Quantum gravitational interaction between two objects induced by external gravitational radiation fields
QQuantum gravitational interaction between two objects inducedby external gravitational radiation fields
Yongshun Hu, Jiawei Hu a , Hongwei Yu b1 Department of Physics and Synergetic InnovationCenter for Quantum Effects and Applications,Hunan Normal University, Changsha, Hunan 410081, China
Abstract
We explore, in the framework of linearized quantum gravity, the induced gravitational interactionbetween two gravitationally polarizable objects in their ground states in the presence of an externalquantized gravitational radiation field. The interaction energy decreases as r − in the near regime,and oscillates with a decreasing amplitude proportional to r − in the far regime, where r is thedistance between the two objects. The interaction can be either attractive or repulsive dependingon the propagation direction, polarization and frequency of the external gravitational field. That is,the induced interaction can be manipulated by varying the relative direction between the orientationof the objects with respect to the propagation direction of the incident gravitational radiation. a [email protected] b [email protected] a r X i v : . [ g r- q c ] J un . INTRODUCTION It is well known that in a quantum sense, there inevitably exist quantum vacuum fluc-tuations, which may induce some novel effects. One of the most famous examples is theelectromagnetic Casimir-Polder (CP) interaction [1]. In general, fluctuating electromagneticfields in vacuum induce instantaneous electric dipole moments in neutral atoms, which thencouple with each other via the exchange of virtual photons to yield an interaction energy.For atoms or molecules in different states, such CP interactions behave differently in termsof distance-dependence [2–15]. For example, the interatomic or intermolecular interactionbehaves as r − and r − in the near and far regimes respectively when the atoms or moleculesare in their ground states [1], while it behaves as r − and r − in the near and far regimesrespectively when they are prepared in a symmetric/antisymmetric entangled state [14].Likewise, one may also expect a gravitational CP-like interaction if one accepts that basicquantum principles are also applicable to gravity. Unfortunately, a full theory of quantumgravity is elusive at present. Even though, one may still study quantum gravitational effectsat low energies in the framework of linearized quantum gravity [16, 17], the basic idea ofwhich is to express the spacetime metric as a sum of the flat background spacetime metricand a linearized perturbation, and quantize the perturbation part in the canonical approach.Based on linearized quantum gravity, the gravitational CP-like interactions between twogravitationally polarizable objects in their ground states, and between one gravitationallypolarizable object and a gravitational boundary, have recently been studied in Refs. [18–23].Similar to the electromagnetic case, the behaviors of gravitational CP-like interactions aresignificantly different when the gravitationally polarizable objects are prepared in differentstates. For example, the gravitational CP-like potential is found to be proportional to r − and r − in the near and far regimes respectively when the two objects are in their groundstates [18–21], while it behaves as r − and r − in the near and far regimes respectively whenthe two objects are in a symmetric/antisymmetric entangled state [24].Naturally, a question arises as to whether such quantum gravitational effects can bemodified or enhanced in certain circumstances. Fortunately, there are similar examples inquantum electrodynamics. For example, the interaction between two ground-state atomsor molecules is found to be modified in the presence of external electromagnetic radiationfields [25–29]. That is, the externally applied electromagnetic field induces dipole moments2n atoms or molecules, which are coupled with each other via the exchange of a single virtualphoton, and an interaction is induced. This process is clearly different from the case withoutexternal electromagnetic fields, which arises from two-photon exchange. Similarly, in thegravitational case, one may expect that the quantum gravitational quadrupole-quadrupoleinteractions will also be modified in the presence of an external gravitational radiation field.In this paper, we explore the quantum gravitational quadrupole-quadrupole interactionbetween a pair of gravitationally polarizable objects in their ground states, which are sub-jected to a weak external gravitational radiation field based on the leading-order perturbationtheory in the framework of linearized quantum gravity. First, we describe in details the sys-tem we deal with. Then, we obtain the general expression for the interaction energy betweenthe two objects. Finally, we discuss our results in specific cases and obtain the correspond-ing interaction potentials. Throughout this paper, the Einstein summation convention forrepeated indices is assumed, and the Latin indices run from 1 to 3 while the Greek indicesrun from 0 to 3. Units with ¯ h = c = 16 πG = 1 are applied, where ¯ h is the reduced Planckconstant, c is the speed of light and G is the Newtonian gravitational constant. II. BASIC EQUATIONS
We consider two gravitationally polarizable objects (labeled as A and B) coupled with abath of fluctuating gravitational fields in vacuum, which are subjected to a weak externalgravitational radiation field. The objects A and B are modeled as two-level systems withtwo internal energy levels, ± ω , associated with the eigenstates | g (cid:105) and | e (cid:105) , respectively.The total Hamiltonian is H = H F + H R + H S + H I , (1)where H F is the Hamiltonian of the fluctuating vacuum gravitational field, H R the Hamilto-nian of the external gravitational radiation field, H S the Hamiltonian of the two-level systems(A and B), and H I the interaction Hamiltonian between the objects and the gravitationalfields. Here H I takes the form H I = − Q Aij [ (cid:15) ij ( (cid:126)x A ) + E ij ( (cid:126)x A )] − Q Bij [ (cid:15) ij ( (cid:126)x B ) + E ij ( (cid:126)x B )] , (2)where Q A ( B ) ij is the induced quadrupole moment of the object A (B), (cid:15) ij is the gravitoelectrictensor characterizing the weak external gravitational radiation field, and E ij is the grav-3toelectric tensor of the fluctuating vacuum gravitational fields defined as E ij = C i j byan analogy between the linearized Einstein field equations and the Maxwell equations [30],where C µναβ is the Weyl tensor. We write the spacetime metric g µν as a sum of the flatspacetime metric η µν and a linearized perturbation h µν , then the gravitoelectric tensor E ij can be expressed as (in the transverse traceless gauge) E ij = 12 ¨ h ij . (3)Suppose that the linearized perturbation h µν is quantized, in this regard, we can decompose h µν into positive and negative frequency parts h + µν and h − µν , respectively, and define thegravitational vacuum state | (cid:105) as h + µν | (cid:105) = 0 , (cid:104) | h − µν = 0 . (4)It follows immediately that (cid:104) | h µν | (cid:105) = 0. In general, however, (cid:104) | ( h µν ) | (cid:105) (cid:54) = 0, wherethe expectation value is understood to be suitably renormalized. In the transverse tracelessgauge, the quantized gravitational perturbations have only spatial components h ij , whichtakes the standard form h ij = (cid:88) (cid:126)p,λ (cid:115) ω (2 π ) (cid:104) a λ ( (cid:126)p ) e ( λ ) ij e i ( (cid:126)p · (cid:126)x − ωt ) + H.c. (cid:105) , (5)where a λ ( (cid:126)p ) is the annihilation operator of the gravitational vacuum field with wave vector (cid:126)p and polarization λ , e ( λ ) ij are polarization tensors, ω = | (cid:126)p | = ( p x + p y + p z ) / , and H.c.denotes the Hermitian conjugate. As for the weak external gravitational radiation field, weassume that it can be described as a quantized monochromatic gravitational wave containing N gravitons. Then, the corresponding gravitoelectric tensor (cid:15) ij can be given as (cid:15) ij = − (cid:118)(cid:117)(cid:117)(cid:116) ω R ρ n N (2 π ) (cid:104) b ( (cid:126)k ) e ( ε ) ij e i ( (cid:126)k · (cid:126)x − ω R t ) + H.c. (cid:105) , (6)where ρ n is the number density of gravitons, b ( (cid:126)k ) and e ( ε ) ij are respectively the correspondingannihilation operator and the polarization tensors with | (cid:126)k | = ω R , and ε labels the polariza-tion state.In the absence of an external gravitational field, the interaction between a pair of ground-state objects coupled with a bath of fluctuating gravitational fields in vacuum is a fourth-order effect [18–20]: The gravitational vacuum fluctuations induce quadrupole moments in4he two objects, which are correlated and an interaction energy is thus induced. Physi-cally speaking, such an induced interaction originates from vacuum fluctuations and arisesthrough the exchange of a pair of virtual gravitons between the two objects. In the presentcase, the leading interaction between quadrupole moments induced by the external gravita-tional radiation field will also be a fourth-order effect. However, the difference is that thequadrupole moments are now induced by the external gravitational field, which are thencorrelated to each other through gravitational vacuum fluctuations. That is, a real gravi-ton will be scattered by a pair of objects which are coupled via the exchange of a virtualgraviton, and an interaction is then induced, which is analogous to the electromagnetic case[25].We choose the initial state of the system to be | φ (cid:105) = | g A g B (cid:105)| (cid:105)| N (cid:105) , (7)where | g A g B (cid:105) is the ground state of the objects, | (cid:105) is the vacuum state of the fluctuat-ing gravitational field, and | N (cid:105) is the number state of the external gravitational radiationfield. The initial energy of the whole system is E φ = E + N ω R , where E denotes theground-state energy of the objects and fluctuating gravitational field in vacuum. The lead-ing contribution to the interaction energy can be obtained from fourth-order perturbationtheory, which contains 48 possible Feynman diagrams in our case, and a typical one isshown in Fig. 1. However, the calculations can be greatly simplified by collapsing the twoone-graviton interaction vertices in the time-ordered diagrams, which can be described as aneffective two-graviton interaction Hamiltonian. To do this, we introduce the gravitationalpolarizability of the objects, and, for simplicity, assume that the objects are isotropicallypolarizable. Then, the induced quadrupole can be expressed as Q A ( B ) ij = α ( ε ) A ( B ) (cid:15) ij , (8)where α ( ε ) A ( B ) is the isotropic polarizability of object A(B). In order to calculate the interac-tion, we only keep the corresponding terms after substituting Eq. (8) into Eq. (2). Then,the effective Hamiltonian takes the form H effI = − α ( ε ) A (cid:15) ij ( (cid:126)x A ) E ij ( (cid:126)x A ) − α ( ε ) B (cid:15) ij ( (cid:126)x B ) E ij ( (cid:126)x B ) . (9)The interaction energy can be calculated based on the second order perturbation theory∆ E = − (cid:88) I (cid:104) φ | H effI | I (cid:105)(cid:104) I | H effI | φ (cid:105) E I − E φ , (10)5 IG. 1. A typical time-ordered diagram for the calculation of inter-object interaction in the exis-tence of an external quantized gravitational field. The blue solid line represents a real graviton,while the dotted one represents a virtual one. with only four contributing time-ordered diagrams as shown in Fig. 2. Summing up all the
FIG. 2. Four time-ordered diagrams represent the four contributing terms in the second orderperturbation theory. E AB = − ω R ρ n π ) α ( ε ) A α ( ε ) B e ( ε ) ij e ( ε ) kl cos ( (cid:126)k · (cid:126)r ) (cid:90) d (cid:126)p (cid:88) λ e ( λ ) ij e ( λ ) kl ω ω − ω R e i(cid:126)p · (cid:126)r − ( N + 1) ω R ρ n N (2 π ) α ( ε ) A α ( ε ) B e ( ε ) ij e ( ε ) kl cos ( (cid:126)k · (cid:126)r ) (cid:90) d (cid:126)p (cid:88) λ e ( λ ) ij e ( λ ) kl ω ω + ω R e i(cid:126)p · (cid:126)r , (11)where (cid:126)r = (cid:126)x A − (cid:126)x B . Here the summation of polarization tensors in the transverse tracelessgauge gives [17] (cid:88) λ e ( λ ) ij e ( λ ) kl = δ ik δ jl + δ il δ jk − δ ij δ kl − ω H ijkl + 1 ω P ijkl , (12)where H ijkl = ∂ i ∂ j δ kl + ∂ k ∂ l δ ij − ∂ i ∂ k δ jl − ∂ i ∂ l δ jk − ∂ j ∂ k δ il − ∂ j ∂ l δ ik , P ijkl = ∂ i ∂ j ∂ k ∂ l . (13)For convenience, we define a gravitational radiation intensity I R in analogy to the electro-magnetic case [31] as I R = (cid:104) N | (cid:15) ij | N (cid:105) = ω R ρ n N (2 π ) (2 N + 1) , (14)since the intensity of the radiation field should be proportional to the number of gravitons.For a large graviton number N (cid:29)
1, we have I R (cid:39) ω R ρ n π ) . (15)Thus, the interaction energy (11) can be expressed as∆ E AB = − I R π α ( ε ) A α ( ε ) B e ( ε ) ij e ( ε ) kl cos ( (cid:126)k · (cid:126)r ) V ijkl , (16)where V ijkl = (cid:104) ( δ ik δ jl + δ il δ jk − δ ij δ kl ) ω R − ω R H ijkl + P ijkl (cid:105) cos ω R rr . (17)After some algebraic manipulations, the full form of V ijkl is given by V ijkl = 1 r (cid:20) ( δ ik δ jl + δ il δ jk − δ ij δ kl + ˆ r i ˆ r j δ kl + ˆ r k ˆ r l δ ij − ˆ r i ˆ r k δ jl − ˆ r i ˆ r l δ jk − ˆ r j ˆ r k δ il − ˆ r j ˆ r l δ ik + ˆ r i ˆ r j ˆ r k ˆ r l ) r ω R cos ω R r +2( − δ ik δ jl − δ il δ jk + δ ij δ kl − ˆ r i ˆ r j δ kl − ˆ r k ˆ r l δ ij + 2ˆ r j ˆ r k δ il +2ˆ r j ˆ r l δ ik + 2ˆ r i ˆ r k δ jl + 2ˆ r i ˆ r l δ jk − r i ˆ r j ˆ r k ˆ r l ) r ω R sin ω R r +( − δ ik δ jl − δ il δ jk + δ ij δ kl + 3ˆ r i ˆ r j δ kl + 3ˆ r k ˆ r l δ ij + 9ˆ r j ˆ r k δ il r j ˆ r l δ ik + 9ˆ r i ˆ r k δ jl + 9ˆ r i ˆ r l δ jk − r i ˆ r j ˆ r k ˆ r l ) r ω R cos ω R r +3( δ ik δ jl + δ il δ jk + δ ij δ kl − r i ˆ r j δ kl − r k ˆ r l δ ij − r j ˆ r k δ il − r j ˆ r l δ ik − r i ˆ r k δ jl − r i ˆ r l δ jk + 35ˆ r i ˆ r j ˆ r k ˆ r l ) rω R sin ω R r +3( δ ik δ jl + δ il δ jk + δ ij δ kl − r i ˆ r j δ kl − r k ˆ r l δ ij − r j ˆ r k δ il − r j ˆ r l δ ik − r i ˆ r k δ jl − r i ˆ r l δ jk + 35ˆ r i ˆ r j ˆ r k ˆ r l ) cos ω R r (cid:21) , (18)where ˆ r i is a component of the unit vector (cid:126)r/r . The above result shows that the totalinteraction energy depends on the polarization, frequency and propagation direction of theexternal gravitational radiation field. In the following, we consider two explicit examples.First, when the propagation direction of the external gravitational radiation field is par-allel to the orientation of the two objects, or equivalently, the polarization plane is perpen-dicular to (cid:126)r , i.e., (cid:126)k · (cid:126)r = ω R r and e ( ε ) ij ˆ r i = 0, Eq. (16) can be rewritten as∆ E AB = − I R πr α ( ε ) A α ( ε ) B e ( ε ) ij e ( ε ) ij (cid:16) r ω R cos ω R r − r ω R sin ω R r cos ω R r − r ω R cos ω R r +3 rω R sin ω R r cos ω R r + 3 cos ω R r (cid:17) , (19)where e ( ε ) ii = 0 and e ( ε ) ij = e ( ε ) ji have been applied. In the near regime, i.e., ω R r (cid:28)
1, theleading term takes the form ∆ E AB (cid:39) − I R πr α ( ε ) A α ( ε ) B e ( ε ) ij e ( ε ) ij , (20)while in the far regime, i.e., ω R r (cid:29)
1, it becomes∆ E AB (cid:39) − ω R I R πr α ( ε ) A α ( ε ) B e ( ε ) ij e ( ε ) ij cos ( ω R r + φ ) cos ω R r, (21)where φ = arcsin √ ω R r . This shows that the gravitational interaction between two objectsin the presence of an external gravitational field decreases as r − in the near regime, whilein the far regime it oscillates with a decreasing amplitude proportional to r − . Moreover,from Eqs. (20)-(21), we observe that in the near regime, the interaction is always attractive,while in the far regime, it can be attractive or repulsive depending on the frequency of theexternal gravitational field and the interobject distance.Second, if the propagation direction of the incident external gravitational radiation fieldis perpendicular to the orientation of the two objects, i.e., (cid:126)k · (cid:126)r = 0, then Eq. (16) yields∆ E AB = − I R πr α ( ε ) A α ( ε ) B (cid:104) (cid:16) e ( ε ) ij e ( ε ) ij − e ( ε ) i e ( ε ) i + e ( ε )11 e ( ε )11 (cid:17) r ω R cos ω R r (cid:16) − e ( ε ) ij e ( ε ) ij + 8 e ( ε ) i e ( ε ) i − e ( ε )11 e ( ε )11 (cid:17) r ω R sin ω R r +3 (cid:16) − e ( ε ) ij e ( ε ) ij + 12 e ( ε ) i e ( ε ) i − e ( ε )11 e ( ε )11 (cid:17) r ω R cos ω R r +3 (cid:16) e ( ε ) ij e ( ε ) ij − e ( ε ) i e ( ε ) i + 35 e ( ε )11 e ( ε )11 (cid:17) rω R sin ω R r +3 (cid:16) e ( ε ) ij e ( ε ) ij − e ( ε ) i e ( ε ) i + 35 e ( ε )11 e ( ε )11 (cid:17) cos ω R r (cid:105) , (22)where we have taken ˆ r i = (1 , ,
0) and (cid:126)k = (0 , , k ). In the near regime, i.e., ω R r (cid:28)
1, theleading term of Eq. (22) becomes∆ E AB (cid:39) − I R πr α ( ε ) A α ( ε ) B (cid:16) e ( ε ) ij e ( ε ) ij − e ( ε ) i e ( ε ) i + 35 e ( ε )11 e ( ε )11 (cid:17) . (23)So, the interaction energy decreases as r − in the near regime. Remarkably, it can be eitherattractive or repulsive depending on the polarization of the external gravitational radiationfield. For example, the interaction is attractive if the polarization tensor contains onlydiagonal elements which may correspond to the + mode of gravitational waves, while itbehaves as repulsive when there are only off-diagonal elements which may correspond to the × mode. In the far regime, i.e., ω R r (cid:29)
1, Eq. (22) reduces to∆ E AB (cid:39) − ω R I R πr α ( ε ) A α ( ε ) B (cid:16) e ( ε ) ij e ( ε ) ij − e ( ε ) i e ( ε ) i + e ( ε )11 e ( ε )11 (cid:17) cos ω R r. (24)That is, when the propagation direction of external gravitational radiation field is perpendic-ular to the orientation of the objects, the interaction energy in the far regime oscillates witha decreasing amplitude which is proportional to r − . The interaction can be attractive orrepulsive depending on the polarization and frequency of the external gravitational radiationfield, and the interobject distance. For a given external field, the interaction periodicallybehaves between attractive and repulsive as the interobject distance varies. III. DISCUSSION
In this paper, we investigate the gravitational quadrupole-quadrupole interaction betweentwo gravitationally polarizable objects coupled with a bath of fluctuating gravitational fieldsin vacuum in the presence of a weak quantized gravitational radiation field, based on theleading order perturbation theory in the framework of linearized quantum gravity. Our resultshows that the interaction energy behaves as r − in the near regime and oscillates with adecreasing amplitude proportional to r − in the far regime. The interaction can be either9ttractive or repulsive, depending on the polarization, frequency and direction of propagationof the external gravitational field. When the orientation of the two objects is parallel tothe propagation direction of the incident gravitational radiation field, the interaction isalways attractive in the near regime, while in the far regime it can be attractive or repulsivedepending on the frequency of the external gravitational field and the interobject distance.When the orientation of the objects is perpendicular to the propagation direction of theincident gravitational radiation field, the attractive or repulsive property of the interactiondepends on the polarization of the incident gravitational radiation in the near regime, whilein the far regime it also depends on the frequency of the external gravitational field andthe interobject distance. To conclude, the induced gravitational interaction due to a weakexternal gravitational field can be manipulated by changing the relative orientation of theobjects with respect to the propagation direction of the incident gravitational field.Finally, let us note that there are contributions from other multipole moments to the inter-object interactions (such as monopole-quadrupole cross terms). In the presence of gravita-tional waves, a mass monopole oscillates, and an effective mass quadrupole is formed as seenby a distant observer in analogy to the electromagnetic case [32]. Therefore, the monopole-monopole and monopole-quadrupole interactions due to gravitational vacuum fluctuationsand in the presence of external gravitational waves can also be investigated in the presentformalism. We hope to turn to these issues in the future. ACKNOWLEDGMENTS
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