PHYSICAL REVIEW RESEARCH 2, 013281 (2020) Analogues of gravity-induced instabilities in anisotropic metamaterials Caio C. Holanda Ribeiro * and Daniel A. Turolla Vanzella † Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970, São Carlos, São Paulo, Brazil (Received 27 November 2019; accepted 11 February 2020; published 9 March 2020) In the context of field theory in curved space-times, it is known that suitable background space-time geometries can trigger instabilities of fields, leading to exponential growth of their (quantum and classical) fluctuations, a phenomenon called vacuum awakening in the quantum context, which in some classical scenarios seeds spontaneous scalarization or vectorization. Despite its conceptual interest, an actual observation in nature of this effect is uncertain since it depends on the existence of fields with appropriate masses and couplings in strong-gravity regimes. Here, we propose analogues for this gravity-induced instability based on nonlinear optics of metamaterials which could, in principle, be observed in laboratory. DOI: 10.1103/PhysRevResearch.2.013281 I. INTRODUCTION spin-1 fields through appropriate nonminimal couplings [18] and, in analogy with the scalar case, the stabilization pro- The influence of a background material medium on the cess was termed “spontaneous vectorization.” To the best of propagation of mechanic and electromagnetic waves is well our knowledge, condensed-matter and optical analogues of known to be formally analogous to that of an effective curved these gravity-induced instabilities have not been proposed space-time geometry. This idea was first presented, in the to this date. In this work, we propose and explore possible electromagnetic context, by Gordon in 1923 [1] and it has analogues of gravity-induced instabilities in the context of since been developed in a number of different scenarios, par- electromagnetism in polarizable and magnetizable anisotropic ticularly after Unruh’s [2] and Visser’s [3] works on acoustic (meta)materials. analogues of black holes and their associated Hawking-type Electromagnetic instabilities in flat space-time are ex- radiation. More recent applications of this formal analogy pected to occur in some materials. One celebrated example include mimicking in material media quantum light-cone appeared in the context of plasma physics in the late 1950s fluctuations [4] and anisotropy in cosmological space-times and became known as Weibel instability [19]. The system, a [5]. The most appealing feature of these condensed-matter neutral plasma whose components have anisotropic velocity analogues of gravitational backgrounds is the possibility of distribution, possesses growing electromagnetic transverse observing in laboratory subtle but conceptually interesting waves. Related effects have been studied since then, with effects which can be virtually unobservable in their original recent applications to solar plasma instability [20] and solid- contexts, Hawking radiation being certainly the most emblem- state devices [21]. Moreover, causal aspects of classical prop- atic among them, with claims of having already been observed agation in active materials were discussed in Ref. [22], where in laboratory [6–8]. properties of the refractive index were established. Neverthe- An interesting effect in the context of (quantum) fields less, aside from the fairly recurrence in the literature, usually in curved space-times is the triggering of field instabili- quantization in such scenarios is not considered [23–25]orit ties due to the background space-time geometry, a phe- is regarded as inconsistent [26,27]. nomenon called vacuum awakening in the quantum context It is noteworthy that instability of the electromagnetic [9–12]. These gravity-induced instabilities exponentially am- field is always accompanied by evolution of the background, plify vacuum fluctuations to the point they decohere and seed ending with the stabilization of the system as a whole. In classical perturbations [13], which, depending on field pa- the case of gravity-induced instability, the gravitational field rameters, eventually evolve to a nonzero classical field con- changes with time, whereas electromagnetic instability in the figuration (“spontaneous scalarization” in the case of scalar presence of plasmas involves growing plasmons. In the case fields [14–17]), stabilizing the whole system. More recently, of electromagnetic fields in the presence of matter, for what- this mechanism was also predicted to occur for massless ever form of the interaction with the background, the field’s evolution is ruled by Maxwell’s equations in the presence of polarizable and magnetizable media, and the interaction with *[email protected] the background is encapsulated in the functional dependence †[email protected] of the electric displacement (magnetic) vector field D (H) with the true (microscopic) fields E and B. If the magnitudes Published by the American Physical Society under the terms of the involved are small (e.g., in the beginning of the instability Creative Commons Attribution 4.0 International license. Further action), these functional relations become linear and one may distribution of this work must maintain attribution to the author(s) find the form of the coefficients for such systems. For the case and the published article’s title, journal citation, and DOI. of Weibel instability, for instance, if the velocity anisotropy is 2643-1564/2020/2(1)/013281(20) 013281-1 Published by the American Physical Society RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020) taken in the z direction, the instability is modeled by a negative where ∂a is the derivative operator compatible with the flat squared refractive index in the direction perpendicular to z. metric ηab (but in arbitrary coordinates) and the square brack- We apply Gordon’s method to propose a family of optical- ets denote antisymmetrization over the indices enclosed by based analogue models for electromagnetic fields presenting them. These equations must be supplemented by medium- ab instabilities in curved space-times. We show how anisotropies dependent constitutive relations between Fab and G ,aswell of the background enter the effective equations in the form of as initial and boundary conditions, in order to provide a well- nonminimal couplings, and in the case of strong anisotropy posed problem. These constitutive relations are usually set at a a (just like for the Weibel instability), this coupling results in the level of (observer-dependent) fields Ea, B , D , and Ha, ab unstable solutions. We also discuss that for these systems the related to Fab and G through stabilization process occurs through the nonlinear nature of = b, the background, which may seed spontaneous vectorization Ea Fabu (3) a ab in analogy to the Einstein’s field equations in the gravitational D = G ub, (4) scenario. Ba =−1 abcd F u , (5) The paper is organized as follows. In Sec. II, we present 2 bc d the covariant formalism of electromagnetism in anisotropic =−1 bc d , Ha 2 abcd G u (6) polarizable and magnetizable materials, establishing the for- a mal analogy with nonminimally coupled electromagnetism where u is the four-velocity of the observer measuring these = in curved space-times. In Sec. II A, we consider a particular √fields and abcd is the Levi-Civita pseudotensor [with 0123 type of nonminimal coupling inspired by one-loop quantum −η, η := det(ημν )]. Moreover, the constitutive relations electrodynamics (QED) corrections to electromagnetism in usually take a simpler form in the reference frame in which curved space-times. In Sec. III, we apply the formalism the medium is (locally and instantaneously) at rest. presented in the previous section to the scenario of a plane- Here, we consider a polarizable and magnetizable medium symmetric anisotropic medium at rest in an inertial frame. whose constitutive relations in its instantaneous rest frame Although plane-symmetric curved space-times (in four di- take the form mensions) are not really (physically) appealing, we consider Da = εabE , (7) this scenario for its simplicity and for its possible implications b b for the physics of the material medium. We construct the Ha = μabB , (8) ˆ electromagnetic quantum-field operator A (in the generalized εab μ Coulomb gauge) in the standard-vacuum representation, dis- where the tensors and ab may depend on space-time cuss the conditions for appearance of instabilities and their coordinates, and the system is assumed dispersionless. We types (Sec. III A), and present a concrete example (homoge- return to this point later. The fact that Eqs. (7) and (8) are valid in the medium’s instantaneous rest frame means that the fields neous medium; Sec. III B) where calculations can be carried a a ab over to the end. In Sec. IV, we repeat the treatment of the Ea, B , D , and Ha appearing in them are related to Fab and G a = a previous section, but now for a more appealing scenario through Eqs. (3)–(6) with u v , the medium’s four-velocity field. We proceed by splitting the “spatial” [30] tensors εab and on the gravitational side: spherically symmetric, stationary μ anisotropic media. Conditions for triggering instabilities and ab into isotropic and traceless anisotropic parts, their types are shown to be very similar to those in the ab ab ab ε = ε h + χ ε , (9) plane-symmetric case (Sec. IV A). As a concrete applica- ( ) μ = μ−1 + χ (μ), tion, in Sec. IV B we show how to mimic QED-inspired ab hab ab (10) nonminimally coupled electromagnetism in the background a = δa + a space-time of a Schwarzschild black hole. Then, Sec. V is where h b : b v vb is the projection operator orthogonal a a = a dedicated to discuss possible stabilization mechanisms which to v . Inverting Eqs. (4) and (6) (with u v ), might bear analogy to some curved-space-time phenomena, ab [a b] abcd G = 2v D − Hcvd , (11) such as spontaneous vectorization [18] and particle bursts due to tachyonic instability [28]. Finally, in Sec. VI we present and substituting Eqs.