Cosmology in the Laboratory: an Analogy Between Hyperbolic Metamaterials and the Milne Universe

Cosmology in the laboratory: an analogy between hyperbolic metamaterials and the Milne universe

David Figueiredo Departamento de F´ısica, CCEN, Universidade Federal da Para´ıba, Caixa Postal 5008, 58051-900, Jo˜aoPessoa, Para´ıba, Brazil

Fernando Moraes∗ Departamento de F´ısica, Universidade Federal Rural de Pernambuco, 52171-900, Recife, Pernambuco, Brazil

SébastienFumeron and Bertrand Berche Statistical Physics Group, Laboratoire de Physique et Chimie Théoriques, Universitéde Lorraine, B.P. 70239, 54506 Vandœuvre les Nancy, France (Dated: November 21, 2017) This article shows that the compactified Milne universe geometry, a toy model for the big crunch/big bang transition, can be realized in hyperbolic metamaterials, a new class of nanoengi- neered systems which have recently found its way as an experimental playground for cosmological ideas. On one side, Klein-Gordon particles, as well as tachyons, are used as probes of the Milne geometry. On the other side, the propagation of light in two versions of a liquid crystal-based metamaterial provides the analogy. It is shown that ray and wave optics in the metamaterial mimic, respectively, the classical trajectories and wave function propagation, of the Milne probes, leading to the exciting perspective of realizing experimental tests of particle tunneling through the cosmic singularity, for instance.

I. INTRODUCTION in the entire four-dimensional spacetime [2]. The question we want to address in this work is: can we simulate Initial conditions are always a trouble in cosmology but the passage through the singularity in the laboratory? can be circumvented by cyclic universe models, like an Recent advances in the field of metamaterials suggest endless repetition of big crunches followed by big bangs, this possibility. Arising from a pioneering idea by Vese- for instance. This is in fact an old idea that can be traced lago [5] and developed by Pendry [6], metamaterials are back to ancient mythologies. Even without referring to artificial media structured at subwavelength scales, such initial conditions some issues remain in these models, no- that their permittivity and permeability values can be tably the passage through the singularity, the transition taylored quasiarbitrarily (for instance, they may exhibit from big crunch to big bang. Recently, safe transition negative refractive index). Because analogy is a power- has been proposed [1, 2], where the singularity is noth- ful tool that a physicist possesses to arrive at an under- ing more than the temporary collapse of a fifth dimen- standing of the properties of nature, along with the fact sion. The three space dimensions remain large and time that visualizations of celestial object features in the lab- keeps flowing smoothly. A toy model for the geometry oratory have been a charming subject for human beings of this transition is the compactified 2D Milne universe over centuries, analogue gravity became an active field in arXiv:1706.05470v2 [gr-qc] 19 Nov 2017 [3] which is essentially a double cone in 3D Minkowski physics with the help of metamaterials. Therefore, sev- spacetime. Because there is still a singularity in one spa- eral works based on different results of general relativity tial dimension, a physically correct model should be able were done, including topics like analog spacetimes [7, 8], to describe the propagation of a particle through it. An time travel [9], cosmic strings [10], celestial mechanics example of such a model [4] revealed that the compacti- [11], black holes [12, 13] and wormholes [14], just to cite fied Milne universe seems to model the cosmic singularity a few. However, most of the approaches aforementioned in a satisfactory way. It is important to stress that the rely on transformation optics [15], where the permittiv- collapse here is far less severe than in ordinary 4D general ity ij and permeability µij tensors must be numerically relativity, because it happens just in one spatial dimen- equal (impedance matched). It turns out that sometimes sion, whereas in the ordinary case the collapse happens it is difficult to obtain such a constraint experimentally. Nevertheless, a particularly promising class of such media is that of hyperbolic metamaterials, for which one of the ∗ Also at Departamento de F´ısica, Universidade Federal da eigenvalues of either the permittivity or the permeability Para´ıba, Caixa Postal 5008, 58051-900, JoãoPessoa, Para´ıba, tensors do not share the same sign with the two others Brazil (thus, they do not need to be impedance matched). Hy- 2 perbolic metamaterials are now extensively studied, both for practical purpose (enhanced spontaneous emission, hyperlenses [16–18]) but also for modeling cosmological phenomena (metric signature transition [19], modeling of time [20, 21], and even inflation [22]). Therefore, since inflationary models of the universe can be mimicked through hyperbolic metamaterials, it can be interesting to see if the cyclic/ekpyrotic ones also can. As far as light propagation is concerned, a (2+1) Minkowski spacetime can be simulated with such materials. As will be seen below, the quantum dynamics of Klein-Gordon particles through the big crunch/big bang transition may be experimentally verified with light propagating in a suitable hyperbolic metamaterial. FIG. 1. Plane sector which can be bent around a cone. The deficit angle δ = 2π(1 − sin θ) reduces the range of the polar angle ϑ to [0, 2π sin θ). II. THE MC UNIVERSE

In this section we will summarize the definition and one. Making these changes in Eq. (6), we obtain the 2D features of the Milne universe and the compactified Milne induced line element for the double cone universe, MC . However, first it is useful to analyze the geometry of a rectangular cone in R3 to get some insight. ds2 = dl2 + l2 sin2 θdφ2. (8) Therefore, let θ be the angle between the generatrix and the axis. In Cartesian coordinates (x, y, z) the cone sur- Concerning the cosmological model, consider the usual face will have the following form Robertson-Walker line element with negative spatial curvature [24] x2 + y2 = z2 tan2 θ, (1) ds2 = −dt2 + a(t)2 dχ2 + sinh2 χ dΩ2 , (9) with parametric equations where a(t) is the expansion factor of the universe and x = r sin θ cos (ϑ/ sin θ) , (2) dΩ2 = dθ2 + sin2 θdφ2 the solid angle. For a linear ex- y = r sin θ sin (ϑ/ sin θ) , (3) pansion factor a(t) = t, we obtain the Milne universe metric z = r cos θ, (4) ds2 = −dt2 + t2 dχ2 + sinh2 χ dΩ2 , (10) and position vector which was proposed by Edward Arthur Milne in 1933 r = [r sin θ cos (ϑ/ sin θ) , r sin θ sin (ϑ/ sin θ) , r cos θ] , and represents a homogeneous, isotropic and expanding (5) model for the universe [25] which grows faster that simple due to the fact that it is locally isometric to a piece of the cold matter dominated or radiation dominated universes. plane [23]. In the parametrization above 0 ≤ ϑ < 2π sin θ We are interested in the hypersurfaces where dΩ = 0. is the polar angle and 2π(1 − sin θ) is the deficit angle Then, Eq. (10) reduces to (see Fig. 1). Rescaling the polar angle as φ = ϑ/ sin θ one recovers the parametrization in terms of spherical ds2 = −dt2 + t2dχ2. (11) coordinates (r, θ, φ). Thus, the line element at the cone surface is, in terms of φ, Next, we make a coordinate transformation to new variables (T,X) given by ds2 = dr2 + r2 sin2 θdφ2. (6) T = t cosh χ, (12) Then, because θ = constant we have an induced 2D met- X = t sinh χ, (13) ric on the cone with the following metric tensor which leads to the following form for the line element in 1 0 g = , (7) Eq. (11), namely ij 0 r2 sin2 θ ds2 = −dT 2 + dX2, (14) i 1 2 where gij = ei · ej (ei = ∂r/∂x , x = r, x = φ). To extend the above case for the double cone, we mod- which is the usual Minkowskian metric in two dimen- ify the radial coordinate r making r → l, so that l ∈ R. sions. However, one must stress that the Milne universe Thus, l > 0 for the upper cone and l < 0 for the lower only covers half of the Minkowski spacetime. To see why, 3 consider the lines of constant values χ = χ0 in Eqs. (12) Taking into account all previous parametrizations, the and (13), parametric equations (17)–(19) become

X = T tanh χ0. (15) x = t sinh θ cos φ, (21) y = t sinh θ sin φ, (22) As a result, we have straight lines in a Minkowskian di- z = t cosh θ, (23) agram. Taking the limits χ0 → ±∞ in the equation above, one gets the light rays X = ±T , which form the √ where sinh θ = κ, cosh θ = 1 + κ2 and φ ∈ S1. As for boundaries of the past and future light cones from the the metric in M , origin (0, 0). Thus, one is confined in this region where C X2 −T 2 < 0, since the slope of the line given by Eq. (15) ds2 = −dt2 + κ2t2dφ2. (24) goes from 0 (for χ0 = 0) to ±1 (for χ0 → ±∞). Another important point to stress is that the three- The presence of κ2 in the above metric indicates a conic dimensional space for the comoving Milne observers has singularity of the curvature at the origin. As we will see infinite extension. The reason is due to the fact that the below, this has important implications to the geodesics lines of constant t = t0 are hyperbolas, namely and wave functions of particles approaching the singularity, which acts as a filter for classical particles and a 2 2 2 T − X = t0, (16) phase eraser for quantum ones. Due to the fact that the ordinary cone surface is embedded in three-dimensional and each one of the hyperbolas has infinite length. This Euclidean space and the MC universe has a cone surface fact is expected since the Milne universe has a negative embedded in three-dimensional Minkowski space, they spatial curvature [25]. share some similarities. For instance, the Milne metric Therefore, in order to compactify the Milne universe, tensor we follow the usual approach [1–3, 26] and let the vari- −1 0 able χ acquire some period Π. The meaning of this is gij = 2 2 . (25) as follows, in the Minkowski diagram (T,X) the lines 0 κ t X = 0 and X = T tanh Π should be identified as one, for is similar to the two-dimensional induced metric tensor instance. Therefore, because one is constrained between given by Eq. (7). Also, as a last comparison, let us con- these two lines, the Milne universe now has a finite length sider the Laplace-Beltrami operator, which will be used and is called compactified Milne universe, MC (see Fig. later in this paper. Thus, 2). Following Refs. [4, 27], one can visualize the MC i 1 p ij ∆ = ∇i∇ = ∂i |g|g ∂j . (26) universe through a mapping into a three-dimensional p|g| Minkowski space with ds2 = −dz2 + dx2 + dy2, with ij where g is the inverse metric tensor and g = det(gij). x = tκ cos (χ/κ) , (17) From Eqs. (8) and (24), we will have y = tκ sin (χ/κ) , (18) 1 ∂ ∂ 1 ∂2 p 2 ∆Cone = l + , (27) z = t 1 + κ , (19) l ∂l ∂l l2sin2θ ∂φ2 1 ∂ ∂ 1 ∂2 where t ∈ 1 and 0 < κ ∈ 1 is a constant parameter R R ∆MC = − t + 2 2 2 , (28) for compactifications (in Refs. [27, 28] it is shown that t ∂t ∂t κ t ∂φ −1 κ is related to the rapidity tanh v = 2πκ of a finite which also look similar due to the resemblances between Lorentz boost). Thus, without loss of generality, we take the two metrics, aside from the timelike behavior in t and the period of χ as 2πκ. Moreover, as in the previous case the hyperbolic cone angle factor κ2 = sinh2 θ. of the ordinary cone, we rescale χ by φ = χ/κ (and hence giving a period of 2π for φ). Solving Eqs. (17)–(19) one gets the following constraint equation III. CLASSICAL PARTICLE IN MC κ2 x2 + y2 = z2. (20) In this section we will obtain the path followed by free 1 + κ2 classical particles (timelike geodesics) in MC . Thus, we start with the relativistic action (in natural units c = 1) This equation is similar to Eq. (1) because the space is given by [29] Euclidean for the planes z = constant and also due to the periodicity of φ. As x, y, z can assume both positive and Z q i j negative values, Eq. (20) represents a double cone with S = −m −gijx˙ x˙ dλ, (29) a vertex at (0, 0, 0) in the 3D Minkowski space (see Fig. 3). However, due to the timelike aspect of z, the cone where the dot notation stands for a derivative with re- angle θ is a hyperbolic angle with tanh2 θ = κ2/1 + κ2. spect to an affine parameter λ along the curve (which can 4

which for the Milne metric (24) gives the following equations X = ± T X = T tanh 2 φ¨ + t˙φ˙ = 0, (34) t t¨+ κ2t φ˙2 = 0. (35) From the equations above, one can see that the nonva- T φ φ nishing Christoffel symbols are Γtφ = Γφt = 1/t and t 2 Γφφ = κ t. Furthermore, since φ is a cyclic coordinate in the Lagrangian, ∂Lkin/∂φ = 0 and the angular momen- ˙ 2 2 ˙ tum pφ = ∂Lkin/∂φ = κ t φ is conserved. This fact is expressed in Eq. (34). 2 X Substituting pφ = κ J = constant in Eq. (35) and after some manipulations, the second geodesic equation becomes FIG. 2. Minkowski diagram showing the M universe (gray C κ2J 2 region) delimited by the identification between the T -axis and t˙2 − = 2E, (36) the blue line X = T tanh Π. The orange lines are showing t2 the light rays X = ±T while the hyperbolas are surfaces for where E is a constant of integration. The left-hand side of constant t in Eq.(16). Eq. (36) is nothing else but minus the square modulus of the velocity ui =x ˙ i. As a result, recalling that in special 8 i relativity one has u ui = −1 and λ is an affine parameter, i 4 it is useful to choose u ui = −2E = −1. Then, Eq. (36) becomes z 0 κ2J 2 t˙2 − = 1. (37) -4 t2 -8 Solving Eq. (37) for t˙, we get -8 8 r -4 4 t2 + κ2J 2 0 ˙ 0 t = ± 2 , (38) y 4 -4 t 8 -8 x which leads to a simple integration of the form FIG. 3. Double cone surface corresponding to Eqs. (1) and Z tdt Z (20) for 3D Euclidean space and 3D Minkowski space, respec- √ = ± dλ, (39) 2 2 2 tively. The upper half part corresponds to l, t > 0 while the t + κ J lower one is for values of l, t < 0. and therefore to the parametric equation t = t(λ) given by i i p i j be the proper time) and L(x , x˙ , λ) = −m −gijx˙ x˙ is p 2 2 2 the Lagrangian. The variation of the action, δS, will be t(λ) = ± ∆λ − κ J , (40)

Z q where ∆λ = λ − λ0, with λ0 being a constant of integra- i j δS = −m δ −gijx˙ x˙ dλ, (30) tion. In order to obtain the parametric equation φ = φ(λ), thus, the least action principle δS = 0 demands that we substitute Eq. (40) in the relation t2φ˙ = J and solve Z q for φ˙. Namely, i j δ −gijx˙ x˙ dλ = 0, (31) J φ˙ = , (41) from which the geodesic equations follow ∆λ2 − κ2J 2 i i j k x¨ + Γjkx˙ x˙ = 0, (32) leading to the following integration i Z Z dλ where Γjk are the Christoffel symbols of the second kind. dφ = J . (42) An alternative, and more direct route to get the ∆λ2 − κ2J 2 geodesic equations and Christoffel coefficients is through As a result, the parametric equation φ = φ(λ) is the Euler-Lagrange equations for the kinetic Lagrangian 1 1 −1 ∆λ L = g x˙ ix˙ j, (33) φ(λ) = φ0 − coth , (43) kin 2 ij κ κJ 5 where φ0 is a constant. Combining Eqs. (40) and (43), the same reasoning applies for particles coming from the one gets the equation of the trajectory orange curve. For the purpose of completeness, we perform the same t t = ± 0 , (44) calculations for spacelike geodesics, which can be world- |sinh κ∆φ| lines of tachyons [32]. The procedure is the same, the major change being the choice of the constant of inte- where t0 = κ |J| and ∆φ = φ−φ0. The above equation is a Poinsot spiral [30], with the + (−) sign corresponding gration in Eq. (36). Therefore, because the momen- to the upper (lower) cone. Furthermore, for J 6= 0 one tum and velocity must be spacelike, it is useful to choose i can see that depending on the choice of the signal (or u ui = −2E = 1. Thus, Eq. (36) becomes sheet of the cone), the particle always remains in the up- κ2J 2 per or lower region with no link between those regions of t˙2 − = −1, (46) spacetime (see Fig. 4), but for J = 0, which corresponds t2 to φ = φ0 and t = ± ∆λ, the geodesics are straight lines. the rest of the procedure being rather straightforward On the other hand, from Eqs. (12) and (13) one can and leading to the following parametric equations see that all the trajectories given by Eq. (44) are straight lines in Minkowski spacetime, p t(λ) = ± −∆λ2 + κ2J 2, (47) t0 1 ∆λ X = ± + T tanh κφ0, (45) φ(λ) = φ + tanh−1 , (48) cosh κφ0 0 κ κJ where for J = 0 one recovers Eq. (15). As a result, the where ∆λ = λ − λ , with λ, λ and φ denoting an affine particle indeed can travel from one cone to the other, but 0 0 0 parameter and constants of integration, respectively. As such trajectories are very unstable since small perturba- for the trajectory t = t(φ), tions in the value of J = 0 cause large deviations on the trajectories. A similar result was found for a classical t0 particle in a double cone in Ref. [31], where a classical t = ± , (49) cosh κ∆φ nonrelativistic particle constrained to a double cone only crosses the vertex through straight lines. where t0 = κ |J| and ∆φ = φ−φ0. Equation (49) is also a However, we are dealing with a toy model for a cyclic Poinsot spiral, with the + (−) sign corresponding to the universe with contraction and expansion phases joined upper (lower) cone. Furthermore, as in the previous case by a cosmic singularity. Thus, as pointed out in Refs. the trajectories are straight lines in Minkowski spacetime, [4, 27], due to the fact that timelike geodesics coincide with the trajectories of test particles, which do not dis- t0 X = ∓ + T coth κφ0, (50) tort the spacetime around them, there is no obstacle for sinh κφ0 such geodesics to reach (leave) the singularity. Further- more, if one postulates that such a particle arriving at the where the − (+) corresponds to the upper (lower) sheet. singularity coming from the lower cone is “annihilated” From Eq. (49) the variable t has t0 as its limiting value. at the singularity, while another one is “created” at the To clarify this result, we remark that it was pointed out upper cone, and considering that the Cauchy problem is by Feinberg [32] that tachyons lose energy√ as they speed i not well defined at t = 0, there are several types of prop- up. Thus, from Eq. (49) the velocity dx dxi/dt (with agation depending on the way a particle travels towards i = 1, 2, 3), the singularity (see Sec. III of Ref. [27]). Although all of them must be consistent both with the constraint given dφ t0 κt = (51) p 2 2 by Eq. (37) and with the fact that the angular momen- dt t0 − t tum J is constant. It is out of the scope of this work to discuss the details and properties of such propagations. ranges from (1, ∞) as t ranges between 0 and t0, respec- Next we present the timelike geodesics in a polar plot, tively (note that for a lightlike interval, κt dφ/dt = 1). which provides a clearer way to visualize the trajectories Therefore, as the tachyon comes accelerating from the (see Fig. 5). As already shown in Fig. 4, the trajecto- singularity, we can see from Eq. (46) that the time com- ries in the lower cone are going towards the cosmic sin- ponent of the momentum p0 = t˙ (which is associated gularity, which means that particles in the red (orange) with the energy) decreases, reaching its minimum value curve have positive (negative) angular momentum J and at t0. From that point on it starts to increase as the therefore are spinning in the counterclockwise (clockwise) tachyon decelerates towards the singularity (see Fig. 6). direction. For the upper cone, particles traveling along This leads to an interpretation of tachyons being created the green (blue) curve have positive (negative) values of and annihilated at the same sheet of the cone, which can J and are spinning counterclockwise (clockwise). As a be the upper sheet or the lower one. Every annihilation result, in a transition from the red to the green curve on a sheet creates a tachyon on the other sheet, in an the angular momentum is conserved, while for a transi- endless cycle. This can be seen in a polar plot as closed tion to the blue one, J would not be conserved. Clearly, curves in spacetime (see Fig. 7). 6

3 t + t + t 2 t 1 y 0 0 arrow of time t / 1

3 3 2 1 0 1 2 3 x

FIG. 4. Graph for the geodesic equation given by Eq. (44) FIG. 5. Timelike geodesics (Poinsot spirals) with the radial with t in units of t0. The blue and green (orange and red) time t given in units of t0 and κ = 1/3, corresponding to the lines correspond to trajectories in the upper (lower) cone. The curves in the previous figure. The blue and green (orange and arrow of time pointing up shows that time always increases, red) lines are moving away (towards) from (to) the singularity. which means that in the first and third quadrants the angle φ Furthermore, particles following the trajectories in the first decreases with time (J < 0), while in the second and fourth (third) and second (fourth) quadrants are spinning clockwise ones it increases with time (J > 0). (counterclockwise). The blank point at the origin is just to emphasize that the curves do not reach the singularity.

The purpose of this section was to calculate the classical trajectories for both ordinary particles as well as t + t tachyons through the least action principle, showing the transitions which may occur between the two cones. Even though we were dealing with classical particles, it happens as if the particles of either kind go through a process 0 of annihilation/creation in order to cross the singularity. arrow of time In the next section we propose a geometric optics model t / that emulates the trajectories of the particles in MC in a hyperbolic metamaterial.

IV. A METAMATERIAL MODEL FOR THE MC UNIVERSE FIG. 6. Graph for the geodesic equation given by Eq. (49) In order to simulate Milne particles in a condensed with t in units of t0, with the blue curve for the upper cone matter system, we study the propagation of light in a hy- and the orange curve for the lower one. A tachyon coming perbolic liquid crystal metamaterial, HLCM, presented from the singularity (from the left of the graph) with J > 0 accelerates until the end of its time t = t0 and then decelerates in Ref. [33]. Our study on light propagation lies in towards the singularity being ultimately annihilated. This the realm of geometrical (or ray) optics, which essen- creates a tachyon on the other sheet which goes through the tially involves the application of Fermat’s principle along same process. with the variational principle that determines the path followed by light (geodesics). Therefore we seek an ex- tremum of the integral called “optical path.”1 Because we are dealing with an anisotropic medium, there are two distinct polarizations Z B S = Nedl, (52) A 1 It was shown in Ref. [34] that the refractive index is dependent where dl is the element of arclength along the path be- of the observer. Here we consider “static observers,” which are tween points A and B, Ne is the eﬀective refractive index at rest with respect to the space coordinates with the time com- ponent of the velocity ut tangent to the coordinate time axis. of the material and the product Nedl between them is 7

y 0

-1

-1 0 1 x

FIG. 7. Spacelike geodesics (Poinsot spirals) with radial time t given in units of t0 and κ = 1/3, corresponding to the curves in the previous ﬁgure. The closed curves show that tachyons spiral outward from the singularity and inward to the singularity on the same sheet of the cone.

FIG. 8. Curve described by the position vector r showing the for the light rays, namely, the ordinary and extraordi- path followed by the extraordinary light rays. The Poynting nary rays. In the former, the polarization (electric field) vector S is tangent to the curve and makes an angle of β with is perpendicular to both the director ˆn (i.e. the unit vec- the director field ˆn (optical axis). tor along the average orientation of the molecular rods constituting the nematic medium) and the wave vector k. In this case, light propagates as in an isotropic medium λ is a parameter along the curve, represents the light of refractive index no with velocity c/no. As for the ex- trajectory, then traordinary ray, the polarization lies in the plane formed by ˆn and S. Further, the direction of the Poynting vector dr S t(λ) = = (55) S differs from the direction of k, which means that the dλ ||S|| energy velocity differs from the phase velocity, leading to is the tangent vector at each position parametrized by two different refractive indexes: the ray index Nr, asso- λ. If the parameter λ is the arclength l, then from the ciated to the energy velocity, and the phase index Np, associated with the phase velocity [35]. In this work, we theory of the differential geometry of curves [37], ||t|| = 1. discuss only the extraordinary ray. Thus, fixing this choice λ = l and also using the fact that The application of Fermat’s principle to the extraor- ||ˆn|| = 1, dinary light grants us the path followed by the energy. cos β = t · ˆn. (56) Therefore, the effective refractive index Ne in Eq. (52) is the ray index Nr and is given by [36] To proceed further, one has to know the expression for 2 2 2 2 2 the director ˆn, which depends on the system in question. Nr = nocos β + nesin β, (53) In what follows, we show the form for ˆn that is suitable for our cosmological analog model. The hyperbolic where β is the angle between the director ˆn and the behavior of the metamaterial is characterized by a topo- Poynting vector S (see FIG. 8). logical defect called disclination. In our case, the liquid Fermat’s principle essentially sums up to the geodesic crystal is in a nematic phase and such disclinations are R B i j determination by requiring δ A γijdx dx = 0. There- classified according to the topological index (or strength) fore, we may think of the curved trajectories of light k which gives a measure of how much the director ˆn ro- rays in an anisotropic material as geodesics which can tates as one goes around the defect. Thus, following the be found from the identification same approach as Refs. [38–40], the director configura-

2 2 i j tions (in the xy plane) are given by Nr dl = γijdx dx , (54) ϑ(φ) = kφ + c, (57) where we introduce the γij notation to emphasize that we are dealing with a three-dimensional metric. The an- where ϑ is the angle between the molecular axis and the gle β is determined from the specific configurations of x-axis, φ is the usual azimuthal angle in cylindrical or the director field ˆn as follows. If the curve r(λ), where spherical coordinates and c is a constant parameter. In 8 the case discussed here, the disclinations are such that the system presents a translational symmetry along the z-axis. Therefore, we have effectively a two-dimensional Glass problem and the director ˆn is given in Cartesian coordinates by [38–40] Glass HLCM ˆn = (cos ϑ, sin ϑ, 0). (58)

In order to simulate Klein-Gordon (KG) particles in a Milne universe, let us consider a radial director ﬁeld (see Fig. 9)

ˆn = (cos φ, sin φ, 0), (59) where in this case k = 1 and c = 0 in Eq. (57). For FIG. 9. Director configurations for HLCM in a cylindrical convenience we will use cylindrical coordinates (ρ, φ, z), shell, with optical axis represented in cylindrical coordinates where r(l) = ρρˆ+ zzˆ and ˆn = ρˆ by Eq. (59). Therefore, (ρ, φ, z) as ˆn = ρˆ. The walls of the inner and outer cylinder are made of glass, while the liquid crystal together with the tangent vector t will be metallic nanorods (aligned in the direction of ˆn) fill the space dr in between. t = =ρ ˙ρˆ + ρφ˙φˆ +z ˙zˆ, (60) dl where the dot stands for d/dl. Then, from Eq. (56) we As pointed out before, the system in question has a have translational symmetry along the z-axis, and therefore we are interested in the extraordinary ray propagat- cos β =ρ. ˙ (61) ing in z = constant planes. Thus, dz = 0 in Eq. (67). Furthermore, the metric tensor γij describes a Next, let us consider the Euclidean line element “real” three-dimensional space, having only spatial coor- 2 2 2 2 2 dinates. However, light propagates along null geodesics dl = dρ + ρ dφ + dz , (62) in a four-dimensional spacetime. Therefore, we use the which leads to the following relation fact that the application of Fermat’s principle in a three- dimensional metric with only spatial coordinates is equiv- ρ˙2 + ρ2φ˙2 +z ˙2 = 1. (63) alent to calculating null geodesics in a four-dimensional spacetime with a pseudo-Riemannian metric gij. Thus, Therefore, combining Eqs. (61) and (63) one gets the relation between the metrics γij and gij is as follows (see p. 1108 of Ref. [24]) q 2 ˙2 2 sin β = ρ φ +z ˙ . (64) g γ = − ij . (69) ij g The ray index Nr can now be obtained with the help 00 of Eqs. (53), (61) and (64) as Taking g00 = 1, Eq. (67) becomes N 2 = n2ρ˙2 + n2(ρ2φ˙2 +z ˙2). (65) r o e 2 2 2 2 2 ds = dT − ⊥dρ − kρ dφ , (70) 2 2 2 i j Then, the line element ds = Nr dl = γijdx dx will have the following form where T is the Minkowskian time. In what follows, we explore the property of metama- 2 2 2 2 2 2 2 2 ds = nodρ + neρ dφ + nedz . (66) terials to produce negative permittivities. Due to the fact the director field is radial (ˆn = ρˆ) and the metal- Because we are dealing with metamaterials, it is more lic nanorods are aligned in the same direction, we have useful to write the refractive indexes no and ne as func- a metallic behavior along the radial coordinate and can tions of the components of the permittivity tensor ij of therefore obtain < 0. Furthermore, considering > 0 the material. Therefore, introducing the usual notation, k √ ⊥ 2 2 and rescaling the radial coordinate by r = ρ ⊥ one gets as in Ref. [41], where no = ⊥ and ne = k, Eq. (66) becomes ds2 = dT 2 − dr2 + α2r2dφ2, (71) ds2 = dρ2 + ρ2dφ2 + dz2, (67) ⊥ k k 2 where α = |k|/⊥ is the disclination parameter associ- where in this case ⊥ = φφ = zz and k = ρρ. Further- ated to a hyperbolic (imaginary) deficit angle of i × 2πα more, the permittivity tensor is given by [35, 36] [42]. Also note that the spatial part of the metric above is equivalent to the MC metric given by Eq. (24), with ˆ ˆ = kρˆ ⊗ ρˆ + ⊥φ ⊗ φ + ⊥zˆ ⊗ zˆ. (68) r behaving as the timelike variable T . 9

By the same procedure developed in the previous section, Eq. (71) gives the Lagrangian Glass T˙ 2 r˙2 α2r2φ˙2 Lkin = − + , (72) 2 2 2 Glass which leads to the following geodesic equations HLCM dT = ν, (73) dλ 2 φ¨ + r˙φ˙ = 0, (74) r r¨ + α2rφ˙2 = 0, (75) where λ is an affine parameter and ν is a constant. The FIG. 10. Director configurations for HLCM in a cylindrical first equation just states the conservation of the canonical shell for a circular field, with optical axis represented in cylin- momentum pT , while the other two are equivalent to the drical coordinates (ρ, φ, z) as ˆn = φˆ. geodesic equations for classical particles in MC given by Eqs. (34) and (35). It is possible to follow the same steps 2 to integrate the equations as we did in Sec. III. How- where α = ⊥/|k|. Therefore, the Lagrangian will be ever, as in this case we are dealing with null geodesics, g x˙ ix˙ j = 0 and therefore one gets the following useful T˙ 2 r˙2 α2r2φ˙2 ij L = + − , (80) constraint kin 2 2 2 α2L2 r˙2 − = ν2, (76) which also gives Eqs. (73)–(75) as geodesic equations. r2 However, as in Sec. III we will have a different constraint. i j with the constant L = r2φ˙ being the angular momentum. Thus, for gijx˙ x˙ = 0, The previous equation is essentially the same as Eq. (37). 2 2 2 α L 2 The geodesic equations for this case were already solved r˙ − 2 = −ν , (81) in Ref. [42] and the solution for the trajectory is the r following which is the analogue of Eq. (46). By the same procedure α r done in the previous cases, the trajectory will be r = 0 , (77) | sinh α∆φ| α r r = 0 , (82) cosh α∆φ where r0 = |L|/ν. As a result, the path followed by light rays in the three-dimensional space in the metamaterial where r0 = |L|/ν again. Therefore, we obtain the ana- are Poinsot’s spirals like the timelike geodesics in M C logue of the spacelike geodesics in MC found in Sec. III. found in Sec. III. Comparing Eqs. (44) and (77), the ra- We remark that in Ref. [17] both trajectories given dial coordinate r behaves like the time coordinate t and by Eqs. (77) and (82) were obtained through numeri- α is the analogue of the constant κ. As pointed out in cal simulations in a scattering experiment in a hyperlens Ref. [42], the hyperbolic behavior of the material gen- system. This is interesting to represent and visualize the erates a force towards the defect and the parameter α double cone geometry; since light going to the disclina- is directly related to it, with 1/α being understood as tion can be viewed as trajectories in the lower cone and the vorticity of the defect (the smaller the value of α, light coming out of the disclination as the trajectories in the stronger is the attraction towards the defect). Fur- the upper one. thermore, since the defect generates an attraction, the Having shown that the classical behavior can be emu- trajectories given by Eq. (77) can be interpreted as the lated, in the following section it will be shown that the equivalent to geodesics in M going to the cosmic sin- C quantum behavior of KG particles in MC also has an gularity (big crunch). analogy in the framework of wave optics. To simulate tachyons, we consider the director field as ˆn = φˆ (see Fig. 10), meaning that k = 1 and c = π/2 in Eq. (58). In this case, our four-dimensional line element V. MIMICKING QUANTUM PARTICLES IN will be MC 2 2 2 2 2 ds = dT − kdρ − ⊥ρ dφ , (78) In this last section, our goal is to show how to mimic where now k = φφ and ⊥ = ρρ. Thus, considering a KG particle in MC through the system presented in k < 0, ⊥ > 0 and rescaling the radial coordinate by the previous section. Therefore, let us consider the KG p r = ρ |k| one gets equation for a scalar field ϕ (in natural units c, ~ = 1) ds2 = dT 2 + dr2 − α2r2dφ2, (79) ∆ − m2 ϕ = 0, (83) 10 √ where ∆ is the Laplace-Beltrami operator as presented where we have used the fact that kr = kρ/ ⊥ due to in Sec. II. For the MC universe, ∆MC was given in Eq. the change of scale passing from ρ to r. Equation (91) is (28). Thus, consistent with Eq. (89) since, in terms of operators, it takes the following form 1 ∂ ∂ 1 ∂2 − t + − m2 ϕ = 0. (84) " # 2 2 2 Lˆ2 t ∂t ∂t κ t ∂φ kˆ2 − z − ω2 ψ = 0, (92) r α2r2 Concerning the metamaterial model, we examine the propagation of light in the scalar wave approximation, ˆ2 ˆ2 where kr and Lz are the usual operators allowing the use of the covariant d’Alembert wave equation 1 ∂ ∂ kˆ2 = − r , (93) r r ∂r ∂r i 1 √ ij ∇i∇ Φ = √ ∂i −gg ∂jΦ = 0, (85) ∂2 −g Lˆ2 = − . (94) z ∂φ2 where Φ is the wave function. Therefore, according to the metric given by Eq. (70), one gets Therefore, separating the variables ψ(r, φ) = f(r)g(φ) leads to the equations 2 2 1 1 ∂ ∂Φ 1 1 ∂ Φ ∂ Φ 2 − ρ − + = 0. (86) d g 2 2 2 2 + l g = 0, (95) ⊥ ρ ∂ρ ∂ρ k ρ ∂φ ∂T dφ2 Assuming a harmonic dependence in time, we make a and −iωT separation of variables Φ(t, ρ, φ) = e ψ(ρ, φ). Thus, d2f df l2 r2 + r + ω2r2 + f = 0. (96) 2 dr2 dr α2 1 1 ∂ ∂ψ 1 1 ∂ ψ 2 − ρ − 2 2 − ω ψ = 0, (87) ⊥ ρ ∂ρ ∂ρ k ρ ∂φ Thus, the solution for Eq. (95) is where ω is the propagation frequency of the light or ef- g(φ) = Aeilφ + Be−ilφ, (97) fective mass of an analogue KG particle. Furthermore, where A, B are constants of integration. As for Eq. (96), recalling our previous choice of ⊥ = φφ > 0 and = < 0, one gets it is a Bessel differential equation of imaginary order il/α k ρρ with solutions [43, 44] 1 1 ∂ ∂ 1 1 ∂2 ˜ ˜ 2 f(r) = CJl/α(ωr) + D Yl/α(ωr), (98) − ρ + 2 2 − ω ψ = 0. (88) ⊥ ρ ∂ρ ∂ρ |k| ρ ∂φ ˜ ˜ which C,D being constants of integration and Jl/α, Yl/α The equation above is similar to Eq. (84) and it was al- are linearly independent solutions defined in Ref. [43] as ready used to mimic KG particles in plasmonic metama- √ πl terials [20, 22]. In terms of r = ρ ⊥ and the parameter J˜ = sech Re J (ωr), (99) 2 l/α il/α α = |k|/⊥, it becomes 2α ˜ πl 1 ∂ ∂ 1 1 ∂2 Yl/α = sech Re Yil/α(ωr), (100) − r + − ω2 ψ = 0, (89) 2α r ∂r ∂r α2 r2 ∂φ2 where Re Jil/α and Re Yil/α are the real parts of Bessel where ω is treated as an effective mass (in natural units). and Neumann functions, respectively. Following the Furthermore, there is a contribution α2 from the discli- same procedure for Eq. (84), one gets the same solu- nation, which by Eq. (27) is associated to a cone angle tions for the scalar field ϕ (as obtained in Ref. [4]) with θ = sin−1 α, whereas in Eq.(84) θ = sinh−1 κ. the variable r replaced by t and the constants ω, α inter- To solve Eq. (89), we remark that the hyperbolic meta- changed by m, κ, respectively. material obeys the dispersion relation An interesting feature [43, 44] of Eqs. (99) and (100) is that they oscillate rapidly near the origin, as one can k2 k2 see from its behavior as r → 0+ ρ − φ = ω2 (90) ⊥ |k| 1/2 ˜ tanh(πl/2α) l ωr Jl/α(ωr) = cos ln − γl/α and the angular momentum conservation l = kφρ, as πl/2α α 2 shown in Ref. [18]. In terms of the angular momentum + O(ω2r2), (101) quantum number l and the radial variable r, Eq. (90) coth(πl/2α)1/2 l ωr becomes Y˜ (ωr) = sin ln − γ l/α πl/2α α 2 l/α l2 k2 − − ω2 = 0, (91) + O(ω2r2), (102) r α2r2 11 where γl/α is a constant defined as γl/α ≡ arg Γ(1+il/α), or, in the operator form with Γ being the Gamma function. The rapid oscillations " # are due to the logarithmic argument of the trigonomet- Lˆ2 kˆ2 − z + ω2 ψ = 0, (107) ric functions. Also, for a fixed l, the smaller the value r α2r2 of α, the stronger the oscillations become (reducing the 2 value of α “squeezes” the period of the trigonometric where α = ⊥/|k| = ρρ/|φφ| and ω is the analogue of functions). This analogous of the classical behavior in Im m = µ. The dispersion relation in this case takes the Sec. IV where 1/α is related to the vorticity. form Furthermore, for l 6= 0 Eqs. (99) and (100) are not con- k2 k2 tinuous through the origin, which is similar to the classi- − ρ + φ = ω2. (108) cal geodesics that in general do not cross the singularity. |k| ⊥ In Ref. [4] it was given an interesting interpretation for (−) (+) Hence, in terms of the angular momentum l, the constant this fact constructing a Hilbert space H = H ⊕ H p as a direct sum of two Hilbert spaces H(−), H(+). The α and kr = kρ/ |k|, Eq. (108) becomes elements of H(−) are solutions of Eq. (84) in the presin- l2 gularity era (t < 0) while the ones of H(+) are solutions in 2 2 kr − + ω = 0, (109) the postsingularity era (t > 0). That is, H = H− ⊕H+ is α2r2 a vector space whose elements ϕ have the following form which is consistent with Eq. (107). [45] Thus, separating the variables as ψ(r, φ) = f(r)g(φ) one gets the same solution given by Eq. (97) for g(φ). ϕ = ϕ(−), ϕ(+) ∈ H(−) × H(+), (103) As for the radial part, d2f df l2 with an inner product r2 + r − ω2r2 − f = 0, (110) dr2 dr α2 D (−) (−)E D (+) (+)E hϕ|ψi = ϕ ψ + ϕ ψ . (104) which is the modified Bessel differential equation with imaginary order il/α. The solution for this case will be Hence, vectors like (ϕ(−), 0) and (0, ϕ(+)) describe states ˜ ˜ of annihilation and creation of particles at the singularity f(r) = CIl/α(ωr) + DKl/α(ωr), (111) t = 0, respectively. By Eq. (104) an inner product be- ˜ tween those kinds of states yields zero, which means no where we kept the notation of Ref. [43], with Il/α = ˜ correlation between them. This reflects the loss of phase Re Iil/α and Kl/α = Kil/α. The functions Iil/α,Kil/α of the wave function due to the strong oscillations around are the modified Bessel functions of first and second kind, the singularity. respectively. As in the previous case, their behavior near The l = 0 case is rather straightforward; from Eqs. the origin is characterized by rapid oscillations. However, (97), (99) and (100), g(φ) is constant and the Bessel their asymptotic behavior is exponential [43, 44], functions of imaginary order reduce to the usual ones J (ωr),Y (ωr). Moreover, due to the fact that in the 1 1/2 1 0 0 I˜ (ωr) = eωr 1 + O , (112) classical case the geodesics are straight lines crossing the l/α 2πωr ωr singularity and knowing that Y0 diverges at the origin, a 1/2 ˜ π −ωr 1 more satisfactory physical solution is given by J0, as it Kl/α(ωr) = e 1 + O . (113) is continuous and well defined at the origin (singularity) 2ωr ωr r = 0 (t = 0). Therefore, since the classical trajectories shown in Figs. Next, we consider briefly the case of tachyons. There- 6 and 7 indicate the idea of bound states, a more appro- fore, as pointed in Ref. [32] tachyons can be regarded as ˜ priate physical solution is Kl/α. having an imaginary “rest” mass m = iµ, with µ ∈ R. We are conscious that the model presented here has As a result, Eq. (84) becomes limitations concerning the cosmic singularity, particularly the radius of the inner core cylinder. However, it 1 ∂ ∂ 1 ∂2 − t + + µ2 ϕ = 0. (105) could be possible to use this as an advantage with an ad- t ∂t ∂t κ2t2 ∂φ2 ditional cost of a more developed metamaterial design. To see this, suppose that for the case in Fig. 9 our new To mimic tachyons in the metamaterial we consider √ ˆ permittivities εk, ε⊥ are now functions of r = ρ ⊥ and again the case of a circular director field ˆn = φ as in given by the previous section. Therefore, substituting the metric (79) in the covariant d’Alembert wave equation (85) and δ2 −iωT ε (r) = 1 + , (114) considering Φ(t, r, φ) = e ψ(r, φ), one gets k k r2 2 2 2 α δ 1 ∂ ∂ 1 1 ∂ 2 ε (r) = 1 + , (115) − r + + ω ψ = 0, (106) ⊥ ⊥ 2 2 r ∂r ∂r α2 r2 ∂φ2 r + δ 12 where k, ⊥ are the previous values of the permittivities, lower cone and created on the upper one, in agreement 2 α = |k|/⊥ and δ is the radius of the inner core, which with the classical result of Sec. III. The same happens is a small value. Substituting Eqs. (114) and (115) in to tachyons with the difference that they are in bound the metric given by Eq. (70) leads to states, in accord with the trajectories obtained in Sec. III. α2δ2 As a last remark, we advise that there was an unfor- ds2 = dT 2 − 1 + dr2 + α2 r2 + δ2 dφ2. δ r2 + δ2 tunate error in Ref. [42], where an equation similar to (116) Eq. (89) was solved. The solution for that case also will Therefore, the wave equation becomes be in the form of ψ(r, φ) = f(r)g(φ), with f(r) and g(φ) given by Eqs. (98) and (97), respectively. r2 + δ2 ∂2ψ r[r2 + δ2(1 + 2α2)] ∂ψ − − r2 + δ2(1 + α2) ∂r2 [r2 + δ2(1 + α2)]2 ∂r 1 ∂2ψ + − ω2ψ = 0. (117) VI. CONCLUSIONS α2(r2 + δ2) ∂φ2

Separating the variables ψ(r, φ) = f(r)g(φ), one gets The possibility of doing experiments in condensed mat- again Eq. (97) for the angular solution. As for the radial ter systems that simulate cosmological cyclic/ekpyrotic part, scenarios was proposed here with the particular focus on the Milne compactified universe MC , which is the (r2 + δ2)2 d2f r(r2 + δ2)[r2 + δ2(1 + 2α2)] df simplest version of spacetime that can model the big + r2 + δ2(1 + α2) dr2 [r2 + δ2(1 + α2)]2 dr crunch/big bang transition. This is the main goal of this work. Thus, we have shown that both Klein-Gordon par- l2 + ω2(r2 + δ2) + f = 0, (118) ticles and tachyons in MC can be nicely represented by α2 ordinary light propagating in specially engineered materials known as hyperbolic metamaterials. Within the realm which is more difficult to solve comparing with Eq. (96). of geometrical optics, we pointed out that the classical However, it can be done numerically as shown in Ref. trajectories of those particles can be perfectly matched [4]. Together with Eq. (97) it represents the solution for to light ray paths in the metamaterial, while the quantum a KG particle moving in a hyperboloid embedded in a wave functions may be realized by wave optics. Further- 3D Minkowski space (with the usual change of variables more, concerning the latter case, we pointed out that is r → t, α → κ as in the previous cases). This regulariza- possible to attenuate the problem of the singularity, de- tion was suggested in Ref. [46] and, in fact, used in Ref. signing a material whose permittivity tensor has compo- [4] to remove the Cauchy problem at the singularity. The nents which are functions of the radial variable r. We re- physical motivation is that, as the particle approaches mark the exciting possibility of experimentally checking, the singularity, its own gravitational field modifies the not only the trajectories, but also the lack of correlation spacetime, slightly deforming the double cone into a one between the wave function of particles on both sides of sheeted hyperboloid (or similar surface). Thus, the extra the cosmological transition, through the analogue model space dimension does not contract to a point, but to some presented here. Finally, further theoretical results can be small value (represented above by δ). As a result, the found through the study of wave scattering, applying the propagation is uniquely defined in the entire spacetime method of partial waves [40] to the present model. This [the wave functions are continuous at the origin (singu- is presently under investigation and will be the theme of larity)] in the same manner as for the case with l = 0 a separate publication. described previously. We remark that the regularization was introduced in an ad hoc manner and therefore ex- plicit gravitational backreaction calculations in the spirit of Refs. [47, 48] are necessary to find the actual per- ACKNOWLEDGMENTS turbed geometry of the Milne cone. With the simple regularization introduced above we see that, through a more developed metamaterial design, it could be possi- D.F. and F.M. are thankful for the financial sup- ble to circumvent or at least minimize the problem of the port and warm hospitality of the group at Université singularity in the optical analogy. de Lorraine where this work was conceived and partly In this section we have seen that Klein-Gordon parti- done. This work has been partially supported by CNPq, cles are annihilated upon reaching the singularity in the CAPES and FACEPE (Brazilian agencies).

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