arXiv:1502.07429v1 [gr-qc] 26 Feb 2015 rprisof properties a such the finding to transparent in [3]. are is field materials gravitational difficulty ordinarily The as gravitational be the medium, gravitons. then of or would quantisation ff e ect the field, gravi- Casimir of the the manifestation to that a opaque so are field, which tational plates Therefore significant imagine [2]). be may ff e ect should Casimir one plates the weakens the fact quickly to massless In (mass any opaque from is ff e ect ff e which ect. Casimir pro- field Casimir the quan- that to the the field contribution of to the only fields contribute all the vacuum theory not in tum [1]. is ff e ect; Casimir temperature course, the zero of duces at field formula EM Lifshitz The re- the field, (EM) in electromagnetic ones sulting the realistic for more derived generalisation to been the conditions have boundary and how- ideal reflecting, bodies these modes Real perfectly of allowed not 1). of (Fig. are restriction them ever between the vacuum reflecting of the perfectly result in two a Casimir between the as form, attraction surfaces basic the most its is theory, In ffe ect field ff e ect. quantum Casimir by the is predicted energy vacuum zero l fterdtcin[6.I ih ftecontroversial under- the an of interaction, light gravitational In enhanced the [16]. of with detection status capa- coupled their environment of ff e ects the ble theorised magni- producing in the small challenges the of practical of some because of perhaps tude unambiguous is and interac- This clear GW produced outcomes. have enhanced none have proposed 16], experiments [15, the tion few test without a to met Although been attempted not 14]. have [13, idea controversy These 11] supercon- [10, [12]. transducers [8], mirrors [9], antennae and detectors GW gravitational as superfluids circuits for ducting proposing medium of 7], a interaction [6, as the superconductors into with made were GW analyses Following further superconductor. the this, in field current Lense-Thirring a a induce that should [4] calculated DeWitt who by [5], investigated Papini first and field was gravitational fluid a quantum of a interaction with the (GW). of waves ff e ects novel gravitational The with interaction the may enhance condensates) Bose-Einstein fluids, Hall quantum ids, n ftems eakbecneune ftenon- the of consequences remarkable most the of One eetyhwvr hr aebe ugsin htthe that suggestions been have there however, Recently c nodnr m speculated ordinary the in by eect ff Casimir produced Casimi gravitonic ect eff the the of to meagreness contribution the gravitonic the of quantification ASnmes 14.70.Kv,04.30.-w,74.20.Fg conse and numbers: theory, H-C PACS the of validity the for test a providing unu fluids quantum c ihnnielsdb non-idealised with ect ff e Casimir gravitonic the derive We nttt o oi tt hsc,Uiest fTko Kas Tokyo, of University Physics, State Solid for Institute sprodcos superflu- (superconductors, rvttoa aii effect Casimir Gravitational Heisenberg-Couloumb ae .Quach Q. James asv etpitpril neatn ihafluctu- a with interacting particle point for test calculated massive been has a The also [18]. potential systems Casimir terrestrial gravitonic realistic for idealised suitable are not conditions and boundary the how- works, context; these cosmological in ever a in Casimir considered gravitonic gravitonic been This the has matter ffe ect to spacetime. flat rise the gravi- in give with ff e ect, to the Casimir interacts conditions how vacuum boundary we at the the what of look the of from we form field where different that tational here, is matter as- considering This is the are field by gravitational conditions. unaffected the boundary be en- works, Casimir to these the In sumed curved on slightly [17]. have the ergy would ff e ect the background see spacetime to fields gravitational en- with in system interactions. theorised contribution gravitational any the hanced in quantify and accurately materials to ordinary bod- needed realistic is for ff e ect ies Casimir gravitonic the of standing normal. surface the to respect and with waves, angle gravitation gravitational corresponding the reflected causes and plates transmitted, the refract. change of to The index wave setup. plate refractive two a the with in ect eff Casimir The 1. FIG. c a lobe netgtdi weak in investigated been also has ff e ect Casimir The c nsprodcos thereby superconductors, in eect ff (H-C) ∗ y γ’’’ reff e c tf r o mr e a lb o d i e s .W eq u a n t i f y unl h xsec fgravitons. of existence the quently te.W loqatf h enhanced the quantify also We atter. udr odtos hsalw the allows This conditions. oundary k’’’ x ia hb 7-51 Japan 277-8581, Chiba hiwa, z k ersnstewv etro h incident, the of vector wave the represents k k’’ γ γ γ γ k’’ k k’ γ’ γ sthe is al 2 ating mass distribution [19]. In this letter we derive the the energy of all allowed modes of the field. The opaque gravitonic Casimir effect for real bodies. In effect we give boundary of the parallel plates in the xy-plane means the Lifshitz formula for the gravitonic contribution to the that the kz components of the field are discrete, whereas Casimir effect at zero temperature. the k∥ ≡ (kx,ky) components remain continuous. In The contribution of small perturbations to the flat terms of the graviton eigenfrequencies, the vacuum en- spacetime metric, ηµν , is well described by the linearised ergy of gravitons between the plates at zero temperature Einstein field equations. We begin with a Maxwell- is given by [25], like formulation of the linearised Einstein field equations ! ∞ known as gravitoelectromagnetism (GEM) [20, 21], + × E0 = k dk (ω + ω )σ, (5) 4π ! ∥ ∥ n n E (E) 0 "n ∇· = κρ , (1) where σ is the surface area of the plate. The allowed (M) + × ∇·B = κρ , (2) modes (ωn , ωn ) between the plates are found by consid- ∂B ering the boundary conditions, as follows. ∇×E = − − κJ(M) , (3) E ∂t A naive application of Stokes’ theorem to the and B ∂E fields at the plate interface will produce an overde- ∇×B = + κJ(E) . (4) ∂t termined problem. The reason for this is that the extra components of these second-order tensors would intro- where κ ≡ 8πG/c4. In this formulation, components of duce extra constraints, as compared to vector fields, such the Weyl tensor (Cαβµν ) play roles analogous to the elec- as the EM field. Thus E and B cannot be considered tric and magnetic fields in electromagnetism: Eij ≡ C0i0j completely smooth across the interface. Instead only the and Bij ≡ ⋆C0i0j ,(⋆ denotes Hodge dualisation [22]). traceless part of the tangential components of the tensor The rhs of the GEM equations contain components of fields are considered smooth across the interface - this is 1 the matter current, Jµνρ ≡−Tρ[µ,ν] + 3 ηρ[µT,ν] where known as the smoothness principle [24]. This gives rise µ Tµν (T ≡ Tµ )arethestressesduetotheperturba- to the boundary conditions, tion: ρ(E) ≡−J , ρ(M) ≡−⋆J , J (E) ≡ J ,and i i00 i i00 ij i0j TT (M) ∆[(1 + κχ/2)E ]=0, (6) J ≡ ⋆Ji0j (we use the convention that Greek letters ij TT go from 0 to 3 and Roman letters go from 1 to 3, unless ∆B =0, (7) otherwise stated). TT TT i(k·r−ωt) E We are interested in the macroscopic properties of the where Eab = Eab e is the traceless part of E B T TT system, so , , will represent Russakoffspatial aver- after it has been projected onto the interface, and Bab = TT i(k·r−ωt) B aged values [23], as is used in the macroscopic Maxwell’s Bab e is the traceless part of after it has been equations. We assume that Tij = χ(ω)Eij ,whereχ(ω) is projected onto the interface (subscripts a, b =1, 2). In TT TT the frequency dependent gravitational susceptibility [24]; the interface frame as shown in Fig. (1), E11 = −E22 = 2 TT TT TT TT we will later show a specific model of matter where α(1 − S /2), E12 = E21 = βC,andB11 = −B22 = 2 TT TT the induced stresses are of this form. We consider the −β(1 − S /2), B12 = B21 = αC,whereS ≡ sin γ = gravitoelectromagnetic fields to be of plane wave form, k∥/k and C ≡ cos γ = kz/k.∆Q ≡ Q1 − Q2 refers to the i(k·r−ωt) i(k·r−ωt) Eij (r,t)=Eij e , Bij (r,t)=Bij e .From change in quantity Q at the interface between medium 1 the GEM equations, E and B are transverse waves with and medium 2. two independent polarisations, ‘+’ and ‘×’, respectively We now adapt van Kampen et al.’s [26] contour in- defined by the only non-vanishing components E11 = tegral method, except with different boundary condi- −E22 = B12 = B21 = α and −B11 = B22 = E12 = E21 = tions, to get the gravitonic Casimir energy. Applying the β, in the proper frame of the plane wave. Here we define boundary conditions of Eq. (6) and (7) at the two plate the proper frame of the plane wave as that in which k is interfaces, we get the following system of linear homo- along the positive z-axis. geneous equations of variables α,α′,α′′,α′′′ for the ‘+’- The vacuum energy of any quantum field between par- polarisation (primes indicate the region of operation, as allel plates (separated by distance a) is a summation of represented in Fig. 1),

′ α′(1 + κχ/2)(1 − S′2/2)e−q a/2 = α(1 − S2/2)e−qa/2 + α′′(1 − S2/2)eqa/2 , (8) ′ α′′′(1 + κχ/2)(1 − S′2/2)e−q a/2 = α(1 − S2/2)eqa/2 + α′′(1 − S2/2)e−qa/2 , (9) ′ α′C′e−q a/2 = αCe−qa/2 − α′′Ceqa/2 , (10) ′ −α′′′C′e−q a/2 = αCeqa/2 − α′′Ce−qa/2 , (11) 3

2 2 2 2 2 ′2 ′2 2 where q ≡−kz = k∥ − ω /c ,andq ≡−kz = k∥ − The boundary conditions Eq. (6) and (7) at an in- (1 + κχ)ω2/c2. A non-trivial solution exists when the terface yield Eq. (8) and (10). Simultaneously solving determinant of the corresponding matrix is zero, Eq. (8) and (10) one gets

′ + −aq ′ 2 ′2 2 −aq α′′ −C′(S2 − 2) + (1 + κχ/2)C(S′2 − 2) ∆ ≡ e {[C (S − 2) − (1 + κχ/2)C(S − 2)] e = eqa . (18) α C′(S2 − 2) + (1 + κχ/2)C(S′2 − 2) − [C′(S2 − 2) + (1 + κχ/2)C(S′2 − 2)]2eaq} =0. (12) The boundary conditions also yield a similar set of linear Comparison of Eq. (18) with Eq. (16) shows that, |r+| = ′′ homogeneous equations of variables β,β′,β′′,β′′′,from |α /α|, i.e. r+ is the magnitude of the gravitational which we get for the ‘×’-polarisation, reflection coefficient of the ‘+’-polarisation. Similarly ′′ for the ‘×’-polarisation, |r×| = |β /β| is the magni- ′ ∆× ≡ e−aq {[(1 + κχ/2)C′(S2 − 2) − C(S′2 − 2)]2e−qa tude of the gravitational reflection coefficient of the ‘×’- polarisation. This tells us that Eq. (16) and (17) are − [(1 + κχ/2)C′(S2 − 2) + C(S′2 − 2)]2eqa} =0. (13) the Fresnel reflection coefficients for the two polarisations Solutions of Eq. (12) and (13) give the allowed modes of the gravitational wave. Therefore Eq. (15) has the + × between the plates, i.e. the eigenfrequencies (ωn , ωn ). same form as the Lifshitz formula for the EM Casimir There will be an infinite number of these eigenfrequen- energy at zero temperature, except the EM reflection cies. We can sum them in Eq. (5), using the argument coefficients have been replaced with their gravitational principle of complex analysis [27], equivalent. It is important to point out that this was not a priori obvious, as our starting point was the lin- −i∞ earised Einstein equations, which is fundamentally dif- 1 ferent from Maxwell’s equation of electromagnetism, al- ωn = ⎡ ωdln ∆+ ωdln ∆⎤ , (14) 2πi ! ! though GEM provides useful analogies between the two "n ⎣ i∞ C+ ⎦ theories. A marked difference is that the EM fields are where the closed contour of integration has been taken vectors whereas E and B are tensor fields. over a semicircle C+ of infinite radius in the right half We need now to find an estimate for χ(ω). A number of of the complex plane of ω with its centre at the origin, models of the interaction of GW with bulk matter exists. and the imaginary axis [i∞, −i∞], in a counterclockwise For example, Ref. [21, 28] considered a medium consisting manner. of molecules modelled as individual harmonic oscillators In the high frequency limit, the system does not to calculate the quadrupole moment induced by an inci- have time to respond to the rapid oscillations of the dent GW; Ref. [29] studied the interaction and dispersion field, and therefore will effectively act with time av- of GW in a hot gas. Here we use Peters’ [30] model who eraged behaviour. Analogous to the EM case, we considered the scattering of GW by the gravitational field thus make the natural assumption limw→∞ χ(ω)= of individual free particles of a thin sheet of matter. Pe- ′ limw→∞ dχ(ω)/dω =0andthereforeγ = γ in this limit ters derive a gravitational refractive index n which was i.e. at very high frequencies the plates are effectively much larger than that generated by just considering the transparent to the GW. With this assumption, the sec- induced quadrupole moments, suggesting that his model ond integral in Eq. (14) is independent of a. encapsulates the dominant GW interaction with matter. The summation over the infinite number of allowed Peters gives the gravitational refractive index as, modes will of course give an infinite value for E0.Toget the finite Casimir energy per unit area we take away the 2πGρ n =1+ 2 , (19) energy at infinite separation. Employing the argument ω principle to Eq. (5) and then integrating by parts, one where ρ is the density of the medium. µα α arrives at the gravitonic Casimir energy (ξ ≡−iω), Under the Lorentz gauge (h ,α =0),∂ ∂α hµν =

E0 E0 −2κTµν (note hµν is traceless). Using this, Eq. (19), E(a)= − lim 2 σ a→∞ σ and the relation Eij = −ω hij /2 , result in gravitational ! ∞ ∞ stresses of the previously assumed form, Tij = χ(ω)Eij 2 −2qa (15) = 2 k∥dk∥ [ln(1 − r+e ) where, 4π !0 !0 2 −2qa 2 + ln(1 − r×e )]dξ, 1 − n(ω) χ(ω)= 2 , (20) where, κc C′(S2 − 2) − (1 + κχ/2)C(S′2 − 2) with the high frequency limits: limw→∞ χ(ω)= r+ ≡ , (16) lim dχ(ω)/dω =0. C′(S2 − 2) + (1 + κχ/2)C(S′2 − 2) w→∞ Except for the lowest frequencies, for ordinary mate- ′ 2 ′2 (1 + κχ/2)C (S − 2) − C(S − 2) 4 r× ≡ . (17) rial parameters, χ ≪ c /G, and therefore the reflection (1 + κχ/2)C′(S2 − 2) + C(S′2 − 2) coefficients and Casimir pressure [P (a)=−∂E(a)/∂a] 4 are negligible. Specifically, if we take a typical O[ρ]= true for EM waves). We can use Eq. (21) in Eq. (15) 104kg/m3 and O[l]=1A,˚ then P (10−6) ≈ 10−21nPa. to calculate the gravitonic Casimir pressure between two This value is of course beyond detection. For the Casimir superconducting films. pressure to be measurable, the density of the material We calculate the gravitonic contribution to Casimir would have to be at the very least O[ρ]=O[c2/G] ≈ pressure for superconducting lead (Pb) of thickness d =2 1027kg/m3. This material density is clearly not achiev- nm at zero temperature. The EM skin depth of Pb is δ = able, at least terrestrially. 37 nm. We compare this with the photonic contribution Up until now we have only considered materials with to the Casimir pressure of superconducting lead. The classical properties. Recently, Minter et al. [12] pro- EM reflection coefficient is [12] pose that the quantum mechanical properties of super- conducting films can give rise to specular reflection of 2λδ2 −1 r = 1+ ξ , (22) GW. Their claim is that in an ordinary metal plate, the E ' cd2 ( ions and normal electrons locally co-move together along the same geodesics in the presence of a GW. However, where λ = 83 nm is the coherence length. The photonic when the plate becomes superconducting, the quantum- contribution to the Casimir pressure is calculated by us- mechanical non-localisability of the negatively charged ing Eq. (22) in the EM Lifshitz formula [1], which has Cooper pair undergoes non-geodesic motion, whereas the same form as Eq. (15). the positive charged ions of the lattice remains on the Eq. (22) and (21) are most valid when the driving fre- geodesic path. Specifically, the authors couple the grav- quency is less than the BCS gap frequency, as for higher itational field to the superconductor through DeWitt’s frequencies the dissipative component of the complex minimal coupling scheme [4]. To first-order, the ground conductivities would need to be taken into account. How- state of the delocalised Cooper pairs do not change in the ever, as higher frequency modes contribute exponentially presence of a GW whose frequency is less than the BCS less to the Casimir energy, one may still use the simple re- gap frequency. In comparison, the shift in the momentum flection coefficients derived by Minter et al. to obtain the of the localised ions is proportional to the gravitational first-order estimate of the Casimir pressure for supercon- vector potential. Because the Cooper pairs and ions are ductors at zero temperature. Fig. 2 compares the gravi- oppositely charged, a strong Coulomb force will resist this tonic to photonic contribution of the Casimir pressure separation of charge caused by the GW, resulting in its as a function of separation of plates of superconducting reflection. The authors dub this the Heisenberg-Coulomb Pb. It shows that gravitons can have a significant con- effect. A similar effect had been proposed in Ref. [6]. tribution to the Casimir pressure, via the H-C effect. In The origins of the arguments employed by Minter et fact, for the superconducting Pb film considered here, the al. are heuristical in nature, some of which we believe gravitons will dominate the Casimir pressure by an order require a much more formal approach to be convinc- of magnitude over photons. ing. This is echoed in a review article [14] on theories The magnitude of the Casimir pressure and plate sepa- of enhanced gravitational interaction with quantum flu- ration distance that we are talking about here, is compa- ids, which predates Ref. [12]; in particular the authors rable in size to what has already been achieved in current urge that a formal derivation of the quantum mechanical experiments [31, 32]. Few experiments however, have coupling between an electron and both the electromag- been conducted at low temperatures [33], as room tem- netic and gravitational field is needed. Nevertheless, the perature setups are a more experimentally accessible en- work by Minter et al. do yield results which can be used vironment. Of these low temperature investigations, only to falsify their theory. The H-C effect should enhance the the ALADIN project has experimented with supercon- Casimir pressure between superconducting plates. Here ducting aluminium (Al) film, separated by a thin oxide we quantify the size of this effect. layer from a thick gold plate, to observe how the Casimir Minter et al. give the reflection coefficient of a super- energy influences the superconducting phase transition conducting film from an incident GW as, (preliminary experimental results are reported in [34]). One could imagine modifying such an experiment to test 2δ2 −1 for the gravitonic contribution to the Casimir effect as r = 1+ ξ , (21) G ' cd ( described in this letter. We have derived here a Lifshitz-type formula for the where δ is the EM skin depth of the superconducting gravitonic Casimir effect for real bodies. Besides com- film and d the film thickness. In the thin superconduct- pleting a theoretical gap in our understanding of the ing film, the current flows in the x − y plane; therefore Casimir effect, this formula is important in light of re- only the normal incident component of the GW will drive cent models of enhanced gravitational interaction, as it the current. At normal incidence the magnitude of the allows us to quantify the gravitonic contribution to the reflection coefficient of the two polarisation modes are Casimir effect predicted by these theories. If measure- the same, as they only differ by a rotation (this is also ments of the Casimir pressure of the setup described in 5

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