Analogous Hawking Radiation in Dispersive Media

Abstract

In the framework of the analogous Hawking effect, we significantly improve our previous analysis of the master equation that encompasses very relevant physical systems, like Bose–Einstein condensates (BECs), dielectric media, and water. In particular, we are able to provide two significant improvements to the analysis. As our main result, we provide a complete set of connection formulas for both the subluminal and superluminal cases without resorting to suitable boundary conditions, first introduced by Corley, but simply on the grounds of a rigorous mathematical setting. Moreover, we provide an extension to the four-dimensional case, showing explicitly that, apart from obvious changes, adding transverse dimensions does not substantially modify the Hawking temperature in the dispersive case. Furthermore, an important class of exact solutions of the so-called reduced equation that governs the behavior of non-dispersive modes is also provided.

1. Introduction

We again consider analytical calculations for the analog Hawking effect by developing and deepening our analysis in [1,2,3]. In the dispersive case, one can find several papers dedicated to analytical calculations (see, e.g., the following (non-exhaustive) list of papers: [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]), albeit not within a unified general theoretical framework. There are also several papers describing experiments for the analogous Hawking effect [28,29,30,31,32,33,34,35,36,37,38].

In [1,2,3], we took into account a fourth-order ordinary differential equation that allowed us to treat the analogous Hawking effect in condensed matter systems (Bose–Einstein condensates (BECs), dielectric media, and water) in a systematic way, as far as weak dispersion effects are considered. The weak dispersion parameter $ϵ$ appears in the aforementioned master equation as the expansion parameter for the physical problem at hand. Strong dispersion effects, which, for example, may occur in some experimental settings, like those involving fiber optics (see, e.g., [36]), cannot be discussed in the same framework and are still beyond the possibility of a full analytical calculation. Moreover, the proposed models, from the theoretical point of view, are 2D, in the sense that two spatial dimensions are suppressed. The latter simplification is, in some sense, natural, as providing robust theoretical models for a non-trivial physical situation is simpler in 2D.

In the present analysis, still in the framework of weak dispersive effects, we are able to provide, in a rigorous way, a complete set of connection formulas on purely mathematical grounds, thus complementing and completing, on the mathematical side, our analysis in [1,2,3]. In particular, we show that in place of the connection formulas originally introduced by Corley [5] using the so-called Corley’s diagrams (see, e.g., [1,2]) and then further developed in [11] for a specific and elementary toy model (the so-called Corley model), one can provide a set of rigorously obtained connection formulas that are at the root of the dispersive Hawking effect and are substantially associated with the behavior of the solutions near the turning point. This represents a fundamental contribution to the existing literature, as the analysis for the near-turning-point region we discussed in [1,2] is universal, i.e., holds for any physical model quoted above, being the same in form as the near-horizon equation governing the physical process near the turning point.

We will refer to the following papers from the mathematical literature: Ref. [39] for the subluminal case and [40,41] for the superluminal one. We will show rigorously that, as a consequence of the aforementioned connection formulas and due to the structure of the Stokes matrices that allow us to transition from a basis for modes inside the black hole region to a basis for modes in the exterior region, there is a mode (which corresponds to the trivial, constant solution of (5)) that decouples from the other three in the near-horizon region and then remains decoupled in a 2D framework, at least in the leading order in the weak dispersion parameter characterizing each model, and a blackbody spectrum is ensured both for the Hawking particle and for the Hawking partner. One can also obtain exact solutions for the near-turning-point equation, which can be expressed in terms of the generalized hypergeometric functions ${}_{1}F_2({\alpha }_1;{\beta }_1,{\beta }_2;z)$.

Then, we will also provide insights into the contribution of transverse dimensions, albeit in very simple but significant geometrical situations. We will show that transverse dimensions do not affect the main picture of the 2D case and leave the temperature of the Hawking effect unaltered. Still, of course, the possibility of obtaining and then detecting particles traveling with a non-vanishing transverse wavenumber is gained, and this can shed light on realistic experimental situations (occurring, e.g., by using laser pulses in dielectric media).

A further important contribution appearing in this paper is the calculation, for specific but physically meaningful velocity profiles, of exact solutions of the reduced equation governing non-dispersive modes in the WKB approximation. This is also an important improvement to our general approach, as, even in the WKB approximation, at the leading order in the weak dispersion parameter $ϵ$, one would be forced to find exact solutions of the reduced equation to provide solutions for non-dispersive modes, to be considered together with the leading-order WKB solutions that can be obtained for the dispersive modes. This improvement is important, as it provides us with a further neat improvement for analytical calculations of physical amplitudes for the Hawking phenomenon.

2. The Master Equation: An Orr–Sommerfeld-Type Fourth-Order Equation

As discussed in [1,2,3], three significant cases of wave equations in dispersive analog gravity can be studied using the following master equation:

$${ϵ}^2{\displaystyle \frac{d^4\Phi }{d{x}^4}}\pm \left[p_3(x,ϵ){\displaystyle \frac{d^2\Phi }{d{x}^2}}+p_2(x,ϵ){\displaystyle \frac{d\Phi }{dx}}+p_1(x,ϵ)\Phi \right]=0,$$

(1)

where the upper sign occurs in the case of subluminal dispersion and the lower one in the case of superluminal dispersion. The latter case is considered in Nishimoto’s works (see, e.g., [40,41,42] and references therein). This fourth-order equation can be obtained by suitably manipulating the equations of motion in the Corley model in the case of dielectric media [1] and also in the case of a BEC and water [2], providing a unified framework for a systematic study of the analogous Hawking effect in weakly dispersive media and allowing a thermal spectrum to be found in all the cases at hand in the same limit, as expected.

A basic assumption of the present method is the analyticity of the coefficients in the following sense:

$$p_i(x,ϵ)=\sum _{n=0}^{\infty }{p}_{in}\left(x\right){ϵ}^n,$$

(2)

where $ϵ$ is the parameter expressing the presence of weak dispersive effects. Solutions of

$$p_{30}\left(x\right)=0$$

(3)

define the turning points (TPs) of the equation, as usual (cf., e.g., [41]), and one obtains the so-called reduced equation as $ϵ\to 0$:

$$p_{30}\left(x\right){\displaystyle \frac{d^2\Phi }{d{x}^2}}+p_{20}\left(x\right){\displaystyle \frac{d\Phi }{dx}}+p_{10}\left(x\right)\Phi =0.$$

(4)

In [41], it is assumed that the reduced equation displays a Fuchsian singularity at the TP (nothing actually prevents the general equation in itself from admitting regular behavior).

It can be shown [41] that near the TP $x=a$, the original equation is replaced by the fourth-order differential equation

$${\displaystyle \frac{d^{4}w}{d{z}^4}}\pm \left(z{\displaystyle \frac{d^{2}w}{d{z}^2}}+\lambda {\displaystyle \frac{dw}{dz}}\right)=0,$$

(5)

where, as discussed in [1], $w\left(z\right)$ represents the (rescaled [1]) wavefunction in the so-called near-horizon approximation to then be matched in the so-called linear region, with WKB solutions that hold far from the turning point. We refer to [1] for further details. In (5), the upper sign is for the subluminal case, and the lower one is for the superluminal case, and

$$z={\left(p_{30}^{\prime }\left(a\right)\right)}^{1/3}{ϵ}^{-2/3}(x-a),$$

(6)

and

$$\lambda ={\displaystyle \frac{p_{20}\left(a\right)}{p_{30}^{\prime }\left(a\right)}}.$$

(7)

As we discussed in [1,2], Equation (5) is universal in form and is at the root of the analogous Hawking effect in dispersive media, as far as they are governed by the above Orr–Sommerfeld-like equation. In the following, without loss of generality, we limit ourselves to the case of a single TP (monotonic profile for the velocity field or the refractive index in the dielectric case), identified with $x=0$.

3. Exact Solutions of the Near-Horizon Equation (5) and Connection Formulas

The problem concerning Equation (5) consists of not only finding solutions to the equation itself but also describing complete bases of solutions in different regions of the complex space and relating these bases (connection formulas). We point out that there exists a further connection problem, represented by the matching of solutions in the so-called linear region around the turning point, where both the near-horizon solutions and the WKB solutions are defined. This second problem (sometimes called the central connection problem [41]) can be solved very simply in the case where suitable bases of solutions in the near-horizon equation are assumed (see [41] and, for specific applications, [1,2]). The former problem (sometimes called the lateral connection problem [41]) is more difficult and is associated with the presence of the Stokes phenomenon, which does not allow us to find diagonal connection matrices. We will comment further on the Stokes phenomenon in the following.

Solutions to Equation (5) are found, e.g., by adopting the method of the Laplace transform, as discussed in [41] for the superluminal case, by means of Laplace integrals:

$$w_j\left(z\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{C_j}dt{t}^{\lambda -2}exp(zt\pm {\displaystyle \frac{1}{3}}{t}^3),$$

(8)

with a suitable choice of paths $C_j$ in the complex t-plane. It must be stressed that such solutions are regular at the TP and admit a series expansion there (see [40]). Of course, there is also a constant solution, which appears to be a Dirac delta in the space of the Laplace transform. This further solution is fundamental to obtaining a complete basis for the scattering problem, also in the region near the TP (i.e., near the horizon) [1]. Solutions of Equation (5) will be indexed by NH (near-horizon solutions, from a physical point of view).

3.1. The Subluminal Case

To be more specific, we first explore the subluminal case, i.e., the case where the plus sign appears both in (5) and in (8). We point out that it is possible to find exact solutions of Equation (5). For example, we can find the general solutions using Mathematica, and the result is

$$\begin{array}{cc}\hfill w\left(z\right)& =C_1+C_{2}z{}_{1}F_2\left({\displaystyle \frac{\lambda }{3}};{\displaystyle \frac{2}{3}},{\displaystyle \frac{4}{3}};-{\displaystyle \frac{z^3}{9}}\right)+C_3{z}^2{}_{1}F_2\left({\displaystyle \frac{1+\lambda }{3}};{\displaystyle \frac{4}{3}},{\displaystyle \frac{5}{3}};-{\displaystyle \frac{z^3}{9}}\right)\hfill \\ \hfill & +C_4{z}^3\left(1-{}_{1}F_2\left({\displaystyle \frac{-1+\lambda }{3}};{\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}};-{\displaystyle \frac{z^3}{9}}\right)\right).\hfill \end{array}$$

(9)

The generalized hypergeometric functions ${}_{1}F_2$ are analytic functions everywhere. They correspond, e.g., to the Airy functions one deals with in standard nonrelativistic quantum mechanics in a neighborhood of turning points, where the semiclassical WKB solutions fail. Even if they are analytic, their asymptotic expansion involves functions that are not analytic, and the Stokes phenomenon emerges. Furthermore, the main problem with these solutions is the fact that they are not in a simple relation with the physical modes involved in the problem in the following sense. As we recalled in the previous discussion, one has to match, in the linear region, WKB solutions and near-horizon solutions (9). This matching requires considering asymptotic expansion as $x\to 0$ (i.e., at the turning point) with asymptotic expansions of (9) as $z\to \infty$. The point is that, even if the asymptotic expansion of ${}_{1}F_2\left(z\right)$ is known, they represent quite cumbersome linear combinations of the physical modes, so it is not so easy to use exact solutions in this sense. Still, the analysis by Langer in [39] allows us to find a basis of solutions and connection formulas. We prefer to adopt another strategy, mimicking the analysis in [40,41] for the superluminal case. We limit ourselves to notice that, with respect to the bases described in [40], the bases described below are obtained simply by a rotation $z\mapsto exp(-i\pi /3)z$, so we have the replacement $arg\left(z\right)\mapsto arg\left(z\right)+{\displaystyle \frac{\pi }{3}}$. These bases are also confirmed by the analysis in [39]. We have the following representations for the non-constant subluminal solutions:

$$w_j\left(z\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{C_j}dt{t}^{\lambda -2}exp(zt+{\displaystyle \frac{1}{3}}{t}^3),$$

(10)

where the relevant paths for the bases are indicated in Figure 1, Figure 2 and Figure 3 below.

The first basis, say B1, which is relative to the sector $-{\displaystyle \frac{2\pi }{3}}<arg\left(z\right)<{\displaystyle \frac{2\pi }{3}}$, is $[1,w_4\left(z\right),$$w_1\left(z\right),w_2\left(z\right)]$.

The second basis, say B2, which is relative to the sector $0<arg\left(z\right)<{\displaystyle \frac{4\pi }{3}}$, is $[1,w_6\left(z\right),$$w_3\left(z\right),w_1\left(z\right)]$.

The third basis, say B3, which is relative to the sector ${\displaystyle \frac{2\pi }{3}}<arg\left(z\right)<2\pi$, is $[1,w_5\left(z\right),$$w_2\left(z\right),w_3\left(z\right)]$.

From a physical point of view, we adopt, substantially as in [1,2] for the subluminal Corley model and for the dielectric case with a single resonance frequency and water (both are subluminal), the basis $[1,w_4\left(z\right),w_1\left(z\right),w_2\left(z\right)]$ to represent the near-horizon expressions for the physical modes (Hawking particle $w_4\left(z\right)$, positive-norm dispersive mode $w_1\left(z\right)$, negative-norm dispersive mode $w_2\left(z\right)$, and 1 for the disconnected regular entering mode), participating in the scattering process in the external region $x>0$, and the basis $[1,w_5\left(z\right),w_2\left(z\right),w_3\left(z\right)]$ to represent the Hawking partner (solution $w_5\left(z\right)$), the exponentially decaying mode ($w_2\left(z\right)$), the exponentially growing mode ($w_3\left(z\right)$, for a comparison see, e.g., [11]), and the regular disconnected mode, represented by 1. Notice that the basis B2 could also be used to describe physical states in the black hole interior.

The present choice of bases in the different sectors has the following advantages. First of all, from the Cauchy formula, for B1, we have

$$w_1\left(z\right)+w_2\left(z\right)+w_3\left(z\right)=w_4\left(z\right),$$

(11)

as is evident from Figure 1. Furthermore, by defining, as in [40],

$$\psi :=exp\left(i{\displaystyle \frac{2}{3}}\pi \right),$$

(12)

we find the following relations:

$$\begin{array}{cc}\hfill w_1\left(z\right)& ={\psi }^{\lambda -1}{w}_3\left(\psi z\right)={\psi }^{2(\lambda -1)}{w}_2\left({\psi }^{2}z\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill w_4\left(z\right)& ={\psi }^{\lambda -1}{w}_6\left(\psi z\right)={\psi }^{2(\lambda -1)}{w}_5\left({\psi }^{2}z\right),\hfill \end{array}$$

(14)

which represent the substantial advantage of using such choices. Let us also introduce the following notation:

$$\begin{array}{cc}\hfill {\overline{w}}_i\left(z\right)& :={\psi }^{-3\lambda }{w}_i\left(z\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\tilde{w}}_i\left(z\right)& :={\psi }^{3\lambda }{w}_i\left(z\right),\hfill \end{array}$$

(16)

for $i=1,2,3,4,5,6$. Given (11), we can see that, for B2,

$${\overline{w}}_1\left(z\right)+w_2\left(z\right)+w_3\left(z\right)=w_6\left(z\right)$$

(17)

holds true, and for B3, the corresponding relation is

$${\overline{w}}_1\left(z\right)+w_2\left(z\right)+{\overline{w}}_3\left(z\right)=w_5\left(z\right).$$

(18)

We can provide the following connection formulas:

$$\begin{array}{cc}\hfill [1,w_4\left(z\right),w_1\left(z\right),w_2\left(z\right)]& =[1,w_6\left(z\right),w_3\left(z\right),w_1\left(z\right)]P_3,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill [1,w_5\left(z\right),w_2\left(z\right),w_3\left(z\right)]& =[1,w_4\left(z\right),w_1\left(z\right),w_2\left(z\right)]P_2,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill [1,w_6\left(z\right),w_3\left(z\right),w_1\left(z\right)]& =[1,w_5\left(z\right),w_2\left(z\right),w_3\left(z\right)]P_1,\hfill \end{array}$$

(21)

where the matrices $P_i,i=1,2,3$ connect the bases above and, in the following, will be explicitly written at the leading order in the series in $ϵ$. It is not difficult to see that, in considering the relation between B1 and B2,

$$\begin{array}{cc}\hfill w_4\left(z\right)& =w_6\left(z\right)+w_1\left(z\right)-{\overline{w}}_1\left(z\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill w_2\left(z\right)& =w_6\left(z\right)-w_3\left(z\right)-{\overline{w}}_1\left(z\right),\hfill \end{array}$$

(23)

hold true, so

$$P_3=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 1\\ 0& 0& 0& -1\\ 0& 1-{\psi }^{-3\lambda }& 1& -{\psi }^{-3\lambda }\end{array}\right).$$

(24)

Analogously, as far as the relation between B3 and B1 is concerned, one finds

$$\begin{array}{cc}\hfill w_5\left(z\right)& ={\overline{w}}_4\left(z\right)+w_2\left(z\right)-{\overline{w}}_2\left(z\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill w_3\left(z\right)& =w_4\left(z\right)-w_1\left(z\right)-w_2\left(z\right),\hfill \end{array}$$

(26)

and then

$$P_2=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\psi }^{-3\lambda }& 0& 1\\ 0& 0& 0& -1\\ 0& 1-{\psi }^{-3\lambda }& 1& -1\end{array}\right).$$

(27)

As to the relation between B2 and B3, we obtain

$$\begin{array}{cc}\hfill w_6\left(z\right)& =w_5\left(z\right)+w_3\left(z\right)-{\overline{w}}_3\left(z\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill w_1\left(z\right)& ={\tilde{w}}_5\left(z\right)-{\tilde{w}}_2\left(z\right)-w_3\left(z\right),\hfill \end{array}$$

(29)

and then

$$P_1=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& {\psi }^{3\lambda }\\ 0& 0& 0& -{\psi }^{3\lambda }\\ 0& 1-{\psi }^{-3\lambda }& 1& -1\end{array}\right).$$

(30)

3.2. The Superluminal Case

In the superluminal case, which, from a physical point of view, is very relevant because the dispersion relation of a BEC is superluminal, both in (5) and in (8), a minus sign appears. Also, in this case, we can find the general solutions:

$$\begin{array}{cc}\hfill w\left(z\right)& =C_1+C_{2}z{}_{1}F_2\left({\displaystyle \frac{\lambda }{3}};{\displaystyle \frac{2}{3}},{\displaystyle \frac{4}{3}};{\displaystyle \frac{z^3}{9}}\right)+C_3{z}^2{}_{1}F_2\left({\displaystyle \frac{1+\lambda }{3}};{\displaystyle \frac{4}{3}},{\displaystyle \frac{5}{3}};{\displaystyle \frac{z^3}{9}}\right)\hfill \\ \hfill & +C_4{z}^3\left(1-{}_{1}F_2\left({\displaystyle \frac{-1+\lambda }{3}};{\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}};{\displaystyle \frac{z^3}{9}}\right)\right).\hfill \end{array}$$

(31)

We do not delve into a discussion, as considerations analogous to those in the previous subsection hold true.

We refer instead to the framework in which generalized Airy solutions are introduced by means of the Laplace transform, as discussed in the previous subsection. For a complete analysis, readers can refer to [40,41]. We present, in a single figure, as in [41], the full set of six non-constant solutions, again expressed in terms of generalized Airy functions:

$$w_j\left(z\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{C_j}dt{t}^{\lambda -2}exp(zt-{\displaystyle \frac{1}{3}}{t}^3),$$

(32)

along the paths indicated in Figure 4.

To construct the three bases

$$\begin{array}{cc}\hfill [1,w_6\left(z\right),w_3\left(z\right),w_1\left(z\right)]& \mathrm{for}-{\displaystyle \frac{\pi }{3}}<arg\left(z\right)<\pi ,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill [1,w_5\left(z\right),w_2\left(z\right),w_3\left(z\right)]& \mathrm{for}{\displaystyle \frac{\pi }{3}}<arg\left(z\right)<{\displaystyle \frac{5\pi }{3}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill [1,w_4\left(z\right),w_1\left(z\right),w_2\left(z\right)]& \mathrm{for}-\pi <arg\left(z\right)<{\displaystyle \frac{\pi }{3}},\hfill \end{array}$$

(35)

one can simply select the corresponding paths in Figure 4, with just one branch cut (i.e., the branch cut involved in the definition of the specific function $w_i\left(z\right)$, $i=4,5,6$, in the basis at hand), and construct figures analogous to Figure 1, Figure 2 and Figure 3. As in [40], the connection formulas are

$$\begin{array}{cc}\hfill [1,w_6\left(z\right),w_3\left(z\right),w_1\left(z\right)]& =[1,w_5\left(z\right),w_2\left(z\right),w_3\left(z\right)]{\Pi }_1,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill [1,w_5\left(z\right),w_2\left(z\right),w_3\left(z\right)]& =[1,w_4\left(z\right),w_1\left(z\right),w_2\left(z\right)]{\Pi }_2,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill [1,w_4\left(z\right),w_1\left(z\right),w_2\left(z\right)]& =[1,w_6\left(z\right),w_3\left(z\right),w_1\left(z\right)]{\Pi }_3,\hfill \end{array}$$

(38)

where, for the matrices ${\Pi }_i$, we have ${\Pi }_i=P_i$ for $i=1,2,3$, with the same $P_i$ as in (24), (27), and (30).

4. Application to the Corley Model

We choose to work in the framework of the simplest possible model, both for brevity and for a more direct comparison with the most quoted literature [5,11]. We note that, even if it allows both a subluminal and a superluminal setting, it allows only a constant sound velocity c, and, moreover, it represents a strong simplification of the picture described in the previous analysis, as the coefficients $p_i(x,ϵ)$ do not actually depend on $ϵ$ in this case. We limit ourselves to the subluminal case, and the superluminal one is sketched in Appendix C. We start with the following action:

$$S={\displaystyle \frac{1}{2}}\int d^{2}x[{\left(({\displaystyle \frac{1}{c}}{\partial }_t+{\displaystyle \frac{v}{c}}{\partial }_x)\varphi \right)}^2+\varphi ({\partial }_x^2+{ϵ}^2{\partial }_x^4)\varphi ],$$

(39)

which was deduced in [5,43]. The model is associated with a conserved inner product (see [5]) and with current conservation. By separating variables as in [5], $\varphi (t,x)=e^{-i\omega t}\phi \left(x\right)$, one obtains the fourth-order ordinary differential equation

$${ϵ}^2{\partial }_x^4\phi +\left(1-{\displaystyle \frac{v^2\left(x\right)}{c^2}}\right){\partial }_x^2\phi +2{\displaystyle \frac{v\left(x\right)}{c^2}}(i\omega -v^{\prime }\left(x\right)){\partial }_x\phi -i{\displaystyle \frac{\omega }{c^2}}(i\omega -v^{\prime }\left(x\right))\phi =0,$$

(40)

where $v\left(x\right)$ is the velocity field, $v^{\prime }\left(x\right)$ stands for its first derivative with respect to x, and c is the constant sound velocity. We assume, as a significant physical scale, as in [5,43], the scale ${ϵ}^2$ associated with nonlinearity, which is to be related to the presence of dispersion in the model itself, and, to allow a better comparison with the literature, we point out that, in [5], one has

$$ϵ\mapsto {\displaystyle \frac{1}{k_0}}$$

(41)

(this is slightly different from the choice in [1], but substantially equivalent results are inferred). As is evident, the model is just 2D. In the limit $ϵ\to 0$, one obtains the so-called reduced equation,

$$\left(1-{\displaystyle \frac{v^2\left(x\right)}{c^2}}\right){\partial }_x^2\phi +2{\displaystyle \frac{v\left(x\right)}{c^2}}(i\omega -v^{\prime }\left(x\right)){\partial }_x\phi -i{\displaystyle \frac{1}{c^2}}\omega (i\omega -v^{\prime }\left(x\right))\phi =0.$$

(42)

In agreement with [5], we assume $v\left(x\right)\le 0$ and a monotonic velocity profile, where $v\left(x\right)$ is asymptotically constant, as in [1,2]. A real turning point occurs for

$$v\left(x\right)+c=0,$$

(43)

and the so-called linear region is the interval where the following approximation holds:

$$v\left(x\right)\simeq -c+\kappa x,$$

(44)

where

$$\kappa :=v^{\prime }(x=0).$$

(45)

For a complete and exhaustive calculation of both the WKB solutions and the near-horizon solutions, we refer to [1]. We simply display the main results obtained in [1] in Section 4.1 and Section 4.2.

4.1. WKB Solutions

The following solutions

$$\begin{array}{cc}\hfill {\phi }_1\left(x\right)& ={\left({\displaystyle \frac{1}{1-\frac{v^2\left(x\right)}{c^2}}}\right)}^{3/4}exp\left({\displaystyle \frac{i}{ϵ}}{\int }^{x}ds\sqrt{1-{\displaystyle \frac{v^2\left(s\right)}{c^2}}}\right)exp\left(i{\displaystyle \frac{\omega }{c}}{\int }^{x}ds{\displaystyle \frac{v\left(s\right)}{c}}{\displaystyle \frac{1}{1-\frac{v^2\left(s\right)}{c^2}}}\right),\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\phi }_2\left(x\right)& ={\left({\displaystyle \frac{1}{1-\frac{v^2\left(x\right)}{c^2}}}\right)}^{3/4}exp(-{\displaystyle \frac{i}{ϵ}}{\int }^{x}ds\sqrt{1-{\displaystyle \frac{v^2\left(s\right)}{c^2}}})exp\left(i{\displaystyle \frac{\omega }{c}}{\int }^{x}ds{\displaystyle \frac{v\left(s\right)}{c}}{\displaystyle \frac{1}{1-\frac{v^2\left(s\right)}{c^2}}}\right),\hfill \end{array}$$

(47)

where (46) is the dispersive (non-normalized) positive-norm mode, and (47) is the dispersive (non-normalized) negative-norm mode. The seemingly weird numbering is useful for matching in the linear region (cf. Figure 1), where

$$\begin{array}{cc}\hfill {\phi }_1\left(x\right)& \simeq {\left({\displaystyle \frac{2\kappa }{c}}x\right)}^{-3/4}{x}^{-{\displaystyle \frac{i\omega }{2\kappa }}}exp\left({\displaystyle \frac{i}{ϵ}}{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2\kappa }{c}}}{x}^{{\displaystyle \frac{3}{2}}}\right),\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\phi }_2\left(x\right)& \simeq {\left({\displaystyle \frac{2\kappa }{c}}x\right)}^{-3/4}{x}^{-{\displaystyle \frac{i\omega }{2\kappa }}}exp\left(-{\displaystyle \frac{i}{ϵ}}{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2\kappa }{c}}}{x}^{{\displaystyle \frac{3}{2}}}\right).\hfill \end{array}$$

(49)

These expansions also hold true for $x<0$: let us assume $x=exp\left(i\pi \right)\left|x\right|$; then, in the linear region, one finds

$$\begin{array}{cc}\hfill {\phi }_g\left(x\right)& \simeq {\left({\displaystyle \frac{2\kappa }{c}}\left|x\right|\right)}^{-3/4}exp(-i{\displaystyle \frac{3}{4}}\pi ){\left|x\right|}^{-{\displaystyle \frac{i\omega }{2\kappa }}}exp\left({\displaystyle \frac{\pi \omega }{2\kappa }}\right)exp\left({\displaystyle \frac{1}{ϵ}}{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2\kappa }{c}}}{\left|x\right|}^{{\displaystyle \frac{3}{2}}}\right),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\phi }_d\left(x\right)& \simeq {\left({\displaystyle \frac{2\kappa }{c}}\left|x\right|\right)}^{-3/4}exp(-i{\displaystyle \frac{3}{4}}\pi ){\left|x\right|}^{-{\displaystyle \frac{i\omega }{2\kappa }}}exp\left({\displaystyle \frac{\pi \omega }{2\kappa }}\right)exp\left(-{\displaystyle \frac{1}{ϵ}}{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2\kappa }{c}}}{\left|x\right|}^{{\displaystyle \frac{3}{2}}}\right),\hfill \end{array}$$

(51)

where d indicates the decaying mode and g the growing mode (cf. [11]). We do not need to give a general WKB expansion for these modes, as they do not enter the asymptotic expansion of the S-matrix (as the decaying mode vanishes and the growing one must be set to zero).

Two further solutions can be obtained from the reduced equation, and, in order to maintain the same order of approximation in our WKB expansion, one would need exact solutions to avoid the introduction of a further expansion parameter. Nevertheless, near the regular singular point $x=0$, which represents the real turning point, i.e., the horizon, we can provide the following series expansions:

$$\begin{array}{cc}\hfill {\phi }_s\left(x\right)& =1+\sum _{n=1}^{\infty }{c}_n{x}^n,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\phi }_4\left(x\right)& =x^{i{\displaystyle \frac{\omega }{\kappa }}}\left(1+\sum _{n=1}^{\infty }{d}_n{x}^n\right),\hfill \end{array}$$

(53)

where ${\phi }_4$ represents the Hawking mode, and ${\phi }_s$ represents a further short wavenumber mode, which, eventually, simply plays the role of a disconnected spectator mode, at least as far as the leading-order process is concerned. It is useful to provide approximate solutions to the reduced equation even for large x (in the external region with respect to the black hole). It is easy to show that for large x in the above sense, we have $v\left(x\right)\sim$ const, and then $v^{\prime }=0$. As a consequence, e.g., under the conditions of theorem 1.9.1 in [44], we obtain, as $x\to \infty$,

$$\begin{array}{cc}\hfill {\phi }_s\left(x\right)& \sim exp(-i\omega {\displaystyle \frac{1}{c-v_r}}x),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\phi }_4\left(x\right)& \sim exp\left(i\omega {\displaystyle \frac{1}{c+v_r}}x\right).\hfill \end{array}$$

(55)

For $x<0$, the reduced equation provides us with two further solutions:

$$\begin{array}{cc}\hfill {\phi }_d\left(x\right)& =1+\sum _{n=1}^{\infty }{e}_n{x}^n,\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill {\phi }_5\left(x\right)& =x^{i{\displaystyle \frac{\omega }{\kappa }}}\left(1+\sum _{n=1}^{\infty }{f}_n{x}^n\right),\hfill \end{array}$$

(57)

with the asymptotic behavior

$$\begin{array}{cc}\hfill {\phi }_d\left(x\right)& \sim exp(-i\omega {\displaystyle \frac{1}{c-v_l}}x),\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\phi }_5\left(x\right)& \sim exp\left(i\omega {\displaystyle \frac{1}{c+v_l}}x\right),\hfill \end{array}$$

(59)

with ${lim}_{x\to -\infty }v\left(x\right)=:v_l<-c<0$. These solutions correspond to left-moving modes in the superluminal region, and they are the only propagating modes in that region. We notice that the mode ${\phi }_5\left(x\right)$ is a negative-norm mode and that the regular mode ${\phi }_d\left(x\right)$ is analogous to ${\phi }_s\left(x\right)$ on the negative real axis.

As useful interpolating formulas inherited by [5] (WKB-like, but they cannot be rigorously obtained by using the $ϵ$-expansion as in the above framework), we could also use

$$\begin{array}{cc}\hfill {\phi }_j^{int}\left(x\right)& \sim exp(-i\omega {\int }^{x}dy{\displaystyle \frac{1}{c-v\left(y\right)}}),j=s,d\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {\phi }_k^{int}\left(x\right)& \sim exp\left(i\omega {\int }^{x}dy{\displaystyle \frac{1}{c+v\left(y\right)}}\right),k=4,5\hfill \end{array}$$

(61)

which still display the correct behavior both in the linear region and in the asymptotic one.

In the appendix, we also provide exact solutions of the reduced equation for very specific velocity profiles.

4.2. Solutions Near the Turning Point

Let us introduce

$$z={\left({\displaystyle \frac{2\kappa }{c}}\right)}^{1/3}{ϵ}^{-2/3}x,$$

(62)

and point out that, in our case, we obtain

$$\lambda =1-i{\displaystyle \frac{\omega }{\kappa }}.$$

(63)

In order to find physically relevant solutions of (5), we adopt the strategy utilized in [1]. We refer to Figure 1, and we choose to directly construct the relevant physical states by exploiting the method of the steepest descents [45,46,47] for approximating, as $z\to \infty$, the solutions $w_1\left(z\right),w_2\left(z\right),w_4\left(z\right)$ occurring in B1. We need to consider, of course, the solutions (10), which, by setting $t=\sqrt{\left|z\right|}u$, can be rewritten as

$$w_j\left(z\right)={\displaystyle \frac{1}{2\pi i}}{\left|z\right|}^{{\displaystyle \frac{\lambda -1}{2}}}{I}_j\left(z\right),$$

(64)

where

$$I_j\left(z\right)={\int }_{{\overline{C}}_j}{dug\left(u\right)exp\left(\right|z|}^{3/2}{h}_{\pm }\left(u\right)),$$

(65)

and

$$\begin{array}{cc}\hfill g\left(u\right)& :=u^{\lambda -2},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill h_{\pm }\left(u\right)& :=\pm u+{\displaystyle \frac{u^3}{3}};\hfill \end{array}$$

(67)

where $\pm =\mathrm{sign}\left(x\right)$ and $C_j$ are the same as in Figure 1. We notice that the constant solution indicated as 1, as in [40,41], is the same on both sides of the linear region. The following analysis involves just the non-trivial solutions of (5) in the asymptotic portions of the linear region, i.e., as $z\to \pm \infty$, as this is the behavior we need to know for matching with WKB solutions. This is by no means different from what is implemented in standard quantum mechanics near a TP.

For $x>0$, the solutions $w_1$ and $w_2$ can be found through the steepest descents passing through the saddle points $u_{\pm }=\pm i$, respectively:

$$\begin{array}{cc}\hfill w_1\left(z\right)\simeq & {\displaystyle \frac{1}{2\sqrt{\pi }}}{e}^{-{\displaystyle \frac{3}{4}}\pi i}{e}^{{\displaystyle \frac{\pi \omega }{2\kappa }}}{\left|z\right|}^{-{\displaystyle \frac{i\omega }{2\kappa }}-{\displaystyle \frac{3}{4}}}{e}^{i{\displaystyle \frac{2}{3}}{\left|z\right|}^{3/2}},\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill w_2\left(z\right)\simeq & {\displaystyle \frac{1}{2\sqrt{\pi }}}{e}^{{\displaystyle \frac{1}{4}}\pi i}{e}^{-{\displaystyle \frac{\pi \omega }{2\kappa }}}{\left|z\right|}^{-{\displaystyle \frac{i\omega }{2\kappa }}-{\displaystyle \frac{3}{4}}}{e}^{-i{\displaystyle \frac{2}{3}}{\left|z\right|}^{3/2}}.\hfill \end{array}$$

(69)

As to the cut contribution for $x>0$, i.e., $w_4\left(z\right)$, it represents the Hawking mode. As shown in [1], the branch cut lies along the steepest descent, and this allows us to find [1]

$$w_4\left(z\right)\simeq -{\displaystyle \frac{1}{i\pi }}{\left|z\right|}^{i{\displaystyle \frac{\omega }{k}}}\Gamma \left(-i{\displaystyle \frac{\omega }{\kappa }}\right)sinh\left({\displaystyle \frac{\pi \omega }{\kappa }}\right).$$

(70)

For $x<0$, we must refer to the basis B3 and to Figure 3. The decaying mode passing through the saddle point $u=1$ is

$$w_3\left(z\right)\simeq {\displaystyle \frac{1}{2\sqrt{\pi }}}{\left|z\right|}^{-{\displaystyle \frac{i\omega }{2\kappa }}-{\displaystyle \frac{3}{4}}}{e}^{-{\displaystyle \frac{2}{3}}{\left|z\right|}^{3/2}}.$$

(71)

The mode $w_2\left(z\right)$ corresponds to the growing mode:

$$w_2\left(z\right)\simeq {\displaystyle \frac{1}{2\sqrt{\pi }}}{\left|z\right|}^{-{\displaystyle \frac{i\omega }{2\kappa }}-{\displaystyle \frac{3}{4}}}{e}^{{\displaystyle \frac{2}{3}}{\left|z\right|}^{3/2}}.$$

(72)

And one may also simply consider the contribution (70) by choosing a suitable analytical continuation for $x<0$. It turns out that, by choosing the branch where $-1=e^{i\pi }$, the further solution one obtains,

$$w_5\left(z\right):\simeq -{\displaystyle \frac{1}{i\pi }}{e}^{-\pi {\displaystyle \frac{\omega }{\kappa }}}{\left|z\right|}^{i{\displaystyle \frac{\omega }{k}}}\Gamma \left(-i{\displaystyle \frac{\omega }{\kappa }}\right)sinh\left({\displaystyle \frac{\pi \omega }{\kappa }}\right),$$

(73)

is such that it corresponds to the Hawking partner (an antiparticle state, i.e., a state with a negative norm).

4.3. Matching: Complete Solutions

A careful comparison with the WKB expansion displayed in the previous section provides us with the connection formulas (cf. the so-called central connections in [41]). We have to match, in a single solution, the WKB part and the near-horizon part of the modes introduced above in such a way as to obtain basis functions that are defined in the whole domain (cf. also [1]). Let us use ${\phi }_i^{WKB}\left(x\right)$ and ${\phi }_i^{NH}\left(x\right)$ to denote the parts to be joined for the i-mode in a specific sector of the complex plane, with indexes that match those in B1, B2, or B3, according to the sector one chooses, and with x taking complex values.

In the matching region, which, in our case, will be considered to be the linear region, where the two approximations coexist, we have

$${\phi }_i^{WKB}\left(x\right)\sim a_i^{WKB}{h}_i\left(x\right),$$

(74)

and also

$${\phi }_i^{NH}\left(x\right)\sim b_i^{NH}{h}_i\left(x\right),$$

(75)

with the same functional dependence $h_i\left(x\right)$ and with known constants $a_i^{WKB},b_i^{NH}$. We remark that, in a rigorous approach, one should also take into account the dependence on the parameter $ϵ$ of both expansions. We limit ourselves to the leading order. For details, see [41]. Let us consider the region $x>0$. As we must connect ${\phi }_i^{WKB}\left(x\right)$ and ${\phi }_i^{NH}\left(x\right)$ in the linear region, where there exists a coexistence region for both the solutions, we find

$${\phi }_i^{NH}\left(x\right)=H_i^{WKB}{\phi }_i^{WKB}\left(x\right);$$

(76)

i.e., matching in the linear region requires

$$H_i^{WKB}={\displaystyle \frac{b_i^{NH}}{a_i^{WKB}}}.$$

(77)

As can be seen, the aforementioned matching is diagonal (in [41], this is indicated as the central connection problem) in the sense that the WKB and NH expansions of each single i-esime mode are connected. We notice that, for the complete solution, which is indicated as ${\phi }_i\left(x\right)$, in the aforementioned sector of the complex plane, we must obtain

$$\begin{array}{cc}\hfill {\phi }_i\left(x\right)& \sim {\phi }_i^{NH}\left(x\right)\mathrm{for}x\in LR\cap U\left(TP\right),\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill {\phi }_i\left(x\right)& \sim H_i^{WKB}{\phi }_i^{WKB}\left(x\right)\mathrm{for}x\in LR\cap D_{WKB},\hfill \end{array}$$

(79)

where $LR$ identifies part of the linear region where both expansions coexist, $U\left(TP\right)$ represents the neighborhood of the TP where the near-horizon approximation holds, and $D_{WKB}$ indicates the region where the WKB approximation holds. As a consequence, either ${\phi }_i^{NH}\left(x\right)$ or $H_i^{WKB}{\phi }_i^{WKB}\left(x\right)$ equivalently identifies the complete solution ${\phi }_i\left(x\right)$. $H_i^{WKB}{\phi }_i^{WKB}\left(x\right)$ is the most interesting for scattering problems, where $x\to \pm \infty$ has to be taken into account.

Analogously, for $x<0$, we will find

$$K_j^{WKB}={\displaystyle \frac{m_j^{NH}}{n_j^{WKB}}},$$

(80)

with an obvious notation (where the replacements $b_j^{NH}\mapsto m_j^{NH}$ and $a_j^{WKB}\mapsto n_j^{WKB}$ are made to distinguish between outer and inner constants), which takes into account that we have different bases for the black hole region $x<0$ and for the exterior one $x>0$ so that such matching will involve solutions on both sides of the horizon (TP).

Herein, we deepen and develop our results obtained in [1,2,3] by taking into account rigorously, from a mathematical point of view, what happens in the scattering process associated with the analog (weakly dispersive) Hawking effect. We consider, in the near-horizon region and then in the matching with WKB solutions, the basis B1 for representing physical states for $x>0$ and the basis B3 for representing states for $x<0$ (notice that, in line with principles, one could also choose the basis B2 for the states at $x>0$). The horizon corresponds to $x=0$ in our assumed setting. Formally, let us define $I\left(B1\right),I\left(B2\right),I\left(B3\right)$ as the sets of indices associated with the bases in the near-horizon region. We choose the index 0 for the constant solution, and then, for example, we have $I\left(B1\right)=\{0,4,1,2\}$.

The first obvious observation is that, as mode (72) explodes at $x\to -\infty$, when one considers the generic states that can be constructed with the basis B3,

$$\varphi \left(x\right):=\sum _{i\in I\left(B3\right)}{a}_i{K}_i^{WKB}{\phi }_i^{WKB}\left(x\right),$$

(81)

where $a_i$ is a constant, one must set $a_2=0$. As the lateral connection formulas are not diagonal, this does not imply that for $x>0$, the element ${\phi }_2^{WKB}\left(x\right)$, which corresponds to a negative-norm state, disappears. Indeed, the Stokes phenomenon (cf. [41]) arises so that the different bases in the different regions of the complex plane are related through non-diagonal matrices. This happens because, although exact solutions of (5) are analytical everywhere, their asymptotic expansions as $z\to \infty$ (which are needed because of the matching in the linear region with the WKB solutions) are not analytical and, in particular, at infinity an irregular singularity appears for the WKB solutions. The finding of Stokes lines, for a higher-order linear ordinary differential equation, represents a highly non-trivial problem, as pointed out in [48], and often requires numerical calculations (see also [49]). Their presence, in our case, is implicit in the non-triviality of the connection matrices $P_i$ and the non-diagonality of the matrices connecting the bases of states for $x<0$ and $x>0$. Following the mathematical literature, we will refer to the aforementioned matrices as Stokes matrices.

For the Stokes matrices connecting asymptotic states, we have the following situation, for which we adopt a general notation, to be specified in the next discussion. Let $\alpha ,\beta$ indicate the specific basis one is referring to. Let $C_i^{\alpha }$ be the central connection coefficient for the basis $B_{\alpha }$ (we mean $B1,B2,B3$, discussed in the previous section), indexed by $\alpha$, and $C_i^{\beta }$ be the analogous coefficient for the basis $B_{\beta }$ indexed by $\beta$, and let $P^{(\alpha \mapsto \beta )}$ represent the (lateral) connection matrix associated with the transition from $B_{\alpha }$ to $B_{\beta }$ (we mean $P_1,P_2,P_3$, discussed in the previous section). Then, we have

$$\sum _{i\in I\left(B_{\alpha }\right)}{a}_i{\phi }_i^{WKB}\left(x\right)=\sum _{i\in I\left(B_{\alpha }\right)}\sum _{j\in I\left(B_{\beta }\right)}{a}_i{M}_{ij}^{(\alpha \mapsto \beta )}{\phi }_j^{WKB}\left(x\right),$$

(82)

where the coefficients $a_i$ are a priori arbitrary coefficients for constructing a generic state, and the Stokes connection matrix $M^{(\alpha \mapsto \beta )}$ has the elements

$$M_{ij}^{(\alpha \mapsto \beta )}={\left(C_i^{\alpha }\right)}^{-1}{P}_{ij}^{(\alpha \mapsto \beta )}{C}_j^{\beta },$$

(83)

i.e., $M^{(\alpha \mapsto \beta )}=\mathrm{diag}{\left(C^{\alpha }\right)}^{-1}{P}^{(\alpha \mapsto \beta )}\mathrm{diag}\left(C^{\beta }\right)$ (see also the discussion in [41]).

Herein, we start by discussing Stokes matrices for our specific case of interest. As $x\to -\infty$, i.e., in the black hole region, there are two non-dispersive WKB solutions of the reduced equation, representing the Hawking partner and the spectator mode, which are the only propagating ones, with central connections in the linear region, with $w_5\left(z\right)$ and with the constant solution 1, respectively. The other two dispersive WKB solutions represent the decaying mode, connected with $w_3\left(z\right)$ in the linear region, and the growing mode, connected with $w_2\left(z\right)$. In the opposite region, as $x\to \infty$, i.e., in the external region, all four solutions propagate. There are, as can be seen, two dispersive and two non-dispersive WKB solutions, to be connected in the linear region with $w_1\left(z\right),w_2\left(z\right),w_4\left(z\right),1$. Let us introduce $I^{\prime }\left(B3\right)=\{0,5,3\}$, i.e., the set $I\left(B3\right)$ without the index of the diverging mode (the growing mode).

As discussed in detail in [1] and inherited by the analysis in [5], in the situations involved with the analog Hawking effect for $x\to \infty$, the propagating modes behave as plane waves:

$${\phi }_j^{WKB}\left(x\right)\sim d_j^{WKB}{e}^{i{k}_j\left(\omega \right)x},$$

(84)

where $d_j^{WKB}$ defines (known) constants emerging in the asymptotic expansion of WKB solutions such that

$${\phi }_j\left(x\right)\sim H_j^{WKB}{d}_j^{WKB}{e}^{i{k}_j\left(\omega \right)x}.$$

(85)

For a direct comparison with the coefficients $c_j:=c_j^{corley}$ appearing in [5], one has $c_j^{corley}=H_j^{WKB}{d}_j^{WKB}$. As to the situation for $x\to -\infty$, we have, analogously,

$${\phi }_i\left(x\right)\sim K_i^{WKB}{g}_i^{WKB}{e}^{i{k}_i\left(\omega \right)x},$$

(86)

where the constants $g_i^{WKB}$ are known and emerge in the asymptotic expansion of the WKB solutions. Then, for the generic state for $x<0$, we obtain (cf. (81))

$$\varphi \left(x\right)\sim a_0{K}_0^{WKB}{g}_0^{WKB}{e}^{i{k}_0\left(\omega \right)x}+a_5{K}_5^{WKB}{g}_5^{WKB}{e}^{i{k}_5\left(\omega \right)x}+a_3{K}_3^{WKB}{g}_3^{WKB}{e}^{-k_3\left(\omega \right)\left|x\right|}$$

(87)

As a consequence, we have, for $x\to -\infty$,

$$\varphi \left(x\right)\sim \sum _{i\in I^{\prime }\left(B3\right)}{a}_i{K}_i^{WKB}{g}_i^{WKB}{e}^{i{k}_i\left(\omega \right)x},$$

(88)

and, as $x\to \infty$,

$$\varphi \left(x\right)\sim \sum _{i\in I^{\prime }\left(B3\right)}\sum _{j\in I\left(B1\right)}{a}_i{P_2}_{ij}{H}_j^{WKB}{d}_j^{WKB}{e}^{i{k}_j\left(\omega \right)x},$$

(89)

where the lateral connection matrix $P_2$ enters in a fundamental way. By comparison with (82), we infer, in our case,

$$M_{ij}^{(B3\mapsto B1)}={\displaystyle \frac{1}{K_i^{WKB}{g}_i^{WKB}}}{P_2}_{ij}{H}_j^{WKB}{d}_j^{WKB}.$$

(90)

We refer, for a more general discussion, involving n-th-order ordinary differential equations and their WKB solutions, to [48], specifically Section 7 therein.

As to the scattering amplitudes, it is interesting to introduce

$$H_j^{WKB}{d}_j^{WKB}=:{\overline{H}}_j{N}_j,$$

(91)

where $N_i$ denotes the normalizations of the propagating modes in the asymptotic region, as in [5,11] (see [1] and references therein). In particular, we have

$$N_j={\displaystyle \frac{1}{\sqrt{4\pi |v_g\left(k_j\left(\omega \right)\right)(\omega -v{k}_j\left(\omega \right))\left|\right)}}},$$

(92)

where $v_g\left(k_j\left(\omega \right)\right)$ is the group velocity of the $j-th$ mode. We note that, as to normalizations, the above formula also holds for the propagating modes for $x<0$ (i.e., in the black hole region), i.e., one can, e.g., define $K_j^{WKB}{g}_j^{WKB}=:{\overline{K}}_j{N}_j$ as well. We note that the decaying mode is not normalizable, but this is irrelevant. See the discussion in the following subsection.

The coefficients ${\overline{H}}_j$ are given by

$${\overline{H}}_j={\displaystyle \frac{H_j^{WKB}{d}_j^{WKB}}{N_j}}={\displaystyle \frac{b_j^{NH}}{N_j{a}_j^{WKB}}}{d}_j^{WKB}.$$

(93)

For $x<0$, we find analogously

$${\overline{K}}_j={\displaystyle \frac{K_j^{WKB}{g}_j^{WKB}}{N_j}}={\displaystyle \frac{m_j^{NH}}{N_j{n}_j^{WKB}}}{g}_j^{WKB}.$$

(94)

These quantities are very relevant, as, in the calculation of the ratio of currents whose conserved flux is measured at infinity, one has, for example,

$${\displaystyle \frac{|J_x^i|}{|J_x^j|}}={\displaystyle \frac{|{\overline{H}}_i{|}^2}{|{\overline{H}}_j{|}^2}}$$

(95)

if both the currents are relative to the external region, with an obvious change if currents in the black hole region are involved. See also the following subsection.

4.4. S-Matrix Revisited: A Further Labeling of States, Stokes Matrices Elements, and Particle Creation

Let us adopt the following physical notation for states: in the external region $x>0$, we set, for the indices, the following correspondence: $0\mapsto V$, $1\mapsto P$, $2\mapsto N$, $4\mapsto H$, where P and N represent the dispersive modes with positive and negative norms, respectively, and H and V are the modes that correspond to solutions of the reduced equation, representing Hawking particles and regular modes, respectively. For $x<0$, the correspondence is $0\mapsto V^{\prime }$, $5\mapsto \overline{H}$ for the propagating modes, where $V^{\prime }$ is the regular mode propagating toward $x\to -\infty$, and $\overline{H}$ is associated with the Hawking partner; furthermore, we consider $3\mapsto D$ for the decaying mode. It is simpler to write the connection matrix by referring to physical modes instead of using the basis elements described in the previous section, as it makes the process associated with the elements of the Stokes matrix more explicit in a trivial way. A subtlety could occur, as some modes, particularly the decaying mode, do not admit normalization; still, this does not represent a problem at all. As is seen, in regard to the growing mode, its amplitude must be set to zero.

The transition from (87) to (85) involves the following non-vanishing elements of the Stokes matrix (90):

$$\begin{array}{cc}\hfill M_{\overline{H}H}& ={\displaystyle \frac{1}{K_{\overline{H}}^{WKB}{g}_{\overline{H}}^{WKB}}}{\psi }^{-3\lambda }{H}_H^{WKB}{d}_H^{WKB},\hfill \end{array}$$

(96)

$$\begin{array}{cc}\hfill M_{\overline{H}N}& ={\displaystyle \frac{1}{K_{\overline{H}}^{WKB}{g}_{\overline{H}}^{WKB}}}(1-{\psi }^{-3\lambda })H_N^{WKB}{d}_N^{WKB},\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill M_{V^{\prime }V}& ={\displaystyle \frac{1}{K_{V^{\prime }}^{WKB}{g}_{V^{\prime }}^{WKB}}}{H}_V^{WKB}{d}_V^{WKB},\hfill \end{array}$$

(98)

$$\begin{array}{cc}\hfill M_{DH}& ={\displaystyle \frac{1}{K_D^{WKB}{g}_D^{WKB}}}{H}_H^{WKB}{d}_H^{WKB},\hfill \end{array}$$

(99)

$$\begin{array}{cc}\hfill M_{DP}& =-{\displaystyle \frac{1}{K_D^{WKB}{g}_D^{WKB}}}{H}_P^{WKB}{d}_P^{WKB},\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill M_{DN}& =-{\displaystyle \frac{1}{K_D^{WKB}{g}_D^{WKB}}}{H}_N^{WKB}{d}_N^{WKB}.\hfill \end{array}$$

(101)

These matrix elements tell us how the various solutions, and thus how the physical modes, are related to each other. As in a Feynman diagram, we consider that there is a so-called mode conversion ([43]) only in the case of non-zero non-diagonal matrix elements.

All the constants appearing in (99), (100), and (101) are known, and, explicitly, in particular, we have

$$\begin{array}{cc}\hfill H_P^{WKB}& =e^{-{\displaystyle \frac{3}{4}}\pi i}{\displaystyle \frac{e^{\frac{\pi \omega }{2\kappa }}}{2\sqrt{\pi }}}{\left({\displaystyle \frac{2\kappa }{c}}\right)}^{-{\displaystyle \frac{i\omega }{6\kappa }}+{\displaystyle \frac{1}{2}}}{ϵ}^{{\displaystyle \frac{i\omega }{3\kappa }}+{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill H_N^{WKB}& =e^{{\displaystyle \frac{1}{4}}\pi i}{\displaystyle \frac{e^{-\frac{\pi \omega }{2\kappa }}}{2\sqrt{\pi }}}{\left({\displaystyle \frac{2\kappa }{c}}\right)}^{-{\displaystyle \frac{i\omega }{6\kappa }}+{\displaystyle \frac{1}{2}}}{ϵ}^{{\displaystyle \frac{i\omega }{3\kappa }}+{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

(103)

$$\begin{array}{cc}\hfill H_H^{WKB}& =-{\displaystyle \frac{sinh\left(\frac{\pi \omega }{\kappa }\right)}{\pi i}}\Gamma \left(-{\displaystyle \frac{i\omega }{\kappa }}\right){\left({\displaystyle \frac{2\kappa }{c}}\right)}^{{\displaystyle \frac{i\omega }{3\kappa }}}{ϵ}^{-{\displaystyle \frac{2i\omega }{3\kappa }}},\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill K_D^{WKB}& =e^{i{\displaystyle \frac{3}{4}}\pi }{\displaystyle \frac{e^{-\frac{\pi \omega }{2\kappa }}}{2\sqrt{\pi }}}{\left({\displaystyle \frac{2\kappa }{c}}\right)}^{-{\displaystyle \frac{i\omega }{6\kappa }}+{\displaystyle \frac{1}{2}}}{ϵ}^{{\displaystyle \frac{i\omega }{3\kappa }}+{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

(105)

$$\begin{array}{cc}\hfill K_{\overline{H}}^{WKB}& =-{\displaystyle \frac{sinh\left(\frac{\pi \omega }{\kappa }\right)}{\pi i}}\Gamma \left(-{\displaystyle \frac{i\omega }{\kappa }}\right){\left({\displaystyle \frac{2\kappa }{c}}\right)}^{{\displaystyle \frac{i\omega }{3\kappa }}}{ϵ}^{-{\displaystyle \frac{2i\omega }{3\kappa }}}.\hfill \end{array}$$

(106)

It is important to stress that the modes $V^{\prime }$ and V correspond to particle states (positive norm), and there is no association between them and negative particle states, so no possible involvement in the process of particle creation from the vacuum is foreseeable. Moreover, the corresponding solutions are regular and analytic at the TP, in contrast to every other solution at hand. The only physically sensible choice is to connect these solutions to each other at $x=0$, and this amounts to setting $M_{V^{\prime }V}=1$, as in a standard scattering in nonrelativistic quantum mechanics. $V^{\prime }$ is simply V inside the horizon, at least as far as the leading order in $ϵ$ is considered in the basis construction, as previously discussed. Furthermore, it is correct to consider, in the same order, the modes V and $V^{\prime }$ as giving rise to a sort of disconnected line of a disconnected Feynman diagram. A corroboration of this picture is found in Appendix B.1, where we have explicit solutions to the reduced equation for a specific but physically meaningful monotonic transcritical velocity profile, and we can see that the regular solution is analytic and describes both V and $V^{\prime }$ asymptotically. As to the disconnected mode V, one has

$$|J_x^{V^{\prime }}|=|J_x^V|,$$

(107)

where $J_x^L$ represents the flux of L-modes at $x=\infty$, with $L=H,P,N,V$, and where (107) holds true at least at the leading order in $ϵ$.

From (11), one obtains the following current conservation:

$$|J_x^H|=|J_x^P|-|J_x^N|,$$

(108)

where no contribution from the decaying mode arises, being $|J_D|\to 0$ as $x\to -\infty$, of course.1 Then, by considering (99), (100), and (101), we obtain

$$1-{\left|P\right|}^2+{\left|N\right|}^2=0,$$

(109)

where

$$\begin{array}{cc}\hfill {\left|P\right|}^2& :={\displaystyle \frac{|J_x^P|}{|J_x^H|}}={\displaystyle \frac{|{\overline{H}}_P{|}^2}{|{\overline{H}}_H{|}^2}},\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill {\left|N\right|}^2& :={\displaystyle \frac{|J_x^N|}{|J_x^H|}}={\displaystyle \frac{|{\overline{H}}_N{|}^2}{|{\overline{H}}_H{|}^2}}.\hfill \end{array}$$

(111)

Explicitly, we find

$$\begin{array}{cc}\hfill |{\overline{H}}_N{|}^2& =2\kappa c{e}^{-\pi {\displaystyle \frac{\omega }{\kappa }}},\hfill \end{array}$$

(112)

$$\begin{array}{cc}\hfill |{\overline{H}}_P{|}^2& =2\kappa c{e}^{\pi {\displaystyle \frac{\omega }{\kappa }}},\hfill \end{array}$$

(113)

$$\begin{array}{cc}\hfill |{\overline{H}}_H{|}^2& =4\kappa csinh\left(\pi {\displaystyle \frac{\omega }{\kappa }}\right).\hfill \end{array}$$

(114)

The ingoing modes moving from ∞ toward the horizon (TP), namely, P and N, at initial times correspond to the only mode H going to ∞ at final times (again, at infinity, the decaying mode cannot contribute, as above):

$$a_{\omega }^H={\alpha }_{\omega }{a}_{\omega }^P+{\beta }_{\omega }{a_{\omega }^N}^{†},$$

(115)

where the various amplitudes are obtained as follows: the ratio giving the particle-creation rate is

$$|{\beta }_{\omega }{|}^2:={\left|N\right|}^2,$$

(116)

and

$$|{\alpha }_{\omega }{|}^2:={\left|P\right|}^2.$$

(117)

It is very important to notice that, if $|0\rangle$ is the vacuum state, then for the number $n_H\left(\omega \right)$ of Hawking particles (created at frequency $\omega$), as a consequence of (115), one obtains

$$n_H\left(\omega \right)=\langle 0|{a_{\omega }^H}^{†}{a}_{\omega }^H|0\rangle =|{\beta }_{\omega }{|}^2={\left|N\right|}^2.$$

(118)

Thermality, as is well known, is associated with the ratio

$$\begin{array}{c}\hfill {\displaystyle \frac{|{\beta }_{\omega }{|}^2}{|{\alpha }_{\omega }{|}^2}}={\displaystyle \frac{|J_x^N|}{|J_x^P|}}=e^{-{\beta }_h\omega },\end{array}$$

(119)

where

$${\beta }_h:={\displaystyle \frac{2\pi }{\kappa }}$$

(120)

is the inverse of the expected Hawking temperature. This ratio and (109) imply a blackbody spectrum,

$${\left|N\right|}^2={\displaystyle \frac{1}{exp\left({\beta }_h\omega \right)-1}},$$

(121)

that, at least at the leading order in $ϵ$, is purely Planckian, because no contribution of the disconnected mode V is present. We notice that this result can also be obtained from a direct calculation of $|J_x^N|/|J_x^H|$, as it is easy to verify.

Finally, let us discuss the state $\overline{H}$ associated with the Hawking partner. Let us recall that we have the set $-1=e^{i\pi }$. Then, we obtain $g_5=e^{-{\displaystyle \frac{{\beta }_h}{2}}\omega }$ (see (57) but also compare (70) with (73)). Furthermore,

$$|{\overline{K}}_{\overline{H}}{|}^2=4\kappa c{e}^{-{\beta }_h\omega }sinh\left(\pi {\displaystyle \frac{\omega }{\kappa }}\right).$$

(122)

As a consequence, we have

$$\begin{array}{cc}\hfill {a_{\omega }^{\overline{H}}}^{†}& ={\displaystyle \frac{{\overline{H}}_H}{{\overline{K}}_{\overline{H}}}}{\psi }^{-3\lambda }{a}_{\omega }^H+{\displaystyle \frac{{\overline{H}}_N}{{\overline{K}}_{\overline{H}}}}(1-{\psi }^{-3\lambda }){a_{\omega }^N}^{†}\hfill \end{array}$$

(123)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{{\overline{H}}_H}{{\overline{K}}_{\overline{H}}}}{\psi }^{-3\lambda }({\alpha }_{\omega }{a}_{\omega }^P+{\beta }_{\omega }{a_{\omega }^N}^{†})+{\displaystyle \frac{{\overline{H}}_N}{{\overline{K}}_{\overline{H}}}}(1-{\psi }^{-3\lambda }){a_{\omega }^N}^{†}\hfill \end{array}$$

(124)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{{\overline{H}}_P}{{\overline{K}}_{\overline{H}}}}{\psi }^{-3\lambda }{a}_{\omega }^P+{\theta }_{\omega }{a_{\omega }^N}^{†},\hfill \end{array}$$

(125)

where we have used (115) for $a_{\omega }^H$ and ${\alpha }_{\omega }={\overline{H}}_P/{\overline{H}}_H$. Notice that the term proportional to ${a_{\omega }^N}^{†}$ cannot contribute to the vacuum expectation value of ${a_{\omega }^{\overline{H}}}^{†}{a}_{\omega }^{\overline{H}}$, so we do not need to make the factor ${\theta }_{\omega }$ explicit. We find

$$\langle 0|{a_{\omega }^{\overline{H}}}^{†}{a}_{\omega }^{\overline{H}}|0\rangle ={\displaystyle \frac{|{\overline{H}}_P{|}^2}{|{\overline{K}}_{\overline{H}}{|}^2}}{\psi }^{-6\lambda }={\displaystyle \frac{1}{exp\left({\beta }_h\omega \right)-1}},$$

(126)

i.e., the same rate of production from the vacuum for $\overline{H}$ as for H, as is expected if one is the antiparticle of the other one. Also, in this case, the latest result is a pure consequence of the rigorous mathematical setting adopted, and no particular effort to find a suitable Corley’s diagram must be made.

One might wonder which results could be obtained if, instead of the basis B3 for describing the near-horizon physical states in the black hole region, one were to use the basis B2. The following (obvious) changes occur in the construction of the scattering matrix:

$$\sum _{i\in I^{\prime }\left(B2\right)}{a}_i{K}_i^{WKB}{g}_i^{WKB}{e}^{i{k}_i\left(\omega \right)x}=\sum _{i\in I^{\prime }\left(B2\right)}{a}_i{K}_i^{WKB}{g}_i^{WKB}\sum _{j\in I\left(B1\right)}{M}_{ij}^{(B2\mapsto B1)}{H}_j^{WKB}{d}_j^{WKB}{e}^{i{k}_j\left(\omega \right)x},$$

(127)

with

$$M_{ij}^{(B2\mapsto B1)}={\displaystyle \frac{1}{K_i^{WKB}{g}_i^{WKB}}}{P_3}_{ij}^{-1}{H}_j^{WKB}{d}_j^{WKB},$$

(128)

where the following holds:

$$P_3^{-1}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ 0& -1+{\psi }^{-3\lambda }& -1& 1\\ 0& 0& -1& 0\end{array}\right).$$

(129)

As a consequence, it is easy to see that, as far as the decaying mode is concerned, which is represented by $w_3\left(z\right)$ in the linear region, the standard calculation of the Hawking particle production occurs, exactly as above. Instead, for the Hawking partner, the lateral connection formula

$$w_6\left(z\right)=w_4\left(z\right)+(-1+{\psi }^{-3\lambda })w_1\left(z\right),$$

(130)

leads to

$$\begin{array}{cc}\hfill {a_{\omega }^{\overline{H}}}^{†}& ={\displaystyle \frac{{\overline{H}}_H}{{\overline{K}}_{\overline{H}}}}{a}_{\omega }^H+{\displaystyle \frac{{\overline{H}}_P}{{\overline{K}}_{\overline{H}}}}(-1+{\psi }^{-3\lambda })a_{\omega }^P\hfill \end{array}$$

(131)

$$\begin{array}{cc}\hfill & =\left({\displaystyle \frac{{\overline{H}}_H}{{\overline{K}}_{\overline{H}}}}{\alpha }_{\omega }+{\displaystyle \frac{{\overline{H}}_P}{{\overline{K}}_{\overline{H}}}}(-1+{\psi }^{-3\lambda })\right)a_{\omega }^P+{\displaystyle \frac{{\overline{H}}_H}{{\overline{K}}_{\overline{H}}}}{\beta }_{\omega }{a_{\omega }^N}^{†}\hfill \end{array}$$

(132)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{{\overline{H}}_P}{{\overline{K}}_{\overline{H}}}}{\psi }^{-3\lambda }{a}_{\omega }^P+{\displaystyle \frac{{\overline{H}}_H}{{\overline{K}}_{\overline{H}}}}{\beta }_{\omega }{a_{\omega }^N}^{†},\hfill \end{array}$$

(133)

and then the same result as in the above discussion is obtained.

In concluding this section, we stress that the present analysis also holds not only for dielectric media and water, which are subluminal, but also for BECs (superluminal case), as discussed in [1,2,3], and we considered the subluminal Corley model just for simplicity. For completeness, in Appendix C, we sketch the basic calculations for the superluminal Corley model as well, which is just the one most directly related to a BEC (see [2] for calculations for BECs in the present framework).

5. The 4D Extension of the Corley Model

We take into account the simplest possible model, i.e., the Corley model, and limit our considerations to the subluminal case, which is considered in [5,11,17]. In contrast to, e.g., BEC and water waves [2], it does not allow a variable speed of sound velocity $c\left(x\right)$. Furthermore, it cannot be related to the dielectric model, which is discussed in the following section. It represents the reference model for analytical studies of the dispersive analog Hawking effect, and thus, it is a useful benchmark for the discussion of a 4D extension and also for the search for exact solutions of the reduced equation. A possible extension consists of both extending the model to transverse directions and considering the following extension of the action. In particular, we add transverse coordinates $y,z$, both in the kinetic term and in the dispersive one, in the simplified hypothesis that the velocity field still depends only on x. This is necessary in order to allow variable separation and analytic studies of the model itself. The present restriction, which is admittedly strong, is not necessary in the Hopfield model that we discuss in the following section.

The extension we propose is the following:

$$S={\displaystyle \frac{1}{2}}\int d^{4}x[{\left(({\displaystyle \frac{1}{c}}{\partial }_t+{\displaystyle \frac{v}{c}}{\partial }_x)\varphi \right)}^2+\varphi ({\partial }_x^2+{\partial }_{\perp }^2+{ϵ}^2{({\partial }_x^2+{\partial }_{\perp }^2)}^2)\varphi ],$$

(134)

where ${\partial }_{\perp }^2:={\partial }_y^2+{\partial }_z^2$. With the separation ansatz $\varphi (t,x,y,z)=e^{i\omega t}\psi (x,y,z)$, we obtain the modified equation of motion

$$\begin{array}{c}{ϵ}^2({\partial }_x^4+2{\partial }_x^2{\partial }_{\perp }^2+{\partial }_{\perp }^4)\psi (x,y,z)+\left(1-{\displaystyle \frac{v^2\left(x\right)}{c^2}}\right){\partial }_x^2\psi (x,y,z)+{\partial }_{\perp }^2\psi (x,y,z)\hfill \\ +2{\displaystyle \frac{v\left(x\right)}{c^2}}(i\omega -v^{\prime }\left(x\right)){\partial }_x\psi (x,y,z)-i{\displaystyle \frac{\omega }{c^2}}(i\omega -v^{\prime }\left(x\right))\psi (x,y,z)=0,\hfill \end{array}$$

(135)

where we have enhanced the spatial dependence of $\psi$. We further adopt the variable separation ansatz:

$$\psi (x,y,z)=e^{i(k_{y}y+k_{z}z)}\phi \left(x\right).$$

(136)

Let us define $k_{\perp }^2:=k_y^2+k_z^2$ and assume that $|k_{\perp }|ϵ\le O\left(1\right)$, i.e., that transverse momenta are not big with respect to the scale ϵ (they are not order ${\displaystyle \frac{1}{ϵ}}$ as the wavenumbers associated with the dispersive modes). Then, we obtain

$$\begin{array}{c}{ϵ}^2({\partial }_x^4-2k_{\perp }^2{\partial }_x^2+k_{\perp }^4)\phi \left(x\right)+\left(1-{\displaystyle \frac{v^2\left(x\right)}{c^2}}\right){\partial }_x^2\phi \left(x\right)-k_{\perp }^2\phi \left(x\right)\hfill \\ +2{\displaystyle \frac{v\left(x\right)}{c^2}}(i\omega -v^{\prime }\left(x\right)){\partial }_x\phi \left(x\right)-i{\displaystyle \frac{\omega }{c^2}}(i\omega -v^{\prime }\left(x\right))\phi \left(x\right)=0,\hfill \end{array}$$

(137)

and then, in terms of the Orr–Sommerfeld-type equation we discussed in the previous section, we obtain the following coefficients:

$$\begin{array}{ccc}\hfill p_3(x,ϵ)& =& \left(1-{\displaystyle \frac{v^2\left(x\right)}{c^2}}-2{ϵ}^2{k}_{\perp }^2\right),\hfill \end{array}$$

(138)

$$\begin{array}{ccc}\hfill p_2(x,ϵ)& =& 2{\displaystyle \frac{v\left(x\right)}{c^2}}(i\omega -v^{\prime }\left(x\right)),\hfill \end{array}$$

(139)

$$\begin{array}{ccc}\hfill p_1(x,ϵ)& =& {\displaystyle \frac{\omega }{c^2}}(\omega +i{v}^{\prime }\left(x\right))-k_{\perp }^2+{ϵ}^2{k}_{\perp }^4.\hfill \end{array}$$

(140)

With respect to the original model, the only corrections we obtain concern an $O\left({ϵ}^2\right)$ correction of $p_3$ and an $O\left({ϵ}^2\right)$ correction of $p_1$, and, as a consequence, we can again corroborate the general picture, in the sense that we confirm, once again, the need to use a more general and less model-dependent approach than the one discussed in [11].

As to the near-turning-point approximation, at the leading order, nothing changes, as it involves only the leading-order coefficient $p_{30},p_{20}$, which remains trivially unaltered.

Also, the leading-order WKB approximation for the above-separated equation is only affected by the shift ${\omega }^2\mapsto {\omega }^2-c^2{k}_{\perp }^2$, and, again, nothing substantial changes, as the leading-order WKB wavefunctions remain unaltered.

There is an important change in the classification of the modes; indeed, one finds that the new quantum number $k_{\perp }$ appears, and then we have to replace the modes $P,N,H,V,\overline{H},V^{\prime },$ and D with a one-parameter family of the modes $P\left(k_{\perp }\right)$, $N\left(k_{\perp }\right)$, $H\left(k_{\perp }\right)$, $V\left(k_{\perp }\right)$, $\overline{H}\left(k_{\perp }\right)$, $V^{\prime }\left(k_{\perp }\right)$, and $D\left(k_{\perp }\right)$. The scattering is, in this situation, by no means as simple as in the 2D setting because of the presence of the transverse contributions, albeit in the most simple choice to allow a trivial separation of variables.

As to thermality, again, it is straightforward to show that the temperature is unaffected and is universal, as expected (cf. the discussion in [51] about the contributions beyond the s-waves; herein, we could say that we consider contributions beyond the case $k_{\perp }=0$).

It is also remarkable that the possible presence of massive modes [11,52] in a 4D framework, but even in the 2D case, does not modify the black hole temperature and can also be treated in the present framework. Indeed, a mass contribution in the action functional corresponds to the term $\propto m^2{\varphi }^2$, which is a non-derivative term. As a consequence, by imposing $m^2\ll 1/{ϵ}^2$ (cf. [11]), it may only contribute a further additive term $-m^2$ to $p_1(x,ϵ)$ above. Under this assumption, no modification to the temperature occurs (still, modifications to the dispersion relation arise). We refer again to the discussion in [51].

6. The 4D Extension of the Hopfield Model and the Analogous Hawking Effect

The electromagnetic Lagrangian for the full Hopfield model is quite involved and has been discussed, using different theoretical tools, in [21]. A simplified model is then a better choice. Our model, introduced in [18], is related to the two-dimensional reduction of the Hopfield model adopted in [53] and is such that the electromagnetic field and the polarization field are simulated by a pair of scalar fields, $\phi$ and $\psi$, respectively, in the so-called $\phi \psi$–model. We also provided a new approach, in a perturbative framework, for the subcritical case in [26] and even exact solutions for a specific monotonic profile for the refractive index in [27], with a full description of connection formulas. Despite its simplification, it is still set up in such a way that we obtain exactly the same dispersion relation, and, moreover, we can simulate the same coupling as in the full case. Its Lagrangian is

$$\begin{array}{c}\hfill {\mathcal{L}}_{\phi \psi }={\displaystyle \frac{1}{2}}\left({\partial }_{\mu }\phi \right)\left({\partial }^{\mu }\phi \right)+{\displaystyle \frac{1}{2\chi {\omega }_0^2}}\left[{\left(v^{\alpha }{\partial }_{\alpha }\psi \right)}^2-{\omega }_0^2{\psi }^2\right]-{\displaystyle \frac{g}{c}}\left(v^{\alpha }{\partial }_{\alpha }\psi \right)\phi ,\end{array}$$

(141)

where $\chi$ plays the role of the dielectric susceptibility, $v^{\mu }$ is the usual four-velocity vector of the dielectric, ${\omega }_0$ is the proper frequency of the medium, and g is the coupling constant between the fields. The latter constant is henceforth set equal to one, as its original motivation (see [18]) can be relaxed without problems in a more advanced discussion (cf. also [1]).

We adopt a phenomenological model where we can leave room for a spacetime dependence of the microscopic parameters $\chi ,{\omega }_0$ in such a way that $\chi {\omega }_0^2$ is a constant. In particular, our parameter for a perturbative expansion is

$${ϵ}^2:={\displaystyle \frac{1}{\chi {\omega }_0^2}},$$

(142)

which corresponds to the parameter appearing in the Orr–Sommerfeld-like equation (master equation). The equations of motion are

$$\begin{array}{cc}\hfill \square \phi +{\displaystyle \frac{1}{c}}\left(v^{\alpha }{\partial }_{\alpha }\psi \right)=& 0,\hfill \end{array}$$

(143)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\chi {\omega }_0^2}}{\left(v^{\alpha }{\partial }_{\alpha }\right)}^2\psi +{\displaystyle \frac{1}{\chi }}\psi -{\displaystyle \frac{1}{c}}{v}^{\alpha }{\partial }_{\alpha }\phi =& 0.\hfill \end{array}$$

(144)

We can separate the above system, obtaining equations involving only one of the fields $\phi ,\psi$. As in [1], we focus on the electromagnetic field $\phi$, obtaining

$$\left[\square +{\displaystyle \frac{1}{c^2}}\left(v^{\alpha }{\partial }_{\alpha }\right)\left(\chi v^{\beta }{\partial }_{\beta }\right)+{ϵ}^2\left(v^{\alpha }{\partial }_{\alpha }\right)\left(\chi \left(v^{\beta }{\partial }_{\beta }\right)\square \right)\right]\phi =0.$$

(145)

As we wish to discuss the 4D setting, we assume that the velocity of the perturbation is along the x-axis. The latter assumption is quite natural, as the velocity in the model is assumed to be spatially homogeneous and constant. We, again, designate ${\mathbf{x}}_{\perp }$ as the transverse spatial directions $y,z$. As a consequence, in the above formula (145), the only explicit dependence on the transverse directions appears in the terms involving the operator □. One may observe that, in the Fourier space, the transverse directions are associated with the contribution ${\mathbf{k}}_{\perp }$ to the wavenumber and are constants in a separation-of-variables process, which is allowed in our settings, and that this contribution affects only the coefficient $p_{10}$ of the master equation, at the leading order. This has a very interesting consequence: the near-horizon equation is totally unaffected by the presence of transverse contributions, as the $p_{10}$ coefficient does not participate in its construction at the leading order. Also, the temperature remains unaffected, but $\omega$ satisfies a different dispersion relation due to the presence of transverse modes. This finding, which is a natural consequence of our general approach, is, of course, in perfect agreement with statements in [21,51].

7. Conclusions

Our main result concerns the analytical calculations of the various interesting rates of particle production for the analog Hawking effect. We have first deduced suitable bases of solutions in the near-horizon region in different sectors of the complex plane and then calculated the relative connection matrices, both in the subluminal case and the superluminal one, in agreement with the analysis in [39,40,41]. Then, for simplicity, we have applied our general analysis, which can also be applied to BECs, dielectrics, and water (cf. [1,2]), to the most well-known theoretical model, the so-called Corley model [5,11], and we have shown how to deduce physical amplitudes without introducing hypotheses ad hoc, like the Corley’s boundary condition appearing in [5], but just on rigorous mathematical grounds. See also [11], where a refinement of Corley’s analysis is found, but not a complete and rigorous analysis of the connection formulas. The subtle point is that the main features of the analog Hawking effect are associated with the Stokes phenomenon, i.e., the appearance of non-trivial Stokes matrices. The Stokes phenomenon, on rigorous mathematical grounds, was discussed before in the context of the dispersive Hawking effect only in [27] for dielectrics, using the Hopfield model and thanks to exact solutions associated with a specific but physically very relevant choice of the refractive index profile. A general setting, albeit for weak dispersive effects, as discussed in [1,2,3], was still lacking.

In our view, our analysis in the present paper represents a strong corroboration, in terms of analytical calculations, of the analog Hawking effect in the framework of weak dispersive effects and completes the analysis in previous papers [1,2,3].

Then, we have taken into account the problem of extending, at least in a preliminary way, our framework to 4D cases, albeit allowing for the separation of variables. We have shown that the expression of the Hawking temperature remains unaltered, as it happens in the standard Hawking effect (see [51]). As a final (but still relevant) contribution, we provide, in Appendix B, exact solutions for the so-called reduced equation that governs the behavior of the short-wavelength modes in the WKB approximation (non-dispersive modes).

Future investigations should involve a more detailed analysis of the 4D case, as well as a better comprehension of the subcritical case, which, from the point of view of analytical calculations, remains a challenging problem (see [22,26]).

There are further possible lines of investigation. The first one concerns the explicit evaluation of the contributions arising from higher-order terms in the expansion parameter $ϵ$. In particular, as discussed in previous works (see, e.g., [1,2] and see also [54]), there is, for small but non-zero $ϵ$, a maximum value of the frequency beyond which the Hawking process is not allowed, and, moreover, its thermal appearance at low frequencies must be corrected. This may happen either when considering a model where transverse dimensions (and/or massive modes) are also allowed because of the possible arising of a non-trivial gray-body factor, or as a consequence of higher-order contributions. Going beyond the leading order in epsilon is possible but is non-trivial (we refer to the papers quoted by Nishimoto, in particular, [40]) and requires further analytical studies, which we intend to develop in the future.

A further line of investigation might be represented by the Unruh effect [55], which is, as is well known, a phenomenon with strong similarities to the Hawking effect. A very important issue from the experimental point of view consists of measuring the effect itself in analog condensed matter systems. See, for example, [56,57,58,59] and references therein. BECs coupled with lasers are, e.g., involved in this picture [56]. As far as a fourth-order equation of the Orr–Sommerfeld kind can be obtained, as is standard in BECs (see, e.g., [2,60]), the present framework could help in performing analytical calculations. We reserve the analysis of this problem for future investigations.