Physics Department, Faculty of Arts and Sciences, Eastern Mediterranean University, North Cyprus, Via Mersin 10, Famagusta, Turkey

    İzzet Sakallı & Ali Övgün

  2. Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo, Macedonia

    Kimet Jusufi

  3. Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4950, Valparaiso, Chile

    Ali Övgün

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  1. İzzet Sakallı
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Correspondence to İzzet Sakallı.

Appendix

Appendix

The confluent Heun equations is obtained from the general Heun equation [80, 111, 112] through a confluence process, that is, a process where two singularities coalesce, performed by redefining parameters and taking limits, resulting in a single (typically irregular) singularity. The confluent Heun equation is given by

$$\begin{aligned} {\frac{d^{2}U\left( y\right) }{d{y}^{2}}}+\left( {\widetilde{a}}+\frac{1+ {\widetilde{b}}}{y}-\frac{1+{\widetilde{c}}}{1-y}\right) {\frac{dU\left( y\right) }{dy}}+\left( \frac{{\widetilde{m}}}{y}-\frac{{\widetilde{n}}}{1-y} \right) U\left( y\right) =0. \end{aligned}$$
(A1)

and thus (A1) has three singular points: two regular ones: \(y=0\) and \( y=1\), and one irregular one: \(y=\infty \). Solution of (A1) is called the confluent Heun’s function: \(U\left( y\right) =\text{ HeunC }({\widetilde{a}} ,{\widetilde{b}},{\widetilde{c}},{\widetilde{d}},{\widetilde{e}};y)\), which is regular around the regular singular point \(y=0\). It is defined as

$$\begin{aligned} \text{ HeunC }({\widetilde{a}},{\widetilde{b}},{\widetilde{c}},{\widetilde{d}}, {\widetilde{e}};y)=\sum _{n=0}^{\infty }u_{n}({\widetilde{a}},{\widetilde{b}}, {\widetilde{c}},{\widetilde{d}},{\widetilde{e}})y^{n}, \end{aligned}$$
(A2)

which is the convergent in the disk \(|y|<1\) and satisfies the normalization \( \text{ HeunC }({\widetilde{a}},{\widetilde{b}},{\widetilde{c}},{\widetilde{d}}, {\widetilde{e}};0)=1\). The parameters \({\widetilde{a}},{\widetilde{b}},\widetilde{c },{\widetilde{d}},{\widetilde{e}}\) are related with \({\widetilde{m}}\) and \( {\widetilde{n}}\) as follows

$$\begin{aligned} {\widetilde{m}}= & {} \frac{1}{2}({\widetilde{a}}-{\widetilde{b}}-{\widetilde{c}}+ {\widetilde{a}}{\widetilde{b}}-{\widetilde{b}}{\widetilde{c}})-{\widetilde{e}}, \end{aligned}$$
(A3)
$$\begin{aligned} {\widetilde{n}}= & {} \frac{1}{2}({\widetilde{a}}+{\widetilde{b}}+{\widetilde{c}}+ {\widetilde{a}}{\widetilde{c}}+{\widetilde{b}}{\widetilde{c}})+{\widetilde{d}}+ {\widetilde{e}}. \end{aligned}$$
(A4)

The coefficients \(u_{n}({\widetilde{a}},{\widetilde{b}},{\widetilde{c}},\widetilde{ d},{\widetilde{e}})\) are determined by three-term recurrence relation:

$$\begin{aligned} A_{n}u_{n}=B_{n}u_{n-1}+C_{n}u_{n-2}, \end{aligned}$$
(A5)

with initial conditions {\(u_{-1}=0\,\),\(\,u_{0}=1\)} and we have

$$\begin{aligned} A_{n}= & {} 1+{\frac{{\widetilde{b}}}{n}}\,\rightarrow 1,\,\,\,\text { as}\,\,\,n\rightarrow \infty , \nonumber \\ B_{n}= & {} 1+{\frac{-{\widetilde{a}}+{\widetilde{b}}+{\widetilde{c}}-1}{n }}+{\frac{{\widetilde{e}}+({\widetilde{a}}-{\widetilde{b}}-{\widetilde{c}})/2+ {\widetilde{b}}/2\left( {\widetilde{c}}-{\widetilde{a}}\right) }{n^{2}}} \,\rightarrow 1,\,\,\,\text {as}\,\,\,n\rightarrow \infty , \nonumber \\ C_{n}= & {} {\frac{{\widetilde{a}}}{n^{2}}}\left( {\frac{{\widetilde{d}} }{{\widetilde{a}}}}+{\frac{{\widetilde{b}}+{\widetilde{c}}}{2}}+n-1\right) \,\rightarrow 0,\,\,\,\text {as}\,\,\,n\rightarrow \infty . \end{aligned}$$
(A6)

\(\text{ HeunC }({\widetilde{a}},{\widetilde{b}},{\widetilde{c}},{\widetilde{d}}, {\widetilde{e}};y)\) reduces to a polynomial of degree \(N\left( =0,1,2,\ldots \right) \) with respect to the variable y if and only if the following two conditions are satisfied [82]:

$$\begin{aligned} {\frac{{\widetilde{d}}}{{\widetilde{a}}}}+{\frac{{\widetilde{b}}+{\widetilde{c}}}{2}} +N+1= & {} 0, \end{aligned}$$
(A7)
$$\begin{aligned} \Delta _{N+1}({\widetilde{m}})= & {} 0. \end{aligned}$$
(A8)

(A7) is known as “\(\delta _{N}\)-condition” and (A8) is called “\( \Delta _{N+1}\)-condition”. In fact, the \(\delta _{N}\)-condition is nothing but \(C_{N+2}=0\) and the \(\Delta _{N+1}\)-condition corresponds to \(u_{N+1}( {\widetilde{a}},{\widetilde{b}},{\widetilde{c}},{\widetilde{d}},{\widetilde{e}})=0\).

Since the confluent Heun equation thus has two regular singularities and one irregular singularity, it includes the \(_{2}F_{1}\) hypergeometric equation [78]:

$$\begin{aligned} \left( -z+{z}^{2}\right) {\frac{d^{2}}{d{z}^{2}}}Y\left( z\right) +\left[ \left( 1+{\widehat{a}}+{\widehat{b}}\right) z-{\widehat{c}}\right] {\frac{d}{dz}} Y\left( z\right) +{\widehat{a}}{\widehat{b}}Y\left( z\right) =0 \end{aligned}$$
(A9)

which can be expressed in terms of \(\text{ HeunC }\) functions as follows [112]

$$\begin{aligned} Y\left( z\right)= & {} C _{ 1 }\,\left( 1-z\right) ^{-{\widehat{a}}} \text{ HeunC }\left( 0,{\widehat{a}}-{\widehat{b}},{\widehat{c}}-1,0,\frac{1}{2}\, \left[ \left( {\widehat{c}}-2\,{\widehat{a}}\right) {\widehat{b}}\right. \right. \nonumber \\&\quad \left. \left. -\,{\widehat{c}} \left( 1-{\widehat{a}}\right) +1\right] ,\left( 1-z\right) ^{-1}\right) \nonumber \\&\quad +\, C _{ 2 }\,\left( 1-z\right) ^{-{\widehat{b}}}\text{ HeunC }\left( 0,{\widehat{b}}-{\widehat{a}},{\widehat{c}}-1,0,\frac{1}{2}\,\left[ \left( {\widehat{c}}-2\,{\widehat{a}}\right) {\widehat{b}}\right. \right. \nonumber \\&\quad \left. \left. -\,{\widehat{c}}\left( 1-\widehat{ a}\right) +1\right] ,\left( 1-z\right) ^{-1}\right) \end{aligned}$$
(A10)

In fact, the \(_{2}F_{1}\) hypergeometric function is related to \(\text{ HeunC }\) function by [112]

$$\begin{aligned} \begin{aligned} _{2}F_{1}({\widehat{a}},{\widehat{b}};\,{\widehat{c}};\,z)&=\left( 1-z\right) ^{- {\widehat{b}}}\text{ HeunC }\left( 0,{\widehat{c}}-1,{\widehat{b}}-{\widehat{a}},0, \frac{1}{2}\,\left( -1+{\widehat{a}}+{\widehat{b}}\right) {\widehat{c}}\right. \\&\qquad \left. -\,{\widehat{a}} {\widehat{b}}+\frac{1}{2},{\frac{-z}{1-z}}\right) , \\ \,\,\,\text {where}\,\,\,z&\ne 1. \end{aligned} \end{aligned}$$
(A11)

Inversely, \(\text{ HeunC }\) function can be rewritten in terms of the \( _{2}F_{1}\) hypergeometric function as follows [112,113,114]

$$\begin{aligned} \text{ HeunC }\left( 0,{\widehat{b}},{\widehat{c}},0,{\widehat{e}},z\right)= & {} \left( 1-z\right) ^{-\frac{1}{2}\left( 1+{\widehat{b}}+{\widehat{c}}+\sqrt{1+ \widehat{{b}}^{2}+\widehat{{c}}^{2}-4{\widehat{e}}}\right) }\nonumber \\&\times {_{2}F_{1}}\left[ \frac{1}{2}\left( 1+{\widehat{b}}+{\widehat{c}}+\sqrt{1+ {\widehat{b}}^{2}+{\widehat{c}}^{2}-4{\widehat{e}}}\right) \right. ,\, \nonumber \\&\left. \frac{1}{2}\left( 1+{\widehat{b}}-{\widehat{c}}+\sqrt{1+{\widehat{b}}^{2}+ {\widehat{c}}^{2}-4{\widehat{e}}}\right) ;\,1+{\widehat{b}};\,{\frac{z}{-1+z}} \right] , \nonumber \\ \,\,\,\text {where}\,\,\, 1+{\widehat{b}}\ne & {} 0\,\,\,\text {and}\,\,\,z\ne 1. \end{aligned}$$
(A12)

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Sakallı, İ., Jusufi, K. & Övgün, A. Analytical solutions in a cosmic string Born–Infeld-dilaton black hole geometry: quasinormal modes and quantization. Gen Relativ Gravit 50, 125 (2018). https://doi.org/10.1007/s10714-018-2455-4

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