Casimir Force Induced by Electromagnetic Wave Polarization in Kerr, Gödel and Bianchi–I Spacetimes

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 produces 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 effective 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± . (14) for electromagnetic waves with right- or left-polarization. On the other hand, cosmological anisotropy with ij ≈ i δij This equation can be readily solved by using a new change ± and μ j = 0, produces birefringence that also modifies the of variables j±(x) = exp(ik2x2 + ik x3)φ j±(x1), where ± 3 propagation of polarized waves. k2 and k3 are the wavevectors in directions x2 and x3 respec- Thereby, these effects induce changes in the magnitude of tively, for the corresponding polarizations. Each component Casimir force, which are directly associated to the interaction of φ j± satisfies different equations. The component along the φ (ξ) ∝ (−ξ 2/ ) ± (ξ) of light polarization and gravitational fields. x3-direction has the solution 3±√ exp 2 Hn [35], where ξ = (2ηωx1 + k2)/ 2ηω, and Hn± is the Her- ± 3.1 Electromagnetic waves propagating in a slowly Kerr mite polynomial, with n = 0, 1, 2,.... For this solution, black hole background the dispersion relation

ω2 − ± 2 ± η ± = ηω( ± + ), For the sake of simplicity, we focus in the case of an electro- k3 2 k3 2 2n 1 (15) magnetic wave travelling in the exterior of the gravitational field of a slowly rotating Kerr black hole with mass M and must be satisfied. Solutions for the remaining components spin J. This problem was solved by Mashhoon [35], show- φ1± and φ2± are also known, and we refer the reader to ing that these propagating electromagnetic waves experience Ref. [35] in order to find their thorough mathematical study. chiral effects in such a metric due to coupling with their polar- Dispersion relation (15) defines the propagation of an elec- ization. tromagnetic wave in Kerr metric (and in a Gödel universe as + = − For a slowly rotating black hole, the stationary Kerr metric we will see). As k3 k3 , this relation shows that right– and in isotropic coordinates becomes [35,44] left–circularly polarized waves move with different phase velocities when propagating along the direction of the rota- 2M 4J ds2 ≈− 1 − dt2 − (x dx − x dx ) dt tion axis of rotation of curved spacetime. In general, we find r r 3 1 2 2 1 that the difference between the wavevectors is i j +δijdx dx , (11) + − − = δ = η + (ω − η)2 − ηω + k3 k3 2 k 2 4 n for a rotation in the x3-direction, and with r =|x|. Its spin (angular momentum of magnitude J)isJ j = δ j3 J in these − (ω − η)2 − 4ηωn− , (16) coordinates. This intrinsic angular momentum is measured which is a function of the frequency of the wave. However, far from the black hole [44]. Using metric (11)in(5), we get because of our previous assumption on the nature of the black that hole and the scales of electromagnetic waves, we always ω/η = ω ( / )−1( / 2)−1 2M J k xm find that M M D J D 1, describing ij ≈ δij 1 + ≈ δij ,μ≈−2ε , (12) |x| j 0 jkm |x|3 waves in the high-frequency limit. In this case, the physical solutions for the electromagnetic wave components φ1± and φ + − = − + are the permittivity and permeability of the effective medium 2± imply that n 1 n 1[35]. All these allow us to for an electromagnetic wave propagating far from the black find that the final wavelength difference becomes simply | |∼ hole, with x D M, where D is the given distance 2δk =−2η. (17) where the light propagates. For our slowly rotating metric, |μ|∼J/D2 1. Therefore, Maxwell equations (10) reduce Difference (17) is a physical consequence of the reaction to [35] of wave polarization to the rotation axis of the spacetime. Each spin of the circularly polarized wave couples differ- ±ωδjk = ε0 jkm ∂ − ωη ( δ − δ ) , fk± [ k i x2 1k x1 2k ] fm± ently to curvature. It is not an effect of coordinate choice, (13) as the angular momentum of Kerr metric is well-defined as a 123 1005 Page 4 of 7 Eur. Phys. J. C (2020) 80 :1005

ij i ij measure of the frame dragging of this spacetime [44,45]. The μ j = 0 and = δ , with above result shows that these left– and right–circularly polarized electromagnetic waves propagate in an fashion analo- i = a1a2a3 . 2 (22) gous to waves experiencing Faraday rotation effect, which ai occurs in materials [46,47] and magnetized plasmas [48]. This cosmology behaves in analogue fashion to a birefrin- 1 = 2 = 3 = 1 3.2 Electromagnetic waves propagating in a slowly rotating gent medium, as in general [37]. Gödel universe background Friedmann–Lemaître–Robertson–Walker cosmology is the particular case when a1 = a2 = a3 = 1 = 2 = 3. Let us consider a Gödel universe, with the stationary metric Using the general metric (21), Eqs. (10) are now written in cartesian coordinates [35,49] as 2 2 2 2 i ij 0ijm ds =−dt + dx + dx ± ∂ δ ζ ± = ε k ζ ± . (23) √ 1 √ 3 0 j j m 2x1 +2 2 1 − e dx2dt √ √ ( , ) = ζ ( ) ik·x where we have now considered that Fi± t x j± t e , + 4e 2x1 − e2 2x1 − 2 dx2 , (18) 2 with constant k = (k1, k2, k3). The three coupled Eqs. (23) can be solved for any component. As an example, let us obtain where is a constant related to the angular velocity of the the wave equation for polarization ζ ±. First, manipulating rotating universe. 3 (23), the following condition is obtained In the case of a slowly rotating universe, we have x1 1, and from (5) we simply ﬁnd 3 3 −k2k3 ζ3± ± k1∂ξ ( ζ3±) ζ ± = , (24) 2 1 2 + 2 2 ij ij k1 k2 ≈ δ ,μj =−2 x1δ2 j , (19)

where the derivative ∂ξ is in terms of the effective cosmolog- Using this in Eq. (10), it is straightforward to show that elec- ical time tromagnetic waves propagates satisfying the following equation dt ξ = . (25) 1 j 0 jkm 0 jkm ∓i ω± =−i ε ∂ j m± +2ωx1ε δ2km± , (20) Using again (24)inEqs.(23), we finally can get −iωt with the transformation Fj±(t, x) = j±(x)e and wave 1 2 k1 3 frequency ω. ∂ξ ∂ξ ζ3± 1k2 + 2k2 Equations (20) is exactly the same as Eq. (14) (replacing 1 2 η 1 by ). Hence, we can find the same solutions, satisfying 2 3 k1k2k3 3 −∂ξ ζ ± ∓ ∂ξ ζ ± the dispersion relation (15), and wavelength difference (17). 3 1 2 + 2 2 3 k1 k2 Anew, the polarized behavior of light is a consequence of 3 2 1 2 k3 its coupling with the spacetime of Gödel universe, whose = k 1 + ζ ± . (26) 2 1 2 + 2 2 3 rotation is a consequence of the total vorticity of the flow k1 k2 that generates it [49]. From this equation can be readily seen that different polarizations propagate at different speed when k1 = 0, k2 = 0 3.3 Electromagnetic waves propagating in anisotropic and k3 = 0. Equations similar to (26) can be found for ζ1± cosmology and ζ2±. In order to obtain a sensible and simple solution, let us Let us consider an anisotropic Bianchi–I cosmological model consider the case of an almost–isotropic cosmology, such [50], with an interval given by that a2 = a, and a1 = a3 = a(1 + ϑ/2), with ϑ 1. The anisotropy is in the direction of k2. In this case, we get 2 =− 2 + 2 i i , 1 3 2 ds dt ai dx dx (21) = = a and ≈ a(1 + ϑ). Furthermore, let us assume for simplicity the particular case when k1 = k2 = k3 = k. in cartesian coordinates (i = 1, 2, 3). Here, ai = ai (t) are Thus, Eq. (26) reduces to arbitrary time dependent functions, such as a1 = a2 = a3 = 2 1 ± a ∂ξ (a ζ3±) + ∂ξ ϑ∂ξ (a ζ3±) + λ a ζ3± = 0 , (27) 1, in general. With this metric, we find from Eq. (5) that 2 123 Eur. Phys. J. C (2020) 80 :1005 Page 5 of 7 1005 where propagating in a Kerr and Gödel background metrics and in anisotropic cosmologies, respectively. ϑ ± 2 k λ = 3 − k ∓ ∂ξ ϑ. (28) As a consequence, we can treat these curved spacetimes 2 2 as analogue media for photon propagation with the sole purpose of calculating the associated Casimir force [33,34]. In The different behavior of polarizations is due to the difference our case, polarized high–frequency electromagnetic waves of the variations of anisotropy parameters only. Eq. (27) can fulfilling differences (17) and (32), modify the Casimir force be solved for ζ ±,as[51] 3 decreasing its intensity, so it becomes less attractive. ζ ϑ ± 3o W3 ζ ±(t) = exp − exp −i dt , (29) 4.1 Casimir force due to a rotating spacetime 3 a 4 a In this case, the wavelenght difference is δk =−η for Kerr ζ ± where 3o is a constant, and W3 fulfill the equations sacetimes, and δk =− for Gödel spacetimes. Now, let ± 2 2 ± us consider two parallel plates with a length separation l,at ∂ξ W ∂ξ W ± 2 = 3 3 − 3 some distance D l far from the rotating black hole. Plates W3 ± ± 4 W3 2 W3 size are negligible compared to black hole dimensions. The 2 ∂2ϑ plates are perpendicular to the rotation axis of the rotating ± (∂ξ ϑ) ξ +λ − − . (30) spacetime, and in between them, polarized electromagnetic 8 4 waves propagate. The Casimir force per unit area (2) expe- Similar solutions can be found for ζ1± and ζ2±. ± rienced between two plates is then The behavior of ζ3± depends on W through its evolution 3 ∞ ∞ given in Eq. (30). This equation can be considered the disper- 1 2 FC = d d k sion relation for that particular propagation in this anisotropic 2π 3 −∞ 0 cosmology. Hence, the wave frequency for each polarization e−4l − e−2l cos(2ηl) − ηe−2l sin(2ηl) ± ± × , ω = W /a − − can be defined as 3 3 . This definition takes into 1 + e 4 l − 2e 2 l cos(2ηl) account the cosmological redshift. An approximated solu- ± (33) tion can be obtained assuming that variations of W3 are negligible compared with variations of ϑ. In this case, we where we have used η for both Kerr and Gödel cases. get Although this result is general, we must notice that for Kerr metric ηl = 2J/D2 (l/D) 1, whereas for a Gödel uni- √ ± ± k verse l = (D) (l/D) 1. Using this, we can explicitly a ω ≈ λ ≈ 3k2 ∓ ∂ξ ϑ. (31) 3 2 evaluate (2) to obtain √ ± π 2 ω = ± δ 10 2 2 Wave frequency can be written as a 3 3k k,asa F ≈− 1 − η l , (34) C π 2 4 deviation from a light–like behavior when k ∂ξ ϑ. Thereby 240l Casimir force per unit area (34) shows that spacetimes a (ω3+ − ω3−) = 2δk describing rotating bodies (such as a Kerr black hole and k k = 3k2 − ∂ξ ϑ − 3k2 + ∂ξ ϑ Gödel universe) induce a reduction on the magnitude of 2 2 attractive Casimir forces. This effect is due to the chiral a ∂ ϑ ≈− √0 . (32) behavior of polarized electromagnetic waves triggered by 2 3 the coupling of different circularly polarized waves and the Under the above assumption, it is clear that different polar- rotation of spacetime, in an analogue fashion to what occurs izations propagate differently only due to their interaction in a chiral medium. with the anisotropic characteristic of this cosmology. Strictly speaking, in order to obtain the result (34)we have used the flat–spacetime calculation for electromagnetic waves in a medium, developed in Ref. [4]. However, forces 4 Casimir force due polarized electromagnetic waves (33) and (34) are evaluated on curved spacetimes. This may be done because we have taken into consideration the effect In Ref. [4] it was shown that polarized electromagnetic waves of spacetime curvature into the electromagnetic wave propa- can give origin to repulsive or attractive Casimir forces when gation, by defining the effective medium described by the per- the wavevector of their left and right polarizations fulfill mittivity and permeability (12). Therefore, a medium treat- k+ = k− in chiral media. This is exactly the behavior of dif- ment for Casimir energy becomes completely analogous to ferences (17) and (32) for polarized electromagnetic waves waves in curved spacetime. On the other hand, Casimir effect 123 1005 Page 6 of 7 Eur. Phys. J. C (2020) 80 :1005 has already been studied in this limit on Kerr spacetimes for when electromagnetic waves are not treated as light rays (in scalar fields [29–31,52], and electromagnetic fields [31]. For the eikonal limit). It is the coupling between the spacetime both kind of fields, the gravitational correction to Casimir curvature with the electromagnetic wave extended proper- energy depends on O(M/D), producing a reduction of the ties which modify the strength of Casimir forces, in addition attractive force. The above used assumptions in our calcu- to the curvature effects produced by the gravitational field. lations require that M/D 1, as it can be seen in (12). In this way, the spacetime background has a direct physi- This implies that we are neglecting the gravitational potential cal consequence that cannot be replicated by scalar fields or (Schwarzschild–like) correction to the Casimir force in our electromagnetic fields in the eikonal limit. calculations. However, the net effect of the rotation spacetime Finally, forces (34) and (35) hint that a more general spec- does not vanish, as η is independent of the M. Previous results trum of Casimir forces can be found for propagating electro- for the Casimir force produced by electromagnetic fields in magnetic waves in these kinds of curved backgrounds with a weak Kerr gravitational field have shown the M/D depen- inequivalent directions. The results presented in this article dence without considering the polarization coupling with the can be generalized to any spacetime with μ j = 0 or non– rotation of the black hole [31]. As soon as this effect is taken constant jk. In general, for high–frequency electromagnetic into account, new corrections of order η2l2, not previously waves in rotating spacetimes, the difference between the envisaged, emerge as a source of reduction of the magnitude propagation of different polarizations should be proportional of this force. to |μ|, or variations of | |. Those effects create the small corrections to the Casimir force. However, one should expect to 4.2 Casimir force due to anisotropic cosmology have larger effects if more general solutions of wave equation (10) can be found. Whenever the current assumptions can be In this case, the difference between left– and right–polarization relaxed, the effect induced by the different propagation for is produced by anisotropies√ in spacetime, and is equal to different polarizations of electromagnetic waves with longer δk =−a∂0ϑ/(4 3). This induces a modification on the wavelengths could be solved. Then it would be possible to Casimir force (2) for two parallel plates with a proper length study the repulsive effect on Casimir forces due to curved separation al. spacetimes, in an analogue fashion to what occurs in general Consider a separation smaller than the scale of variation chiral materials [4]. These endeavors are left as the subject of the anisotropy, al∂ξ ϑ 1. In this case, force (2)forthe for future research. ζ3± polarizations is simplified to Acknowledgements F.A.A. thanks Fondecyt-Chile Grant No. 1180139. 4 π 2 5a 2 FC ≈− 1 − (l ∂ ϑ) , (35) 24π 2 0 240 a4l4 Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This manuscript showing how anisotropic cosmology variations reduces the presents only theoretical results.] strength of Casimir force due to electromagnetic interactions, Open Access This article is licensed under a Creative Commons Attri- as this expanding spacetime behaves as a medium. In the case bution 4.0 International License, which permits use, sharing, adaptation, of an isotropic cosmology with ϑ = 0, Casimir force (35) distribution and reproduction in any medium or format, as long as you reduces to its known ∝ a−4 behavior [53]. give appropriate credit to the original author(s) and the source, pro- vide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indi- 5 Discussion cated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended Results described in this work belong to a family of differ- use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copy- ent behaviors of Casimir forces due to spacetime curvature. right holder. 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