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provided by Heriot Watt Pure PHYSICAL REVIEW D 83, 024015 (2011) Dielectric black holes induced by a refractive index perturbation and the Hawking effect
F. Belgiorno,1 S. L. Cacciatori,2,3 G. Ortenzi,4 L. Rizzi,2 V. Gorini,2,3 and D. Faccio5 1Dipartimento di Fisica, Universita` di Milano, Via Celoria 16, IT-20133 Milano, Italy 2Department of Physics and Mathematics, Universita` dell’Insubria, Via Valleggio 11, IT-22100 Como, Italy 3INFN sezione di Milano, via Celoria 16, IT-20133 Milano, Italy 4Dipartimento di Matematica e Applicazioni, Universita` di Milano-Bicocca, Via Cozzi 53, IT-20125 Milano, Italy 5School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh, Scotland EH14 4AS, UK (Received 22 March 2010; revised manuscript received 18 November 2010; published 13 January 2011) We consider a 4D model for photon production induced by a refractive index perturbation in a dielectric medium. We show that, in this model, we can infer the presence of a Hawking type effect. This prediction shows up both in the analogue Hawking framework, which is implemented in the pulse frame and relies on the peculiar properties of the effective geometry in which quantum fields propagate, as well as in the laboratory frame, through standard quantum field theory calculations. Effects of optical dispersion are also taken into account, and are shown to provide a limited energy bandwidth for the emission of Hawking radiation.
DOI: 10.1103/PhysRevD.83.024015 PACS numbers: 04.62.+v, 04.70.Dy
I. INTRODUCTION experiments involving Hawking analogue radiation has been explored [11–23]. Recently Philbin et al. proposed One of the most intriguing predictions of quantum fields an optical analogue in which a soliton with intensity I, in curved spacetime geometries is the production of propagating in an optical fibre, generates through the non- Hawking radiation. In 1974 S. Hawking predicted that linear Kerr effect a refractive index perturbation (RIP), black holes emit particles with a thermal spectrum. ¼ Therefore a black hole will evaporate, shedding energy n n2I, where n2 is the Kerr index [24]. The same under the form of a blackbody emission [1–3]. However, mechanism has also been generalized to a full 4D geometry it turns out that for a typical stellar mass black hole the by Faccio et al. [25] and has led to the first experimental temperature of this radiation is so low ( 10 nK) that it has observation of quanta emitted from an analogue horizon no hope of being directly detected. Notwithstanding, [26]. The RIP modifies the spacetime geometry as seen by Hawking radiation does not actually require a true gravi- copropagating light rays and, similarly to the acoustic tational event horizon but, rather, it is essentially a kine- analogy, if the RIP is locally superluminal, i.e. if it locally matical effect, i.e. it requires some basic ‘‘kinematical’’ travels faster than the phase velocity of light in the ingredients but no specific underlying dynamics (see e.g. medium, an horizon is formed and Hawking radiation is [4–6]). What suffices is the formation of a trapping horizon to be expected. in a curved spacetime metric of any kind, and the analysis Here we take into consideration the Hawking effect in of the behavior of any quantum field therein (see also the dielectric black holes induced by a RIP which propagates discussion in [6]). The quanta of this field will then be with constant velocity v. We first show that, under suitable excited according to the prediction of Hawking. On this conditions, we can remap the original Maxwell equations basis, a number of analogue systems have been proposed, for nonlinear optics into a geometrical description, in anal- for the first time by W. Unruh [7] and later by other ogy to what is done in the case of acoustic perturbations in researchers (see e.g. [8] and references therein), that aim condensed matter. Then we provide a model in which the at reproducing the kinematics of gravitational systems. presence of the Hawking radiation can be deduced even Most of these analogies rely on acoustic perturbations without recourse to the analogous model characterized by a and on the realization of so-called dumb holes: a liquid curved spacetime geometry. Hawking radiation is a new or gas medium is made to flow faster than the velocity of phenomenon for nonlinear optics, never foreseen before, acoustic waves in the same medium so that at the transition which could legitimately meet a sceptical attitude by the point between sub and supersonic flow, a trapping horizon nonlinear optics community, because of a missing descrip- is formed that may be traversed by the acoustic quanta, viz. tion of the phenomenon by means of the standard tools of phonons, only in one direction [8]. Unfortunately the pho- quantum electrodynamics. As a consequence, it is impor- non blackbody spectrum is expected to still have very low tant to provide also a corroboration of the analogue picture temperatures, thus eluding direct detection (see e.g. [9]). A simply by using ‘‘standard’’ tools of quantum field theory parallel line of investigation involves effective geometry (without any reference to the geometrical picture). for light, which has been introduced by Gordon [10] and We then take into consideration the effects of optical extended also to nonlinear electrodynamics; black hole dispersion, which give rise to relevant physical consequen- metrics have been introduced and the possibility to perform ces on the quantum phenomenon of particle creation.
1550-7998=2011=83(2)=024015(17) 024015-1 Ó 2011 American Physical Society F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) We first introduce a 2D reduction of the model in the group velocity horizons, which are shown to be involved presence of dispersion, and see how the dispersion relation with different spectral bandwidths, and with very different is affected by the frequency-dependence of the refractive qualitative behavior regarding their action on photons. index. Then we discuss both phase velocity horizons and Section V is devoted to the conclusions. group velocity horizons, which can both play a relevant Finally, we have added three Appendixes. The first es- role in the physical situation at hand; their qualitative tablishes a correspondence between our black hole metric difference and the possibility to discriminate between and the acoustic one. The second one relates the tempera- them in experiments (numerical and/or laboratory-based) ture to the standard conical singularity of the Euclidean is considered. version of the metric. The third establishes the relevant The structure of the paper is as follows. In Sec. II we analytic properties of the Bogoliubov coefficients. introduce our idealized model of a RIP propagating, in a nonlinear Kerr medium, with a constant velocity v in the x II. STATIC DIELECTRIC BLACK HOLE direction and infinitely extended in the transverse y and z directions. In the eikonal approximation, the model is We consider a model-equation which is derived from embodied in a suitable wave equation for the generic nonlinear electrodynamics (with 2 ¼ 0 and 3 0)in component of the electric field propagating in the nonlinear the eikonal approximation for a perturbation of a full- medium affected by the RIP. We describe this propagation nonlinear electric field propagating in a nonlinear Kerr as taking place in an analogue spacetime metric, written in medium. In order to make the forthcoming analysis as the pulse frame, where the metric is static and displays two simple as possible we replace the electric field with a scalar field (cf. e.g. Schwinger’s analysis of sonoluminescence horizons xþ and x for propagating photons, analogues to a black hole and a white hole horizons, respectively. By [27]), for which we obtain the wave equation [28] 2 assigning to the black hole horizon a surface gravity in a n ðxl vtlÞ 2 2 2 2 @t @x @y @z ¼ 0: (1) standard way, we calculate the evaporation temperature Tþ c2 l l of the RIP in the latter frame, which turns out to be proportional to the absolute value of the derivative of the Coordinates in the lab frame are denoted by tl, xl, y, z (the RIP’s profile, evaluated at x . We also show that T is labels of y, z are omitted because they will not be involved þ þ in the boost relating the lab frame with the pulse frame). conformally invariant, as expected for the consistency of Here, nðx vt Þ is the refractive index, which accounts the model. l l for the propagating RIP in the dielectric. The latter can be In Sec. III, we tackle the model from a different per- obtained by means of an intense laser pulse in a nonlinear spective, namely, within the framework of quantum field dielectric medium (Kerr effect). In our model the RIP does theory. We construct a complete set of positive frequency not depend on the transverse coordinates, namely, it is solutions k of the wave equation in the lab frame, in l infinitely extended in the transverse dimensions. This ap- terms of which we quantize the field in the standard way. proximation is necessary in order to carry out the calcu- Then, we compare the k with the corresponding plane l lations below, which allow us to draw a tight analogy wave solutions of the wave equation in the absence of the between the Hawking geometrical description and the RIP. This enables us to calculate the Bogoliubov coeffi- quantum field theoretical treatment. Such an infinitely cients, in terms of which we express the expectation value extended RIP is clearly an idealization, which cannot be of the number operator of the outgoing quanta in the in obtained in an actual experiment. However, relatively flat, vacuum state. As expected, this average is a thermal-like e.g. super-Gaussian-like pulses may be produced that distribution displaying the typical cos temperature profile would, at least locally, fall within the approximations in the lab frame, where is the emission angle with respect adopted here. to the direction of motion of the RIP. We then display the Equation (1) arises also in the eikonal approximation for transformation law which relates the temperatures in a scalar field in the metric the pulse and in the lab frame, respectively, by means of 2 the usual Doppler formula. 2 c 2 2 2 2 ds ¼ 2 dtl dxl dy dz : (2) The results worked out in the preceding sections apply in n ðxl vtlÞ the approximation in which dispersion is neglected. In To carry out the analysis in the context of the analogue Sec. IV we study the modifications we expect when dis- metric, we pass from the laboratory frame to the pulse persion is taken into account. In particular, we show that, frame (refractive index perturbation frame) by means of a whereas in the dispersionless case one should expect to v boost: t ¼ ðtl 2 xlÞ, x ¼ ðxl vtlÞ, and we obtain observe all the blackbody spectrum, in the presence of c 1 nv nv v dispersion only a limited spectral region, which depends ds2 ¼ c2 2 ð1 þ Þð1 Þdt2 þ 2 2 ð1 n2Þdtdx on v, will be excited, and the blackbody spectrum shape n2 c c n2 v v will not be discernible. Moreover, one can introduce two 2ð1 þ Þð1 Þdx2 dy2 dz2: (3) different concepts of horizon: phase velocity horizons and nc nc
024015-2 DIELECTRIC BLACK HOLES INDUCED BY A ... PHYSICAL REVIEW D 83, 024015 (2011) v We assume 1 nðxÞ ¼ 0; (9) c nðxÞ¼n þ n ¼ n þ I ðxÞ; (4) 0 0 which, by Eq. (4), becomes n0 þ IðxÞ¼c=v. Because of where is meant to be positive, with 1 due to the the specific shape assumed for I there will be two horizons: actual smallness of the Kerr index (a negative could be one, denoted by xþ, located on the rising edge of the RIP j easily taken into account); I denotes the normalized inten- (i.e. dn=dx xþ < 0) and one, denoted by x , on the falling sity of the pulse, is taken to be a C1 function, rapidly edge (i.e. dn=dxjx > 0). Since 0 I 1, the condition decaying at infinity and with a single maximum at x ¼ 0, for the occurrence of the event horizons is of height 1. A scheme of the RIP is shown in Fig. 1. The 1 v 1 form of I implies that both @t ¼: and @ are Killing < : (10) n þ c n vectors for the given metric. Then the surface g00 ¼ 0 is 0 0 lightlike and corresponds to an event horizon. It is also The external region corresponds to x
024015-3 F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) 2 represents the overall spatial extension of the RIP, at least a c dn þ ¼ 2 ðxþÞ ; (15) further scale that corresponds to the ‘‘thickness’’ of the ðn0 þ k Þ 1 dx region where the refractive index undergoes its most rapid ℏc 1 dn variation. Then we can adopt a profile which is analytically T ¼ ðx Þ : (16) þ 2 k ðn þ k Þ2 dx 0 described as follows: let us define b 0 1 þ x x HðxÞ :¼ 1 þ tanh tanh ; (22) B. Gaussian pulse wh bh
As an example of a refractive index perturbation I ðxÞ where wh, bh > 0 are length scales describing the consider a Gaussian normalized intensity of the pulse, ‘‘thickness’’ of the region over which a rapid variation of the refractive index occurs, and 2 >0 corresponds to the x2 overall spatial extension of the RIP. Compare also the IðxÞ¼exp 2 (17) 2 choice of the velocity profile for acoustic geometries and a small parameter. These choice reflects a typical chosen in [33]. Then for the refractive index profile we situation in experimental optics. Then the horizons given choose by Eq. (9)) occur for HðxÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðxÞ¼n0 þ : (23) c 1 maxHðxÞ x ¼ 2 log n ; (18) x v 0 It is evident that asymptotically n converges to n0 and the and x ¼ xþ. The surface gravity and the temperature RIP is correctly normalized. This model can be also useful are, respectively, given by in view of exploring the possibility to obtain a bh-wh laser sffiffiffiffiffiffiffiffiffiffiffiffiffi [33]. 2 2 k 1 From a physical point of view, we point out that, in the þ ¼ v 2 log ; (19) k presence of a shock front typical values of wh; may be 6 3 order of wh 10 m and 5 10 respectively, and by leading to T 1K. sffiffiffiffiffiffiffiffiffiffiffiffiffi þ ℏ ℏc 1 k 1 T ¼ þ 2 log : (20) þ 2 III. QUANTUM FIELD THEORETICAL 2 kbc 2 kb n 1 k 0 TREATMENT: HAWKING RADIATION Typical values of the parameters are: 10 5m, IN THE LABORATORY FRAME 10 3, n 1:45. Then, for k ¼ 1 , say, we get 0 2 The deduction of Hawking radiation emission by a (non- 2 Tþ 2 10 K; (21) extremal) black hole can also be carried out in a dynamical situation where the effects of the onset of a black hole which is greater than in standard cases of sonic black holes. horizon on quantum field theory modes is taken into ac- This temperature value is not to be meant as characteristic, count. See e.g. the seminal paper by Hawking [2] and also somewhat larger values may be obtained. See the following [34], where the equivalence between the static and the subsection. dynamical picture for Hawking radiation is considered. In Note that this is the temperature in the frame of the the following section, we develop a dynamical picture for pulse, with which the geometry is associated. A further particle production, with the aim of not relying only on the transformation is needed in order to recover the tempera- analogue gravity picture but also obtaining the phenome- ture in the laboratory frame (see the following section). non in a nongeometrical setting by means of standard tools Note also that the temperature is proportional to the de- of quantum field theory. rivative with respect to x of the refractive index, evaluated Consider a massless scalar field propagating in a dielec- at the black hole horizon. This is in agreement with the tric medium and satisfying Eq. (1). It is not difficult to results of Ref. [24]. show that this equation arises in the eikonal approximation for the components of the electric field perturbation in a C. Shockwave model nonlinear Kerr medium [12,15]. nðxl vtlÞ is the refrac- Let us assume that the rear part (trailing edge) of the tive index, which accounts for the propagating refractive filament is characterized by a shockwave profile (see e.g. index perturbation in the dielectric. [24]). This is a typical e.g. of spontaneous laser pulse In order to develop the model we adopt the following dynamics in transparent media with third order nonlinear- strategy: a) we look first for a complete set of solutions of ity. An analogous behavior is expected for the refractive Eq. (1); b) then we employ these solutions to perform a index. As a consequence, in our model it is necessary to comparison between an initial situation, consisting of an introduce, beyond the scale 2 , which, roughly speaking, unperturbed dielectric without a laser pulse, with a uniform
024015-4 DIELECTRIC BLACK HOLES INDUCED BY A ... PHYSICAL REVIEW D 83, 024015 (2011) and constant refractive index n0, and a final situation where Then, we find the following dispersion relation: a laser pulse inducing a superluminal RIP is instead n2v2 present. This scenario is analogous to the original situation ðk k Þ2 ðk þ k Þ2 k2 ¼ 0; (30) 2 u w u w ? envisaged and treated by Hawking [2] in which one con- c siders an initial spherically symmetric astrophysical ob- 2 2 2 where k? ¼ ky þ kz . By solving for ku as a function of kw, ject, with no particles present at infinity, followed by a k? one arrives at a second degree equation whose solutions collapsing phase leading eventually to an evaporating are of the form: Schwarzschild black hole. In analogy with the original 2 uvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 treatment by Hawking, our aim is to evaluate the mean 2 2 u 2 n2v2 k 6 n v nvu k 1 2 7 value of the number of quanta (calculated by means of final k ¼ w 41þ 2 t1 ? c 5: (31) u n2v2 c2 c k2 n2v2 creation and annihilation operators) on the initial vacuum 1 c2 w 4 c2 state. The constraint
A. Out states 2 n2v2 k 4 2 ? c ; (32) Recast Eq. (1) in terms of the new retarded and advanced k2 n2v2 w 1 c2 variables, respectively, u ¼ xl vtl and w ¼ xl þ vtl: must hold in order that propagating solutions are available. n2ðuÞv2 2 2 This seemingly would imply the existence of a limiting 2 ð@u þ@w 2@u@w Þ c angle for the emission, but one has to take into considera- 2 2 2 2 ð@u þ@w þ2@u@w Þ @y @z ¼ 0; (24) tion that, in approaching the horizon, the right-hand side of the latest equation becomes infinite, leaving room for no and look for monochromatic solutions of the form real limitation on the emission angle. ik wþik yþik z þ ðu; w; y; zÞ¼AðuÞe w y z : (25) The root ku is singular at the horizons, whereas ku is þ regular there. We focus on the behavior of ku in the neigh- Then, the ansatz of Eq. (25) leads to borhood of uþ. Near uþ the dependence on k? is washed 2 2 2 out, in agreement with the analysis in [5], and we have 00 c þ n ðuÞv 0 A ðuÞþ2ikw 2 2 2 A ðuÞ c n ðuÞv 2kw ku : (33) 2 2 1 n v 2 ky þ kz c kw þ 2 v2 AðuÞ¼0: (26) 1 n ðuÞ c2 Then, Assuming nðuÞ to be analytic (see e.g. (17)) we see that the 2ck AðuÞ i w ðu u Þ : 0ð Þ ð Þ exp 0 log þ (34) coefficients of A u and of A u have a first order pole at vn ðuþÞ the roots of The logarithmic divergence in the phase as u # u is entirely v þ 1 nðuÞ ¼ 0: (27) analogous to the one experienced by the covariant wave c equation at the horizon of a Schwarzschild black hole [2,3] Then, assuming n to be of the form (4), the condition for and represents the typical behavior of outgoing modes the occurrence of the event horizons is given by (10). Since approaching the horizon [5]. It is worth to remark that this x ¼ u, the black hole and white hole horizons, when they divergence appears here without any reference to geometry. exist, are located, respectively, at In addition, as shown in Appendix C, it emerges naturally as an exact asymptotic (as u # u ) consequence of Eq. (24). On 1 þ u ¼ x ; (28) the other hand, the same asymptotic behavior can be in- ferred also in the context of the geometric approach. Indeed, and we have a second order linear differential equation such behavior is characteristic also of the monochromatic solutions of the covariant wave equation in the metric (2), with Fuchsian singular points at u ¼ u (with u
024015-5 F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) ? or negative frequency components ffk ;f g. These form a l kl complete set of solutions which are mutually orthogonal Fþ eiðkuþkwÞxlþikyyþikzz ivtlðku kwÞ: (36) with respect to the index kl relative to the Klein-Gordon This can be written as product in the retarded variables: þ þ k x Z F ¼ Fk expði l l i!ltlÞ; (37) @ @ l ð ; Þ¼i dudydz : (46) R3 @w @w where kl ¼ðkxl;ky;kzÞ, with The above product is a function of w (due to the factor ! ¼ vðk k Þ; (38) 2 l u w n ðxl vtlÞ it is indeed impossible to define in a standard way a conserved inner product for Eq. (1)). Expanding an ? kxl ¼ ku þ kw; (39) arbitrary solution of Eq. (1) over the modes ffk ;f g we l kl so that quantize the field by promoting to a field operator according to the expansion 1 ! k ¼ k þ l ; (40) Z u xl 3 ? ? 2 v ¼ d k ðak fk þ a f Þ; (47) l l l kl kl 1 ! ? l where the creation and annihilation operators ak , ak kw ¼ kxl : (41) l l 2 v satisfy the usual commutation relations We look for solutions which are asymptotically of posi- ? ½ 0 ¼ 0 akl ;ak klk : (48) tive frequency and outgoing with respect to the dielectric l l perturbation, which is right moving. In other words, we Comparison between the creation and annihilation opera- choose !l > 0 and kxl > 0. Moreover, for definiteness we tors of the in and out states will allow us to calculate the fix kw < 0. As a consequence, (38) and (39) are satisfied Bogoliubov coefficients relating one set to the other, thus for ku > kw.Asto!l, it satisfies the asymptotic dis- leading us to evaluate the average number of emitted 2 2 k2 2 persion relation n0!l ¼ l c . A similar analysis can be quanta. performed for the solution which is regular at u ¼ uþ, which we denote by Fþ . Similarly, we denote by F kl;reg kl D. Greybody factor and by Fk the singular and, respectively, regular solu- l;reg In our four dimensional problem, a priori we cannot tions at u . neglect the effect of backscattering, i.e., the fact that, once Introducing the angle of emission , defined by kx ¼ emitted near the horizon, photons can undergo with a k j lj cosð Þ, Eq. (41) can be written as certain probability, reflection back into the horizon due to the presence of a nonvanishing potential that they encoun- !l v kw ¼ 1 n0 cosð Þ : (42) ter in their propagation [2]. This originates the so-called 2v c greybody factor, which can be calculated as the square modulus of the transmission coefficient. B. In states Consider a massless scalar field propagating in a dielec- tric medium and satisfying the massless Klein-Gordon Prior to the formation of the pulse the wave equation is equation in the pulse metric (7), and let us call gðxÞ the n2 component g appearing in (8). We get 0 @2 @2 @2 @2 ¼ 0; (43) 2 tl xl y z c n2ðxÞ 1 1 @2 nðxÞ@ gðxÞ@ whose independent positive frequency plane wave solu- c2 gðxÞ x nðxÞ x tions are trivially given by (with !l > 0) 2 2 @y @z ¼ 0: (49) in Fk ¼ expðikl xl i!ltlÞ; (44) l We consider solutions of the form where the obvious dispersion relation holds ð ; x; y; zÞ¼ei! ’ðxÞe ikyy ikzz: (50) n2 0 !2 k2 k2 ¼ 0; (45) As a consequence, we obtain an equation of the form c2 l xl ? 0 0 2 2 2 2 ðpðxÞ’ Þ þ ! kðxÞ’ qðxÞ’ ¼ 0; (51) and where k? ¼ ky þ kz. where the prime stays for the derivative with respect to x C. Quantization and þ þ g n 1 We can separate the monochromatic solutions fF ;Fregg pðxÞ :¼ ; kðxÞ :¼ ; qðxÞ :¼ k2 : (52) n 2 ? n corresponding to all possible values of kl, into positive and c g
024015-6 DIELECTRIC BLACK HOLES INDUCED BY A ... PHYSICAL REVIEW D 83, 024015 (2011) The following change of variable: We have to translate the latter result in the laboratory sffiffiffiffiffiffiffiffiffiffi Z Z frame; the simple substitution x kðuÞ x n sðxÞ :¼ du ¼ du ; (53) ¼ð Þ pðuÞ cg ! !l vkxl ; (62) where s results to coincide with the tortoise coordinate, where the label l indicates laboratory frame quantities, leads to the following Schrodinger-like form of the equa- leads to the following expression for the greybody factor: ’ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion for : 2 2 2 4ð!l vkxlÞ ð!l vkxlÞ k?q0 d2’ ð! ; k Þ’ ; (63) þð!2 QðsÞÞ’ ¼ 0; (54) l ? ½ð! vk Þþk q 2 ds2 l xl ? 0 where where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2g c v2 QðsÞ :¼ k2 : (55) 2 ? q0 ¼ 1 n0 2 : (64) n n0 c It can be noticed that, as x ! x , i.e. as s ! 1, one þ One can also take into account that obtains QðsÞ!0, and that 2 2 v 2 c 2 v !l vkxl ¼ !l 1 n0 cosð Þ ; (65) limQðsÞ¼k 1 n ¼: Q1: (56) c s!1 ? 2 0 2 n0 c so that QðsÞ turns out to be monotonically increasing from 0 to the aforementioned constant value as s !1, and the transi- 4ð1 n v cosð ÞÞjn v cosð Þj ’ 0 c 0 c tion from the zero value and the asymptotically constant v v 2 ½1 n0 c cosð Þþjn0 c cosð Þj value for a bump is very fast, in such a way that, as a ! v plausible approximation for the calculation of the trans- ð ð1 n cosð ÞÞ k Þ: (66) H q 0 c ? mission coefficient, we replace the above potential QðsÞ 0 1 with a steplike effective potential with height Q1: Being 1, and v=c ¼ðn0 þ k Þ , k 2ð0; 1Þ,we 2 obtain d ’ 2 þð! Q1Þ’ ¼ 0: (57) ds2 v k 1 n0 cosð Þ 1 cosð Þþ cosð Þ; (67) In this approximation, we can refer to well-known results c n0 of standard quantum mechanics (see e.g. [35], where the step barrier appears as a subcase of problem 37), in order to v k n0 cosð Þ 1 cosð Þ ; (68) infer that: c n0 (a) the transmission coefficient is zero if which leads to the conclusion that s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi c v2 ’ 1: (69) !< Q ¼ k 1 n2 : (58) 1 ? n 0 c2 0 It is worth pointing out that, on the grounds of the previous result, a 90-degree emission would not suffer any substan- (b) the transmission coefficient is tial suppression by backscattering. This is particularly relevant in connection to recent measurements of 90- pffiffiffiffiffiffiffiffi ! Q degree photon emission from RIP-induced horizon [26]. 2 ffiffiffiffiffiffiffiffi1 jTj ¼ 4 p 2 ¼ ; (59) ð! þ Q1Þ E. Thermal spectrum for In analogy with the corresponding black hole calcula- pffiffiffiffiffiffiffiffi tions, in order to find the evaporation temperature of the !> Q : (60) 1 RIP we must compute the expectation value of the number in We point out that the dispersion relation operator of the outgoing photons evaluated in the vacuum. In this connection, it is important to remark n2 that, contrary to the standard black hole scenario, in the k2 þ k2 ¼ 0 !2 (61) x ? c2 case of a RIP propagating in a nonlinear Kerr medium we have, strictly speaking, no real collapse situation. implies that a) cannot occur. As a consequence, the Nevertheless, we can ‘‘simulate a collapse’’ by building Heaviside function which, in principle, should multiply up a refractive index perturbation starting from an initial the aforementioned jTj2, is always equal to 1. situation in which no signal is present in the dielectric.
024015-7 F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) We perform a coordinate shift which moves the black gives the wavelength of maximum emission of a blackbody hole horizon in u ¼ 0. A photon starting at u<0 is as a function of the temperature trapped and cannot reach the front observer. As a conse- T ¼ b ¼ 2:9 10 3m K: (77) quence, states with u<0 cannot be available to the front max observer and he will be led to consider in his description Converting to the frequency of maximum emission gives only the u>0 part of the Fþ solution, thus multiplying the 2 c latter by a Heaviside function ðuÞ. The Bogoliubov co- ! ¼ T: (78) l max b efficients k k0 and k k0 relating, respectively, the positive l l l l and negative frequency components between the initial in Now, under the boost connecting the lab frame to the pulse and final out states are analyzed in D, and satisfy the frame, the frequency transforms according to the relativis- fundamental relation tic Doppler formula in a medium with refractive index n0, X X 2 ð4 ck =n0ðu ÞvÞ 2 v j k k0 j ¼ e w þ j k k0 j ; (70) l l l l ! ¼ !l 1 n0 cos ; (79) k0 k0 c l l which is also proved in the Appendix. Moreover, the where ! is the frequency in the pulse frame, and is the expectation value of the number operator of the outgoing emission angle with respect to the direction of motion of photons in the initial vacuum state is the pulse in the lab frame. Combining Eq. (79) with X Eq. (78) gives Eq. (75). out 2 h0 injNk j0 ini¼ j k k0 j : (71) l l l k0 l IV. EFFECTS OF OPTICAL DISPERSION Then, by using the completeness relation for the In the framework of analogue models for black holes Bogoliubov coefficients and taking into account backscat- (dumb holes, bec models) there is a number of studies tering, X devoted to the analysis of the actual generation of 2 2 Hawking radiation (see e.g. [9,33,36–43]). In particular, a ðj k k 0 j j k k 0 j Þ¼ ð! ; k Þ; (72) l l l l l ? k 0 suitable dispersion law, which involves the fourth power of l the momentum k in the comoving frame is considered, and Eq. (70), we obtain the following thermal-like distri- since the original model by Unruh [36]. Calculations, in bution written in terms of asymptotic (physical) frequen- particular, in presence of a black hole—white hole system, cies: are quite involved and require also numerical simulations. We do not deal herein quantitatively with the problem of hNk i¼ ℏ (73) mode conversion, and we do not consider the problem of l expð !lÞ 1 kbT black hole lasers (see [9,33,36–44]), which will be taken into account in further developments of the present model. where Still, we aim to point out the main physical consequences ℏ 1 dn due to optical dispersion, and we shall stress that optical T ¼ v2 ðu Þ : (74) v þ dispersion behaves as a fundamental ingredient for dielec- 2 kBc 1 c n0 cos du tric models. In this respect, there is agreement with dis- By comparing this temperature with the temperature Tþ persive models in sonic black holes. derived in the analogue Hawking model in the pulse frame We start by recalling how optical dispersion affects the (see Eq. (13)), we find the relation dispersion relation for the quantum electromagnetic field in 1 1 linear homogeneous dielectric media (see e.g. [45]): T ¼ Tþ: (75) 1 v n cosð Þ 2 2 k2 2 c 0 n ð!lÞ!l l c ¼ 0: (80) For the same values of the parameters leading to Eq. (21), This dispersion relation is obeyed by the monochromatic 2 for ¼ 0 and for a typical value v 3 c, we would obtain components of the field, and in [45] phenomenological quantization of the electromagnetic field in a dielectric T 78K; (76) medium is justified on the grounds of a more rigorous which is again much greater than the values of T for typical approach. In the presence of optical dispersion acoustic black holes. Moreover, in the specific case of a Z 1 shock front, this temperature may increase significantly so ðxl;tlÞ¼ d!l ð!l; xl;tlÞ; (81) that T 2000K (cf. Subsection II C). 0 i!ltl i!ltl Equation (75) gives the correct transformation law for where ð!l; xl;tlÞ¼’þð!l; xlÞe þ ’ ð!l; xlÞe the temperature in going over from the pulse frame to the are the monochromatic components. Then it is easy to laboratory frame. Indeed, to find how the temperature show that, even by allowing a spatial dependence for the transforms, start from Wien’s displacement law which refractive index
024015-8 DIELECTRIC BLACK HOLES INDUCED BY A ... PHYSICAL REVIEW D 83, 024015 (2011) n2ð! ; x Þ We may also wonder if the above relation can be trusted r’ð! ; x Þþ!2 l l ’ð! ; x Þ¼0: (82) l l l c2 l l in our case, as we started from a nonlinear medium which is also nonhomogeneous because of the RIP. We recall that Moreover, in the case of an homogeneous medium, by we assume to be involved with linearized quantum fields x k x letting ’ð!; Þ/expði l Þ, one finds that Eq. (82)is around our effective geometry, and then it is reasonable to satisfied only if explore what happens in the presence of optical dispersion by considering linear dispersion effects. Moreover, we nð! Þ jk j¼! l ; (83) make the following ansatz: l l c nðx vt ;!Þ¼n ð! Þþ fðx vt Þ; (86) i.e. only if each monochromatic component travels at its l l l 0 l l l phase velocity (as obvious). i.e. we neglect optical dispersion effects in the RIP, and As a typical example, in a so-called 1-resonance model, keep trace of it only in the background value n0, which we get the Sellmeier equation does not depend on spacetime variables, and passes from the status of constant to that of a function depending on !l. 2 2 !c In order to justify a posteriori this ansatz we can take n ð!lÞ¼1 þ 2 2 ; (84) !0 !l into account what happens in the usual modelization of dielectric media when inhomogeneities are considered. A where !0 is the resonance frequency and !c is the cou- possibility is to account for inhomogeneities by including pling constant, which is also called plasma frequency. It is space-dependent density, polarizability, resonant frequen- 2 worth pointing out that the function n ð!lÞ is a Lorentz cies, and so on (see e.g. [48]). By neglecting dissipation in scalar and it is invariant under boosts [it may be written in a a polariton model, one can obtain a dielectric susceptibility 2 manifestly covariant form as n ðv k Þ, where v stays for which depends on space and frequency, in such a way that the velocity 4-vector and k is the four-momentum whose the Sellmeier formula changes only because of an explicit zeroth covariant component in the lab coincides with !l; dependence of !c, !0 on space variables. In our case, cf. [46]]. For the case of interest in Ref. [26], which because of the properties of fused silica in the visible concerns optical dispersion in fused silica, it can be shown region, where the Cauchy formula works well, we can that a more suitable formula involves the summation of even neglect the spatial dependence of !0 (which should three terms like the frequency-dependent ones appearing be considered much greater than !l and !c), and then, on the right-hand side of (84). The refractive index arising recalling that u :¼ xl vtl, on the grounds of Ref. [48]we from the complete Sellmeier for the case discussed in assume that Ref. [26] is displayed in Subsection IV B. Still, we note 2 2 2 that, although complete expressions can be provided for !cðuÞ¼!c0 þ !c1ðuÞ; (87) the dispersion relation, for practical purposes it is very where 1 is a small parameter; if we also define useful to use the so-called Cauchy formula sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 nð! Þ¼n þ B !2; (85) c0 l 0 0 l n0ð!lÞ¼ 1 þ 2 2; (88) !0 !l where n is the would-be refractive index in absence of 0 the Sellmeier Eq. (84) then leads to optical dispersion and B0 is a suitable constant; this ap- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi proximation works well for fused silica in the visible and 1 !2 ðuÞ near-infrared (below 1 m wavelengths) spectral region nð! ;uÞ¼n ð! Þ 1 þ c1 n ð! Þ l 0 l 2ð Þ 2 2 0 l where it is practically indistinguishable from the full n0 !l !0 !l Sellmeier relation. Differences may become more evident 2 þ 2 !c1ðuÞ; (89) between the Cauchy and Sellmeier relations if one attempts 2n0!0 to calculate the gradients of the curves, e.g. in order to where we have used the Cauchy approximation and ne- evaluate the group index or group velocity in the medium. 2 However, once the RIP velocity is fixed we are only con- glected terms order of !l . This validates our ansatz (86), cerned with the phase velocities and the Cauchy formula with straightforward identifications. Another possibility is approximation is sufficiently precise. to adopt in the visible frequency region the following perturbative ansatz:
2 A. Dispersion and RIP nð!l;uÞ¼n0 þ A1ðuÞþðB0 þ B1ðuÞÞ!l n0 We note that the dispersion relation for a linear homo- 2 þ B0! þ A1ðuÞ; (90) geneous dielectric medium is quite different from the one l 2 which has been hitherto considered in the sonic model by where n0 þ B0!l n0ð!lÞ and where we neglect terms 2 Unruh and its subsequent developments (see also analo- Oð !l Þ. The latter approximation is very useful, because gous comments in Ref. [47]). we can write
024015-9 F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) 1.5 vl;’ ¼ v , v’ ¼ 0: (97)
In the lab frame the calculation is elementary, and leads to c 16 0 v ¼ ; (98) l;’ nð! ;uÞ
(10 1/s) l ω which, due to (97), implies the horizon condition c -1.5 nð! ;uÞ¼ ; (99) -4 -2 0 2 4 l v 8 k (10 1/m) it is worth noticing that condition (99) is a straightforward FIG. 2 (color online). Branches relative to the dispersion rela- generalization to the dispersive case of the condition ob- tion (94) in the comoving frame. The dashed line corresponds to tained in the absence of dispersion. It is also important to ! ¼ vkx. Comoving frequencies corresponding to the experi- point out that the horizon condition in the lab in presence of ments in Ref. [26] are of the order of 1013 rads=s. dispersion can be also expressed as follows: v nð! ;uÞ¼n ð! Þþ fðuÞ¼nðuÞþB !2; (91) 1 nð! ;uÞ ¼ 0; (100) l 0 l 0 l l c which isolates the contribution of the optical dispersion. As for the dispersion relation, we obtain in the lab which e.g. in the Gaussian model provides sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nð!l;uÞ!l ¼ jkljc; (92) c 1 u ¼ 2 log n0ð!lÞ ; (101) i.e. v 3 k and, in general, a modified horizon condition appears: nðuÞ!l þ B0!l j ljc ¼ 0: (93) This can be carried into the comoving (pulse) frame, where c n0ð!lÞ < n0ð!lÞþ : (102) one obtains v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 This condition is displayed in Fig. 3 (where wavelength 2 v 2 nðxÞ ð! þ vkxÞ kx þ 2 ! þ k?c replaces !l) and the velocity of the RIP is chosen to match c the experimental conditions of Ref. [26]. Explicitly, by 3 3 þ B0 ð! þ vkxÞ ¼ 0; (94) using the Cauchy formula (85), we obtain (we consider c only positive frequencies) for n0 þ !l ¼ ð! þ vkxÞ; (95) n 1.453 and then positive frequencies in the lab correspond to the region above the dashed line ! ¼ vk : (96) x 1.449 0.7 0.8 0.9 1 1.1 wavelength (µm) B. Dispersion and horizons The definition of the horizon which is suitable for this FIG. 3 (color online). Prediction of the Hawking emission kind of situation, where one cannot recover a metric by spectral range accounting for the RIP velocity and the material reinterpreting the dispersion relation as an eikonal approxi- refractive index n from a complete Sellmeier equation. The two mation of the wave equation (due to the presence of more curves show, for the case of fused silica, n0 (in blue) and n0 þ n (in green) with n ¼ 1 10 3. The shaded area indicates than quadratic terms), can a priori involve phase velocity the spectral emission region predicted for a Bessel filament that and/or group velocity; cf. e.g. a short discussion in [49]. has an effective refractive index n ¼ c=v where the Bessel Let us first consider what happens when one involves the B peak velocity is vB ¼ vG= cos with cone angle ¼ 7 deg and phase velocity, indicated as vl;’ in the lab reference frame vG ¼ d!=dk is the usual group velocity. The values and pa- and as v’ in the comoving frame: rameters used correspond to the experiments shown in Ref. [26]. 024015-10 DIELECTRIC BLACK HOLES INDUCED BY A ... PHYSICAL REVIEW D 83, 024015 (2011) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c n interaction of photons with the dielectric material. See v n0 v ffiffiffiffiffiffi 0 pffiffiffiffiffiffi !l < p ; (103) e.g. [50]. This is per se a sufficient reason for expecting B B 0 0 deviations from the standard Planckian distribution. A very i.e., in terms of wavelength, naive inclusion of dispersion in our case would also lead to pffiffiffiffiffiffi pffiffiffiffiffiffi T ¼ Tð!Þ, which would make even less plausible a pure 2 c B 2 c B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 : (104) Planckian spectrum. c c v n0 v n0 Regarding the width of the spectral emission window we note that this is determined by the value of . For a typical c If, instead, n0 þ v , i.e. for a substantially larger per- case we consider a Gaussian pump pulse in fused silica, turbation, we obtain n ð800 nmÞ¼1:467 and we find that for n ¼ 10 3, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g c n Eq. (102) is satisfied over a 15 nm bandwidth at v ffiffiffiffiffiffi 0 0 !l < p ; (105) 435 nm. This window may become substantially large in B0 lower dispersion regions as shown in Fig. 3. i.e., in terms of wavelength, The phase velocity horizon condition may also be calcu- pffiffiffiffiffiffi lated in the comoving frame, where one obtains v’ ¼ 0 if 2 c B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 ! ¼ 0 (for kx 0), i.e. for v>0: c < <1: (106) n0 v c nð ð! þ vk Þ;xÞj ¼ : (108) For the parameters of Fig. 3 (see caption), Eq. (105)is x f!ðkx;xÞ¼0;kx 0g v verified for >0:013, a rather large yet not completely unrealistic refractive index variation. We may evaluate the Notice that ! ¼ !ðkx;xÞ is solution of the 2D reduction of influence of material dispersion in relation to the event Eq. (94). The above condition requires that, as in absence horizon condition. In the absence of dispersion a horizon is c c of dispersion, n0 < v . Moreover, if also n0 þ Moreover, one has to take into account that, even in homo- 12 0 geneous transparent dielectrics, optical dispersion affects (10 1/s) the spectral density of photons which is actually to be taken ω into account, by introducing a phase space factor multi- plying the standard Planckian distribution term which -2 nð!Þ -4 -2 0 2 4 depends on both the refractive index and on the group 7 velocity vgð!Þ [50]: k (10 1/m) 2 ð ℏ!3 Þ n ð!Þ FIG. 4 (color online). The dispersion relation (96) in the 2c2 v ð!Þ ð!Þ¼ g : (107) comoving frame for a 2D model at a fixed point x (only the expðℏ!Þ 1 kbT branch relevant to horizon formation is shown). The shaded regions delimit the bands representing the values of kx for which This formula is intended to hold true in the reference of the a phase horizon exists. See Eqs. (110) and (111), in which we thermal bath. These optical dispersion contributions chose values of the velocity v and for illustrative purposes and amounts to a sort of greybody factor arising from the that are not related to the experimental values used in Fig. 3. 024015-11 F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c n c n the comoving frame the dispersion equation in the follow- v ffiffiffiffiffiffi0 v ffiffiffiffiffiffi0 p 024015-12 DIELECTRIC BLACK HOLES INDUCED BY A ... PHYSICAL REVIEW D 83, 024015 (2011) TABLE I. Comparison between the analogue Hawking and QFT descriptions of quantum vacuum emission by a moving RIP. Black Hole—Pulse Frame Quantum Field Theory—Lab Frame propagation equation (eikonal approximation) propagation equation (eikonal approximation) ## v appearance of a bh horizon onset of the surface s.t. 1 nðuÞ c ¼ 0 ## logarithmic singularity in the phase of quantum modes logarithmic singularity in the phase of quantum modes ## thermal spectrum with temperature Tþ thermal spectrum with temperature T frequencies that form the wave packet fall within the phase hole) for a generic but static dielectric perturbation. horizon frequency window, then all of these will be re- Despite the fact that the analogue metric is determined flected from the RIP. Each frequency will be reflected from up to an overall conformal factor, the temperature is con- a different point within the RIP (each corresponding to the formally invariant. Then we have taken into account quan- horizon at the relative frequency) and this may severely tum field theory in the lab frame, in a way that is distort the wave packet during reflection. Nevertheless, it is independent of the analogous geometric framework we clear that if all spectral components are 100% reflected have again shown that there is a photon production with then there will be zero transmission. What is expected is a thermal spectrum, corresponding to the Hawking effect. that, in general, an incoming wave packet is converted into These findings are summarized in Table I. We have also a reflected and possibly very broad and irregular wave provided the transformation law of the temperature be- train. Clearly, if part of the wave packet spectrum falls tween the given frames and taken into account the effects outside the phase horizon window, then these frequencies of dispersion on the horizon condition. will be transmitted through the RIP and we can no longer talk of perfect blocking. ACKNOWLEDGMENTS The action of a group velocity horizon appears to be F. B. wishes to thank Sergio Serapioni and Lesaffre Italia different, in the sense that, although the existence of a S.p.A. for financial support to the Department of window of allowed frequencies seems to be analogous to Mathematics of the Universita` degli Studi di Milano, where the one of a phase velocity horizon, we have to point out part of this work was performed. that in the present case !l refers to the carrier wave in the wave train. The group velocity horizon appears to be less selective than the phase velocity one: it limits itself to APPENDIX A: FORMAL MAPPING cause a reflection of wave packets which have carrier TO AN ACOUSTIC BLACK HOLE frequencies in the allowed window, without distinguishing It is interesting to notice that our black hole metric in the between monochromatic components composing the pulse frame can be mapped into a form of acoustic black packet itself. The outgoing wave train is expected to be hole metric, provided that suitable identifications are made. still in the form of a compact and relatively undisturbed In particular, we refer to the acoustic black hole metrics wave packet. Both numerical simulations and experimental taken into consideration in [51], and limit ourselves to the results could be able to reveal this different character of the analysis of the x t part of the metric (indeed, only 2D two above horizon versions, and, in particular, measure- metrics are studied in [51]). In particular, we are looking ments should be able to discern which definition is relevant for a transformation allowing to carry for the physics at hand. It is worth pointing out that, in Ref. [26], the main role in photon production appears to be 2 ds2 ¼ ðc2 n2v2Þdt2 2vð1 n2Þdtdx related to the phase velocity horizon rather than to the ð2Þ n2 group velocity horizon. v2 þ n2 dx2 (A1) c2 V. CONCLUSIONS into the form We have studied the photon production induced in a dielectric medium by a refractive index perturbation, 2 2 2 2 2 2 dsacoustic ¼ ½ ðc~ v~ Þdt 2vdtd~ x~ þ dx~ ; (A2) which is created in a nonlinear medium by a laser pulse through the Kerr effect. The pulse has been assumed to where v~ plays the role of fluid velocity (in general depend- have constant velocity v. In the pulse frame, we have ing on t, x~) and c~ is the local speed of sound, assumed to be investigated the analogous metric, and we have shown constant as in [51]. This remapping of the optical metric that two Killing horizons appear (black hole and white into an acoustic one is implemented by the identifications 024015-13 F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) choice of an angle parameter, to be related to the inverse of the Hawking temperature, is chosen. The following rescaling to dimensionless variables is performed: 2 c 2ð0; Þ ° c ¼ c 2ð0; 2 Þ (B1) 2 2 x 2ðx ; 1Þ ° x ¼ x 2 x ; 1 ; (B2) þ þ where stays for the Euclidean time. Moreover, the fol- lowing (local) diffeomorphism on the ct x part of the metric is introduced: 2 1 1 ds2 ¼ g ðx Þdðc2 2Þþ dx 2 ð2Þ 2 n2ðx Þ g ðx Þ 2 ¼ H2ðyÞðy2dc 2 þ dy2Þ; (B3) FIG. 5 (color online). Penrose diagram for the analogue metric 2 (2). whence c~ ¼ c; (A3) 1 g ðx Þ¼y2H2ðyÞ; (B4) 2ð Þ n2 1 n x v~ ¼ 2v ; (A4) n 1 dx 2 ¼ H2ðyÞ: (B5) 1 1 g ðx Þ dy 2 ¼ ; (A5) 2 n2 v2 c2 The above diffeomorphism is such that the ct x part of the metric near the horizon can, at least locally, be made n2 v2 dx~ 2 c2 conformal to a 2D plane with a suitable choice of , with ¼ : (A6) 2 2 dx n conformal factor ð2 Þ H ðyÞ. As a consequence, one finds It is straightforward to check that the horizon condition (taking the positive sign) v~ ¼ c is equivalent to the condition n ¼ c , i.e. the solution 2 dx 1 1 v ¼ g ðxÞ; (B6) of (9). By defining dy y nðxÞ dx~ du~ ¼ dt ; (A7) i.e. c þ v ~ ~ Z 2 x 1 dx~ y ¼ A exp dznðzÞ ; (B7) dw~ ¼ dt þ ; (A8) g ðzÞ c~ v~ as in [51], one can also easily draw the Penrose diagram of where A is an integration constant. The diffeomorphism is our spacetime (see Fig. 5), which is analogous to that of defined to be regular if we can include in the manifold also [51] Fig. 28. It is also straightforward to verify that the the point y ¼ 0, and then we have to require surface gravity can be calculated also by means of the 2 1 2 formula lim H ðyÞ¼ lim 2 g ðxðyÞÞ ¼ h0; (B8) y!0þ y!0þ y @ þ ¼ c~ ðc~ v~Þ (A9) where h2 is a finite positive constant. The above limit is @x~ x¼x 0 þ equivalent to (cf. [32]). 1 lim 2 g ðxÞ: (B9) x!xþ y ðxÞ APPENDIX B: THE HAWKING TEMPERATURE AND ITS CONFORMAL INVARIANCE Near the horizon one finds We consider the Euclidean version of the metric and find 2 ð2 = Þðc2=v2Þð1= 2Þð1= dnðx ÞÞ y ðx x Þ dx þ (B10) that near the horizon the ct x part of the metric behaves þ like a cone which becomes a flat plane only if a special and, as a consequence, one has to choose 024015-14 DIELECTRIC BLACK HOLES INDUCED BY A ... PHYSICAL REVIEW D 83, 024015 (2011) 2 0 c 1 1 F ðu; w; y; zÞ¼ ðuÞeið2kwc=vn ðuþÞÞ logðu uþÞþikwwþikyyþikzz; : ℏ 2 ¼ 2 2 2 dn ¼ c h: (B11) v dx ðxþÞ (C6) The above method can be used also to confirm that the where ðuÞ is holomorphic in the neighborhood of temperature does not depend on the (static) conformal u ¼ uþ and vanishes as u ! uþ: indeed, we have reab- factor. Indeed, an overall conformal factor 2ðxÞ which sorbed in it the factor (u uþ) associated with the real 2ð Þ¼ term in 2, in such a way that is finite and nonvanishing at the horizons (limx!x x b2 > 0) does not modify Eq. (B10), as may be realized ðuÞ¼ðu u Þ ðuÞ; (C7) using Eqs. (B8) and (B9). þ where ðuÞ is holomorphic in the neighborhood of u ¼ uþ APPENDIX C: EXACT ANALYSIS OF and ðuÞ¼c0 þ c1ðu uþÞþ... as u ! uþ. We stress EQUATION (26) NEAR THE SINGULARITY the presence of the logarithmic divergence of the phase as u approaches uþ even in this approach. Also, by comparison We confine ourselves to discuss the root uþ, since all with the study of the solutions contained in Sec. III A,we considerations can be trivially extended to u . Then, we can infer that the above solution corresponds to the (exact) rewrite Eq. (26)as þ expansion of Fk ðu; w; y; zÞ near u ¼ uþ. PðuÞ QðuÞ A00ðuÞþ A0ðuÞþ AðuÞ¼0; (C1) u uþ u uþ APPENDIX D: ANALYTIC CONTINUATIONS where we have introduced the functions PðuÞ and QðuÞ We are interested in the Bogoliubov coefficient kk0 , (holomorphic in the disk ju u j < ju u j): which relates the positive frequencies between the initial in þ þ state: 2 2 2 c þ n ðuÞv in ik xþik yþik z i!t PðuÞ¼2ik ; (C2) Fk ¼ e x y z ; (D1) w c2 n2ðuÞv2 þ and final out state Fk : 2 2 þ þ ið logðu uþÞþkwwþkyyþkzzÞ 2 ky kz Fk ¼ ðu uþÞ kðuÞe ; (D2) QðuÞ¼ kw þ 2 : (C3) 1 n2ðuÞ v c2 where we have introduced the shorthand notation ¼ 2c In a neighborhood of u Eq. (26) writes, to leading order in 0 k and, in agreement with the results obtained in þ n ðuþÞv w u uþ,as Appendix C, we have introduced also the analytic part ðuÞ. It will be evident that, at least in the large frequency 2ik c 1 00 w 0 limit, this ðuÞ cannot affect the thermal character of A ðuÞ 0 A ðuÞ vn ðuþÞ u uþ particle emission. 2 2 ðky þ kzÞc Let us start with the computation of kk0 , which, apart þ 0 AðuÞ¼0; (C4) from a factor which will not affect our goal, which consists 2vn ðuþÞðu uþÞ in the deduction of (D7), is given by where the prime denotes derivation with respect to the lab Z 1 0 2 k k0 i ikuu variable u. The indicial equation is ð 1Þ kk0 / ð ? ?Þ ðuÞu e du; (D3) 0 2ikwc 2ikwc 0 ¼ 0, with roots 1 ¼ 0 and 2 ¼ 1 þ 0 . vn ðuþÞ vn ðuþÞ where we have shifted the variable u so that u is mapped Therefore, in the neighborhood of u , Eq. (26) has two þ þ on 0. Since 2k0 ¼ðk0 þ !0=vÞ > 0, we see that the second linearly independent solutions of the form u x exponential factor is rapidly decreasing at infinity when X1 ImðuÞ < 0. Consider the path starting from 0 to R>0 ðiÞ ðiÞ n A ðuÞ¼ðu uþÞ i cn ðu uþÞ ; (C5) along the real line, then following the arc of radius R n¼0 clockwise until iR, and finally coming back from iR to 0 along the imaginary axis. As the integrand is analytic i ¼ 1, 2, where the series define holomorphic functions in inside the region bounded by the path, the integral along the disc ju uþj < juþ u j and whose coefficients ðiÞ vanishes for any positive value of R. Then, taking the limit can be obtained recursivelyP from the equation cn ð i þ R !þ1we then see that the integral along the positive n 1 nÞð i þ n 1 þ p0Þþ r¼0 ðð i þ rÞpn r þ real axis is equal to the integral along the negative imagi- ðiÞ qn r 1Þcr ¼ 0, where the pk, uk (k ¼ 0; 1; 2; ...) are the nary axis and coefficients of the expansion about uþ of PðuÞ and QðuÞ, 2 k k0 Z 1 ð =2Þ ð ? ?Þ it i t respectively. Of particular interest is the solution (25) kk0 / e dt t e : (D4) ð 0 Þ1þi k0 corresponding to ¼ 2, which for u>uþ has the form i ku 0 u 024015-15 F. BELGIORNO et al. PHYSICAL REVIEW D 83, 024015 (2011) 0 We can consider the limit as ku 1 and approximate for tion on the arc vanishes for ImðuÞ > 0. Thus, rotating the small values of its argument. Then we obtain path counterclockwise, we obtain 2 k k0 2ðk þ k0 Þ ð =2Þ ð ? ?Þ ð =2Þ ? ? 0 kk0 / e ð þ i Þ: kk / e 0 2þi ð2 þ i Þ; (D5) 0 2þi 2 (D6) ðkuÞ ðkuÞ where is the Euler Gamma function. By comparing Eqs. 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