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

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 nanoengineered 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.

Figure 1

Plane sector which can be bent around a cone. The deficit angle $\delta =2\pi (1-\sin \theta )$ reduces the range of the polar angle $\vartheta$ to $[0,2\pi \sin \theta )$.

Figure 2

Minkowski diagram showing the ${\mathcal{M}}_C$ universe (gray region) delimited by the identification between the $T$-axis and the blue line $X=T\tanh \mathrm{\Pi }$. The orange lines are showing the light rays $X=\pm T$ while the hyperbolas are surfaces for constant $t$ in Eq. (16).

Figure 3

Double cone surface corresponding to Eqs. (1) and (20) for 3D Euclidean space and 3D Minkowski space, respectively. The upper half part corresponds to $l,t>0$ while the lower one is for values of $l,t<0$.

Figure 4

Graph for the geodesic equation given by Eq. (44) with $t$ in units of $t_0$. The blue and green (orange and red) lines correspond to trajectories in the upper (lower) cone. The arrow of time pointing up shows that time always increases, which means that in the first and third quadrants the angle $\varphi$ decreases with time ($J<0$), while in the second and fourth ones it increases with time ($J>0$).

Figure 5

Timelike geodesics (Poinsot spirals) with the radial time $t$ given in units of $t_0$ and $\kappa =1/3$, corresponding to the curves in the previous figure. The blue and green (orange and red) lines are moving away (towards) from (to) the singularity. Furthermore, particles following the trajectories in the first (third) and second (fourth) quadrants are spinning clockwise (counterclockwise). The blank point at the origin is just to emphasize that the curves do not reach the singularity.

Figure 6

Graph for the geodesic equation given by Eq. (49) with $t$ in units of $t_0$, with the blue curve for the upper cone and the orange curve for the lower one. A tachyon coming from the singularity (from the left of the graph) with $J>0$ accelerates until the end of its time $t=t_0$ and then decelerates towards the singularity being ultimately annihilated. This creates a tachyon on the other sheet which goes through the same process.

Figure 7

Spacelike geodesics (Poinsot spirals) with radial time $t$ given in units of $t_0$ and $\kappa =1/3$, 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.

Figure 8

Curve described by the position vector $\mathbf{r}$ showing the path followed by the extraordinary light rays. The Poynting vector $\mathbf{S}$ is tangent to the curve and makes an angle of $\beta$ with the director field $\widehat{\mathbf{n}}$ (optical axis).

Figure 9

Director configurations for HLCM in a cylindrical shell, with optical axis represented in cylindrical coordinates $(\rho ,\varphi ,z)$ as $\widehat{\mathbf{n}}=\widehat{\mathit{\rho }}$. The walls of the inner and outer cylinder are made of glass, while the liquid crystal together with metallic nanorods (aligned in the direction of $\widehat{\mathbf{n}}$) fill the space in between.

Figure 10

Director configurations for HLCM in a cylindrical shell for a circular field, with optical axis represented in cylindrical coordinates $(\rho ,\varphi ,z)$ as $\widehat{\mathbf{n}}=\widehat{\mathit{\varphi }}$.