Eur. Phys. J. C (2020) 80:1005 https://doi.org/10.1140/epjc/s10052-020-08593-5 Regular Article - Theoretical Physics Casimir force induced by electromagnetic wave polarization in Kerr, Gödel and Bianchi–I spacetimes Felipe A. Asenjo1,a , Sergio A. Hojman2,3,4,b 1 Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago 7491169, Chile 2 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago 7491169, Chile 3 Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago 7800003, Chile 4 Centro de Recursos Educativos Avanzados, CREA, Santiago 7500018, Chile Received: 28 April 2020 / Accepted: 21 October 2020 / Published online: 30 October 2020 © The Author(s) 2020 Abstract Electromagnetic waves propagation on either when electromagnetic waves propagate in chiral material rotating or anisotropic spacetime backgrounds (such as Kerr media between two conducting parallel plates. In such cases, and Gödel metrics, or Bianchi–I metric) produce a reduction the behavior of the Casimir force can be controlled by vary- of the magnitude of Casimir forces between plates. These ing the electric permittivity and magnetic permeability of curved spacetimes behave as chiral or birefringent materi- the material. In this chiral material, right– and left–circularly als producing dispersion of electromagnetic waves, in such polarized electromagnetic waves propagate with wavevectors a way that right– and left–circularly polarized light waves k+ and k− respectively. In this material, both wavevectors propagate with different phase velocities. Results are explic- are described by k± = k ± δk, meaning that the difference itly calculated for discussed cases. The difference on the between the wavevectors of the two polarized waves is given wavevectors of the two polarized electromagnetic waves pro- by k+ − k− = 2δk. This difference is responsible for pro- duces an abatement of a Casimir force which depends on ducing tunable repulsive or attractive Casimir forces when the interaction between the polarization of electromagnetic the separation l between plates is varied. waves and the properties of the spacetime. In such cases, the Casimir energy is sensitive to this phase difference, and it turns out to be [4] ∞ ∞ d d2k 1 Introduction E = C π π 2 0 2 −∞ 4 − − The Casimir force, which acts on two (uncharged) conduct- ln 1 + e 4 l − 2e 2 l cos(2δkl) , (1) ing parallel plates in vacuum is, in general, attractive [1,2]. = ω = ( 2 + 2)1/2 = ( 2 + 2)1/2 Casimir force can be explained as a consequence of quantum where i , k , k k1 k2 , vacuum fluctuations, but it can be alternatively understood in with the wave frequency ω, and the wavevector components terms of Van der Waals interactions [3]. Although this force ki (i = 1, 2, 3). From here, the Casimir force per unit area is attractive in vacuum and depends only in the plates separa- FC =−dEc/dl experienced between two plates is readily tion, a different behavior can be manifested under other con- found to be ditions. When a medium made of different materials (or meta- ∞ ∞ 1 2 F = d d k materials), with diverse properties or geometrical shapes, is C 3 2π 0 −∞ inserted between the plates, a repulsive Casimir force may e−4l − e−2l cos(2δkl) − δke−2l sin(2δkl) appear (see for instance, Refs. [4–18]). The Casimir force can × . + −4l − −2l ( δ ) even produce a torque in the plates in optically anisotropic 1 e 2e cos 2 kl materials [19]. (2) Recently, Jiang and Wilczek [4] have shown that a tunable This force can be either attractive or repulsive depending on repulsive and attractive Casimir force can also be obtained the magnitude of 2δkl, and it has been shown in Ref. [4] that repulsive Casimir force may appear in Faraday and Optical a e-mail: [email protected] (corresponding author) active materials. On the contrary, in a non-chiral material, b e-mail: [email protected] as there is no difference between phase velocities of right– 123 1005 Page 2 of 7 Eur. Phys. J. C (2020) 80 :1005 and left–circularly polarized waves δk = 0, the force (2) space coordinates. The above vector fields are related by [35– becomes simply −π 2/(240l4), recovering the value for vac- 37,39,40,43] uum Casimir force. Di = ijE − ε0ijkμ H In Ref. [4], only material media were considered. The j j k i = ij + ε0ijkμ , main purpose of this work is to show that gravitational fields B H j j Ek (4) described by stationary spacetimes (metrics with rotation, explicitly showing the analogy with electric and magnetic such as Kerr black holes or Gödel universe), or by anisotropic fields in the presence of a medium. Here cosmological spacetimes (such as Bianchi–I metric), produce a behavior similar to the one appearing in chiral media when √ ij g ij =− − g ,μ=− 0 j , electromagnetic waves propagate on it. g j (5) g00 g00 Related gravitational effects due to spacetime curvature on Casimir force have been explored previously [20–28]. Also, are the effective permittivity and the effective vector perme- the effect of spacetime curvature correction due to Kerr met- ability of the curved spacetime, respectively. Both are defined μν ric on Casimir forces has been studied for both scalar [29– in terms of spacetime metric (and its inverse g ). Using the 33] and electromagnetic fields [34]. Our work explores how above electric and magnetic fields, Maxwell equations (3) the coupling of polarization with the angular momentum or can now be written in a vectorial fashion as anisotropy of spacetime modifies the Casimir force. We show ∂ Di = 0 ,∂Di = ε0ijk∂ H , this effects by describing the analogy between electromag- i 0 j k ∂ i = ,∂ i =−ε0ijk∂ , netic wave propagation in curved spacetimes with the elec- i B 0 0 B j Ek (6) tromagnetic fields in media. We obtain solutions for Kerr, where ∂0 and ∂i stands for the time and spatial partial deriva- Gödel and Bianchi–I spacetimes, where this coupling cre- tives. Equation (6) are completly equivalent to the covariant ates a difference between right– and left–circularly polar- Maxwell equations (3). ized electromagnetic waves. We finally show how the chiral An alternative and enlightening representation of electro- behavior of light on those spacetimes modifies the Casimir magnetic fields in curved spacetime [35,41] can be obtained force. by defining the (generalized) Riemann–Silberstein vectors Fj± = E j ± iHj , j j j 2 Maxwell equations in curved spacetime S± = D ± iB , (7) where the + (−) symbol represents right (left) polarization We start by describing the dynamics of electromagnetic fields of the electromagnetic fields. Using relations (4), we obtain on a gravitational background field (from now on we use that natural units c = h¯ = 1). In general, covariant Maxwell equations in curved spacetime may be written as j ij 0ijk S± = Fj± ± iε μ j Fk± , (8) αβ ∗αβ ∇α F = 0 , ∇α F = 0 , (3) and therefore, Maxwell equations (6) reduce to the simplest form in terms of the antisymmetric electromagnetic field tensor αβ ∗αβ j F and its dual F . Here, ∇α is the covariant derivative ∂ j S± = 0 , defined by a metric gμν. j 0 jkm ±i ∂0 S± = ε ∂k Fm± . (9) For the case of electromagnetic waves, several articles [35–42] have shown that light does not propagate along We can now use Riemann–Silberstein polarization vectors, null geodesics in curved spacetimes, giving rise to a dis- to finally write Maxwell equations curved spacetime as [35] persion relation that depends on their polarization. This jk 0 jkm occurs because gravitational fields behave as effective mate- ±i ∂0 Fk± = ε ∂k Fm± + ∂0 (μk Fm±) . (10) rial media with non–trivial effective permeability and effec- tive permittivity, both of them modifying the (vacuum) Equations (10) are completely equivalent to Eq. (3). From Maxwell equations. Thus, the electromagnetic field ampli- here is clear that, in general, both polarization states (±) tude and polarization (besides its phase), couple to space- propagate differently. Spacetime couples with different com- time curveture. This is explicitly shown√ by defining the cor- ponents of electromagnetic fields (this fact does not occur in = i = − 0i responding electric Ei Fi0, D √ gF , and mag- vacuum), even producing rotation of the polarization state of i 0ijk 0ijk ij netic B = ε Fjk, ε Hk = −gF fields, where light [37,40,42]. Isotropic and symmetric spacetimes (with 0ijk ij ij ε is the Levi-Civita symbol and g the metric determi- μ j = 0 and = δ ) allow null geodesic light propaga- nant with latin indices used to denote (three–dimensional) tion [37,42], while a general curved spacetime behaves as 123 Eur. Phys. J. C (2020) 80 :1005 Page 3 of 7 1005 −iωt an effective medium for travelling electromagnetic waves. where we have used Fj±(t, x) = f j±(x)e , with wave 3 Nevertheless, when μ j = 0 spacetime acts as a chiral mate- frequency ω. Besides, η = 2J/D . Furthermore, the scale rial. Similarly, when ij = i δij, with i = j for i = j, of the electromagnetic wave is much less than the one of the spacetimes produce birefringence [37,42]. black hole, such that ωM 1. Solutions to Eq. (13) represent right– and left–circularly polarized electromagnetic waves propagating in the x2 − x3 3 Polarized electromagnetic waves in rotating and plane, very far from the Kerr black hole (where spacetime is anisotropic spacetimes almost flat). Performing the change of variables f j±(x) = exp(iωηx1x2) j±(x),fromEq.(13) we find It can be straightforwardly shown that Kerr and Gödel curved j 0 jkm 0 jkm spacetimes, both having μ j = 0, produce different solutions ∓i ω± =−i ε ∂ j m± + 2ωηx1ε δ2km± .