# Thermally Induced Effective Spacetimes in Self-Assembled Hyperbolic Metamaterials

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{m}and ε

_{d}of cobalt and kerosene, respectively. The volume fraction of cobalt nanoparticles aligned into nanocolumns by the external magnetic field, α(H,T), depends on the temperature and the field magnitude. At very large magnetic fields, all nanoparticles are aligned into nanocolumns, so that α(∞,T) = p = 8.2%, and a 3D wire array hyperbolic metamaterial [4,5] is formed as shown schematically in Figure 1a. We will assume that, at smaller fields, the difference α(∞,T)–α(H,T) describes cobalt nanoparticles, which are not aligned and distributed homogeneously inside the ferrofluid. Dielectric polarizability of these nanoparticles may be included into ε

_{d}, leading to slight increase in its value. Using this model, the diagonal components of the ferrofluid permittivity may be calculated using the Maxwell-Garnett approximation as follows [12]:

_{2}and ε

_{1}at α(∞,T) = p = 8.2% are plotted in Figure 1b. These calculations are based on the optical properties of cobalt in the infrared range [13]. While ε

_{1}stays positive and almost constant, ε

_{2}changes sign to negative around λ = 1 µm. If the volume fraction of cobalt nanoparticles varies, this change of sign occurs at some critical value α

_{H}:

_{z}obeying the wave equation:

_{ϖ}in 3D Minkowski spacetime:

_{0}/c

^{2}), which propagate in an effective 2 + 1 dimensional Minkowski spacetime. Note that the components of the metamaterial dielectric tensor define the effective metric g

_{ik}of this spacetime: g

_{00}= −ε

_{1}and g

_{11}= g

_{22}= −ε

_{2}.

^{(υ)}of the metamaterial:

^{(2)}

_{ijl}must be equal to zero. It is clear that Equation (8) provides coupling between the matter content (photons) and the effective metric of the metamaterial spacetime. Nonlinear optical effects “bend” this effective Minkowski spacetime, resulting in gravity-like interaction of extraordinary light rays. It appears that, in the weak field limit, the nonlinear optics of hyperbolic metamaterials may indeed be formulated as an effective gravity [5]. In such a limit, the Einstein equation

_{00}is identified with −ε

_{1}, Equation (10) may be re-written as

^{(2)}is the 2D Laplacian operating in the xy plane, γ* is the effective “gravitation constant”, and σ

_{zz}is the zz component of the Maxwell stress tensor of the electromagnetic field in the medium:

_{1}are small, so that we can separate ε

_{1}into the constant background value ε

_{1}

^{(0)}and the weak nonlinear corrections. These nonlinear corrections do indeed look like the Kerr effect assuming that the extraordinary photon wave vector components are large compared to ω/c:

^{(3)}of the hyperbolic metamaterial. Since ε

_{2}< 0, the sign of χ

^{(3)}must be negative for the effective gravity to be attractive. This condition is satisfied naturally in most liquids, and, in particular, in kerosene. Because of the large and negative thermo-optic coefficient inherent to most liquids, heating produced by partial absorption of the propagating beam translates into a significant decrease of the refractive index at higher light intensity. For example, the reported thermo-optic coefficient of water reaches ∆n/∆T = −5.7 × 10

^{−4}K

^{−1}[15]. Moreover, introduction of nanoparticles or absorbent dye into the liquid allows for further increase of the thermal nonlinear response [16]. Therefore, a ferrofluid-based self-assembled hyperbolic metamaterial naturally exhibits a potentially interesting version of effective gravity. The thermal origin of this effective gravity looks interesting in light of the modern advances in gravitation theory [1,2], which strongly indicate that the classic general relativity description of gravity results from thermodynamic effects.

## 3. Results

_{2}and ε

_{1}(which may be understood as the effective metric coefficients g

_{00}=−ε

_{1}and g

_{11}= g

_{22}= −ε

_{2}of the metamaterial spacetime) depend on the volume fraction α(H,T) of cobalt nanoparticles aligned into nanocolumns by the external magnetic field, which, in turn, depends on the local temperature of the ferrofluid. Since ferrofluids subjected to an external magnetic field are known to exhibit classical superparamagnetic behaviour [17], well established results from the theory of magnetism may be used to calculate α(H,T). Superparamagnetism is a form of magnetism that is exhibited by magnetic materials, which consist of small ferromagnetic or ferrimagnetic nanoparticles. Superparamagnetism occurs in nanoparticles which are single-domain, which is possible when their diameter is below ~50 nm, depending on the material. Since the typical size of magnetic nanoparticles in ferrofluids is ~10 nm, magnetic ferrofluids also belong to the class of superparamagnetic materials. When an external magnetic field is applied to an assembly of superparamagnetic nanoparticles, their magnetic moments tend to align along the applied field, leading to a net magnetization. If all the particles may be considered to be roughly identical (as in the case of a homogeneous ferrofluid), and the temperature is low enough, then the magnetization of the assembly is [17]

_{m}>> ε

_{d}and assume that α(H,T) is small (which is typically required for the Maxwell-Garnett approximation to be valid). In such a case, the effective metric coefficients are:

_{m}is negative). The effective spacetime appears to have the correct signature if the temperature is low enough, so that

_{0}. The temperature distribution around the source is defined by the two-dimensional heat conductance equation

_{p}is the heat capacity and ρ is the density of the ferrofluid. If the linear source of heat is weak enough, so that the temperature-dependent terms in the denominator of Equation (22) may be considered constant, Equation (23) may be re-written as an equation for the effective gravitational field $\overrightarrow{G}$:

_{d}, the wave equation (Equation (5)) must be modified. Assuming a spatial soliton-like solution that conserves energy per unit length W~P/c (where P is the laser power), the soliton width ρ and the magnetic field amplitude B of the extraordinary wave are related as

_{1}

^{(0)}is the dielectric permittivity component at P = 0 (note that the nonlinear contribution to ${\epsilon}_{2}\approx \alpha {\epsilon}_{m}$ may be neglected). Equation (26) looks like a Klein-Gordon equation in a spacetime with a metric coefficient ${g}_{00}={\epsilon}_{1}^{(0)}-\frac{\left(-{\chi}^{(3)}\right)P}{c{\rho}^{2}}$, which has a black hole-like horizon at

_{2}) plays the role of a scale factor of the effective spacetime

_{1}is positive and almost constant, as plotted in Figure 3b. The scale factor of the effective spacetime calculated using Equation (18) is plotted in Figure 3a as a function of kT/µH at different values of $-{\alpha}_{\infty}{\epsilon}_{m}$. Let us assume that the temperature distribution inside the ferrofluid may be described as

_{c}is the temperature of the metric signature transition (the ε

_{2}= 0 point). Since the z-coordinate plays the role of time in the effective spacetime described by Equation (28), such a temperature gradient η will result in a Big Bang-like spacetime expansion described by the scale factor –ε

_{2}(T

_{c}− ηz) plotted in Figure 3a. Indeed, as the ferrofluid temperature falls away from the T

_{c}boundary at z = 0, the spacetime scale factor increases sharply as a function of z. Note that, at larger values of $-{\alpha}_{\infty}{\epsilon}_{m}$, expansion of the effective spacetime “accelerates” at lower temperatures (compare the slopes of the red and black curves in Figure 3a near the metric signature transitions marked by the arrows and at lower temperatures). As demonstrated below, this “cosmological” spacetime expansion may be visualized directly using an optical microscope.

_{1}= ε

_{x}= ε

_{y}and ε

_{2}= ε

_{z}, the electromagnetic field separation into the ordinary and the extraordinary components remains well defined [19,20]. Taking into account z derivatives of ε

_{1}and ε

_{2}, Equation (5) becomes

_{1}remains almost constant in a very broad temperature range far from T = 0 (as evident from Figure 3b), its derivatives may be neglected. If we also neglect the second derivative of ε

_{2}, the wave equation for the extraordinary field ϕ

_{ω}= E

_{z}may be re-written as

_{00}= −ε

_{1}and g

_{11}= g

_{22}= −ε

_{2}:

_{2}on the effective time calculated at λ = 1.4 µm is shown in Figure 4d. This wavelength is close enough to the λ = 1.5 µm illumination wavelength used in the experiment, while the experimental data for the electromagnetic properties of cobalt [13] may also be used. Calculated values of –ε

_{2}are based on the Fourier analysis of smaller regions in Figure 4b (shown by the yellow boxes). The measured behaviour of the scale factor in this experiment corresponds to small values of $-{\alpha}_{\infty}{\epsilon}_{m}$, which is expected at λ = 1.4 µm based on the optical properties of cobalt [13]. As evident from Figure 4d, the measured data show good agreement with theoretical calculations based on Equation (18).

## 4. Discussion

## 5. Conclusions

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic geometry of the metal nanowire-based hyperbolic metamaterial; (

**b**) Wavelength dependencies of the real parts of ε

_{z}and ε

_{x}= ε

_{y}for a cobalt nanoparticle-based ferrofluid at α = 8.2% volume fraction of nanoparticles. While ε

_{x}= ε

_{y}stays positive and almost constant, ε

_{z}changes sign to negative around λ = 1 µm; (

**c**) Microscopic image of cobalt nanoparticle-based ferrofluid in external magnetic field reveals nanoparticle alignment along the field direction.

**Figure 3.**(

**a**) Scale factor of the effective de Sitter spacetime g

_{11}= g

_{22}= −ε

_{2}(calculated using Equation (18)) plotted as a function of kT/µH at different values of $-{\alpha}_{\infty}{\epsilon}_{m}$. It is assumed that ε

_{d}= 2; (

**b**) Metric coefficient g

_{00}= −ε

_{1}of the effective de Sitter spacetime calculated using Equation (19).

**Figure 4.**(

**a**) Microscopic transmission image of the cobalt nanoparticle-based ferrofluid taken in external magnetic field H using illumination with λ = 1.5 µm light; (

**b**) Analysis of gradual melting of the effective Minkowski spacetime within a single microscopic image of the ferrofluid. This image was obtained while the magnitude of external magnetic field H was decreased gradually. The effective time direction is indicated by the red arrow. The effective spacetime expansion is shown schematically by the yellow cone: the scale factor is proportional to –ε

_{2}, and it vanishes at the point of metric signature transition (see Equation (1)) where the ferrofluid periodicity is no longer visible; and (

**d**) Measured dependence of the spacetime scale factor –ε

_{2}on the effective time calculated at λ = 1.4 µm. These calculations are based on the Fourier analysis (

**c**) of smaller regions of Figure 4b shown by the yellow boxes. The measured data are compared with theoretical calculations based on Equation (18).

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Smolyaninov, I.I.
Thermally Induced Effective Spacetimes in Self-Assembled Hyperbolic Metamaterials. *Universe* **2017**, *3*, 23.
https://doi.org/10.3390/universe3010023

**AMA Style**

Smolyaninov II.
Thermally Induced Effective Spacetimes in Self-Assembled Hyperbolic Metamaterials. *Universe*. 2017; 3(1):23.
https://doi.org/10.3390/universe3010023

**Chicago/Turabian Style**

Smolyaninov, Igor I.
2017. "Thermally Induced Effective Spacetimes in Self-Assembled Hyperbolic Metamaterials" *Universe* 3, no. 1: 23.
https://doi.org/10.3390/universe3010023