New examples of Hawking radiation from acoustic black holes

TL;DR

This paper investigates Hawking radiation in complex acoustic black holes, including those with corners, expanding understanding of quantum effects in analogue gravity systems.

Contribution

It introduces analysis of Hawking radiation in more complex acoustic black holes, such as those with corner points, beyond previously studied simple models.

Findings

Hawking radiation observed in complex acoustic black holes.

Black holes with corners can emit Hawking radiation.

Extended understanding of analogue black hole quantum effects.

Abstract

Rotating acoustic metrics may have black holes inside the ergosphere. Simple cases of acoustic black holes were studied in the references [21], [4]. In the present paper we study the Hawking radiation for more complicated cases of acoustic black holes including black holes that have corner points.

Equations

$$v=(v^1,v^2)=\frac{A}{r}\hat{x}+\frac{B}{r}\hat{\theta },$$

$${\square }_{g}u(x_0,x_1,x_2)=\frac{1}{\sqrt{g(x_1,x_2)}}\sum _{j,k=0}^2\frac{\partial }{\partial x_k}\left(\sqrt{g(x_1,x_2)}{g}^{jk}(x_1,x_2)\frac{\partial u(x_0,x_1,x_2)}{\partial x_j}\right)=0,$$

$$H(\rho,\varphi,\eta_{0},\eta_{\rho},\eta_{\varphi})=\Big{(}\eta_{0}+\frac{A}{\rho}\eta_{\rho}+\frac{B}{\rho^{2}}\eta_{\varphi}\Big{)}^{2}-\eta_{\rho}^{2}-\frac{1}{\rho^{2}}\eta_{\varphi}^{2},$$

$$H=H^{+}{H}^{-},$$

$$H^{\pm }(\rho ,\phi ,{\eta }_0,{\eta }_{\rho },{\eta }_{\phi })={\eta }_0+\frac{A}{\rho }{\eta }_{\rho }+\frac{B}{{\rho }^2}{\eta }_{\phi }\pm \sqrt{{\eta }_{\rho }^2+\frac{1}{{\rho }^2}{\eta }_{\phi }^2}.$$

$$\displaystyle f_{k}^{+}(x_{0},x_{1},x_{2})\Big{|}_{x_{0}=0}=\gamma_{k}e^{ik\cdot x},$$

$$\displaystyle\frac{\partial}{\partial x_{0}}f_{k}^{+}(x_{0},x_{1},x_{2})\Big{|}_{x_{0}=0}=i\lambda_{0}^{-}(k)\gamma_{k}e^{ik\cdot x},$$

$${\lambda }_0^{-}(k)=-\frac{A}{\rho }{\eta }_{\rho }-\frac{Bm}{{\rho }^2}-\sqrt{{\eta }_{\rho }^2+a^2}.$$

$$f_k^{-}(x_0,x)=\overline{f_{-k}^{+}(x_0,x)}.$$

$$<u,v>=i\underset{x_0=t}{\int }|g{|}^{\frac{1}{2}}\{u,v\}d{x}_{1}d{x}_2,$$

$$<f_k^{+},f_{k^{\prime }}^{+}>=\delta ({\eta }_{\rho }-{\eta }_{\rho }^{\prime }){\delta }_{m{m}^{\prime }},<f_k^{-},f_{k^{\prime }}^{-}>=-\delta ({\eta }_{\rho }-{\eta }_{\rho }^{\prime }){\delta }_{m{m}^{\prime }},<f_k^{+},f_{k^{\prime }}^{-}>=0,$$

$$\Phi=\sum_{m=-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\big{(}\alpha_{k}^{+}f_{k}^{+}(x_{0},x_{1},x_{2})+\alpha_{-k}^{-}f_{-k}^{-}(x_{0},x_{1},x_{2})\big{)}d\eta_{\rho},$$

$${\alpha }_k^{+}=<f_k^{+},\Phi >,{\alpha }_k^{-}=-<f_k^{-},\Phi >.$$

$$[{\alpha }_k^{+},{\alpha }_{-k^{\prime }}^{-}]=\delta ({\eta }_{\rho }-{\eta }_{\rho }^{\prime }){\delta }_{m{m}^{\prime }}I,[{\alpha }_k^{+},{\alpha }_{k^{\prime }}^{+}]=0,$$

$$C=\sum_{m=-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\big{(}C^{+}(k)f_{k}^{+}(x_{0},x_{1},x_{2})+C^{-}(k)f_{-k}^{-}(x_{0},x_{1},x_{2})\big{)}d\eta_{\rho},$$

$$C^{\pm }=\sum _{m=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}{C}^{\pm }(k)f_{\pm k}^{\pm }(x_0,x)d{\eta }_{\rho },$$

$$C=C^{+}+C^{-}.$$

$$<C,\Phi>=\sum_{m=-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\big{(}\overline{C^{+}}(k)\alpha_{k}^{+}-\overline{C^{-}}(k)\alpha_{-k}^{-}\big{)}d\eta_{\rho}.$$

$${\alpha }_k^{+}|0⟩=0\text{mbox}forallk.$$

$$N(C)=<C,\Phi {>}^{\ast }<C,\Phi >,$$

$$⟨0|N(C)|0⟩=\sum _{m=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}|C^{-}(k){|}^{2}d{\eta }_{\rho },$$

$$<C^{-},C^{-}>=-\sum _{m=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}|C^{-}(k){|}^{2}d{\eta }_{\rho }.$$

$$⟨0|N(C)|0⟩=-<C^{-},C^{-}>,$$

$$<C,\Phi >|0⟩=-\sum _{m=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}\overline{C^{-}(k)}{\alpha }_{-k}^{-}|0⟩d{\eta }_{\rho }.$$

$$⟨0|<C,\Phi {>}^{\ast }=-\sum _{m=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}⟨0|C^{-}(k){\alpha }_k^{+}d{\eta }_{\rho },$$

$$\begin{gathered} ⟨0|<C,\Phi {>}^{\ast }<C,\Phi >|0⟩ \\ =\sum _{m=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}⟨0|C^{-}(k){\alpha }_k^{+}d{\eta }_{\rho }\cdot \sum _{m^{\prime }=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}\overline{C^{-}(k^{\prime })}{\alpha }_{-k^{\prime }}^{-}|0⟩d{\eta }_{\rho }^{\prime } \\ =\sum _{m=-\infty }^{\infty }\underset{-\infty }{\overset{\infty }{\int }}|C^{-}(k){|}^{2}d{\eta }_{\rho }, \end{gathered}$$

$$\Big{(}-\eta_{0}+\frac{A}{\rho}S_{\rho}+\frac{B}{\rho^{2}}S_{\varphi}\Big{)}^{2}-S_{\rho}^{2}-\frac{1}{\rho^{2}}S_{\varphi}^{2}=0.$$

$$S_{\rho}=\frac{|A|\big{(}\eta_{0}-\frac{Bm}{|A|^{2}}\big{)}}{\rho-|A|}+O(\rho-|A|).$$

$$S_{1}=-\eta_{0}x_{0}+|A|\Big{(}\eta_{0}-\frac{Bm}{|A|^{2}}\Big{)}\ln(\rho-|A|)+m\varphi$$

$$\displaystyle C(x_{0},\rho,\varphi)\Big{|}_{x_{0}=0}=\theta(\rho-|A|)\frac{(\rho-|A|)^{\varepsilon}}{\sqrt{\rho}}\ e^{-a(\rho-|A|)}\,e^{i\xi_{0}|A|\ln(\rho-|A|)+im\varphi},$$

$$\displaystyle\frac{\partial C(x_{0},\rho,\varphi)}{\partial x_{0}}\Big{|}_{x_{0}=0}=i\beta C(x_{0},\rho,\varphi_{0})\Big{|}_{x_{0}=0},$$

Full text

In memory of Selim Grigorievich Krein on occasion of his 100th birthday

1. Introduction.

In classical general relativity black hole is the region such that no signal or disturbance can escape it. It was a remarkable discovery by S. Hawking [10] that once the quantum effects are added the black holes emit quantum particles. This effect is called the Hawking radiation and it was studied subsequently in many papers (cf. [1], [2], [3], [8], [9], [12], [13], [14] and others). In an astrophysical experiment the Hawking effect is too weak to be detected on the earth.

In quest to find the experimental confirmation of the Hawking radiation effect W. Unruh considered the Hawking radiation for the acoustic black holes [18]. Acoustic black holes are one of the examples of analogue black holes (cf. [21], see also [15], [16]). The Hawking radiation for analogue black hole have been studied in [19], [20], [22] and others.

In all previous works on Hawking radiation the case of spherically symmetric metric or the case of one space dimension was treated.

In present paper, as in [6], we study the case of two space dimensional (rotating) acoustic metric. It was shown in [4] and [5] that there is a rich class of acoustic metrics having black holes. In [6] we consider the case of smooth black holes but as it was shown in [5] there exist black holes that are not smooth and that event horizons may have corners.

The Hawking radiation from such black holes will be considered in this paper.

The plan of the paper is the following:

In §2 we set up the framework of the quantum field theory on curved spacetimes following mostly T. Jacobson [12], [13] (see also [8], [11]). In departure from [6], where the Unruh type vacuum [17] was considered, we use in §2 a more common vacuum state as in S. Hawking [10] and T. Jacobson [12].

In §3 we briefly consider the case of a simple acoustic black hole similar to the one in §3 of [6]. There is a difference in computations with §3 of [6] since we consider a different vacuum state in this paper.

In §4 we study the Hawking radiation for the acoustic metrics when the ergosphere has a finite number of characteristic points. This case is different from the case considered in section 4.1 of [6] where the characteristic points were not allowed. In §5 we consider the Hawking radiation from a black holes having corners. Note that constructions in §4 and §5 requires a localization in the angular coordinate.

2. The number of particles operator

Consider a fluid flow in a vortex with the velocity field

[TABLE]

where $r=|x|,\ \hat{x}=\frac{(x_{1},x_{2})}{|x|},\ \hat{\theta}=\frac{(-x_{2},x_{1})}{|x|},\ A,B$ are functions of $x=(x_{1},x_{2})$.

The acoustic waves in the moving fluid are described by the wave equation (cf. [21])

[TABLE]

$x_{0}$ is the time variable, $g^{00}=1,\ g^{0j}=g^{j0}=v^{j},\ g^{jk}=(-\delta_{jk}+v^{j}v^{k}),\linebreak 1\leq j,k\leq 2,\ g(x_{1},x_{2})=1$ and we, for the simplicity, assume the sound speed and density are equal to 1.

The Hamiltonian corresponding to (2.2) has the following form in polar coordinates $(\rho,\varphi)$:

[TABLE]

where $(\eta_{0},\eta_{\rho},\eta_{\varphi})$ are dual to $(x_{0},\rho,\varphi)$.

Note that (2.3) can be factored

[TABLE]

where

[TABLE]

We refer to [11], [12] to introduce the main facts of quantum field theory on curved spacetimes.

We define $f_{k}^{+}(x_{0},x)$ as the solution of (2.2) in $\mathbb{R}^{2}\times\mathbb{R}$ having the following initial conditions in polar coordinates $(\rho,\varphi)$

[TABLE]

where x=(\rho,\varphi),\ k=(\eta_{\rho},m),\ m\in\mathbb{Z},\ k\cdot x=\rho\eta_{\rho}+m\varphi,\ \gamma_{k}=\frac{1}{2\pi\sqrt{2}\sqrt{\rho}\big{(}\eta_{\rho}^{2}+a^{2}\big{)}^{\frac{1}{4}}},\linebreak a>0 is arbitrary,

[TABLE]

Let

[TABLE]

Let $u,v$ be the solutions of (2.2) and $<u,v>$ be their Klein-Gordon inner product

[TABLE]

where \{u,v\}=\sum_{j=0}^{2}g^{0j}\Big{(}\overline{u}\frac{\partial v}{\partial x_{j}}-\frac{\partial\overline{u}}{\partial x_{j}}v\Big{)}.

Note that (2.8) is independent of $t$ (cf. [12]).

It is easy to see (cf. [6]) that

[TABLE]

where $k=(\eta_{\rho},m),k^{\prime}=(\eta_{\rho}^{\prime},m^{\prime})$.

Thus $f_{k}^{+},f_{-k}^{-}$ form a basis of solutions of (2.2).

Now we introduce the selfadjoint field operator $\Phi$

[TABLE]

where $k=(\eta_{\rho},m),\eta_{\rho}\in\mathbb{R},m\in\mathbb{Z}$,

[TABLE]

Since $\overline{f}_{k}^{+}=f_{-k}^{-}$ we have that $(\alpha_{-k}^{-})^{*}=\alpha_{k}^{+}$.

Operators $\alpha_{k}^{+},\alpha_{-k}^{-}$ are called the annihilation and creation operators, respectively, and they satisfy the following commutation relations (cf. [12]):

[TABLE]

$[\alpha_{k}^{-},\alpha_{k^{\prime}}^{-}]=0,\ I$ is the identity operator.

Let $C(x_{0},x_{1},x_{2})$ be a solution of (2.2) with some initial conditions at $x_{0}=0$ that will be specified later.

Expanding $C(x_{0},x_{1},x_{2})$ in the basis $f_{k}^{+},f_{-k}^{-}$ we get

[TABLE]

where $k=(\eta_{\rho},m),\ C^{+}(k)=<f_{k}^{+},C>,\ C^{-}(k)=-<f_{-k}^{-},C>$.

Let

[TABLE]

Thus

[TABLE]

It follows from (2.10), (2.13) that

[TABLE]

The vacuum state $|0\rangle$ is defined by the conditions

[TABLE]

Let $N(C)$ be the number of particles operator created by the wave packet $C$ (cf. [12])

[TABLE]

and let $\langle 0|N(C)|0\rangle$ be the average number of particles.

As in [6] one have the following theorem

Theorem 2.1**.**

The average number of particles created by the wave packet $C$ is given by the formula

[TABLE]

where $C^{-}$ is the same as in (2.14).

Note that (cf. (2.14))

[TABLE]

Therefore

[TABLE]

where $C^{-}$ is the same as in (2.14).

Proof of Theorem 2.1 We have

[TABLE]

Analogously,

[TABLE]

since $\langle 0|\alpha_{-k}^{-}=0$. Therefore

[TABLE]

since $\alpha_{k}^{+}\alpha_{-k^{\prime}}^{-}=\alpha_{-k}^{-}\alpha_{k}^{+}+I\delta(k-k^{\prime})$ and $\alpha_{k}^{+}|0\rangle=0,\ \forall k$.

3. Hawking radiation from the simple rotating acoustic black hole

Consider the simplest case when $A<0,B\neq 0$ are constants. This case was studied in [6], §3, and we briefly repeat the computations taking into account that the vacuum state in this paper is different from [6], §3.

Let $S(x_{0},\rho,\varphi)$ be the eikonal satisfying the equation

[TABLE]

We are looking for the eikonal of the form $S=-\eta_{0}x_{0}+s(\rho)+m\varphi$ where $s(\rho)\rightarrow+\infty$ when $\rho>|A|$ and $\rho\rightarrow|A|$.

It can be shown (cf. [6]) that when $\rho\rightarrow|A|$

[TABLE]

We will not try to solve the eikonal equation exactly. Instead we use the approximation

[TABLE]

of $S(x_{0},\rho,\varphi)$ to construct the exact solution of (2.2) having the following initial conditions

[TABLE]

where $a>0,\varepsilon>0$ are arbitrary, $\theta(\rho-|A|)=1\ \ \mbox{for}\ \ \rho>|A|\ \ \mbox{and}\ \ \ \theta(\rho-|A|)=0$ for $\rho<|A|,\ \xi_{0}=\eta_{0}-\frac{Bm}{|A|^{2}},$

[TABLE]

For the convenience we take $a>0$ in (3.2) equal to $a$ in (2.6). It is easy to compute the KG norm of $C(x_{0},\rho,\varphi)$ (cf. (3.8) in [6]):

[TABLE]

We shall call $C(x_{0},\rho,\varphi)$ a wave packet.

By Theorem 2.1 to compute the average number of created particles we need to find $C^{-}(k)=-<f_{-k}^{-},C>$. Computing the KG norm we have

[TABLE]

where (cf. (3.11), (3.12) in [6])

[TABLE]

[TABLE]

Integrating in $\varphi$ and using the formula (cf. [6])

[TABLE]

we get

[TABLE]

[TABLE]

where $\delta_{m_{1},m_{2}}=1$ when $m_{1}=m_{2}$ and $\delta_{m_{1},m_{2}}=0$ when $m_{1}\neq m_{2}$. Since \big{|}e^{i\frac{\pi}{2}(i\xi_{0}|A|+\varepsilon)}\big{|}=e^{-\frac{\pi}{2}\xi_{0}|A|},\ \Gamma(i\xi_{0}|A|+\varepsilon+1)=(i\xi_{0}|A|+\varepsilon)e^{-\frac{\pi}{2}\xi_{0}|A|}\Gamma_{1}(i\xi_{0}|A|+\varepsilon),\linebreak\Gamma_{1} is bounded, we have

[TABLE]

Taking the sum in $m^{\prime}$, integrating in $\eta_{\rho}$ and making change of variables $\eta_{\rho}\rightarrow a\eta_{\rho}$, we get

[TABLE]

where

[TABLE]

Denote by $C_{n}$ the normalized wave packet, i.e. $C_{n}=\frac{C}{<C,C>}$. Thus $N(C_{n})=\frac{N(C)}{<C,C>}$. Dividing (3.11) by $<C,C>$ and taking the limit as $a\rightarrow\infty$ we get

[TABLE]

Therefore we proved the following theorem:

Theorem 3.1**.**

The average number of particles created by the normalized wave packet $C_{n}$ is given by (3.13) where $C_{3}(\eta_{\rho})$ is the same as in (3.12).

Now we will analyze the decay of the right size of (3.13) when $\xi_{0}|A|\rightarrow\infty$. Note

[TABLE]

[TABLE]

We have

[TABLE]

[TABLE]

[TABLE]

Note that $\sin^{-1}\frac{1}{\sqrt{\eta_{\rho}^{2}+1}}\leq\frac{\pi}{2}$. Therefore for $\eta_{\rho}>0$ we have

[TABLE]

i.e. $C_{3}$ is exponentially decaying on $(0,+\infty)$.

When $\eta_{\rho}<0$ we have

[TABLE]

If $|\eta_{\rho}|\leq(\xi_{0}|A|)^{\delta},\ 0<\delta<1$, then

[TABLE]

Thus $C_{3}$ is exponentially decaying on $(-(\xi_{0}|A|)^{\delta},0)$. On $(-\infty,-(\xi_{0}|A|)^{\delta})$ we get

[TABLE]

Hence

[TABLE]

Therefore normalizing $C$ to make $<C_{n},C_{n}>=1$, we get (cf. (3.3))

[TABLE]

Remark 3.1 It follows from (3.19) that $\int_{0}^{\infty}C_{3}\,d\eta_{\rho}$ is exponentially decaying in $\xi_{0}|A|$ (cf. [10]) and so it contributes to the Hawking radiation. However, the integral $\int_{-\infty}^{0}C_{3}\,d\eta_{\rho}$ has only decay of order (3.24) and it does not contribute to the Hawking radiation. Note that for $f_{k}^{+}(x_{0},x_{1},x_{2})$ and $f_{-k}^{-}(x_{0},x_{1},x_{2})=\overline{f_{k}^{*}}$ to form a basis of solutions of (2.2) one needs to use all $\eta_{\rho}\in(-\infty,+\infty)$. Therefore $\langle 0|N(C)|0\rangle$ must include also $\int_{-\infty}^{0}C_{3}\,d\eta_{\rho}$. Only when we replace $|0\rangle$ by the Unruh type vacuum $|\Psi\rangle$ (cf. [6]) we get that all terms of $\langle\Psi|N(C)|\Psi\rangle$ contribute to the Hawking radiation.

4. Hawking radiation from rotating acoustic black holes

In this and the next sections we shall continue to study the Hawking radiation from rotating acoustic black holes started in [6]. We shall consider the case of the fluid flow (2.1) where $A(\rho,\varphi),B(\rho,\varphi)$ are functions of $(\rho,\varphi),\ A<0$. The ergosphere of the acoustic metric is $\frac{A^{2}(\rho,\varphi)+B^{2}(\rho,\varphi))}{\rho^{2}}=1$ and we assume that it is a smooth Jordan curve. It was proven in [5] that there are always black holes for the acoustic matrics. In [6] we consider a particular case when the normals to the ergosphere are not characteristic at any point of the ergosphere. In this case there is a smooth black hole inside the ergosphere. When the ergosphere has characteristic points the black hole, in general, may have corners points. We shall study two typical examples. In this section we consider the case of finite number of characteristic points on the ergosphere where the black hole is tangent to the ergosphere, and in the next section we shall consider an example of acoustic black hole having a corner.

Let the velocity field be of the form

[TABLE]

where $A<0$ is a constant and $B(\varphi)$ has finite number of zeros

[TABLE]

Here $\rho=|A|$ is the boundary of the black hole and it touches the ergosphere at points $\theta_{k},1\leq k\leq p$.

Consider the eikonal $S(\rho,\varphi)$:

[TABLE]

such that $S_{\rho}\rightarrow+\infty$ when $\rho\rightarrow|A|$. As in [6] (cf. also §3), for small $\rho-|A|$ it is convenient to use an approximation $S_{1}(\rho,\varphi)$ of eikonal $S(\rho,\varphi)$ that satisfies the equation

[TABLE]

We look for a solution of (4.4) in a form

[TABLE]

where $S_{2}(\rho,\varphi)$ satisfies

[TABLE]

Let $[\varphi_{r},\varphi_{r1}]$ be a closed interval where $B(\varphi)\neq 0$ and $\varphi_{r0}\in(\varphi_{r},\varphi_{r1})$. For $\varphi\in[\varphi_{r},\varphi_{r1}]$ we have

[TABLE]

where $\alpha_{k}$ and $d_{k}$ are arbitrary. Therefore

[TABLE]

where

[TABLE]

for $\varphi\in[\varphi_{r},\varphi_{r1}]$.

Let $\hat{C}_{r}(x_{0},\rho,\varphi)$ be a solution of (2.2) (wave packet) having the following initial conditions

[TABLE]

where

[TABLE]

$a>0,\varepsilon>0$ (cf. §3), $\mbox{supp\,}c_{r}(\varphi)\subset(\varphi_{r},\varphi_{r_{1}})$,

[TABLE]

[TABLE]

Note that although (4.7) is an approximation of the eikonal, the solution $C_{r}(x_{0},\rho,\varphi)$ is an exact solution of the wave equation (2.2).

Finally we define wave packet $\hat{C}(x_{0},\rho,\varphi)$ as the sum

[TABLE]

Similarly to (3.3) the KG norm of $\hat{C}$ is

[TABLE]

To compute the average number of particles created by $\hat{C}(x_{0},\rho,\varphi)$ we use Theorem 2.1. Note that

[TABLE]

We have

[TABLE]

Note that

[TABLE]

[TABLE]

Therefore we need to take care of the extra turns when we take derivatives in $\rho$ and in $\varphi$. These extra terms will dissapear when we will take the limit when the parameter $a\rightarrow\infty$.

Computing $\hat{C}_{r}^{-}(k)$ as in (3.4)-(3.9) in §3 we obtain

[TABLE]

where

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

Note that

[TABLE]

since $c_{r_{1}}(\varphi)c_{r_{2}}(\varphi)\equiv 0.$ Therefore taking into account (4.23) we get, as in §3:

[TABLE]

where

[TABLE]

and we used that

[TABLE]

and that (cf. (4.22)) $|\hat{\gamma}_{m^{\prime}}^{(r)}|\leq\frac{C}{|m^{\prime}|^{2}}$.

Therefore we proved the following theorem.

Theorem 4.1**.**

The average number of particles created by the wave packet (4.9), (4.11) is given by the formula (4.24).

Finally, making the change of variables $\eta_{\rho}\rightarrow a\eta_{\rho}$, replacing $\hat{C}$ by the normalized wave packet $\hat{C}_{n}=\frac{\hat{C}}{<\hat{C},\hat{C}>}$ and taking the limit where $a\rightarrow\infty$ we get

[TABLE]

where $\hat{C}_{3r}(\eta_{\rho})$ is the same as $C_{3}$ in (3.12) with $|\xi_{0}|A|$ replaced by $\eta_{0}|A|+\alpha_{r}$.

Note that when $\alpha_{r}$ is the same for all $1\leq r\leq p$ then the sum $\sum_{r=1}^{p}\int_{0}^{2\pi}|c_{k}(\varphi)|^{2}d\varphi$ in (4.27) cancels.

Remark 4.1 (cf. Remark 3.1). As in the end of §3 we have that $\int_{0}^{\infty}\hat{C}_{3r}(\eta_{\rho})d\eta_{\rho}$ is exponentially decaying when $(\eta_{0}|A|+\alpha_{r})\rightarrow\infty$ and

[TABLE]

(cf. (3.24)).

5. The Hawking radiation in the case of black hole with corners

In this section we study the Hawking radiation from rotating acoustic black holes having corners. It was proven in [4], [5] and [7] that the zero energy null geodesics form two family of smooth curves inside the ergosphere and the boundary of the black hole (the event horizon) consists of segments of zero energy null geodesics belonging to one or another family. In the case when the normal to the ergosphere is not characteristic at any point, the black hole is formed by one family of null geodesics and it is smooth closed curve. When the ergosphere contains the characteristic points then the black hole (or black holes) consists of segments belonging to the different families and therefore when adjacent segments belong to different families they intersect and form a corner.

Let, as in [5], Example 4.2,

[TABLE]

$A$ and $B$ define the velocity field by formula (2.1).

The equation of the ergosphere is

[TABLE]

Points $\alpha_{1}=(r_{0}(\frac{\pi}{2}),\frac{\pi}{2})$ and $\alpha_{2}=(r_{0}(-\frac{\pi}{2}),-\frac{\pi}{2})$ on the ergosphere are characteristic points. Note that both families of zero-energy null-geodesics are tangent to the ergosphere at the characteristic point. For the ergosphere (5.2) we have zero energy null-geodesics $\gamma_{1}$ and $\gamma_{2}$ that starts at $\alpha_{1}=(r_{0}(\frac{\pi}{2}),\frac{\pi}{2})$ and intersected at some point $\alpha_{3}$ forming an angle (see Fig. 1)

\gamma_{2}$$\gamma_{1}$$\alpha_{1}$$\alpha_{2}$$\alpha_{3}

Fig. 1. Null-geodesics $\gamma_{1}$ and $\gamma_{2}$ intersect at point $\alpha_{3}$ and bound a black hole

having corner point $\alpha_{3}$.

Denote by $\gamma_{10}$ the part of $\gamma_{1}$ where $\varphi_{1}<\varphi<\varphi_{11}$ on $\gamma_{1}$. Analogously let $\gamma_{20}$ be the part of $\gamma_{2}$ such that $\varphi_{2}<\varphi<\varphi_{20}$ on $\gamma_{2}$. Let $\rho=\rho_{i}(\varphi)$ be the equation of $\gamma_{i0},\ \varphi_{i}<\varphi<\varphi_{i1},i=1,2$.

Connect $\rho_{1}(\varphi_{1})$ and $\rho_{2}(\varphi_{21})$ and connect $\rho_{1}(\varphi_{11})$ and $\rho_{2}(\varphi_{2})$ by smooth curves to get a smooth periodic curve $\rho=\rho_{0}(\varphi),\ 0<\varphi<2\pi$ such that $\rho_{0}(\varphi)=\rho_{i}(\varphi)$ for $\varphi_{1}\leq\varphi\leq\varphi_{11}$ and $\rho_{0}(\varphi)=\rho_{2}(\varphi)$ for $\varphi$ between $\varphi_{21}$ and $\varphi_{2}$.

The following arguments are not restricted to Example 4.2 and apply to any situation when we have two smooth segments $\gamma_{10}$ and $\gamma_{20}$ of the boundary of the black hole.

As in [6], §4.2, make change of variable

[TABLE]

Denote by $\tilde{f}_{k}^{+},k=(\eta_{\rho},m)$, the solutions of the wave equation in $(x_{0},\tilde{\rho},\varphi)$ coordinates such that

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

The eikonal equation in $(\tilde{\rho},\varphi)$ coordinates takes the form (cf. (4.19) in [6])

[TABLE]

where $\tilde{A},\tilde{B},\tilde{C}$ are the same as in (4.20) in [6]. Since $\tilde{\rho}=0$ is characteristic when $\varphi_{1}<\varphi<\varphi_{11}$ and when $\varphi_{2}<\varphi<\varphi_{21}$ we have that $\tilde{A}(\tilde{\rho},\varphi)=A_{01}(\tilde{\rho},\varphi)\tilde{\rho}$ when $\varphi_{1}<\varphi<\varphi_{11}$ and when $\varphi_{2}<\varphi<\varphi_{21}$.

As in [6] we have that

[TABLE]

where equation (5.8) for $j=1$ holds on $(\varphi_{1},\varphi_{11})$ and (5.8) for $j=2$ holds on $(\varphi_{2},\varphi_{21})$.

Similarly to §4 the solution of (5.8) has the form

[TABLE]

where $\tilde{\alpha}_{j}>0$ and $d_{j}$ are arbitrary, $\varphi_{j0}\in(\varphi_{j},\varphi_{j1})$.

Note that (5.9) for $j=1,2$ holds when $\varphi_{j}<\varphi<\varphi_{j1}$.

Now we shall construct the wave packet $\tilde{C}(x_{0},\tilde{\rho},\varphi)$ in the form

[TABLE]

where

[TABLE]

[TABLE]

where

[TABLE]

,

[TABLE]

Denote

[TABLE]

By the Parseval’s equality

[TABLE]

Therefore as in §3 and §4 computing the average number of created particles we get

[TABLE]

Thus we proved the following theorem:

Theorem 5.1**.**

The average number of particles created by the wave packet (5.11), (5.12) is given by (5.16).

As in §4, replacing $\tilde{C}$ by the normalized wave packet $\tilde{C}_{n}$ and taking the limit as $a\rightarrow\infty$ we get that $\lim_{a\rightarrow\infty}\langle 0|N(\tilde{C}_{n}|0\rangle$ has an expression similar to (4.27) with $\eta_{0}|A|+\alpha_{r}$ replaced by $\tilde{\alpha}_{r}$.