Table Of Content

Analogue black holes for light rays in static dielectrics E. Bittencourt1,2,∗ V. A. De Lorenci3,4,† R. Klippert5,‡ M. Novello1,§ and J. M. Salim1¶ 1Instituto de Cosmologia Relatividade Astrofisica ICRA - CBPF, Rua Dr. Xavier Sigaud 150 Urca, 22290-180, Rio de Janeiro, Brazil 2Sapienza Università di Roma - Dipartimento de Fisica P.le Aldo Moro 5 - 00185 Rome - Italy and ICRANet, Piazza della Repubblica 10 - 65122 Pescara - Italy 3Instituto de F´ısica e Qu´ımica, Universidade Federal de Itajubá, Itajuba´, MG 37500-903, Brazil 4 Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA and 5Instituto de Matema´tica e Computa¸caõ, Universidade Federal de Itajuba´, Itajuba´, MG 37500-903, Brazil (Dated: January 30, 2014) Propagationoflightinnonlinearmaterialsisherestudiedintheregimeofthegeometricaloptics. Itisshownthatasphericallysymmetricmediumatrestwithsomespecificdielectricpropertiescan be used to produce an exact analogue model for a class of space-times which includes spherically 4 symmetric and static black hole solutions. The optical model here presented can be a useful tool 1 to reproduce in laboratory the behavior of optical null geodesics near a compact object with an 0 observable gravitational Schwarzschild radius. 2 n PACSnumbers: 04.20.Cv;04.20.-q;11.10.Lm;42.15.Dp. a J 8 I. INTRODUCTION coefficients (cid:15) and µ, here called electric permittivity and 2 magnetic permeability, respectively. In such case, it can ] Analogue models of general relativity have long ago be shown that the paths of light are geodesic in an effective geometry given by c been considered in the literature. This theme was intro- q duced by Gordon in 1923 [1] when an effective geometric - gµν =ηµν −(1−µ(cid:15))vµvν, (1) r interpretation for light propagation in homogeneous di- g electrics was proposed. In the following six decades few [ where ηµν is the Minkowski metric and vα is the ob- works on this theme were published (for a review, see server’s geodesic four-velocity (assumed to be normal- 1 Refs. [2–5]). However, this research area was warmed ized to unity by simplicity) with respect to which the v up in the 80’s with the proposal of a hydrodynamic ana- 4 material medium is at rest. The above metric is known logue modelof a black hole by Unruh[6]. Since then, lot 4 as the Gordon metric [1, 12]. More general results can of other models were proposed and their consequences 5 be obtained by relaxing the constance of the dielectric investigated. For a review, see Ref. [7] and references 7 coefficients. In this case several models describing dif- . therein. The main expectation is that some tiny effects 1 ferent solutions of general relativity were obtained. For predicted to occur in the realm of semiclassical gravity 0 instance, almost axially symmetric optical analogues of could be tested in such analogous to general relativity 4 Schwarzschildblackholesarealreadyknown[13–15]. For 1 systems. Of particular interest is the issue of Hawking these models, the occurrence of an event horizon is fun- : radiation, which is expected to occur whenever an event v damentally dependent upon the vortical motion of the horizon is brought forth in a physical system, although i dielectric fluid. A model that does not involve mechani- X some authors have different opinion [8]. Such effect was cal motion of the medium was already presented [16] in r unveiled by Hawking in 1975 [9] as a result of the quan- the context of metamaterials, where an effective black a tization of fields in the spacetime of a black hole. This holesolutionwasproposed. Inthispaper, weinvestigate issue was recently examined in laboratory by means of further the possibility of producing a material medium optic [10] and hydrodynamic [11] gravitational analogue at rest with specific dielectric properties in such way systems. In both cases Hawking-like radiation was re- that an exact analogue model for Schwarzschild geome- ported to occur. try becomes available. Particularly, the behavior of light Inthecontextofelectrodynamicsthemostsimpleana- rays propagating inthismedium isstudied. We focus on logue model describing the propagation of light in a con- the theoretical aspects of the phenomenon. Experimen- tinuum media arises by considering constant dielectric tal verification is already known in some tailored media [17, 18]. The outline of the paper is as follows. In Section II, basic aspects of light propagation is reviewed, including ∗Electronicaddress: [email protected] the derivation of the dispersion relations for nonlinear †Electronicaddress: [email protected] material media and the corresponding effective optical ‡Electronicaddress: [email protected] §Electronicaddress: [email protected] metric interpretation. An optical model is presented in ¶Electronicaddress: [email protected] Sec. III. Some kinematical aspects of the propagating 2 optical wave modes in such model are discussed in Sec. where the generalized Fresnel matrix is written in terms IV, which justify calling it an optical analogue model of of the normalized vector ˆlα =Eα/E as spherically symmetric static black hole solutions. Con- 1 cludingremarksarepresentedinSec.V,includingabrief Zα =(cid:15)(δα−vαv )−(cid:15)(cid:48)Eˆlαˆl − (δα−vαv +qˆαqˆ ), β β β β µv2 β β β discussion about the issue of Hawking thermal radiation ph in this model. We assume throughout this work that the (6) physics takes place in the flat Minkowski space-time of where (cid:15)(cid:48) denotes the derivative of the function (cid:15)(E) with special relativity, which is described in a general coor- respect to E. This relation was written in terms of µ dinate system with metric γµν. Geometrical units are (instead of µ0) for latter convenience. The existence of chosen such that the speed of light in vacuum is c = 1. non-trivial eigenvectors eα (cid:54)= 0 then leads [1, 21, 22] to We define the completely skew-symmetric pseudo-tensor the two possible optical geometries ηαβµν such that η0123 = 1 when written in Cartesian gαβ =γαβ −(1−µ(cid:15))vαvβ, (7) coordinates. (+) gαβ =γαβ −[1−µ((cid:15)+(cid:15)(cid:48)E)]vαvβ − (cid:15)(cid:48)Eˆlαˆlβ, (8) (−) (cid:15) II. THE GENERALIZED GORDON OPTICAL METRIC such that the two wave vectors k = k(±) satisfy the λ λ corresponding dispersion relations gαβk(+)k(+) = 0 and (+) α β We briefly recall here the main steps [19] to achieve gαβk(−)k(−) = 0. The former equation is equivalent to theopticalmetricdescriptionofthewavepropagationin (−) α β material media in the limit of geometrical optics. Let vα µ(cid:15)ω2 =q2 , which describes the isotropic propagation (+) (+) √ be the observer’s geodesic four-velocity (assumed to be of the ordinary mode with speed v = 1/ µ(cid:15). The ph normalizedtounityforsimplicity),withrespecttowhich effective optical geometry Eq. (7) is usually referred to thematerialmediumisatrest. Maxwellequationsinside as Gordon geometry . this medium in Minkowski space-time can be written in Our interest here lies mostly in the so-called extraor- covariant notation as dinary mode, the optical geometry Eq. (8), and thus we simply denote it as gαβ = gαβ. Assuming that the de- Pαβ =Jα, (2) (−) ;β terminant of gαβ is non zero, then a simple calculation [22] gives its inverse matrix as the effective metric (∗F)αβ =0, (3) ;β (cid:20) 1 (cid:21) (cid:15)(cid:48)E where (∗F)αβ = ηαβρσFρσ/2 = vαBβ − vβBα + gαβ =γαβ − 1− µ((cid:15)+(cid:15)(cid:48)E) vαvβ + (cid:15)+(cid:15)(cid:48)Eˆlαˆlβ. (9) ηαβ vρEσ stands for the dual of the Maxwell field ρσ strength tensor Fαβ = vαEβ − vβEα − ηαβ vρBσ, The results above can easily be generalized to include ρσ while the Faraday tensor Pαβ = ε(vαEβ − vβEα) − the case for which also the permeability parameter has (1/µ)ηαβ vρBσ describes the field excitations. We as- an arbitrary dependence upon the electric field strength ρσ sume that the permittivity parameter (cid:15) = (cid:15)(E) may as µ=µ(E). Indeed, the Fresnel matrix Eq. (6) and the be dependent upon the magnitude of the electric field two optical geometries Eqs. (7)–(8) can be shown [23] √ strength E ≡ −E Eα, while the permeability param- to hold good in this case as well, provided the magnetic α eter is momentarily being taken as the vacuum constant field B(cid:126) is zero. We will henceforth assume this condition µ=µ . to be satisfied. Let us now seek for static spherically 0 TheelectromagneticfieldstrengthsEα andBα areas- symmetric black hole analogue models to this effective sumed both to be continuous but with possibly non-zero optical metric. finite Hadamard discontinuities [20] in their derivatives at the wave-front hypersurface Σ, as III. ANALOGUE SPHERICAL BLACK HOLES [E ] =e k , (4) α,β Σ α β Let us consider a dielectric medium as above, with [B ] =b k , (5) α,β Σ α β four-velocity vα = δα, subjected to an electric field 0 directed along the radial direction and no magnetic where k = ∂ Φ = (ω,(cid:126)q) = q(v ,qˆ) is the wave vector β β ph field. For the static spherically symmetric situation we (with a phase speed v pointing along the direction of ph are dealing with, the current four-vector Jµ = (ρ,J(cid:126)) the normalized vector qˆ) orthogonal to Σ, while Σ is de- presents only its time component, the charge density ρ. scribed as Φ(xα)=0. The two space-like vectors eα and Maxwell Eqs. (2)–(3), written in flat spherical coordi- bα respectively describe the polarizations of the electric nates (t,r,θ,φ) adapted to the dielectric medium, then and magnetic components of the wave. The discontinu- reduce to ityofEqs.(2)–(3)overΣyieldsalinearlypolarizedwave √ bα = ηαβ k vρeσ/ω, with the electric polarization eα ∂ ( −γ(cid:15)E) ρσ β r √ =ρ, (10) being obtained from the eigenvalue problem Zαβeβ =0, −γ 3 wherethedeterminantofthemetricg isγ =−r4sin2θ reduce to E = ±E R/(r −R) with a quadratic charge αβ 0 as in the flat case. The effective geometry Eq. (8) reads density profile ρ = (cid:15) E R/r2 and a linear electric dis- 0 0 placement D = (cid:15) E R/r (note that ρ/D = 1/r in this (cid:18) (cid:15)+(cid:15)(cid:48)E 1 1 (cid:19) 0 0 gαβ =diag µ((cid:15)+(cid:15)(cid:48)E), − , − , − . case). Therefore, E diverges at the horizon but remains (cid:15) r2 r2sin2θ finite everywhere else, while both D and ρ are finite at (11) the horizon but they both diverge at the center (except Thisformallowsonetoseekforanaloguesphericallysym- for A−1∼r−2, which gives ρ=0 everywhere). The in- metric static black hole solutions ner solution should then be regularized near the center. (cid:18) (cid:19) Our definite proposal to a spherically symmetric black 1 1 1 gαβ =diag , −A, − , − , (12) hole analogue with radius R is thus built with a medium A r2 r2sin2θ whose permittivity is such that where A = A(r) is a given radial function such that  E0 , if r >R, A(r)=1−R/rdescribesaSchwarzschildblackholewith (cid:15)  E+E0 = E0 , if 1R<r <R, (19) a horizon at the Schwarzschild radius R. Equation (12) (cid:15)0 1E,−E0 if r2< 1R. includesallsphericallysymmetricblackholesinaunified 2 form,withthewell-knownsolutionsofEinsteinequations Whenexpressedintermsoftheradialcoordinater,then (suchasSchwarzschild,Reissner-Nordström,de-Sitteror Eq. (19) gives (cid:15)/(cid:15) = |A|, where −1 < A < 1. We 0 combinationsthereof)beingcharacterizedbytheexplicit can compare this result with previous similar proposals form of the radial dependence of the function A. The for the radial behavior of a nonlinear dielectric medium identificationofEqs.(11)and(12)givesthetwopossible at rest: for example, [16] which rely upon postulating a solutions core absorption coefficient; here, no doping is required, (cid:112) but only a variable volumetric density of the medium. (cid:15)+(cid:15)(cid:48)E =± (cid:15)/µ. (13) Moreover, as already noted [24], that was not a consis- tent solution of Einstein field equations; this latter in- Inordertointegratethisequation,atwo-parameterfunc- stead deals with the cylindric case, and proposed a non- tion µ = µ((cid:15),E) can arbitrarily be chosen. Among diagonal structure for both (cid:15) and µ, while we treat these all possibilities, we restrict ourselves here (other possi- two parameters both as scalars. ble choices are left to the concluding discussions) to the mathematically convenient form 3.0 µ= (cid:15)20, (14) 2.5 E(cid:144)E0 E(cid:144)E0 (cid:15)3 wµh(cid:15)ere=(cid:15)01).isEtqhueatviaocnuu(1m3)ptehremnitrteiavditsy((cid:15)cEon)(cid:48)st=ant±(cid:15)(2w/i(cid:15)th, 2.0 m I m II m 0 0 0 u u u whose integration immediately yields (cid:15)=(cid:15)±, where 1.5 ucaV ideM ideM (cid:15) = (cid:15)0 , (15) 1.0 ± E ±1 Ε(cid:144)Ε E0 0.5 0 Ε(cid:144)Ε and A(r) = ±(cid:15) /(cid:15) = 1/(1±E/E ), where E > 0 is 0 ± 0 0 0 a constant of integration; for (cid:15) = (cid:15) one has E = E at 0.0 + 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 A = 1/2 (i.e., at r = 2R for the Schwarzschild model), r while the solution (cid:15) = (cid:15) is limited to E > E (since − 0 R E < E would correspond to A > 1 in this case). The 0 usual range −∞ < A < 1 of an effective black hole can FIG.1: (coloronline). Thebehaviorofthedielectriccoefficient thusbeobtainedbyjoiningthesolution(cid:15)−insidehorizon (cid:15)andtheelectricfieldEareshownintermsoftheradialdistance with the solution (cid:15)+ outside horizon. r. As stated by Eq. (19), three distinct media are considered in Theexpressions oftheelectric fieldE, theelectricdis- thismodel: vacuumfor0<r<0.5R,mediumIfor0.5<r<R placement D =(cid:15)E, and the charge density ρ in terms of and medium II for r>R. Notice that (cid:15) is a C0 function at the A then give horizon, while the electric field diverges at this point. However, theobservablefieldD isaregularfunctionthroughr=R. The E (1−A) dotted vertical line depicts the analogue event horizon. =± , (16) E A 0 D =(cid:15) E (1−A), (17) 0 0 (cid:20) (cid:21) 2(1−A) dA ρ=(cid:15)0E0 r − dr , (18) IV. KINEMATICS which hold for either 0 < A < 1 or A < 0. In the case The usefulness of the optical geometry Eq. (12) is to ofaSchwarzschildanalogueblackhole,theseexpressions express the dispersion relation as gαβk k =0. Written α β 4 in terms of the phase speed v = ω/q of the wave, we scale E is given by E = Q/6π(cid:15) R2 ≈ 6.0 · 105V/m. ph 0 0 0 have k =ωv +q =q(v v +qˆ ), and then Thechargedensityinthisregionofradialdistancesthus λ λ λ ph λ λ ranges between ρ =(cid:15) E /4R≈1.33·10−5C/m3 and min 0 0 v2 =A[1+(A−1)(ˆl·qˆ)2]. (20) ρ = 4(cid:15) E /R ≈ 2.12·10−4C/m3. For the sake of ph max 0 0 comparison, the above values are similar to the ones we The phase velocity (cid:126)v = v qˆ and the group velocity can find in usual electronic capacitors with capacitance ph ph (cid:126)v =dω/d(cid:126)q are rangingaround(1−50)µF andprovidinganelectrostatic gr potentialof1V. Fromtheregularityofthephysicalprop- (cid:126)v =|A|1/2|1+(A−1)(ˆl·qˆ)2|1/2qˆ, (21) erties mentioned above, we see that the proposed dielec- ph (cid:115) tric media can reproduce in laboratory all the classical |A| (cid:126)v = [qˆ+(A−1)(ˆl·qˆ)ˆl]. behavior of optical null geodesics near a compact object gr |1+(A−1)(ˆl·qˆ)2| with a gravitational Schwarzschild radius equal to R. (22) These two velocities coincide for each of the following V. CONCLUSION three situations: ˆl·qˆ= 1, ˆl·qˆ= 0, or ˆl·qˆ= −1. For other directions qˆ of the phase velocity with respect to Suppose a static spherically symmetric dielectric the radial direction ˆl of the background electric field E(cid:126), medium with electric charge density ρ given by Eq. (18), the two velocities (cid:126)vph and (cid:126)vgr given by Eqs. (21)–(22) while its dielectric parameters (cid:15) and µ are real quanti- lie along lines not parallel to one another. Let ψ be the ties (that is, with no ‘by-hand’ absorption) which be- angle between (cid:126)vgr and (cid:126)vph. It then follows from Eqs. have non-linearly according with Eqs. (14) and (19). (21)–(22) that Maxwell equations then yield no magnetic field and a radial electric field given by Eq. (16). With such electro- 1−cosψ (ˆl·qˆ)2 = , (23) magnetic background fields, small electromagnetic field 1−A disturbances propagate as two linearly polarized wave modes. Theextraordinaryoneamongstthesetwomodes whichimpliescosψ ≥A. Inparticular,cosψ >0outside behavesexactlyasanullwaveoftheblackholeanalogue the effective black hole (i.e., for the region 0<A<1). geometry Eq. (12). The ordinary mode propagates dif- It should be remarked that both the phase and group ferently,butinsuchawayastoavoidpropagationacross velocities of this extraordinary mode have a zero limit the same analogue event horizon. Thus, no electromag- when approaching the regular distinguished closed sur- netic field disturbance can propagate from the inner re- faceA=0,lackingpropagationofthiswavemodeacross gionoftheanalogueblackholetotheregionoutsidefrom that surface. it. For the ordinary mode, and taking into account the As well known, quantization of fields on the classi- solution Eq. (15) for (cid:15), we obtain cal Schwarzschild spacetime (semiclassical gravity) leads to the concept of Hawking radiation phenomenon. In v(+) =|A|=v(+), (24) ph gr terms of the surface gravity κ, the Planckian spectrum associated with such radiation defines the temperature where A is the same radial function considered above in T = (cid:126)cκ/(2πk ). For the case of astrophysical candi- Eq. (12). Equation (24) states that the ordinary mode H B datesofblack-holesthemagnitudeofthistemperatureis k(+) also cannot propagate across the surface A = 0. λ toosmalltobeexperimentallyprobed. Analoguemodels Therefore,thissurfaceplaystheroleoftheeffectiveevent of the Schwarzschild solution could thus provide a useful horizon of the above considered optical analogue model arena to investigate Hawking radiation. For instance, if of a black hole. we naively accept that Hawking radiation is produced in ThearbitraryconstantE canbeeliminatedfromEqs. 0 a system described by the model examined in the last (17)–(18), thus yielding dA/dr = (1−A)(2/r −ρ/D), section, we would obtain T ≈ 1.822·10−4[K ·m]/R, H fromwhichthestandard definition ofthe surfacegravity which lies in the accessible range scale of conventional parameter κ=(c2/2)lim dA/dr reads r→R thermometers for a material sample with a few centime- (cid:16) ρ (cid:17) ters in size. However, a proper demonstration of the κ= lim . (25) existence of this phenomenon requires the quantization r→R 2D of the electromagnetic field in the nonlinear media de- Particularization of this result to the Schwarzschild scribed by the dielectric coefficients here proposed. This model gives κ=[ρ/(2D)]| =1/(2R). is an issue that deserves a careful analysis. r=R In order to have some estimates, suppose a dielectric As an apparently simpler alternative to Eq. (14), the probewithanopticalSchwarzschildradiusofR=10cm. choice µ = µ could also be worked out as well, yield- 0 The total amount of electric charge in the volume delim- ing E = E A2/(1−A)2. For such a case, however, the 0 ited by the radius R/2 and 2R of the dielectric would background electric field would vanish at horizon A=0. be of the order Q ∼ 10−6C, while the electric field Thus,thesmallmagnitudeoftheelectricfieldofthewave 5 could not be neglected in Eq. (8) when compared to the room for the optical equivalence beyond the geometrical backgroundelectricfield. ThismeansthatEq.(8)would limit. Other alternative choices for µ((cid:15),E) may possibly not be an effective geometry very near the horizon, but prove useful as well to help understanding better optical it would instead depend as well on the field of the prop- analogue models. agating wave there. That is to say, the analogue event horizon would become rather blurred and undefined in such simple case. Other choices of the form µ = (cid:15)n/(cid:15)n+1 with arbitrary Acknowledgements 0 constant n (cid:54)= 0 yield qualitatively similar results as the ones presented in the preceding sections for n > 0, or E. B. would like to thank Universidade Federal de those mentioned in the previous paragraph for n < 0; Itajubá for the hospitality. This work was partially suchaclassincludestheproportionalitycondition[7]be- supported by CNPq, CAPES (BEX 18011/12-8 and tween (cid:15) and µ claimed to be required in order to give 13956/13-2), FINEP and FAPERJ. [1] W. Gordon, Ann. Phys. (Leipzig) 72, 421 (1923). 1367 (2004). [2] J. Plebanski, Phys. Rev. 118, 1396 (1960). [16] E. E. Narimanov and A. V. Kildishev, Appl. Phys. Lett. [3] L. D. Landau and E. M. Lifshitz, The classical theory of 95, 041106 (2009). fields (Pergamon Press Ltd., Oxford, 1971), p 256. [17] C.Sheng,H.Liu,Y.Wang,S.N.Zhou,andD.A.Genov, [4] F. de Felice, Gen. Rel. Grav. 2, 347 (1971). Nature Photonics 7, 902-906 (2013). [5] M.Schenberg,Revista Brasileira de Fisica,1,91(1971). [18] M. Yin, X. Y. Tian, L. L. Wu, and D. C. Li, Opt. Exp. [6] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). 21 19082 (2013). [7] C. Barcelo´, S. Liberati, and M. Visser, Living Rev. Rel. [19] V.A.DeLorenciandR.Klippert,Phys. Lett. A357,61 8, 12 (2005); ibidem 14, 3 (2011). (2006). [8] V. A. Belinski, Phys. Lett. A 209, 13 (1995). [20] J.Hadamard,Leçons sur la propagation des ondes et les [9] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975). équations de hydrodynamique (Hermann, Paris, 1903); [10] F. Belgiorno et al., Phys. Rev. Lett. 105, 203901 (2010). V. D. Zakharov, Gravitational waves in Einstein’s the- [11] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. ory (John Wiley & Sons, Inc., New York, 1973). G. Unruh, and G. A. Lawrence, Phys. Rev. Lett. 106, [21] F.W.HehlandY.N.Obukhov,Foundations of classical 021302 (2011). electrodynamics: charge, flux, and metric, Progress in [12] M.NovelloandE.Bittencourt,Phys. Rev. D86,124024 MathematicalPhysics,v.33(Birkhäuser,Boston,2003). (2012). [22] M. Novello and J. M. Salim Phys. Rev. D 63, 083511 [13] M.Novello,S.E.Perez-Bergliaffa,J.M.Salim,V.A.De (2001). Lorenci,andR.Klippert,Class.QuantumGrav. 20,859 [23] V. A. De Lorenci and G. P. Goulart, Phys. Rev. D 78, (2003). 045015 (2008). [14] V. A. De Lorenci, R. Klippert, and Yu. N. Obukhov, [24] H.-Y. Chen, R.-X. Miao, and M. Li, Opt. Express 18, Phys. Rev. D 68, 061508R (2003). 15183-15188 (2010). [15] V. A. De Lorenci and R. Klippert, Braz. J. Phys. 34,