1 Introduction

Progress in investigations of critical phenomenas in general relativity and quantum field theory in curved space-time has been lacking of experimental feedback. Strong efforts have been strived in search of non-relativistic systems that experimentally testable in the laboratory. In 1981, Unruh [1] investigated the propagation of sound waves in an inhomogeneous flowing fluid and found out that the governing equation was analogous to a massless scalar field in curved space-time. The surface of the fluid flow’s velocity has an inward pointing normal component that is equal to the local speed of sound preventing any sound wave propagating through this surface in outward direction. Thus, the surface acts analogously as the event horizon of a black hole. This analogy implies that the analogous black hole should also emits Hawking radiation in the form of phonons. This analogous black hole system makes experimental verification of such phenomenon possible to be done in laboratory.

Many different kinds of analogue models has been proposed after Unruh’s prediction such as Bose–Einstein condensates, electromagnetic wave guides, graphene, optical black hole, sonic black hole and ion rings [2,3,4,5,6]. In optical domain, Leonhardt and Piwnicki [7] proposed the idea of an optical black hole for the first time in the year of 2000. The idea is that propagation of light in a moving medium resembles many features of a motion in a curved space-time background. Analogue gravity models have been emerging as a new interest as they suit as test-beds for several aspects relativistic quantum field theory in a curved space-time.

In this letter, we will be focus on photon-fluid system, a non-linear optical system that is represented by hydrodynamic equations of an interacting Bosonic gas [8]. Together with exciton-polariton system and Bose–Einstein condensation of photons in an optical microcavity [9], the photon-fluid system also belongs to the family of the so-called quantum fluids of light [10].

In [11] a photon-fluid with both local and non-local interactions have been investigated, showing that the phonons acquire a finite mass and propagate according to the massive Klein–Gordon equation in a (2+1)-dimensional curved space-time. In this system, with the presence of suitable vortex flows, both quasibound states and stationary states of phonons can be created [12]. These represent the acoustic counterpart of quasibound states and scalar clouds originating from massive fields around Kerr black holes. Scalar clouds in photon fluids have been analytically studied in [ [13] and [14], while the derivation of exact solutions of phonon quasibound states is the subject of the present manuscript.

In general, any inhomogeneous neutrally stable nonlinear system provides an effective curved space-time on which its linear elementary excitations can propagate [15]. For the longer wavelengths, phonons propagating in an inhomogeneous photon flow has a similar evolution with scalar fields in curved space-time that is governed by a Klein–Gordon like equation, obtained via linearization around the background state. By comparing the governing equation with the Klein–Gordon equationin 2+1 dimension, we can extract the analogous space-time metric as follows [11],

$$\begin{aligned} ds^2&=-f(r)d(ct)^2+\frac{dr^2}{g(r)}-2\Omega _H r_H^2 d\theta d(ct) +r^2 d\theta ^2, \end{aligned}$$
(1)
$$\begin{aligned} f(r)&=1-\frac{r_H}{r}-\frac{\Omega ^2_H r_H^4}{r^2}, \end{aligned}$$
(2)
$$\begin{aligned} g(r)&=1-\frac{r_H}{r}, \end{aligned}$$
(3)

where \(r_H\) is the radius of the analog black-hole’s horizon that is geometrically interpreted as the circular ring at which the inward radial velocity of the fluid flow is equal to the speed of sound. The \(\Omega _H\) is the angular velocity at the horizon. The term \(d\theta d(ct)\) represents frame dragging and this term vanishes as \(\Omega _H \rightarrow 0\).

In [15], it is mentioned that the metric (1) is comparable with equatorial slice of the rotating Kerr black hole geometry in Boyer–Lindquist coordinates where there exists an analog ergoregion that allows phonons to have negative energies leading to energy extraction. The radius of the ergosphere is given by the vanishing of the temporal component, i.e.,

$$\begin{aligned} r_E=\frac{r_H}{2}\left( 1+\sqrt{1+4r_H^2 \Omega _H^2}\right) . \end{aligned}$$
(4)

The frame dragging effect itself was firstly investigated in detal by [16, 17] in 1918. They consider Coriolis effects up to the first order and calculating such effects near the center of a rotating mass shell and in the far field of a rotating spherical body. In the context of a real black hole possessing accretion disk, the radial precession causes stresses and dissipation and when the torque is strong enough compared to the internal viscous forces, the inner regions of the disk are forced to align with the spin of the central black hole [18]. In these recent years, the frame dragging effect has been successfully measured directly [19,20,21].

In 2015, the gravitational wave signal of a binary black hole merger was directly detected for the first time [22] that makes black hole spectroscopy a new emerging interest. In this research, we are interested to investigate the Klein–Gordon equation of phonon quasibound states in the vortex flow of photons that resembles a single horizoned black hole [12]. The phonon fields in the analog black hole background is governed by the covariant Klein–Gordon equation. Quantum mechanically, the phonon states have discrete complex spectrum as they tunnel through the potential barrier and captured by the horizon. Thus, the quasibound states are decaying localized solutions lying in the black hole potential wells.

However, due to the complexity of the equations involved, especially the radial equation, analytical methods were used less often and only for certain problems. The vast majority of these studies made use of numerical techniques such as the asymptotical analysis, WKB, and continued fraction to investigate the specific task at hand. Fortunately, very recently, [23,24,25,26,27] successfully finds novel exact scalar quasibound state solutions respectively around an analog systems to the Schwarzschild black hole, charged and chargeless Lense-Thirring black hole and Reissner Nordstrom black hole where the radial equations of the scalar field are succesfully solved in terms of the confluent Heun functions. The importance of this particular special function in black hole physics was mentioned in [28].

In this present work, we are going to show in detail analytical derivation of exact solutions of relativistic phonon quasibound states around an analog rotating optical black hole. We successfully solve the radial equation exactly in terms of confluent Heun functions and having the exact solutions in the hand, the complex quantized energy levels expression is obtained from the confluent Heun’s polynomial condition. And finally, by applying the Damour–Ruffini method, the Hawking radiation of the apparent black hole’s horizon is investigated and the Hawking temperature is obtained.

2 The Klein–Gordon equation

The Klein–Gordon equation describes the relativistic quantum mechanics of a scalar field in a curved space-time. The relativistic equation comes from the canonical quantization of the second class constrained of the relativistic classical mechanics [29],

$$\begin{aligned} \left\{ -{\hbar }^2\left[ \frac{1}{\sqrt{-g}}{\partial }_\mu \sqrt{-g}g^{\mu \nu }{\partial }_\nu \right] +k^2c^2\right\} \psi =0, \end{aligned}$$
(5)

where \(E_{0}=kc^2\) is the scalar’s rest energy per unit mass, so, \(k=1\) represents massive particles and \(k=0\) represents massless particles. For the phonon-photon flow system, the governing equation reads as follows [12],

$$\begin{aligned} \left\{ -\left[ \frac{1}{\sqrt{-g}}{\partial }_\mu \sqrt{-g}g^{\mu \nu }{\partial }_\nu \right] +\frac{\Omega _0^2}{\hbar ^2}\right\} \psi =0, \end{aligned}$$
(6)

where \(\frac{\Omega _0}{\hbar }\) is the phonon rest energy per unit mass defined as \(\frac{\Omega _0}{\hbar }=v_s^2\) where \(v_s\) is the speed of sound.

Substituting the metric in (1) and using the ansatz of separation of variables,

$$\begin{aligned} \psi (t,r,\theta )=e^{-i\frac{E}{\hbar c}ct}e^{im\theta }\frac{R(r)}{\sqrt{r}}, \end{aligned}$$
(7)

where m is the azimuthal quantum number. After substitution of the ansatz, the radial equation is successfully isolated as follows,

$$\begin{aligned}{} & {} \Delta {\partial }_x\left( \Delta {\partial }_x\right) R\left( x\right) +\left[ \Delta \left( \frac{\Delta }{4x^2}-\frac{1}{2x^3}-\frac{m^2}{x^2}-{\Omega }^2_0\right) \right. \nonumber \\{} & {} \quad \left. +{\left( \Omega -\frac{m{\Omega }_H}{x^2}\right) }^2\right] R\left( x\right) =0, \end{aligned}$$
(8)

where,

$$\begin{aligned} \Delta= & {} 1-\frac{r_H}{r}=1-\frac{1}{x}, \end{aligned}$$
(9)
$$\begin{aligned} x= & {} \frac{r}{r_H}, \end{aligned}$$
(10)
$$\begin{aligned} \Omega= & {} \frac{Er_H}{\hbar c}. \end{aligned}$$
(11)

Now, operating the derivatives in the first term and multiplying the whole equation by \({\Delta }^{\mathrm{-}\textrm{2}}\), we obtain this following equation,

$$\begin{aligned}{} & {} {\partial }^{\textrm{2}}_xR+\left( \frac{{\partial }_x\Delta }{\Delta }\right) {\partial }_xR+\left[ {\Delta }^{\mathrm{-}\textrm{1}}\left( \frac{\Delta }{4x^2}-\frac{1}{2x^3}-\frac{m^2}{x^2}-{\Omega }^2_0\right) \right. \nonumber \\ {}{} & {} \quad \left. +{\Delta }^{\mathrm{-}\textrm{2}}{\left( \Omega -\frac{m{\Omega }_H}{x^2}\right) }^2\right] R=0. \end{aligned}$$
(12)

And by substituting \(\Delta \left( x\right) \) explicitly as(9), we obtain,

$$\begin{aligned}{} & {} \frac{{\partial }_x\Delta }{\Delta }=\frac{1}{x-1}-\frac{1}{x}, \end{aligned}$$
(13)
$$\begin{aligned}{} & {} {\partial }^{\textrm{2}}_xR+p\left( x\right) {\partial }_xR+q\left( x\right) R=0, \end{aligned}$$
(14)

where,

$$\begin{aligned} p\left( x\right)= & {} \frac{1}{x-1}-\frac{1}{x}, \end{aligned}$$
(15)
$$\begin{aligned} q\left( x\right)= & {} \left( \frac{x}{x-1}\right) \left( \frac{x\mathrm{-}\textrm{1}}{4x^3}-\frac{1}{2x^3}-\frac{m^2}{x^2}-{\Omega }^2_0\right) \nonumber \\{} & {} +{\left( \frac{x}{x-1}\right) }^2{\left( \Omega -\frac{m{\Omega }_H}{x^2}\right) }^2. \end{aligned}$$
(16)

The coefficient of R, i.e. \(q\left( x\right) \), is a polynomial function that contains of mixture between x and \(x-1\). It is possible to express \(q\left( x\right) \) in terms of \(\frac{1}{x}\) and \(\frac{1}{x-1}\) using these following fractional decompositions,

$$\begin{aligned} \frac{1}{x\left( x-1\right) }= & {} \frac{1}{x-1}-\frac{1}{x}, \end{aligned}$$
(17)
$$\begin{aligned} \frac{\textrm{1}}{x^2\left( x-1\right) }= & {} \frac{1}{x-1}-\frac{1}{x}-\frac{1}{x^2}, \end{aligned}$$
(18)
$$\begin{aligned} \frac{\textrm{1}}{x^2{\left( x-1\right) }^2}= & {} -\frac{2}{x-1}+\frac{1}{{\left( x-1\right) }^2}+\frac{2}{x}+\frac{1}{x^2}, \end{aligned}$$
(19)
$$\begin{aligned} \frac{x}{x-1}= & {} \frac{1}{x-1}+1, \end{aligned}$$
(20)

and by making use of the decomposition formulas above, we decompose \(q\left( x\right) \) as follows,

$$\begin{aligned} q\left( x\right)= & {} \left( {\Omega }^{\textrm{2}}-{\Omega }^2_0\right) +\frac{1}{x}\left( \frac{1}{2}+m^2\left( 1+2{\Omega }^2_H\right) \right) \nonumber \\{} & {} +\frac{1}{x-1}\left( -\frac{1}{2}-m^2\left( 1+2{\Omega }^2_H\right) -{\Omega }^2_0+2{\Omega }^{\textrm{2}}\right) \nonumber \\ {}{} & {} +\frac{1}{x^2}\left( \frac{3}{4}+m^2{\Omega }^2_H\right) +\frac{1}{{\left( x-1\right) }^2}{\left( \Omega -m{\Omega }_H\right) }^2. \end{aligned}$$
(21)

We are interested to find solutions in the region outside the horizon, i.e. \(r_H\le r<\infty \) or equivalently \(1\le x<\infty \), that for the sake of notation convenience, we define a new radial variable \(z=-\left( x-1\right) \rightarrow {\partial }_x=-{\partial }_z\) that shifts the domain of interest to \(0\le z<-\infty \). In the new variable, the radial equation becomes as follows,

$$\begin{aligned}{} & {} {\partial }^{\textrm{2}}_zR+p\left( z\right) {\partial }_zR+q\left( z\right) R=0, \end{aligned}$$
(22)
$$\begin{aligned}{} & {} p\left( z\right) =\frac{1}{z}-\frac{1}{z-1}, \end{aligned}$$
(23)
$$\begin{aligned}{} & {} -\frac{1}{2}{\partial }_zp\left( z\right) =\frac{1}{2}\frac{1}{z^2}-\frac{1}{2}\frac{1}{{\left( z-1\right) }^2}, \end{aligned}$$
(24)
$$\begin{aligned}{} & {} -\frac{1}{4}p^2\left( z\right) =-\frac{1}{4}\frac{1}{z^2}-\frac{1}{4}\frac{1}{{\left( z-1\right) }^2}+\frac{1}{2}\frac{1}{x-1}-\frac{1}{2}\frac{1}{x}, \end{aligned}$$
(25)
$$\begin{aligned}{} & {} q\left( z\right) =\left( {\Omega }^{\textrm{2}}-{\Omega }^2_0\right) -\frac{1}{z-1}\left( \frac{1}{2}+m^2\left( 1+2{\Omega }^2_H\right) \right) \nonumber \\ {}{} & {} \quad -\frac{1}{z}\left( -\frac{1}{2}-m^2\left( 1+2{\Omega }^2_H\right) -{\Omega }^2_0+2{\Omega }^{\textrm{2}}\right) \nonumber \\ {}{} & {} \quad +\frac{1}{{\left( z-1\right) }^2}\left( \frac{3}{4}+m^2{\Omega }^2_H\right) +\frac{1}{z^2}{\left( \Omega -m{\Omega }_H\right) }^2. \end{aligned}$$
(26)

Following Appendix A, the normal form of the radial equation is obtained as follows,

$$\begin{aligned}{} & {} {\partial }^{\textrm{2}}_zY\left( z\right) +K\left( z\right) Y\left( z\right) =0, \end{aligned}$$
(27)
$$\begin{aligned}{} & {} Y\left( z\right) =z^{\frac{1}{2}}\left( z-1\right) ^{-\frac{1}{2}}R(r) \end{aligned}$$
(28)

where,

$$\begin{aligned} K\left( z\right)= & {} -\frac{1}{2}{\partial }_zp\left( z\right) -\frac{1}{4}p^2\left( z\right) +q\left( z\right) =\left( {\Omega }^{\textrm{2}}-{\Omega }^2_0\right) \nonumber \\{} & {} -\frac{1}{z-1}\left( m^2\left( 1+2{\Omega }^2_H\right) \right) -\frac{1}{z}\left( -m^2\left( 1+2{\Omega }^2_H\right) \right. \nonumber \\{} & {} \left. -{\Omega }^2_0+2{\Omega }^{\textrm{2}}\right) \nonumber \\{} & {} +\frac{1}{{\left( z-1\right) }^2}\left( m^2{\Omega }^2_H\right) +\frac{1}{z^2}\left( \frac{\textrm{1}}{\textrm{4}}\mathrm{+}{\left( \Omega -m{\Omega }_H\right) }^2\right) . \end{aligned}$$
(29)

2.1 The radial solution

Comparing (27) together with (29) with the normal form of the confluent Heun’s differential equation in Appendix B, we find the exact radial solution of the phonon quasibound states in the vortex flow of photons as follows,

$$\begin{aligned} R= & {} e^{\frac{1}{2}\alpha z}z{\left( z-1\right) }^{\frac{1}{2}\gamma }\left[ Az^{\frac{\beta }{2}}{\text {HeunC}}(z)+Bz^{-\frac{\beta }{2}}{\text {HeunC}}'(z) \right] , \nonumber \\ \end{aligned}$$
(30)
$$\begin{aligned} z= & {} -\frac{r_H-r}{r_H}, \end{aligned}$$
(31)

where the confluent Heun function’s parameters are algebraically solved as follows,

$$\begin{aligned} \alpha= & {} 2\sqrt{{\Omega }^2_0-{\Omega }^{\textrm{2}}}, \end{aligned}$$
(32)
$$\begin{aligned} \beta= & {} 2i\left( \Omega -m{\Omega }_H\right) , \end{aligned}$$
(33)
$$\begin{aligned} \gamma= & {} i\sqrt{4m^2{\Omega }^2_H-1}=i\left| \gamma \right| , \end{aligned}$$
(34)
$$\begin{aligned} \delta= & {} {\Omega }^2_0-2{\Omega }^{\textrm{2}}, \end{aligned}$$
(35)
$$\begin{aligned} \eta= & {} \frac{1}{2}-m^2\left( 1+2{\Omega }^2_H\right) -{\Omega }^2_0+2{\Omega }^{\textrm{2}}. \end{aligned}$$
(36)

Finally, can write the exact solution of the Klein–Gordon equation (7) as follows,

$$\begin{aligned} \psi (t,r,\theta )= & {} e^{-i\frac{E}{\hbar c}ct}e^{im\theta }e^{\frac{1}{2}\alpha z}z{\left( z-1\right) }^{\frac{1}{2}(\gamma -1)}\times \nonumber \\{} & {} \times \left[ Az^{\frac{\beta }{2}}{\text {HeunC}}(z)+Bz^{-\frac{\beta }{2}}{\text {HeunC}}'(z) \right] , \end{aligned}$$
(37)

where \(z=-\frac{r_H-r}{r_H}\).

2.2 Energy levels

Polynomial functions have a very important role in quantum mechanics. In the matter of the scalar’s radial solution, the polynomial function is essentially responsible for the behavior of scalar field between zero and infinity. Moreover, the degree of the respective polynomial determines how many zeroes the radial wave function has [30]. Now, let us apply the polynomial condition of the confluent Heun function (see Appendix B) to obtain the energy levels expression as follows,

$$\begin{aligned}{} & {} \frac{\delta }{\alpha }+\frac{\beta +\gamma }{2}=-(n_r+1)\ \ \, \ \ \ n_r+1=n=1,2,3,\dots \nonumber \\ \end{aligned}$$
(38)
$$\begin{aligned}{} & {} \frac{{\Omega }^2_0-2{\Omega }^{\textrm{2}}}{2\sqrt{{\Omega }^2_0-{\Omega }^{\textrm{2}}}}+i\left( \Omega -m{\Omega }_H+\frac{\left| \gamma \right| }{2}\right) =-n. \end{aligned}$$
(39)

For the case \(\frac{{\Omega }^{\textrm{2}}}{{\Omega }^2_0}\ll 1\), the equation can be solved algebraically as follows,

$$\begin{aligned} {\Omega }_{m,n}\mathrm{=}\frac{\textrm{2}i}{\textrm{3}}{\Omega }_0\mathrm{+}\frac{\textrm{1}}{\textrm{3}}\sqrt{\mathrm{-}\textrm{4}{\Omega }^{\textrm{2}}_0\mathrm{-}\textrm{12}i{\Omega }_Hm\mathrm{+}6i\left| \gamma \right| \mathrm{+12}n\mathrm{+6}}. \end{aligned}$$
(40)

We can compare the exact quasibound states’ frequencies by solving the exact formula (39) with the one presented in [12] that are numerically calculated by Numerov integration method. Following [12], we set \(\Omega _H=1.0224\), \(n_r=0\) and \(m=1\) and varying \(\Omega _0\) following Table 1 in the aforementioned reference. The comparison between the two are presented in Table 1. Comparing both of the real and imaginary parts of the exact and numerical \(\Omega \), we can conclude that the real part of \(\Omega \) are in agreement with the numerically estimated values, while for the imaginary part, which is related to the quasibound states’ lifetime, our analytic formula provides exact results improving the previously published numerical estimations.

Table 1 Comparison of the exact and numerical values of \(\Omega \)
Full size table

2.3 Wave function in extreme limits

In this subsection, investigate the behaviour of the quasibound states solution near the horizon \(r \rightarrow r_H\) and far away from the horizon \(r \rightarrow \infty \) will be investigated. In this \(r \rightarrow r_H\) limit, the confluent Heun functions’ argument \(r-r_H\) is approaching 0, thus, \({\text {HeunC}}(0)={\text {HeunC}}'(0)\approx 1\). Also \(e^{-\frac{1}{2}\alpha \left( \frac{r-r_H }{r_H}\right) }\approx 1\). Thus, we can rewrite the wave function (37) as follows,

$$\begin{aligned} \psi _{\rightarrow r_H} =e^{-i{\frac{E}{\hbar c}}ct}e^{im\theta }\left[ A{\left( \frac{r-r_H}{r_H}\right) }^{\frac{1}{2}\beta }\!\!+B{\left( \frac{r-r_H}{r_H}\right) }^{-\frac{1}{2}\beta }\right] .\nonumber \\ \end{aligned}$$
(41)

Now, for the sake of notation aesthetic, we redefine \(\frac{r-r_H}{r_H}=\zeta r-\zeta _0\) and expressing \(\beta \) explicitly as in (33). This leads to this following equation,

$$\begin{aligned} \psi _{\rightarrow r_H}= & {} e^{i{\frac{E}{\hbar c}}ct}Y_{\ell }^{ m_\ell }\left( \theta ,\phi \right) \nonumber \\{} & {} \left[ A{\left( \zeta r-\zeta _0\right) }^{\frac{i|\beta |}{2}}+B{\left( \zeta r-\zeta _0\right) }^{-\frac{i|\beta |}{2}}\right] , \end{aligned}$$
(42)

and using the complex relation \(z^i = e^{1{\text {ln}}(z)}\) together with \(\cos {z}=\frac{1}{2}\left( e^{iz} + e^{-iz}\right) \), we get,

$$\begin{aligned} \psi _{\rightarrow r_H} =e^{-i{\frac{E}{\hbar c}}ct}e^{im\theta }\left[ C{\cos {\left( \zeta r-\zeta _0\right) }}\right] , \end{aligned}$$
(43)

which represent a purely ingoing wave.

Thus, approaching \(r_H\) the solution represents a purely ingoing wave while approaching \(\infty \), the exponential part of the wave function, i.e. \(e^{-\frac{1}{2}\alpha \left( \frac{r-r_H}{r_H}\right) }\), suppresses the confluent Heun polynomial and quenches the whole wave function, i.e. \(\psi _{\rightarrow \infty }\approx 0\).

3 Hawking radiation

With the exact radial solution in hand, in this section, the Damour–Ruffini [31] method will be applied to calculate the Hawking temperature of black hole’s horizon. We start by expressing the exact radial solution in the near the horizon limit, i.e. \(r \rightarrow r_H\). In this limit, \({\text {HeunC}}(0)={\text {HeunC}}'(0)\approx 1\) and \(e^{-\frac{1}{2}\alpha \left( z\right) }\approx 1\). This condition allows us to rewrite the wave function (37) as follows,

$$\begin{aligned} \psi _{\rightarrow r_H} =e^{-i\frac{E}{\hbar c}ct}e^{im\theta }z \left[ Az^{\frac{\beta }{2}}+Bz^{-\frac{\beta }{2}}\right] , \end{aligned}$$
(44)

where the factor \({\left( -1\right) }^{\frac{1}{2}(\gamma -1)}\) has been absorbed into the normalization constant A and B.

The wave function (44) can be viewed as consisting of two parts, that are the ingoing and outgoing waves,

$$\begin{aligned} \psi =\left\{ \begin{array}{cc} \psi _{+in}=A{z}^{\frac{1}{2}\beta } &{} ingoing \\ \psi _{+out}=B{z}^{-\frac{1}{2}\beta } &{} outgoing \end{array} \right. , \end{aligned}$$
(45)

When there is an ingoing wave hitting the horizon \(r_H\), a particle-antiparticle pair are induced. The particle is reflected and will enhance outgoing wave while the antiparticle counterpart becomes the transmitted wave crossing the horizon. Analytical continuation of the wave function \(\psi \left( z\right) \) can be calculated by using this following trick,

$$\begin{aligned} {z}^{\lambda }\rightarrow {\left[ \left( \frac{r}{r_H }-1\right) +i\epsilon \right] }^{\lambda } =\left\{ \begin{array}{cc} {z}^{\lambda } &{},\ r>r_H \\ {\left|z\right|}^{\lambda }e^{i\lambda \pi } &{},\ r<r_H \end{array} \right. . \end{aligned}$$
(46)

This allows us to obtain the expression for \(\psi _{-out}\) as follows,

$$\begin{aligned}{} & {} \psi _{-out}=\psi _{+out}, \end{aligned}$$
(47)
$$\begin{aligned}{} & {} z\rightarrow ze^{i\pi }, \end{aligned}$$
(48)
$$\begin{aligned}{} & {} z\rightarrow -z=ze^{i\pi }, \end{aligned}$$
(49)

that leads to,

$$\begin{aligned}{} & {} \begin{aligned} \psi _{-out}&=B{\left( ze^{i\pi }\right) }^{-\frac{1}{2}\beta },\\&=\psi _{+out}e^{-\frac{1}{2}i\pi \beta } \end{aligned} \end{aligned}$$
(50)
$$\begin{aligned}{} & {} {\left|\frac{\psi _{-out}}{\psi _{+in}}\right|}^2={\left|\frac{\psi _{+out}}{\psi _{+in}}\right|}^2e^{-i2\pi \beta }={\left|\frac{\psi _{+out}}{\psi _{+in}}\right|}^2e^{4\pi \left[ \Omega -m\Omega _H\right] }.\nonumber \\ \end{aligned}$$
(51)

The normalization of the total probability leads to,

$$\begin{aligned} {\left|\frac{\psi _{-out}}{\psi _{+in}}\right|}^2+{\left|\frac{\psi _{+out}}{\psi _{+in}}\right|}^2=1, \end{aligned}$$
(52)

i.e. the total probablity of the particle wave going out to infinity and the antiparticle wave going inside the black hole must be equal to 1. Near the \(r_H\), thus, we have,

$$\begin{aligned} \psi _{out}=\left\{ \begin{array}{cc} \psi _{+out} &{},\ r>r_H \\ \psi _{-out} &{},\ r<r_H \end{array} \right. , \end{aligned}$$
(53)

or can be rewritten as follows,

$$\begin{aligned} \psi _{out}=B{z}^{-\frac{1}{2}\beta }\left[ \theta \left( -z\right) +\theta \left( z\right) e^{2\pi \left[ \Omega -m\Omega _H\right] }\right] . \end{aligned}$$
(54)

The Hawking temperature \(T_H\) of the corresponding horizon is to be extracted from the thermal spectrum known as radiation distribution function by calculating the modulus square of the ratio between the normalization constant between the outgoing and incoming waves as follows,

$$\begin{aligned}{} & {} \left\langle \frac{\psi _{out}}{\psi _{in}}\Bigg \vert \frac{\psi _{out}}{\psi _{in}}\right\rangle =1={\left|\frac{B}{A}\right|}^2\left|1-e^{4\pi \left[ \Omega -m\Omega _H\right] }\right|, \end{aligned}$$
(55)
$$\begin{aligned}{} & {} \left|\frac{B}{A}\right|^2=\frac{1}{e^{4\pi \left[ \Omega -m\Omega _H\right] }-1}. \end{aligned}$$
(56)

Now, let us consider this following algebraic modification,

$$\begin{aligned} 4\pi \left[ \Omega -m\Omega _H\right]= & {} 4\pi \left[ \frac{Er_H}{\hbar c}-m\Omega _H\right] =\frac{\hslash \left( \omega -{\omega }_H\right) }{\left[ \frac{c\hbar }{4\pi r_H}\right] },\nonumber \\ \end{aligned}$$
(57)
$$\begin{aligned} {\omega }_H= & {} \frac{c m\Omega _H}{r_H}. \end{aligned}$$
(58)

Finally, by comparing with the bosonic distribution function \(e^{\frac{\hbar \omega }{k_B T}}\) we obtain the analog black hole’s horizon temperature as follows,

$$\begin{aligned} T_H=\frac{c\hbar }{4\pi k_B r_H}. \end{aligned}$$
(59)

The expression (59) is equal to the Hawking temperature of chargeless Lense-Thirring black hole \(T_H=\frac{r_Hc\hslash }{4\pi k_Br^2_s}\) in the recent published result [26].

4 Conclusions

In this work, the exact analytical phonon quasibound state’s quantized energy levels and their wave functions in a vortex photon flow analog black hole are derived in detail. The exact angular solution is found in terms of pure harmonics functions while the radial exact solutions are discovered in terms of confluent Heun functions (30). The obtained exact radial solution is valid for all region of interest, i.e. \(r_s \le r < \infty \), a significant improvement of assymptotical method that can only solves for either region very close to the horizon of very far away from the horizon which was presented in [12]. Investigation of the obtained radial solution around the horizon shows a purely ingoing wave while far away from the horizon, the wave function is decaying exponentially.

By making use of the confluent Heun function’s polynomial condition, the quantized energy levels’ expression is succesfully derived and presented in (39). Comparison of the values of \(\Omega \) between the previously published results [12] and the the exact results by solving Eq. (39) is presented in Table 1 where the real parts of the \(\Omega \) are convergent while in this letter, we present for the first time, the exact values of the imaginary part of the \(\Omega \). And in low frequncy limit, we obtain a simplified algebraic expression of the energy levels (40). In the last section, after indentifying the ingoing and outgoing component of the wave funtion, the [31] method is applied to find the Hawking temperature of the black hole’s horizon (59).