Sine-Gordon soliton as a model for Hawking radiation of moving black holes and quantum soliton evaporation

The intriguing connection between black holes' evaporation and physics of solitons is opening novel roads to finding observable phenomena. It is known from the inverse scattering transform that velocity is a fundamental parameter in solitons theory. Taking this into account, the study of Haw\-king radiation by a moving soliton gets a growing relevance. However, a theoretical context for the description of this phenomenon is still lacking. Here, we adopt a soliton geometrization technique to study the quantum emission of a moving soliton in a one-dimensional model. Representing a black hole by the one soliton solution of the sine-Gordon equation, we consider Haw\-king emission spectra of a quantized massless scalar field on the soliton-induced metric. We study the relation between the soliton velocity and the black hole temperature. Our results address a new scenario in the detection of new physics in the quantum gravity panorama.


I. INTRODUCTION
During the last ten years, analogue gravity systems have attracted major interest in the scientific community [1]. These models aim at providing valuable scenarios to test inaccessible features of quantum gravity, as the Hawking radiation emission by black holes (BHs) [2]. Furthermore, the recent observation of gravitational waves (GWs) emitted by colliding BHs [3,4] shaded new light and opened unexplored roads towards the search for quantum effects in gravity [5], as the Hawking's BH evaporation [6]. Indeed, quantum BH emission might be observed by the concomitant monitoring of the BH collisions by gravitational and electromagnetic antennas. However, the collision process changes the original Hawking's framework.
Originally, Hawking considered quantum fields in a stationary BH background, the Schwarzschild metric, and discovered that BHs emit thermal radiation and evaporate. His paper appeared exactly one year after a trailblazing article by Ablowitz, Kaup, Newell and Segur (AKNS), that cast new light on nonlinear waves by establishing the general method to solve classes of nonlinear field equations [7,8]. Surprisingly, AKNS classes generate a metric and define an event horizon (EH). Indeed, it is known in the field of the nonlinear waves that integrable systems, which can be solved exactly by the inverse scattering transform (IST), describe a Riemannian surface with constant negative curvature [9,10].
Recently, Hawking radiation analogues from solitons were considered in a huge variety of physical contexts, including light [11][12][13][14][15], ultracold gases [16][17][18][19], water and sound waves [20,21]. Here, we study the geometrization of soliton equation by considering a canonical field quan- * Corresponding author. Email: marcucci.giulia@gmail.com tization in the classical background of the Sine-Gordon (SG) soliton metric. Indeed, the 1 + 1 dimensional Sine-Gordon (SG) equation is a nonlinear model that exhibits a Riemannian surface with constant negative curvature. In this frame, the SG equation can be considered the AKNS counterpart of a two dimensional gravitational theory. Two dimensional theories of gravity are useful models to understand the quantum properties of higherdimensional gravity. These theories capture essential features of higher-dimensional counterparts, and in particular have black hole solutions and Hawking radiation [22][23][24][25]. The link between the 1+1 dimensional gravity and the SG model introduces further simplifications since the quantum properties of this equation have been largely studied [26,27]. As we shall recall in the next section, the integrability condition of SG equation determines a metric, with a coordinate singularity, which defines an EH. In particular 1+1 dimensional BHs can be realized as solitons of the SG equation [28] and it has been shown with a one loop perturbative computation that this BH emits thermal radiation [29,30].
In this paper, we show that SG soliton emits thermal particles with a specific Hawking temperature, finding the way the temperature changes with the velocity of the SG-BH. Afterward, we perform two different kinds of quantization, one for a massless scalar field and another for the soliton itself, and obtain their Hawking emission spectra. In both cases, we discover that an observer on the soliton tail detects a thermal radiation with a temperature directly proportional to the soliton speed. Furthermore, we analyze the temperature detected by an observer at rest by adding a Doppler effect.
Our paper is organized as follows: in sec. II we review the geometrization of the SG model; we show the connection between a soliton solution of an AKNS system and a metric on a two dimensional surface. In sec. III we study the BH metric induced by the SG equation and introduce suitable coordinate systems for the field quantization. In section IV we quantize massless scalar fields on the soliton background. In section V we quantize the SG soliton following the Faddeev semiclassical quantization [26], and show that the sine-gordon BH evaporates. Conclusions are drawn in section VI. A short appendix furnishes a minimal mathematical background to forms and curvature.

II. SINE-GORDON GEOMETRIZATION
We start reviewing the way integrable nonlinear equations generates surfaces with constant negative curvature [9]. By considering the SG equation defined in Eq. (1), we perform the coordinate transformation and get φ χθ = sin φ.
Eq. (8) defines the constant negative curvature metric induced by the ISM associated to the SG equation (3). By changing the coordinates set as in the following, we write the first fundamental form ds 2 as [9] ds 2 = sin 2 φ 2 dτ 2 + cos 2 φ 2 dξ 2 , which results to be associated with a SG equation of the form where Thus the metric tensor iŝ However, ds 2 in Eq. (9) is not Lorentz invariant and it does not lead to a Schwarzschild-like metric. Following [28], in order to obtain a Minkowski-like metric, we perform a Wick rotation τ → iτ and obtain the elliptic SG (ESG) equation: whose corresponding metric is III.

THE SINE-GORDON SOLITON BLACK HOLE
We show that the one-soliton solution of the ESG equation determines a BH metric.
The well known forward-propagating one-soliton solution of the Eq. (10) is with γ = (1 − β 2 s ) −1/2 and 0 < β s < 1 the soliton velocity [32]. The backward-propagating one-soliton solution gives the same treatise with −1 < β s < 0, by substituting β s in −β s in what follows. For this reason, we can choose solution (15) without loss of generality. Eq. (15) is also solution of Eq. (13) with We adopt Eq. (16) hereafter. Substituting Eq. (15) in Eq. (14), we have with ρ = γ(ξ − β s τ ). Following [28], we adopt various coordinate transformations: first from (τ, ξ) to (T , ρ), with ρ as defined above and Next, we transform (T , ρ) to (T , r) by The result of the transformation is the line element Eq. (20) is the metric of a 1+1 dimensional BH with EH at r g := β s . Figures 1 and 2 show the EH positions ρ g = arcsech(γr g ) on the soliton profiles and energy densities E, respectively for different velocities β s . The energy density, at fixed t, is defined as follows [26]:  It is now convenient to introduce two new sets of coordinates: the modified Regge-Wheeler coordinate, that we call the slug coordinate in analogy with the tortoise coordinate, as usually reported [2,5], and the Kruskal-Szekeres coordinates. We get the slug coordinate r * (r) according to so that Eq. (20) then becomes The slug coordinate is singular at r = β s and it is defined on the exterior of the BH when ρ → ±∞ and r → 0. In fact, as r approaches β s , r * goes to +∞, while far away from the BH r * → 0 as r → 0. Introducing the slug lightcone coordinates we write Eq. (20) as The slug lightcone coordinates are singular and they span only the exterior of the black hole. To describe the entire spacetime, we need another coordinate system. In order to be consistent with literature, we refer to them as the Kruskal-Szekeres (KS) coordinates . From Eq. (22) and Eq. (24) it follows that The BH metric thus becomes In the KS lightcone coordinates, defined as Eq. (27) takes the form and it is regular at r = β s . The singularity occurring in the ESG-soliton metric is, as the Schwarzschild one, a coordinate singularity, which can be removed by a coordinate transformation. The KS coordinates, indeed, span the entire spacetime.

IV. MASSLESS SCALAR FIELD QUANTIZATION
We consider a field quantization on the classical soliton background metric. We first analyze a massless scalar field with the action where g µν represents the inverse of a general metric tensor g µν , g is the determinant of g µν and x = (x 0 , x 1 ). The action in Eq. (30) is conformally invariant, and in terms of lightcone slug coordinates and lightcone KS coordinates (29) it reads We write the solution of the scalar field equation in terms of the lightcone slug coordinates and in the lightcone KS coordinate as where A,Ã and B,B are arbitrary smooth functions. In correspondance of the tail of the soliton, i.e., far away from the EH, the mode expansion of the field iŝ In Eq. (34) the left moving part is given by the terms weighted by e ±iΩṽ in the mode expansion. The vacuum state |0 B , defined byb − Ω |0 B = 0, is the Boulware vacuum (BV) and does not contain particles for an observer located far from the EH. However, as the slug coordinate is singular at horizon, the BV is also singular at the EH.
To obtain a vacuum state defined over the entire spacetime, we expand the field operator in terms of the KS lightcone coordinateŝ The creation and annihilation operatorsâ ± ω determine the Kruskal vacuum (KV) stateâ − ω |0 K = 0. The KV is regular on the horizon and corresponds to true physical vacuum in the presence of the BH. For a remote observer the KV contains particles. To determine their number density, we follow the original calculations of Hawking and Unruh with the only difference in the definition of the KS coordinates (see chapters 8 and 9 of [5] for details). We find that the remote observer moving with the soliton tail sees particles with the thermal spectrum If we consider a finite volume quantization we can put V = δ(0) [5] and we obtain the number density corresponding to the temperature In Fig. 3, we show the radiance B(Ω) = Ω 3 n Ω . We observe that for a static soliton (β s = 0) we get T H = 0. This result may appear in contradiction with the Hawking original work, where he considered the emission from a static BH. However the result in eq. (38) is coherent with the structure of the metric induced by the SG equation, where the singularity occurs for r = r g = β s and no emission can be observable for r g = 0. This dependence of the Hawking radiations on the translation velocity is peculiar of soliton dynamics [33] and it is related to the structure of the spectral parameter in the IST [7,8].

A. Hawking temperature in the laboratory frame
Unlike the Schwarzschild BH, the ESG soliton is not static, but translates with velocity β s . The frequency Ω seen by an observer at rest with respect to the soliton contains a Doppler shift. Letting Ω s be the frequency emitted by the soliton in (36), the frequency measured by an observed moving with velocity −β s with respect to the soliton, and located at an angle θ s with respect to the soliton direction is In the collinear case θ s = 0, and we have The corresponding Hawking temperature is (for small β s ) This calculation also applies to a massive bosonic field, as the number density spectrum depends only on the statistics [34]. In the case of a fermionic field the theory is similar, but the number density spectrum follows the Fermi-Dirac statistics [2].

V. SOLITON QUANTIZATION
Previously we studied the BH evaporation following the works of Hawking and Unruh in [2,35]. Now, we analyze a quantum perturbation of the BH metric given by the classical soliton solution of the ESG equation, and we obtain a BH evaporation without the interaction with a massless scalar field. We start from where φ s is the classical solution in Eq. (15) and φ 1 represents a weak field perturbation. We consider the conformally invariant action which leads to a field equation The solutions of Eqs. (10,13) differ for a Wick rotation.
In other words, one passes from the SG soliton to the ESG one by the transformation We perform the inverse Wick rotation, i.e., τ → −iτ , β s → iβ s , passing from the ESG to the SG, and substitute Eq. (42) into Eq. (44), hence we obtain where we neglect terms O(φ 2 1 ). This equation expresses the interaction between a massive particle and the gravitational field, because the weak quantum field φ 1 obeys a generalized Klein-Gordon equation with squared mass cos(φ s ) depending on the soliton, and thus on the metric. Recalling Eq. (15), we have For an observer located on the tail of the soliton (ρ → ∞), the field equation reduces to while for an observer on the horizon (ρ → ρ g ), we have with F (ρ) given by Eq. (50) truncated at the order zero in ρ−ρ g , i.e., exactly on the horizon, leads to (51) Due to the inverse Wick rotation, even if the action is conformally invariant, the quantization is not straightforward. We need to adapt both the slug and the KS lightcone coordinates in Eqs. (24,28) to the rotated system. We obtain Since the action (43) is conformally invariant, we thus write the field equation as follows Eqs. (53) have exponential solution with the following dispersion relations, From now on, we omit the K and k indices. We write the quantum fields as followŝ where, as in the non interacting case, the annihilation operatorsb − Ω andâ − ω define the Boulware vacuum |0 B and the Kruskal vacuum |0 K , respectively. The operatorsâ ± ω andb ± Ω are related by the Bogolyubov transformationŝ By substituting this in Eq. (56), we find hence we obtain Seemingly for β Ωω , we have (60) Using now the KS coordinate (52), after lengthy but straightforward calculations, we find It follows that α Ωω and β Ωω obey the useful relation |α Ωω | 2 = e 4πΩ/βs |β Ωω | 2 .
Therefore we can compute the expectation value of the b-particle number operatorN Ω =b +

Ωb
− Ω in the Kruskal vacuum [5], and obtain the number density This corresponds to an emitted radiation with twice the frequency with respect to the simple massless case, of which spectral radiance B(Ω) = Ω 3 n Ω is reported in figure 4. We observe that the Hawking temperature is equal to Eq. (38) for the massless scalar field. This is expected since the surface gravity is the same. For a moving observer with respect to the soliton the Hawking temperature, for small β s , reads Eq. (64) provides the Hawking temperature of soliton evaporation in this toy model.

VI. CONCLUSIONS
We adopted the geometrization of the ESG model and reported on the connection between the one-soliton solution of the 1+1-dimensional elliptic sine-Gordon equation and a metric with a Schwarzschild-like coordinate singularity. We determined the BH metric and, by suitable coordinate systems, we eliminated the singularity and obtained a regular metric on the EH. We quantized a massless scalar field and found the thermal radiation detected by an observer far away on the BH exterior. We obtained that the temperature is proportional to the soliton velocity. We analyzed the temperature detected by an observer in the laboratory frame, by a Doppler effect. We studied also the quantum soliton evaporation, and found the corresponding spectrum.
Our analysis allows to predict the Hawking radiation for a moving 1+1 dimensional BH and shows that the velocity affects the temperature and the corresponding emitted thermal spectrum. In a BH collisional process one can hence expect a frequency shift of the emitted photon concomitant with the variation of spiraling velocity of the BHs. The resulting chirp of the emitted photons may have a clear and detectable signature in the electromagnetic spectrum. Analogues of these processes may be eventually simulated in the long-range interactions between optical solitons pairs recently observed over astronomical distances [36], or similar optical experiments [37,38].
Our results may be extended to any metric induced by AKNS systems, hence to many different physical models to conceive experimentally realizable analogues for studying Hawking evaporation of moving black holes.

ACKNOWLEDGMENTS
We are pleased to acknowledge Prof. Fabio Biancalana for invaluable discussions and for a critical reading of the manuscript. We also acknowledge I. M. Deen for technical support with the computational resources. C.C. and G.M. acknowledge support from the Templeton Foundation (grant number 58277), the H2020 Quan-tERA project QUOMPLEX (project ID 731473) and PRIN project NEMO (ref. 2015KEZNYM).

APPENDIX: MINIMAL INTRODUCTION TO FORMS, PFAFF PROBLEMS AND CURVATURE
A 1-form Ω = X dx + T dt is a combination of the differentials dx and dt, which have to be retained as elements of a basis. X and T can be matrices with the same size, or also operators. A 2-form is a combination of the symbols ("exterior products") dx ∧ dt and dt ∧ dx = − dx ∧ dt. One can obtain a 2-form from a 1-form by the differential operator d: which can be kept in mind by letting dx ∧ dx = dt ∧ dt = 0, so that terms like ∂ x X and ∂ t T do not appear in dΩ. One can also obtain a 2-form by the exterior product Ω ∧ Ω again by dx ∧ dx = dt ∧ dt = 0 Ω ∧ Ω = XT dx ∧ dt + T X dt ∧ dx = [X, T ] dx ∧ dt (66) with [X, T ] the commutator. By using forms, the AKNS integrability condition For some authors, using forms has the advantage of a more compact notation as the explicit coordinates x and t do not appear in (68). Eq. (68) is referred to a Pfaffian integrability condition, or Pfaff problem. Forms are directly connected to the curvature of surfaces. If one considers a surface, and a local point vector P on the surface, let e 1 and e 2 the orthogonal tangent vectors. For infinitesimal motion on the surface dP dP = σ 1 e 1 + σ 2 e 2 , where σ 1 and σ 2 contain the differentials of the adopted coordinates and are hence 1-form. σ 1 ∧σ 2 is the elemental area on the surface. When one moves of an amount dP, e 1,2 changes of amounts de 1,2 . One considers a surface such that de 1 = ωe 2 and de 2 = −ωe 1 where ω depends on the shape of the surface, contains the differentials of the coordinate systems, and is a 1-form named the connection one form. One finds the following equation where K is the Gaussian curvature. ω, σ 1 and σ 2 are one forms that fix all the properties of the surface. In the particular case K = −1, one has from (70) dω = σ 1 ∧ σ 2 .