This paper was converted on www.awesomepapers.org from LaTeX by an anonymous user.
Want to know more? Visit the Converter page.

Repulsive quantum gravitoelectric-gravitomagnetic interaction

Di Hao Department of Physics, Synergetic Innovation Center for Quantum Effects and Applications,
and Institute of Interdisciplinary Studies,
Hunan Normal University, Changsha, Hunan 410081, China
   Jiawei Hu jwhu@hunnu.edu.cn Department of Physics, Synergetic Innovation Center for Quantum Effects and Applications,
and Institute of Interdisciplinary Studies,
Hunan Normal University, Changsha, Hunan 410081, China
   Hongwei Yu hwyu@hunnu.edu.cn Department of Physics, Synergetic Innovation Center for Quantum Effects and Applications,
and Institute of Interdisciplinary Studies,
Hunan Normal University, Changsha, Hunan 410081, China
Abstract

We investigate, in the framework of linearized quantum gravity, the quantum gravitational interaction between a gravitoelectrically polarizable object and a gravitomagnetically polarizable object. This interaction originates from the coupling between the instantaneous mass quadrupole moment and the mass-current quadrupole moment of the objects, induced by fluctuating gravitoelectric and gravitomagnetic fields in a vacuum. Using leading-order perturbation theory, we derive the explicit expression of the quantum gravitoelectric-gravitomagnetic interaction energy, which shows a distance dependence of r8r^{-8} in the near regime and r11r^{-11} in the far regime, where rr is the distance between the two objects. Remarkably, this interaction energy is positive, indicating that the force is repulsive. Since interactions between objects polarizable in the same gravitoelectric or gravitomagnetic manner are inherently attractive, for objects which are both gravitoelectrically and gravitomagnetically polarizable, the overall quantum gravitational interaction potential is reduced when the repulsive quantum gravitoelectric-gravitomagnetic interaction is taken into account. However, for two isotropically polarizable objects with identical gravitoelectric and gravitomagnetic polarizabilities and energy level spacing, the repulsive quantum interaction cannot surpass the attractive interactions.

I Introduction

Gravitational waves, ripples in spacetime predicted by general relativity Einstein , were not directly confirmed until the Laser Interferometer Gravitational-Wave Observatory (LIGO) detected signals from black hole mergers LIGO . This detection measured their tiny effects on the length differences between the arms of the interferometer. More recently, this prediction was further validated by the detection of nanohertz stochastic background gravitational wave signals, observed through their influence on the arrival times of radio pulses from arrays of pulsars CPTA ; PPTA ; EPTA ; NANO . These discoveries highlight the classical effects of gravity described by general relativity.

Naturally, one may wonder what happens if, like electromagnetic waves, gravitational waves can also be quantized. Unfortunately, a complete theory of quantum gravity remains elusive. Nevertheless, quantum gravitational phenomena at low-energy scales can still be explored. For example, by treating general relativity as an effective field theory, quantum corrections to the classical Newtonian gravitational interaction between a pair of mass monopoles have been investigated by summing one-loop Feynman diagrams involving off-shell gravitons Donoghue1994prl ; Donoghue1994prd ; Hamber1995 ; Kirilin2002 ; Holstein2003 ; Holstein2005 .

If gravity is indeed quantum, a natural consequence would be the inevitable existence of fluctuating background gravitational fields, even in a vacuum. These gravitational quantum vacuum fluctuations, representing typical low-energy quantum gravitational effects, can be studied using linearized quantum gravity, where the gravitational field is treated as a linear perturbation on flat spacetime. In weak gravity scenarios like the one considered here, the gravitational field equations can be expressed in a form analogous to Maxwell’s equations, known as Weyl gravitoelectromagnetism Campbell1976 ; Matte1953 ; Campbell1971 ; Szekeres1971 ; Maartens1998 ; Ruggiero2002 ; Ramos2010 . In this formalism, the gravitational field is decomposed into two parts: the gravitoelectric field and the gravitomagnetic field, analogous to the electric and magnetic fields, respectively.

Any nonpointlike object would then be polarized by these fluctuating gravitational fields, inducing instantaneous mass quadrupole moments from the gravitoelectric field and mass-current quadrupole moments from the gravitomagnetic field. Consequently, quantum corrections to the classical Newtonian gravitational interaction arise from interactions between these instantaneous quadrupole moments. In this context, quantum corrections due to the gravitational interaction between instantaneous mass quadrupole moments induced by fluctuating gravitoelectric fields have been studied both between a nonpointlike object and a boundary Hu2017 , and between two nonpointlike objects Ford2016 ; Wu2016 ; Holstein2017 . Subsequently, these effects have been explored in several scenarios, including the quantum gravitational interaction between two nonpointlike objects induced by a thermal bath of gravitons Wu2017 , and the interaction between two nonpointlike objects near a gravitational boundary yu2018 . Additionally, studies have investigated the interaction between a pair of nonpointlike objects in symmetric and antisymmetric entangled states yongs2020epjc , as well as the nonadditive interaction among three nonpointlike objects yongs2022prd .

Recently, quantum corrections due to the gravitational interaction between instantaneous mass-current quadrupole moments induced by fluctuating gravitomagnetic fields in a vacuum have also been investigated hao2024 . Interestingly, the quantum gravitational interactions between two gravitoelectrically polarizable objects Ford2016 ; Wu2016 ; Holstein2017 and between two gravitomagnetically polarizable objects hao2024 both exhibit an r10r^{-10} dependence in the nonretarded regime and an r11r^{-11} dependence in the retarded regime, and are both inherently attractive.

Therefore, a gap remains in our understanding of the quantum gravitational interaction between two nonpointlike objects arising from the instantaneous quadrupole moments induced by gravitational vacuum fluctuations, specifically, the quantum corrections due to the gravitational interaction between an instantaneous mass quadrupole moment and a mass-current quadrupole moment. Moreover, it is particularly interesting to determine whether this interaction leads to an attractive force that enhances the classical gravitational interaction, or a repulsive force that causes a suppression, given that previous quantum corrections involving mass and mass-current quadrupole moments have resulted in attractive forces.

To complete this final piece of the puzzle, we now investigate the quantum gravitational interaction energy between a gravitoelectrically polarizable object and a gravitomagnetically polarizable object. Specifically, we focus on the distance dependence of this interaction energy and whether the interaction is repulsive or attractive.

The paper is organized as follows. First, we derive the quantum gravitational energy resulting from the crossed gravitoelectric-gravitomagnetic term using leading-order perturbation theory. Then, we examine its asymptotic behavior in both the near and far regimes, and analyze the sign of the energy to determine whether the interaction is repulsive or attractive. Throughout the paper, we adopt Greek letters for four-dimensional spacetime indices (0-3), and Latin letters for three-dimensional spatial indices (1-3). We also employ the Einstein summation convention for repeated indices. Unless otherwise specified, natural units c==1c=\hbar=1 are adopted.

II The basic formalism

Under the weak-field approximation, the spacetime metric gμνg_{\mu\nu} can be expressed as a linearized perturbation hμνh_{\mu\nu} propagating on a flat background spacetime ημν\eta_{\mu\nu}, i.e., gμν=ημν+hμνg_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}. Within the framework of linearized quantum gravity, hμνh_{\mu\nu} is quantized, and in the transverse-traceless (TT) gauge, it takes the standard form Wu2017 ; Oniga2016

hij(r,t)=k,λ8πG(2π)3ωeij(k,λ)[gk,λeikriωt+gk,λeikr+iωt],h_{ij}(\textbf{r},t)=\sum_{\textbf{k},\lambda}\sqrt{\frac{8\pi G}{(2\pi)^{3}\omega}}e_{ij}(\textbf{k},\lambda)\left[g_{\textbf{k},\lambda}e^{i\textbf{k}\cdot\textbf{r}-i\omega t}+g_{\textbf{k},\lambda}^{\dagger}e^{-i\textbf{k}\cdot\textbf{r}+i\omega t}\right], (1)

where GG is the Newton’s gravitational constant, k the wave vector, ω=|k|\omega=|\textbf{k}| the frequency, eij(k,λ)e_{ij}(\textbf{k},\lambda) the gravitational polarization tensor, λ\lambda the polarization states, and gk,λg_{\textbf{k},\lambda} and gk,λg_{\textbf{k},\lambda}^{\dagger} the annihilation and creation operators of the gravitational field with wave vector k and polarization λ\lambda satisfying the following commutation relations:

[gk,λ,gk,λ]=0,\displaystyle\left[g_{\textbf{k},\lambda},g_{\textbf{k}\prime,\lambda\prime}\right]=0,
[gk,λ,gk,λ]=0,\displaystyle\left[g_{\textbf{k},\lambda}^{\dagger},g_{\textbf{k}\prime,\lambda\prime}^{\dagger}\right]=0,
[gk,λ,gk,λ]=δλλδkk,\displaystyle\left[g_{\textbf{k},\lambda},g_{\textbf{k}\prime,\lambda\prime}^{\dagger}\right]=\delta_{\lambda\lambda\prime}\delta_{\textbf{kk}\prime}, (2)

with δab\delta_{ab} being the Kronecker symbol.

The fluctuating gravitational fields can induce quadrupole moments in a nonpointlike object, and the interaction Hamiltonian between the fluctuating gravitational fields and the induced quadrupole moments can be written as hao2024

Hint\displaystyle H_{\mathrm{int}} =12QijEij13SijBij,\displaystyle=-\frac{1}{2}Q^{ij}E_{ij}-\frac{1}{3}S^{ij}B_{ij}, (3)

where QijQ^{ij} represents the induced mass quadrupole moment, and SijS^{ij} denotes the induced mass-current quadrupole moment of the object. Here, EijE_{ij} and BijB_{ij} are the gravitoelectric and gravitomagnetic fields, respectively, which are defined as Campbell1976 ; Matte1953 ; Campbell1971 ; Szekeres1971 ; Maartens1998 ; Ruggiero2002 ; Ramos2010

Eij=C0i0j,E_{ij}=-C_{0i0j}, (4)

and

Bij=12ϵiflCfl0j,B_{ij}=\frac{1}{2}\epsilon_{ifl}C_{fl0j}, (5)

through an analogy between the Maxwell equations and the linearized gravitational field equations under the weak-field approximation, where CαβγσC_{\alpha\beta\gamma\sigma} is the Weyl tensor, and ϵifl\epsilon_{ifl} is the third-order Levi-Civita tensor. The object-field interaction Hamiltonian given by Eq. (3) describes the gravitational quadrupolar coupling. Specifically, the explicit expressions for the two kinds of gravitational quadrupole moments take the standard form hao2024 ; Flanagan2007 ; Rezzolla1999

Qjk=d3xρ(x)(xjxk13δjkr2),Q_{jk}=\int d^{3}x\rho(x)(x_{j}x_{k}-\frac{1}{3}\delta_{jk}r^{2}), (6)

and

Slj=d3xϵif(lxj)xiρ(x)vf,S_{lj}=\int d^{3}x\epsilon_{if(l}x_{j)}x_{i}\rho(x)v_{f}, (7)

where ρ(x)\rho(x) is the mass density of the nonpointlike object, and ρ(x)vf\rho(x)v_{f} denotes the localized mass-current density. Note that A(ij)=12(Aij+Aji)A_{(ij)}=\frac{1}{2}(A_{ij}+A_{ji}).

Furthermore, based on Weyl gravitoelectromagnetism and the theory of linearized quantum gravity, the explicit expressions for the quantized gravitoelectric and gravitomagnetic fields can be derived as follows. Under the weak-field approximation, the Riemann curvature tensor RαβμνR_{\alpha\beta\mu\nu} can be written in terms of the gravitational perturbation hμνh_{\mu\nu} as

Rαβμν=12(βμhαναμhβνβνhαμ+ανhβμ).R_{\alpha\beta\mu\nu}=\frac{1}{2}\left(\partial_{\beta}\partial_{\mu}h_{\alpha\nu}-\partial_{\alpha}\partial_{\mu}h_{\beta\nu}-\partial_{\beta}\partial_{\nu}h_{\alpha\mu}+\partial_{\alpha}\partial_{\nu}h_{\beta\mu}\right). (8)

Then, according to the definitions of EijE_{ij} and BijB_{ij} shown in Eqs. (4) and (5), and noting that the Riemann curvature tensor is equivalent to the Weyl tensor in the vacuum case, the gravitoelectric and gravitomagnetic fields can be expressed as

Eij=12h¨ij,\displaystyle E_{ij}=-\frac{1}{2}\ddot{h}_{ij}, (9)

and

Bij=12ϵiflfh˙lj,\displaystyle B_{ij}=-\frac{1}{2}\epsilon_{ifl}\partial_{f}\dot{h}_{lj}, (10)

respectively, where a dot denotes the first derivative with respect to time. Substituting Eq. (1) into Eqs. (9) and (10), the quantized fields take the form

Eij(r,t)=12λd3k8πGω3(2π)3eij(k,λ)[gk,λ(t)eikr+gk,λ(t)eikr],\displaystyle E_{ij}(\textbf{r},t)=-\frac{1}{2}\sum_{\lambda}\int d^{3}\textbf{k}\sqrt{\frac{8\pi G\omega^{3}}{(2\pi)^{3}}}e_{ij}(\textbf{k},\lambda)\left[g_{\textbf{k},\lambda}(t)e^{i\textbf{k}\cdot\textbf{r}}+g_{\textbf{k},\lambda}^{\dagger}(t)e^{-i\textbf{k}\cdot\textbf{r}}\right], (11)

and

Bij(r,t)=12λd3k8πGω3(2π)3ϵifle3felj(k,λ)[gk,λ(t)eikr+gk,λ(t)eikr],\displaystyle B_{ij}(\textbf{r},t)=-\frac{1}{2}\sum_{\lambda}\int d^{3}\textbf{k}\sqrt{\frac{8\pi G\omega^{3}}{(2\pi)^{3}}}\epsilon_{ifl}e^{f}_{3}e_{lj}(\textbf{k},\lambda)\left[g_{\textbf{k},\lambda}(t)e^{i\textbf{k}\cdot\textbf{r}}+g_{\textbf{k},\lambda}^{\dagger}(t)e^{-i\textbf{k}\cdot\textbf{r}}\right], (12)

respectively. Here, gk,λ(t)=gk,λeiωtg_{\textbf{k},\lambda}(t)=g_{\textbf{k},\lambda}e^{-i\omega t} and gk,λ(t)=gk,λeiωtg^{\dagger}_{\textbf{k},\lambda}(t)=g^{\dagger}_{\textbf{k},\lambda}e^{i\omega t}. e3=k/|k|\textbf{e}_{3}=\textbf{k}/|\textbf{k}| represents the unit vector along the propagation direction of the gravitational field, and e3f(f=x,y,z)e^{f}_{3}~{}(f=x,y,z) denotes the ffth coordinate component of e3\textbf{e}_{3}.

III The quantum gravitoelectric-gravitomagnetic interaction

We consider a pair of gravitationally polarizable objects A and B coupled to fluctuating gravitational fields in a vacuum. Both objects can be modeled as two-level systems, with the excited and ground states being |eA(B)|e_{A(B)}\rangle and |gA(B)|g_{A(B)}\rangle respectively, and the corresponding energy level spacing is ωA(B)\omega_{A(B)}. The total Hamiltonian of the object-field system can thus be written as

Htot=HA+HB+HF+HAF+HBF,H_{\mathrm{tot}}=H_{A}+H_{B}+H_{F}+H_{AF}+H_{BF}, (13)

where HFH_{F} is the Hamiltonian of the fluctuating gravitational fields, HA(B)H_{A(B)} is the Hamiltonian of object A(B), and HA(B)FH_{A(B)F} represents the interaction between object A(B) and fluctuating gravitational fields. For simplicity, we assume that object A is gravitoelectrically polarizable while object B is gravitomagnetically polarizable; hence, the interaction Hamiltonian of the system reads

Hintcross=HAF+HBF=12QAijEij(rA)13SBijBij(rB).H^{\mathrm{cross}}_{\mathrm{int}}=H_{AF}+H_{BF}=-\frac{1}{2}Q^{ij}_{A}E_{ij}(\textbf{r}_{A})-\frac{1}{3}S^{ij}_{B}B_{ij}(\textbf{r}_{B}). (14)

We assume that both objects are initially in their ground states, |gA|g_{A}\rangle and |gB|g_{B}\rangle, and the gravitational field is in the vacuum state |0|0\rangle. Therefore, the initial state of the whole system can be expressed as

|ϕ=|gA|gB|0.|\phi\rangle=|g_{A}\rangle|g_{B}\rangle|0\rangle. (15)

Then, the quantum gravitational interaction between the two objects induced by gravitational vacuum fluctuations can be obtained using fourth-order perturbation theory, which takes the standard form

ΔEABcross=I,II,IIIϕϕ|Hintcross|II|Hintcross|IIII|Hintcross|IIIIII|Hintcross|ϕ(EIEϕ)(EIIEϕ)(EIIIEϕ).\Delta E^{\mathrm{cross}}_{AB}=-\sum_{\mathrm{I},\mathrm{II},\mathrm{III}\neq\phi}\frac{\langle\phi|H^{\mathrm{cross}}_{\mathrm{int}}|\mathrm{I}\rangle\langle\mathrm{I}|H^{\mathrm{cross}}_{\mathrm{int}}|\mathrm{II}\rangle\langle\mathrm{II}|H^{\mathrm{cross}}_{\mathrm{int}}|\mathrm{III}\rangle\langle\mathrm{III}|H^{\mathrm{cross}}_{\mathrm{int}}|\phi\rangle}{(E_{\mathrm{I}}-E_{\phi})(E_{\mathrm{II}}-E_{\phi})(E_{\mathrm{III}}-E_{\phi})}. (16)

Here, |I|\mathrm{I}\rangle, |II|\mathrm{II}\rangle and |III|\mathrm{III}\rangle are the three intermediate states in the interaction processes. See Table 2 in Appendix A for all possible intermediate states and the corresponding energy denominators in Eq. (16).

Taking case (1) in Table 2 as an example. Substituting |I|\mathrm{I}\rangle , |II|\mathrm{II}\rangle and |III|\mathrm{III}\rangle into Eq. (16) yields

ΔEAB(1)cross(rA,rB)\displaystyle\Delta E^{\mathrm{cross}}_{AB(1)}(\textbf{r}_{A},\textbf{r}_{B}) =\displaystyle= 1360+𝑑ω0+𝑑ω1𝒟(1)[Q^AijS^BabGijabEM(ω,rA,rB)]\displaystyle-\frac{1}{36}\int_{0}^{+\infty}d\omega\int_{0}^{+\infty}d{\omega}^{\prime}\frac{1}{\mathcal{D}_{\mathrm{(1)}}}\left[\hat{Q}^{ij}_{A}\hat{S}^{ab}_{B}G^{\mathrm{E-M}}_{ijab}\left(\omega,\textbf{r}_{A},\textbf{r}_{B}\right)\right] (17)
×[Q^AklS^BcdGklcdEM(ω,rA,rB)].\displaystyle\times\left[\hat{Q}^{*kl}_{A}\hat{S}^{*cd}_{B}G^{\mathrm{E-M}}_{klcd}\left({\omega}^{\prime},\textbf{r}_{A},\textbf{r}_{B}\right)\right].

Here 𝒟(1)\mathcal{D}_{(1)} is the energy denominator corresponding to case (1), Q^Aij=gA|QAij|eA\hat{Q}^{ij}_{A}=\langle g_{A}|Q^{ij}_{A}|e_{A}\rangle is the mass quadrupole transition matrix element, and S^Bij=gB|SBij|eB\hat{S}^{ij}_{B}=\langle g_{B}|S^{ij}_{B}|e_{B}\rangle is the mass-current quadrupole transition matrix element. The conjugate terms are denoted as Q^Aij\hat{Q}^{*ij}_{A} and S^Bij\hat{S}^{*ij}_{B}, respectively. GijabEM(ω,rA,rB)G^{\mathrm{E-M}}_{ijab}(\omega,\textbf{r}_{A},\textbf{r}_{B}) denotes the crossed two-point correlation function of the gravitoelectric and gravitomagnetic fields in the frequency domain, which can be expressed as

GijabEM(ω,rA,rB)=0|Eij(ω,rA)Bab(ω,rB)|0.G^{\mathrm{E-M}}_{ijab}(\omega,\textbf{r}_{A},\textbf{r}_{B})=\langle 0|E_{ij}(\omega,\textbf{r}_{A})B_{ab}(\omega,\textbf{r}_{B})|0\rangle. (18)

Next, we organize the contributions from the remaining 11 intermediate processes into the form of Eq. (17), respectively, and then sum up all the contributions from the 12 intermediate processes. Generally, for a nonpointlike object that obeys time-reversal symmetry, the eigenstates of the object’s Hamiltonian in the position representation can be chosen as real functions Buhmann2013 . Hence, we consider objects with real space-part wave functions ψn(𝐫)\psi_{n}\left(\mathbf{r}\right), where nn denotes the quantum number of the energy eigenstate. On the other hand, the nonpointlike object considered here can be modeled as a system composed of several mass points mm_{\mathbb{P}} (AorB)(\mathbb{P}\in\mathrm{A\ or\ B}). Then, the mass density distribution ρ(x)\rho(x) in the quadrupole moments Eqs. (6) and (7) takes the form of mδ(3)(xx)\sum_{\mathbb{P}}{m_{\mathbb{P}}\delta^{(3)}(x-x_{\mathbb{P}})}, where δ(3)(x)\delta^{(3)}(x) is the three-dimensional Dirac delta function. Therefore, the mass and mass-current quadrupole transition matrix elements can be expressed as

n|QAij|m=d3rkψn(rk)[Am(xixj13δijr2)]ψm(rk),\displaystyle\langle n|Q_{A}^{ij}|m\rangle=\int{d^{3}r_{k}}\psi_{n}^{*}\left(r_{k}\right)\left[\sum_{\mathbb{P}\in\mathrm{A}}{m_{\mathbb{P}}\left(x_{i}^{\mathbb{P}}x_{j}^{\mathbb{P}}-\frac{1}{3}\delta_{ij}r_{\mathbb{P}}^{2}\right)}\right]\psi_{m}\left(r_{k}\right), (19)

and

n|SBij|m=d3rkψn(rk)[Bϵkl(ixj)xk(il)]ψm(rk),\displaystyle\langle n|S_{B}^{ij}|m\rangle=\int{d^{3}r_{k}}\psi_{n}^{*}\left(r_{k}\right)\left[\sum_{\mathbb{P}\in\mathrm{B}}{\epsilon_{kl\left(i\right.}x^{\mathbb{P}}_{\left.j\right)}x^{\mathbb{P}}_{k}(-i\hbar\nabla^{\mathbb{P}}_{l}})\right]\psi_{m}\left(r_{k}\right), (20)

respectively. Note that the momentum mvlm_{\mathbb{P}}v^{\mathbb{P}}_{l} of mass point \mathbb{P} in object B has been replaced by the quantum operator il-i\hbar\nabla^{\mathbb{P}}_{l}. From Eqs. (19) and (20), it is clear that the mass quadrupole transition matrix elements are purely real, satisfying Q^Aij=Q^Aij\hat{Q}^{ij}_{A}=\hat{Q}^{*ij}_{A}, while the mass-current quadrupole transition matrix elements are purely imaginary, leading to S^Bij=S^Bij\hat{S}^{ij}_{B}=-\hat{S}^{*ij}_{B}. Based on this, and with the help of the symmetry of the two-point function (see Appendix B for the proof)

GijklEM(r,r,t,t)=0|Bij(r,t)Ekl(r,t)|0=GijklME(r,r,t,t),\displaystyle G^{\mathrm{E-M}}_{ijkl}(\textbf{r},{\textbf{r}}^{\prime},t,t^{\prime})=-\langle 0|B_{ij}(\textbf{r},t)E_{kl}(\textbf{r}^{\prime},t^{\prime})|0\rangle=-G^{\mathrm{M-E}}_{ijkl}(\textbf{r},{\textbf{r}}^{\prime},t,t^{\prime}), (21)

we find, after an inspection of all the 12 possible intermediate processes, that the contributions to the interaction energy (16) corresponding to cases (2), (3), (7), (8), and (9) can be organized in the same form as Eq. (17) and carry the same sign. In contrast, the contributions from cases (4), (5), (6), (10), (11), and (12) can also be organized in the same form as Eq. (17) but carry the opposite sign. Taking this into account, the total interaction energy between the objects is obtained as follows:

ΔEABcross(rA,rB)\displaystyle\Delta E^{\mathrm{cross}}_{AB}(\textbf{r}_{A},\textbf{r}_{B}) =\displaystyle= 1360+dω0+dω(1𝒟(1)+1𝒟(2)+1𝒟(3)+1𝒟(7)+1𝒟(8)+1𝒟(9)\displaystyle-\frac{1}{36}\int_{0}^{+\infty}d\omega\int_{0}^{+\infty}d{\omega}^{\prime}\Big{(}\frac{1}{\mathcal{D}_{(1)}}+\frac{1}{\mathcal{D}_{(2)}}+\frac{1}{\mathcal{D}_{(3)}}+\frac{1}{\mathcal{D}_{(7)}}+\frac{1}{\mathcal{D}_{(8)}}+\frac{1}{\mathcal{D}_{(9)}} (22)
1𝒟(4)1𝒟(5)1𝒟(6)1𝒟(10)1𝒟(11)1𝒟(12))\displaystyle-\frac{1}{\mathcal{D}_{(4)}}-\frac{1}{\mathcal{D}_{(5)}}-\frac{1}{\mathcal{D}_{(6)}}-\frac{1}{\mathcal{D}_{(10)}}-\frac{1}{\mathcal{D}_{(11)}}-\frac{1}{\mathcal{D}_{(12)}}\Big{)}
×[Q^AijS^BabGijabEM(ω,rA,rB)][Q^AklS^BcdGklcdEM(ω,rA,rB)]\displaystyle\times\left[\hat{Q}^{ij}_{A}\hat{S}^{ab}_{B}G^{\mathrm{E-M}}_{ijab}\left(\omega,\textbf{r}_{A},\textbf{r}_{B}\right)\right]\left[\hat{Q}^{*kl}_{A}\hat{S}^{*cd}_{B}G^{\mathrm{E-M}}_{klcd}\left({\omega}^{\prime},\textbf{r}_{A},\textbf{r}_{B}\right)\right]
=\displaystyle= 1360+𝑑ω0+𝑑ωQ^AijQ^AklS^BabS^BcdGijabEM(ω,rA,rB)GklcdEM(ω,rA,rB)\displaystyle-\frac{1}{36}\int_{0}^{+\infty}d\omega\int_{0}^{+\infty}d{\omega}^{\prime}\hat{Q}^{ij}_{A}\hat{Q}^{*kl}_{A}\hat{S}^{ab}_{B}\hat{S}^{*cd}_{B}G^{\mathrm{E-M}}_{ijab}(\omega,\textbf{r}_{A},\textbf{r}_{B})G^{\mathrm{E-M}}_{klcd}({\omega}^{\prime},\textbf{r}_{A},\textbf{r}_{B})
×4(ωA+ωB+ω)(ωA+ωB)(ωA+ω)(ωB+ω)(1ω+ω1ωω).\displaystyle\times\frac{4\left(\omega_{A}+\omega_{B}+\omega\right)}{\left(\omega_{A}+\omega_{B}\right)\left(\omega_{A}+\omega)(\omega_{B}+\omega\right)}\left(\frac{1}{\omega+{\omega}^{\prime}}-\frac{1}{{\omega}^{\prime}-\omega}\right).

For simplicity, we assume that both objects are isotropically polarizable. As a result, the mass and mass-current quadrupole transition matrix elements satisfy

Q^AijQ^Akl=(δikδjl+δilδjk)α^A,\hat{Q}^{ij}_{A}\hat{Q}^{*kl}_{A}=(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\hat{\alpha}_{A}, (23)

and

S^BabS^Bcd=(δacδbd+δadδbc)χ^B,\hat{S}^{ab}_{B}\hat{S}^{*cd}_{B}=(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})\hat{\chi}_{B}, (24)

respectively, where α^A=|Q^Aij|2\hat{\alpha}_{A}=|\hat{Q}^{ij}_{A}|^{2} and χ^B=|S^Bab|2\hat{\chi}_{B}=|\hat{S}^{ab}_{B}|^{2}. Substituting Eqs. (23) and (24) into Eq. (22), one can obtain

ΔEABcross(rA,rB)\displaystyle\Delta E^{\mathrm{cross}}_{AB}(\textbf{r}_{A},\textbf{r}_{B}) =\displaystyle= 19(ωA+ωB)0+𝑑ω0+𝑑ωGijabEM(ω,rA,rB)GijabEM(ω,rA,rB)\displaystyle-\frac{1}{9(\omega_{A}+\omega_{B})}\int_{0}^{+\infty}d\omega\int_{0}^{+\infty}d{\omega}^{\prime}G^{\mathrm{E-M}}_{ijab}(\omega,\textbf{r}_{A},\textbf{r}_{B})G^{\mathrm{E-M}}_{ijab}({\omega}^{\prime},\textbf{r}_{A},\textbf{r}_{B}) (25)
×α^Aχ^B(ωA+ωB+ω)(ωA+ω)(ωB+ω)(1ω+ω1ωω).\displaystyle\times\frac{\hat{\alpha}_{A}\hat{\chi}_{B}\left(\omega_{A}+\omega_{B}+\omega\right)}{\left(\omega_{A}+\omega)(\omega_{B}+\omega\right)}\left(\frac{1}{\omega+{\omega}^{\prime}}-\frac{1}{{\omega}^{\prime}-\omega}\right).

The frequency-domain two-point function GijabEM(ω,rA,rB)G^{\mathrm{E-M}}_{ijab}(\omega,\textbf{r}_{A},\textbf{r}_{B}) in the equation above can be derived from GijabEM(rA,rB,tA,tB)G^{\mathrm{E-M}}_{ijab}(\textbf{r}_{A},\textbf{r}_{B},t_{A},t_{B}) by a Fourier transform. The crossed two-point function of the gravitoelectric and gravitomagnetic fields can be obtained by using Eqs. (11) and (12) as

GijabEM(r,r,t,t)\displaystyle G^{\mathrm{E-M}}_{ijab}(\textbf{r},{\textbf{r}}^{\prime},t,{t}^{\prime}) =\displaystyle= 0|Eij(r,t)Bab(r,t)|0\displaystyle\langle 0|E_{ij}(\textbf{r},t)B_{ab}(\textbf{r}^{\prime},{t}^{\prime})|0\rangle (26)
=\displaystyle= d3kGω3(2π)2𝒢ijabEM(k)eik(rr)iω(tt),\displaystyle\int d^{3}\textbf{k}\frac{G\omega^{3}}{(2\pi)^{2}}\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k})e^{i\textbf{k}\cdot(\textbf{r}-{\textbf{r}}^{\prime})-i\omega(t-{t}^{\prime})},

where 𝒢ijabEM(k)\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k}) represents the gravitoelectric-gravitomagnetic polarization summation term, which takes the form

𝒢ijabEM(k)\displaystyle\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k}) =\displaystyle= ϵapqe3pλeij(k,λ)eqb(k,λ)\displaystyle\epsilon_{apq}e^{p}_{3}\sum_{\lambda}e_{ij}(\textbf{k},\lambda)e_{qb}(\textbf{k},\lambda) (27)
=\displaystyle= ϵiamk^mδjbϵiamk^mk^jk^b+δiaϵjbnk^nk^ik^aϵjbnk^n,\displaystyle\epsilon_{iam}\hat{k}_{m}\delta_{jb}-\epsilon_{iam}\hat{k}_{m}\hat{k}_{j}\hat{k}_{b}+\delta_{ia}\epsilon_{jbn}\hat{k}_{n}-\hat{k}_{i}\hat{k}_{a}\epsilon_{jbn}\hat{k}_{n},

with k^i\hat{k}_{i} being the iith coordinate component of the unit wave vector k^=k/k\hat{k}=\textbf{k}/k. The detailed derivation of Eq. (27) is shown in Appendix B. Let k^x=sinθcosφ\hat{k}_{x}=\sin\theta\cos\varphi, k^y=sinθsinφ\hat{k}_{y}=\sin\theta\sin\varphi, and k^z=cosθ\hat{k}_{z}=\cos\theta, i.e., transform the equation above to the spherical coordinate:

𝒢ijabEM(k)(θ,φ)𝒢ijabEM(θ,φ),\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k})\xrightarrow{(\theta,\varphi)}\mathcal{G}^{\mathrm{E-M}}_{ijab}(\theta,\varphi), (28)

the time-domain two-point function becomes

GijabEM(r,Δt)=0+𝑑ωGω5(2π)20π𝑑θsinθ02π𝑑φ𝒢ijabEM(θ,φ)eiω(rcosθΔt),G^{\mathrm{E-M}}_{ijab}(r,\Delta t)=\int_{0}^{+\infty}d\omega\frac{G\omega^{5}}{(2\pi)^{2}}\int_{0}^{\pi}d\theta\sin\theta\int_{0}^{2\pi}d\varphi\mathcal{G}^{\mathrm{E-M}}_{ijab}(\theta,\varphi)e^{i\omega(r\cos\theta-\Delta t)}, (29)

where r=|rr|r=|\textbf{r}-{\textbf{r}}^{\prime}| is the distance between the two objects, and Δt=tt\Delta t=t-{t}^{\prime}. Performing the Fourier transform, the two-point function in the frequency domain is obtained as

GijabEM(ω¯,rA,rB)\displaystyle G^{\mathrm{E-M}}_{ijab}(\bar{\omega},\textbf{r}_{A},\textbf{r}_{B}) =\displaystyle= 12π+d(Δt)eiω¯ΔtGijabEM(r,Δt)\displaystyle\frac{1}{2\pi}\int_{-\infty}^{+\infty}d(\Delta t)e^{i\bar{\omega}\Delta t}G^{\mathrm{E-M}}_{ijab}(r,\Delta t) (30)
=\displaystyle= Gω¯5(2π)20π𝑑θsinθ02π𝑑φ𝒢ijabEM(θ,φ)eiω¯rcosθ.\displaystyle\frac{G\bar{\omega}^{5}}{(2\pi)^{2}}\int_{0}^{\pi}d\theta\sin\theta\int_{0}^{2\pi}d\varphi\mathcal{G}^{\mathrm{E-M}}_{ijab}(\theta,\varphi)e^{i\bar{\omega}r\cos\theta}.

Then, by substituting Eq. (30) into Eq. (25) and performing the integration over the variables (θ,φ,θ,φ)(\theta,\varphi,{\theta}’,{\varphi}’), one obtains

ΔEABcross(r)\displaystyle\Delta E^{\mathrm{cross}}_{AB}(r) =\displaystyle= 8G29π2(ωA+ωB)r80+ω𝑑ω0+ω𝑑ωα^Aχ^B(ωA+ωB+ω)(ωA+ω)(ωB+ω)\displaystyle-\frac{8G^{2}}{9\pi^{2}(\omega_{A}+\omega_{B})r^{8}}\int_{0}^{+\infty}\omega d\omega\int_{0}^{+\infty}{\omega}^{\prime}d{\omega}^{\prime}\frac{\hat{\alpha}_{A}\hat{\chi}_{B}(\omega_{A}+\omega_{B}+\omega)}{(\omega_{A}+\omega)(\omega_{B}+\omega)} (31)
×(1ω+ω1ωω)[A1(ωr,ωr)cos(ωr)+B1(ωr,ωr)sin(ωr)],\displaystyle\times\left(\frac{1}{\omega+{\omega}^{\prime}}-\frac{1}{{\omega}^{\prime}-\omega}\right)\Big{[}A_{1}(\omega r,{\omega}^{\prime}r)\cos({\omega}^{\prime}r)+B_{1}(\omega r,{\omega}^{\prime}r)\sin({\omega}^{\prime}r)\Big{]},

where

A1(x,x)\displaystyle A_{1}(x,x’) =\displaystyle= xx(45x2x2+3x2+3x2)cosx\displaystyle xx^{\prime}\left(-45-x^{\prime 2}x^{2}+3x^{2}+3x^{\prime 2}\right)\cos x (32)
x(45+18x22x2x2+3x2)sinx,\displaystyle-x^{\prime}\left(-45+18x^{2}-2x^{\prime 2}x^{2}+3x^{\prime 2}\right)\sin x,

and

B1(x,x)\displaystyle B_{1}(x,x^{\prime}) =\displaystyle= x(4518x23x2+2x2x2)cosx\displaystyle x\left(45-18x^{\prime 2}-3x^{2}+2x^{\prime 2}x^{2}\right)\cos x (33)
(4518x218x2+8x2x2)sinx.\displaystyle-\left(45-18x^{2}-18x^{\prime 2}+8x^{\prime 2}x^{2}\right)\sin x.

Since A1(x,x)=A1(x,x)A_{1}(x,-x’)=-A_{1}(x,x’) and B1(x,x)=B1(x,x)B_{1}(x,-x’)=B_{1}(x,x’), the equation above can be further written as

ΔEABcross(r)\displaystyle\Delta E^{\mathrm{cross}}_{AB}(r) =\displaystyle= 4G29π2(ωA+ωB)r80+𝑑ωωα^Aχ^B(ωA+ωB+ω)(ωA+ω)(ωB+ω)\displaystyle-\frac{4G^{2}}{9\pi^{2}(\omega_{A}+\omega_{B})r^{8}}\int_{0}^{+\infty}d\omega\omega\frac{\hat{\alpha}_{A}\hat{\chi}_{B}(\omega_{A}+\omega_{B}+\omega)}{(\omega_{A}+\omega)(\omega_{B}+\omega)} (34)
×+dω(1ω+ω1ωω)ω[A1(ωr,ωr)iB1(ωr,ωr)]eiωr.\displaystyle\times\int_{-\infty}^{+\infty}d{\omega}^{\prime}\left(\frac{1}{\omega+{\omega}^{\prime}}-\frac{1}{{\omega}^{\prime}-\omega}\right){\omega}’\Big{[}A_{1}(\omega r,{\omega}^{\prime}r)-iB_{1}(\omega r,{\omega}^{\prime}r)\Big{]}e^{i{\omega}^{\prime}r}.

Then, performing the Cauchy principal value integral over the variable ω{\omega}’ in Eq. (34), one obtains

ΔEABcross(r)\displaystyle\Delta E^{\mathrm{cross}}_{AB}(r) =\displaystyle= 4G29π(ωA+ωB)r80+𝑑ωω2α^Aχ^B(ωA+ωB+ω)(ωA+ω)(ωB+ω)\displaystyle-\frac{4G^{2}}{9\pi(\omega_{A}+\omega_{B})r^{8}}\int_{0}^{+\infty}d\omega{\omega}^{2}\frac{\hat{\alpha}_{A}\hat{\chi}_{B}(\omega_{A}+\omega_{B}+\omega)}{(\omega_{A}+\omega)(\omega_{B}+\omega)} (35)
×[A2(ωr)cos(2ωr)+B2(ωr)sin(2ωr)],\displaystyle\times\Big{[}A_{2}(\omega r)\cos(2\omega r)+B_{2}(\omega r)\sin(2\omega r)\Big{]},

where

A2(x)=90x+42x34x5,A_{2}(x)=-90x+42x^{3}-4x^{5}, (36)
B2(x)=4581x2+14x4x6.B_{2}(x)=45-81x^{2}+14x^{4}-x^{6}. (37)

Since A2(x)=A2(x)A_{2}(-x)=-A_{2}(x) and B2(x)=B2(x)B_{2}(-x)=B_{2}(x), the equation above can be further transformed into

ΔEABcross(r)\displaystyle\Delta E^{\mathrm{cross}}_{AB}(r) =\displaystyle= 2G29π(ωA+ωB)r8{0+dωω2α^Aχ^B(ωA+ωB+ω)(ωA+ω)(ωB+ω)[A2(ωr)iB2(ωr)]ei2ωr\displaystyle-\frac{2G^{2}}{9\pi(\omega_{A}+\omega_{B})r^{8}}\Bigg{\{}\int_{0}^{+\infty}d\omega{\omega}^{2}\frac{\hat{\alpha}_{A}\hat{\chi}_{B}(\omega_{A}+\omega_{B}+\omega)}{(\omega_{A}+\omega)(\omega_{B}+\omega)}\Big{[}A_{2}(\omega r)-iB_{2}(\omega r)\Big{]}e^{i2\omega r} (38)
+0dωω2α^Aχ^B(ωA+ωBω)(ωAω)(ωBω)[A2(ωr)iB2(ωr)]ei2ωr}.\displaystyle+\int_{0}^{-\infty}d\omega{\omega}^{2}\frac{\hat{\alpha}_{A}\hat{\chi}_{B}(\omega_{A}+\omega_{B}-\omega)}{(\omega_{A}-\omega)(\omega_{B}-\omega)}\Big{[}A_{2}(\omega r)-iB_{2}(\omega r)\Big{]}e^{i2\omega r}\Bigg{\}}.

Letting ω=iu\omega=iu and performing the integral on the imaginary axis, Eq. (38) can further be simplified as

ΔEABcross(r)=4G29πr80+𝑑uu2αA(iu)χB(iu)F(ur)e2ur,\Delta E^{\mathrm{cross}}_{AB}(r)=\frac{4G^{2}}{9\pi r^{8}}\int_{0}^{+\infty}duu^{2}\alpha_{A}(iu)\chi_{B}(iu)F(ur)e^{-2ur}, (39)

where

F(x)=45+90x+81x2+42x3+14x4+4x5+x6,\displaystyle F(x)=45+90x+81x^{2}+42x^{3}+14x^{4}+4x^{5}+x^{6}, (40)

and

αA(iu)=limϵ0+α^AωAωA2(iu)2iϵ(iu),\alpha_{A}(iu)=\lim\limits_{\epsilon\to 0^{+}}\frac{\hat{\alpha}_{A}\omega_{A}}{\omega^{2}_{A}-(iu)^{2}-i\epsilon(iu)}, (41)
χB(iu)=limϵ0+χ^BωBωB2(iu)2iϵ(iu),\chi_{B}(iu)=\lim\limits_{\epsilon\to 0^{+}}\frac{\hat{\chi}_{B}\omega_{B}}{\omega^{2}_{B}-(iu)^{2}-i\epsilon(iu)}, (42)

denote the ground-state gravitoelectric and gravitomagnetic polarizabilities of the objects, respectively, which satisfy the following relations:

Qij(iu)=α(iu)Eij(iu,r),Q_{ij}(iu)=\alpha(iu)E_{ij}(iu,\textbf{r}), (43)
Sij(iu)=χ(iu)Bij(iu,r).S_{ij}(iu)=\chi(iu)B_{ij}(iu,\textbf{r}). (44)

Now, we examine the distance dependence of the quantum gravitoelectric-gravitomagnetic interaction energy (39) in both the retarded and nonretarded regimes. In the retarded regime, i.e., rωA(B)1r\gg\omega^{-1}_{A(B)}, the exponential decay term in Eq. (39) restricts the uu integral to a range where the polarizabilities of the two objects αA(iu)\alpha_{A}(iu) and χB(iu)\chi_{B}(iu) are well approximated by their static values αA(0)\alpha_{A}(0) and χB(0)\chi_{B}(0), respectively. After performing the integral over the variable uu, the interaction energy in the retarded regime is found to be

ΔEABcross,R(r)=187cG2πr11αA(0)χB(0).\Delta E^{\mathrm{cross,R}}_{AB}(r)=\frac{187\hbar cG^{2}}{\pi r^{11}}\alpha_{A}(0)\chi_{B}(0). (45)

Note that the result is shown in International System of Units (SI units). In the nonretarded regime, i.e., rωA(B)1r\ll\omega^{-1}_{A(B)}, all terms in F(x)F(x) that contain the variable urur can be neglected, and since e2ur1e^{-2ur}\simeq 1, the interaction energy in the nonretarded regime can be expressed as

ΔEABcross,NR(r)=20G2πc2r80+𝑑uu2αA(iu)χB(iu).\Delta E^{\mathrm{cross,NR}}_{AB}(r)=\frac{20\hbar G^{2}}{\pi c^{2}r^{8}}\int_{0}^{+\infty}du{u}^{2}\alpha_{A}(iu)\chi_{B}(iu). (46)

Based on the definitions of the gravitoelectric and gravitomagnetic polarizabilities in Eqs. (41) and (42), the frequency-dependent polarizabilities αA(ω)\alpha_{A}(\omega) and χB(ω)\chi_{B}(\omega) are given by

αA(ω)=αA(0)1(ωωA)2,\alpha_{A}(\omega)=\frac{\alpha_{A}(0)}{1-\left(\frac{\omega}{\omega_{A}}\right)^{2}}, (47)
χB(ω)=χB(0)1(ωωB)2.\chi_{B}(\omega)=\frac{\chi_{B}(0)}{1-\left(\frac{\omega}{\omega_{B}}\right)^{2}}. (48)

Taking Eqs. (47) and (48) into Eq. (46), and performing the integration over the variable uu, the explicit expression of the interaction energy in the nonretarded regime reads

ΔEABcross,NR(r)=10G2c2r8ωA2ωB2ωA+ωBαA(0)χB(0).\Delta E_{AB}^{\mathrm{cross,NR}}\left(r\right)=\frac{10\hbar G^{2}}{c^{2}r^{8}}\frac{\omega^{2}_{A}\omega^{2}_{B}}{\omega_{A}+\omega_{B}}\alpha_{A}\left(0\right)\chi_{B}\left(0\right). (49)

The results above show that the quantum gravitoelectric-gravitomagnetic interaction between a gravitoelectrically polarizable object and a gravitomagnetically polarizable object induced by fluctuating gravitoelectric and gravitomagnetic fields in vacuum exhibits an r8r^{-8} dependence in the near regime and an r11r^{-11} dependence in the far regime. In the retarded regime, the distance dependence of the quantum gravitational interaction from the gravitoelectric-gravitomagnetic cross term follows the same form as those in the gravitoelectric-gravitoelectric and gravitomagnetic-gravitomagnetic cases. This can be seen by comparing Eq. (45) in this paper with Eq. (1) in Ref. Ford2016 and Eq. (46) in Ref. hao2024 . However, in the near regime, the quantum gravitoelectric-gravitomagnetic interaction follows a distinct power law of r8r^{-8}, compared to the r10r^{-10} dependence of the gravitoelectric-gravitoelectric and gravitomagnetic-gravitomagnetic interactions. Interestingly, this is consistent with the electromagnetic case, in which the Casimir-Polder interaction energy between an electrically polarizable atom and a paramagnetically polarizable one in the near and far regimes also differs by r3r^{3} Salam2010 ; Buhmann2013 .

Furthermore, it is noteworthy that the results shown in Eqs. (45) and (49) reveal that the quantum gravitoelectric-gravitomagnetic interaction energy is positive, regardless of whether it is in the retarded or nonretarded regime. This indicates that the vacuum fluctuation induced quantum gravitational interaction between a gravitoelectrically and a gravitomagnetically polarizable object always exhibits a repulsive behavior, which is distinct from the attractive nature in the gravitoelectric-gravitoelectric and gravitomagnetic-gravitomagnetic cases reported in Refs. Ford2016 ; hao2024 .

In Table 1, we summarize the power laws and the attractive or repulsive nature of the quantum gravitoelectric-gravitoelectric (GE-GE), gravitomagnetic-gravitomagnetic (GM-GM), and gravitoelectric-gravitomagnetic (GE-GM) interaction potentials in the asymptotic regimes. In the retarded regime, all three interaction potentials exhibit by a r11r^{-11} power law. In the nonretarded regime, the quantum gravitoelectric-gravitoelectric and gravitomagnetic-gravitomagnetic potentials follow a r10r^{-10} power law, while the crossed gravitoelectric-gravitomagnetic potentials investigated in this work exhibit a weaker r8r^{-8} dependence. For the attractive or repulsive nature, a general rule emerges: interactions between objects polarizable in the same gravitoelectric or gravitomagnetic manner are always attractive, while crossed interactions are always repulsive. This implies that, for a pair of objects which are both gravitoelectrically and gravitomagnetically polarizable, the overall quantum gravitational interaction potential is reduced when the gravitoelectric-gravitomagnetic cross interaction is taken into account.

Table 1: Power laws and attractive/repulsive nature of quantum gravitational quadrupolar interaction potentials
GE-GE 1 GM-GM GE-GM (cross term)
Nonretarded regime 315G22r10ωAωBαA(0)αB(0)ωA+ωB-\frac{315\hbar G^{2}}{2r^{10}}\frac{\omega_{A}\omega_{B}\alpha_{A}\left(0\right)\alpha_{B}\left(0\right)}{\omega_{A}+\omega_{B}} 280G29r10ωAωBχA(0)χB(0)ωA+ωB-\frac{280\hbar G^{2}}{9r^{10}}\frac{\omega_{A}\omega_{B}\chi_{A}\left(0\right)\chi_{B}\left(0\right)}{\omega_{A}+\omega_{B}} 10G2c2r8ωA2ωB2αA(0)χB(0)ωA+ωB\frac{10\hbar G^{2}}{c^{2}r^{8}}\frac{\omega^{2}_{A}\omega^{2}_{B}\alpha_{A}\left(0\right)\chi_{B}\left(0\right)}{\omega_{A}+\omega_{B}}
Retarded regime 3987cG2αA(0)αB(0)4πr11-\frac{3987\hbar cG^{2}\alpha_{A}(0)\alpha_{B}(0)}{4\pi r^{11}} 1772cG2χA(0)χB(0)9πr11-\frac{1772\hbar cG^{2}\chi_{A}(0)\chi_{B}(0)}{9\pi r^{11}} 187cG2αA(0)χB(0)πr11\frac{187\hbar cG^{2}\alpha_{A}(0)\chi_{B}(0)}{\pi r^{11}}
Attractive/repulsive Attractive Attractive Repulsive
  • 1

    The terms “GE” and “GM” denote gravitoelectric and gravitomagnetic, respectively.

Now, an intriguing question naturally arises as to whether a regime exists in which the repulsive quantum gravitational quadrupole interaction investigated here dominates over the attractive quantum gravitational quadrupole interactions. The answer is that, for two isotropically polarizable objects with identical gravitoelectric and gravitomagnetic polarizabilities and energy level spacing, the repulsive quantum gravitational interaction cannot exceed the attractive one, irrespective of the gravitoelectric and gravitomagnetic polarizabilities and the interobject distance. The explicit proof is provided in Appendix C.

IV Summary

In this paper, we have studied the quantum gravitational interaction between a gravitoelectrically polarizable object and a gravitomagnetically polarizable object induced by fluctuating gravitoelectric and gravitomagnetic fields in a vacuum, within the framework of linearized quantum gravity. This interaction originates from the coupling between the induced instantaneous mass quadrupole moment and mass-current quadrupole moment in nonpointlike objects. Our result shows that, the quantum gravitoelectric-gravitomagnetic interaction exhibits a distance dependence of r8r^{-8} in the near regime and r11r^{-11} in the far regime, where rr is the distance between the two objects. Furthermore, the quantum gravitoelectric-gravitomagnetic interaction energy is positive both in the near and far regimes, which indicates that the force is repulsive. Since interactions between objects polarizable in the same gravitoelectric or gravitomagnetic manner are inherently attractive, for objects which are both gravitoelectrically and gravitomagnetically polarizable, the overall quantum gravitational interaction potential is reduced when the repulsive quantum gravitoelectric-gravitomagnetic interaction is taken into account. However, for two isotropically polarizable objects with identical gravitoelectric and gravitomagnetic polarizabilities and energy level spacing, the repulsive quantum interaction cannot surpass the attractive interactions, regardless of the gravitoelectric and gravitomagnetic polarizabilities and the interobject distance.

Acknowledgements.
We would like to thank the anonymous referee for the insightful comments and helpful suggestions. This work was supported in part by the NSFC under Grant No. 12075084, and the innovative research group of Hunan Province under Grant No. 2024JJ1006.

Appendix A Intermediate processes of the quantum gravitoelectric-gravitomagnetic interaction

The twelve possible intermediate states and their corresponding energy denominators 𝒟(n)(n=1,2,3,,12)\mathcal{D}_{(n)}(n=1,2,3,...,12) in Eq. (16) are as in Table 2.

Case |I|\text{I}\rangle |II|\text{II}\rangle |III|\text{III}\rangle       Denominator
(1) |eA|gB|1|e_{A}\rangle|g_{B}\rangle|1\rangle |gA|gB|1,1|g_{A}\rangle|g_{B}\rangle|1,{1}^{\prime}\rangle |gA|eB|1|g_{A}\rangle|e_{B}\rangle|{1}^{\prime}\rangle 𝒟(1)=(ω+ωB)(ω+ω)(ω+ωA)\mathcal{D}_{(1)}=({\omega}^{\prime}+\omega_{B})({\omega}^{\prime}+\omega)(\omega+\omega_{A})
(2) |eA|gB|1|e_{A}\rangle|g_{B}\rangle|1\rangle |gA|gB|1,1|g_{A}\rangle|g_{B}\rangle|1,{1}^{\prime}\rangle |gA|eB|1|g_{A}\rangle|e_{B}\rangle|1\rangle 𝒟(2)=(ω+ωB)(ω+ω)(ω+ωA)\mathcal{D}_{(2)}=(\omega+\omega_{B})({\omega}^{\prime}+\omega)(\omega+\omega_{A})
(3) |eA|gB|1|e_{A}\rangle|g_{B}\rangle|1\rangle |eA|eB|0|e_{A}\rangle|e_{B}\rangle|0\rangle |gA|eB|1|g_{A}\rangle|e_{B}\rangle|{1}^{\prime}\rangle 𝒟(3)=(ω+ωB)(ωB+ωA)(ω+ωA)\mathcal{D}_{(3)}=({\omega}^{\prime}+\omega_{B})(\omega_{B}+\omega_{A})(\omega+\omega_{A})
(4) |eA|gB|1|e_{A}\rangle|g_{B}\rangle|1\rangle |eA|eB|0|e_{A}\rangle|e_{B}\rangle|0\rangle |eA|gB|1|e_{A}\rangle|g_{B}\rangle|{1}^{\prime}\rangle 𝒟(4)=(ω+ωA)(ωB+ωA)(ω+ωA)\mathcal{D}_{(4)}=({\omega}^{\prime}+\omega_{A})(\omega_{B}+\omega_{A})(\omega+\omega_{A})
(5) |eA|gB|1|e_{A}\rangle|g_{B}\rangle|1\rangle |eA|eB|1,1|e_{A}\rangle|e_{B}\rangle|1,{1}^{\prime}\rangle |gA|eB|1|g_{A}\rangle|e_{B}\rangle|1\rangle 𝒟(5)=(ω+ωB)(ωB+ωA+ω+ω)(ω+ωA)\mathcal{D}_{(5)}=(\omega+\omega_{B})(\omega_{B}+\omega_{A}+{\omega}^{\prime}+\omega)(\omega+\omega_{A})
(6) |eA|gB|1|e_{A}\rangle|g_{B}\rangle|1\rangle |eA|eB|1,1|e_{A}\rangle|e_{B}\rangle|1,{1}^{\prime}\rangle |eA|gB|1|e_{A}\rangle|g_{B}\rangle|{1}^{\prime}\rangle 𝒟(6)=(ω+ωA)(ωB+ωA+ω+ω)(ω+ωA)\mathcal{D}_{(6)}=({\omega}^{\prime}+\omega_{A})(\omega_{B}+\omega_{A}+{\omega}^{\prime}+\omega)(\omega+\omega_{A})
(7) |gA|eB|1|g_{A}\rangle|e_{B}\rangle|1\rangle |gA|gB|1,1|g_{A}\rangle|g_{B}\rangle|1,{1}^{\prime}\rangle |eA|gB|1|e_{A}\rangle|g_{B}\rangle|{1}^{\prime}\rangle 𝒟(7)=(ω+ωA)(ω+ω)(ω+ωB)\mathcal{D}_{(7)}=({\omega}^{\prime}+\omega_{A})({\omega}^{\prime}+\omega)(\omega+\omega_{B})
(8) |gA|eB|1|g_{A}\rangle|e_{B}\rangle|1\rangle |gA|gB|1,1|g_{A}\rangle|g_{B}\rangle|1,{1}^{\prime}\rangle |eA|gB|1|e_{A}\rangle|g_{B}\rangle|1\rangle 𝒟(8)=(ω+ωA)(ω+ω)(ω+ωB)\mathcal{D}_{(8)}=(\omega+\omega_{A})({\omega}^{\prime}+\omega)(\omega+\omega_{B})
(9) |gA|eB|1|g_{A}\rangle|e_{B}\rangle|1\rangle |eA|eB|0|e_{A}\rangle|e_{B}\rangle|0\rangle |eA|gB|1|e_{A}\rangle|g_{B}\rangle|{1}^{\prime}\rangle 𝒟(9)=(ω+ωA)(ωB+ωA)(ω+ωB)\mathcal{D}_{(9)}=({\omega}^{\prime}+\omega_{A})(\omega_{B}+\omega_{A})(\omega+\omega_{B})
(10) |gA|eB|1|g_{A}\rangle|e_{B}\rangle|1\rangle |eA|eB|0|e_{A}\rangle|e_{B}\rangle|0\rangle |gA|eB|1|g_{A}\rangle|e_{B}\rangle|{1}^{\prime}\rangle 𝒟(10)=(ω+ωB)(ωB+ωA)(ω+ωB)\mathcal{D}_{(10)}=({\omega}^{\prime}+\omega_{B})(\omega_{B}+\omega_{A})(\omega+\omega_{B})
(11) |gA|eB|1|g_{A}\rangle|e_{B}\rangle|1\rangle |eA|eB|1,1|e_{A}\rangle|e_{B}\rangle|1,{1}^{\prime}\rangle |eA|gB|1|e_{A}\rangle|g_{B}\rangle|1\rangle 𝒟(11)=(ω+ωA)(ωB+ωA+ω+ω)(ω+ωB)\mathcal{D}_{(11)}=(\omega+\omega_{A})(\omega_{B}+\omega_{A}+{\omega}^{\prime}+\omega)(\omega+\omega_{B})
(12) |gA|eB|1|g_{A}\rangle|e_{B}\rangle|1\rangle |eA|eB|1,1|e_{A}\rangle|e_{B}\rangle|1,{1}^{\prime}\rangle |gA|eB|1|g_{A}\rangle|e_{B}\rangle|{1}^{\prime}\rangle 𝒟(12)=(ω+ωB)(ωB+ωA+ω+ω)(ω+ωB)\mathcal{D}_{(12)}=({\omega}^{\prime}+\omega_{B})(\omega_{B}+\omega_{A}+{\omega}^{\prime}+\omega)(\omega+\omega_{B})
Table 2: Twelve intermediate states of the interaction energy and the expressions of their corresponding energy denominators.

Appendix B Summation of gravitoelectric-gravitomagnetic polarization tensors and the symmetry of the two-point functions

Let us introduce a triad of coordinate-independent orthogonal unit vectors, denoted as [e1(k),e2(k),e3(k)][\textbf{e}_{1}(\textbf{k}),\textbf{e}_{2}(\textbf{k}),\textbf{e}_{3}(\textbf{k})]. The vector e3(k)=k/kk^\textbf{e}_{3}(\textbf{k})=\textbf{k}/k\equiv\hat{k} represents the unit vector in the direction of the gravitational field’s propagation. The orthogonal relations and the cross product relations satisfied by this triad can be written in the coordinate system which describes the spacetime metric as

eai(k)eaj(k)=e1ie1j+e2ie2j+k^ik^j=δij,e^{i}_{a}(\textbf{k})e^{j}_{a}(\textbf{k})=e^{i}_{1}e^{j}_{1}+e^{i}_{2}e^{j}_{2}+\hat{k}^{i}\hat{k}^{j}=\delta_{ij}, (50)
ϵijke3je1k=e2i,\epsilon_{ijk}e^{j}_{3}e^{k}_{1}=e^{i}_{2}, (51)
ϵijke3je2k=e1i,\epsilon_{ijk}e^{j}_{3}e^{k}_{2}=-e^{i}_{1}, (52)

and

e1ie2je2ie1j=ϵijke3k=ϵijkk^k.e^{i}_{1}e^{j}_{2}-e^{i}_{2}e^{j}_{1}=\epsilon_{ijk}e^{k}_{3}=\epsilon_{ijk}\hat{k}_{k}. (53)

Here, a=1,2,3a=1,2,3, and i,j=x,y,zi,j=x,y,z, the k^i\hat{k}_{i} is the iith coordinate component of the k^\hat{k}.

With the help of the vectors e1(k)\textbf{e}_{1}(\textbf{k}) and e2(k)\textbf{e}_{2}(\textbf{k}) in the triad, the gravitational polarization tensor eij(k,λ)e_{ij}(\textbf{k},\lambda) in the transverse-traceless (TT) gauge can be expressed as MTW

eij(k,+)=e1i(k)e1j(k)e2i(k)e2j(k),e^{ij}(\textbf{k},+)=e^{i}_{1}(\textbf{k})\otimes e^{j}_{1}(\textbf{k})-e^{i}_{2}(\textbf{k})\otimes e^{j}_{2}(\textbf{k}), (54)
eij(k,×)=e1i(k)e2j(k)+e2i(k)e1j(k).e^{ij}(\textbf{k},\times)=e^{i}_{1}(\textbf{k})\otimes e^{j}_{2}(\textbf{k})+e^{i}_{2}(\textbf{k})\otimes e^{j}_{1}(\textbf{k}). (55)

Thus, the summation of the polarization tensors gives

λeij(k,λ)ekl(k,λ)\displaystyle\sum_{\lambda}e_{ij}(\textbf{{k}},\lambda)e_{kl}(\textbf{k},\lambda) =\displaystyle= eij(k,+)ekl(k,+)+eij(k,×)ekl(k,×)\displaystyle e^{ij}(\textbf{k},+)e^{kl}({\textbf{k},+})+e^{ij}(\textbf{k},\times)e^{kl}(\textbf{k},\times)
=\displaystyle= [e1i(k)e1j(k)e2i(k)e2j(k)][e1k(k)e1l(k)e2k(k)e2l(k)]\displaystyle\left[e^{i}_{1}(\textbf{k})\otimes e^{j}_{1}(\textbf{k})-e^{i}_{2}(\textbf{k})\otimes e^{j}_{2}(\textbf{k})\right]\left[e^{k}_{1}(\textbf{k})\otimes e^{l}_{1}(\textbf{k})-e^{k}_{2}(\textbf{k})\otimes e^{l}_{2}(\textbf{k})\right]
+[e1i(k)e2j(k)+e2i(k)e1j(k)][e1k(k)e2l(k)+e2k(k)e1l(k)].\displaystyle+\left[e^{i}_{1}(\textbf{k})\otimes e^{j}_{2}(\textbf{k})+e^{i}_{2}(\textbf{k})\otimes e^{j}_{1}(\textbf{k})\right]\left[e^{k}_{1}(\textbf{k})\otimes e^{l}_{2}(\textbf{k})+e^{k}_{2}(\textbf{k})\otimes e^{l}_{1}(\textbf{k})\right].

Based on Eq. (B), the gravitoelectric-gravitomagnetic polarization summation term 𝒢ijabEM(k)\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k}) in Eq. (26) can be further expressed as

𝒢ijabEM(k)\displaystyle\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k}) =\displaystyle= ϵapqe3pλeij(k,λ)eqb(k,λ)\displaystyle\epsilon_{apq}e^{p}_{3}\sum_{\lambda}e_{ij}(\textbf{{k}},\lambda)e_{qb}(\textbf{k},\lambda) (57)
=\displaystyle= ϵapqe3p{[e1i(k)e1j(k)e2i(k)e2j(k)][e1q(k)e1b(k)e2q(k)e2b(k)]\displaystyle\epsilon_{apq}e^{p}_{3}\Big{\{}\left[e^{i}_{1}(\textbf{k})\otimes e^{j}_{1}(\textbf{k})-e^{i}_{2}(\textbf{k})\otimes e^{j}_{2}(\textbf{k})\right]\left[e^{q}_{1}(\textbf{k})\otimes e^{b}_{1}(\textbf{k})-e^{q}_{2}(\textbf{k})\otimes e^{b}_{2}(\textbf{k})\right]
+[e1i(k)e2j(k)+e2i(k)e1j(k)][e1q(k)e2b(k)+e2q(k)e1b(k)]}\displaystyle+\left[e^{i}_{1}(\textbf{k})\otimes e^{j}_{2}(\textbf{k})+e^{i}_{2}(\textbf{k})\otimes e^{j}_{1}(\textbf{k})\right]\left[e^{q}_{1}(\textbf{k})\otimes e^{b}_{2}(\textbf{k})+e^{q}_{2}(\textbf{k})\otimes e^{b}_{1}(\textbf{k})\right]\Big{\}}
=\displaystyle= [e1i(k)e1j(k)e2i(k)e2j(k)]\displaystyle\left[e_{1}^{i}\left(\textbf{k}\right)\otimes e_{1}^{j}\left(\textbf{k}\right)-e_{2}^{i}\left(\textbf{k}\right)\otimes e_{2}^{j}\left(\textbf{k}\right)\right]
×{[ϵapqe3pe1q(k)]e1b(k)[ϵapqe3pe2q(k)]e2b(k)}\displaystyle\times\left\{\left[\epsilon_{apq}e_{3}^{p}e_{1}^{q}\left(\textbf{k}\right)\right]\otimes e_{1}^{b}\left(\textbf{k}\right)-\left[\epsilon_{apq}e_{3}^{p}e_{2}^{q}\left(\textbf{k}\right)\right]\otimes e_{2}^{b}\left(\textbf{k}\right)\right\}
+[e1i(k)e2j(k)+e2i(k)e1j(k)]\displaystyle+\left[e_{1}^{i}\left(\textbf{k}\right)\otimes e_{2}^{j}\left(\textbf{k}\right)+e_{2}^{i}\left(\textbf{k}\right)\otimes e_{1}^{j}\left(\textbf{k}\right)\right]
×{[ϵapqe3pe1q(k)]e2b(k)+[ϵapqe3pe2q(k)]e1b(k)}.\displaystyle\times\left\{\left[\epsilon_{apq}e_{3}^{p}e_{1}^{q}\left(\textbf{k}\right)\right]\otimes e_{2}^{b}\left(\textbf{k}\right)+\left[\epsilon_{apq}e_{3}^{p}e_{2}^{q}\left(\textbf{k}\right)\right]\otimes e_{1}^{b}\left(\textbf{k}\right)\right\}.

Then, using the cross product relations shown in Eqs. (51) and (52), the equation above can be simplified as

𝒢ijabEM(k)\displaystyle\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k}) =\displaystyle= [e1ie1je2ie2j][e2ae1b+e1ae2b]\displaystyle\left[e^{i}_{1}\otimes e^{j}_{1}-e^{i}_{2}\otimes e^{j}_{2}\right]\left[e^{a}_{2}\otimes e^{b}_{1}+e^{a}_{1}\otimes e^{b}_{2}\right] (58)
+[e1ie2j+e2ie1j][e2ae2be1ae1b].\displaystyle+\left[e^{i}_{1}\otimes e^{j}_{2}+e^{i}_{2}\otimes e^{j}_{1}\right]\left[e^{a}_{2}\otimes e^{b}_{2}-e^{a}_{1}\otimes e^{b}_{1}\right].

Furthermore, by utilizing Eqs. (50) and (53), the equation above can further be calculated as follows

𝒢ijabEM(k)\displaystyle\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k}) =\displaystyle= e1ie1je2ae1b+e1ie1je1ae2be2ie2je2ae1be2ie2je1ae2b\displaystyle e^{i}_{1}e^{j}_{1}e^{a}_{2}e^{b}_{1}+e^{i}_{1}e^{j}_{1}e^{a}_{1}e^{b}_{2}-e^{i}_{2}e^{j}_{2}e^{a}_{2}e^{b}_{1}-e^{i}_{2}e^{j}_{2}e^{a}_{1}e^{b}_{2} (59)
+e1ie2je2ae2be1ie2je1ae1b+e2ie1je2ae2be2ie1je1ae1b\displaystyle+e^{i}_{1}e^{j}_{2}e^{a}_{2}e^{b}_{2}-e^{i}_{1}e^{j}_{2}e^{a}_{1}e^{b}_{1}+e^{i}_{2}e^{j}_{1}e^{a}_{2}e^{b}_{2}-e^{i}_{2}e^{j}_{1}e^{a}_{1}e^{b}_{1}
=\displaystyle= (e1ie2ae2ie1a)(e1je1b+e2je2b)+(e1ie1a+e2ie2a)(e1je2be2je1b)\displaystyle\left(e^{i}_{1}e^{a}_{2}-e^{i}_{2}e^{a}_{1}\right)\left(e^{j}_{1}e^{b}_{1}+e^{j}_{2}e^{b}_{2}\right)+\left(e^{i}_{1}e^{a}_{1}+e^{i}_{2}e^{a}_{2}\right)\left(e^{j}_{1}e^{b}_{2}-e^{j}_{2}e^{b}_{1}\right)
=\displaystyle= ϵiamk^m(δjbk^jk^b)+(δiak^ik^a)ϵjbnk^n\displaystyle\epsilon_{iam}\hat{k}_{m}\left(\delta_{jb}-\hat{k}_{j}\hat{k}_{b}\right)+\left(\delta_{ia}-\hat{k}_{i}\hat{k}_{a}\right)\epsilon_{jbn}\hat{k}_{n}
=\displaystyle= ϵiamk^mδjbϵiamk^mk^jk^b+δiaϵjbnk^nk^ik^aϵjbnk^n.\displaystyle\epsilon_{iam}\hat{k}_{m}\delta_{jb}-\epsilon_{iam}\hat{k}_{m}\hat{k}_{j}\hat{k}_{b}+\delta_{ia}\epsilon_{jbn}\hat{k}_{n}-\hat{k}_{i}\hat{k}_{a}\epsilon_{jbn}\hat{k}_{n}.

Based on the analysis above, let us consider the form of the two-point functions when the order of the gravitoelectric and the gravitomagnetic field operators is exchanged. In this case, it becomes

GijabME(r,r,t,t)\displaystyle G^{\mathrm{M-E}}_{ijab}(\textbf{r},{\textbf{r}}^{\prime},t,{t}^{\prime}) =\displaystyle= 0|Bij(r,t)Eab(r,t)|0\displaystyle\langle 0|B_{ij}(\textbf{r},t)E_{ab}(\textbf{r}^{\prime},{t}^{\prime})|0\rangle (60)
=\displaystyle= λd3kd3k(Gω3ω3(2π)2)ϵifle3felj(k,λ)eab(k,λ)\displaystyle\sum_{\lambda}\int d^{3}\textbf{k}\int d^{3}{\textbf{k}}^{\prime}\left(\frac{G\sqrt{\omega^{3}{\omega}^{\prime 3}}}{(2\pi)^{2}}\right)\epsilon_{ifl}e^{f}_{3}e_{lj}(\textbf{k},\lambda)e_{ab}({\textbf{k}}^{\prime},\lambda)
×0|[gk,λeikriωt+gk,λeikr+iωt]\displaystyle\times\biggl{\langle}0\bigg{|}\left[g_{\textbf{k},\lambda}e^{i\textbf{k}\cdot\textbf{r}-i\omega t}+g^{\dagger}_{\textbf{k},\lambda}e^{-i\textbf{k}\cdot\textbf{r}+i\omega t}\right]
×[gk,λeikriωt+gk,λeikr+iωt]|0\displaystyle\times\left[g_{\textbf{k}^{\prime},\lambda}e^{i{\textbf{k}}^{\prime}\cdot{\textbf{r}}^{\prime}-i{\omega}^{\prime}{t}^{\prime}}+g^{\dagger}_{\textbf{k}^{\prime},\lambda}e^{-i{\textbf{k}}^{\prime}\cdot{\textbf{r}}^{\prime}+i{\omega}^{\prime}{t}^{\prime}}\right]\bigg{|}0\biggr{\rangle}
=\displaystyle= d3kGω3(2π)2𝒢ijabME(k)eik(rr)iω(tt),\displaystyle\int d^{3}\textbf{k}\frac{G\omega^{3}}{(2\pi)^{2}}\mathcal{G}^{\mathrm{M-E}}_{ijab}(\textbf{k})e^{i\textbf{k}\cdot(\textbf{r}-{\textbf{r}}^{\prime})-i\omega(t-{t}^{\prime})},

where, 𝒢ijabME(k)=λϵifle3felj(k,λ)eab(k,λ)\mathcal{G}^{\mathrm{M-E}}_{ijab}(\textbf{k})=\sum_{\lambda}\epsilon_{ifl}e^{f}_{3}e_{lj}(\textbf{k},\lambda)e_{ab}(\textbf{k},\lambda) is the gravitomagnetic-gravitoelectric polarization summation term, whose explicit expression is

𝒢ijabME(k)\displaystyle\mathcal{G}^{\mathrm{M-E}}_{ijab}(\textbf{k}) =\displaystyle= e1ie1je2ae1be1ie1je1ae2b+e2ie2je2ae1b+e2ie2je1ae2b\displaystyle-e^{i}_{1}e^{j}_{1}e^{a}_{2}e^{b}_{1}-e^{i}_{1}e^{j}_{1}e^{a}_{1}e^{b}_{2}+e^{i}_{2}e^{j}_{2}e^{a}_{2}e^{b}_{1}+e^{i}_{2}e^{j}_{2}e^{a}_{1}e^{b}_{2} (61)
e1ie2je2ae2b+e1ie2je1ae1be2ie1je2ae2b+e2ie1je1ae1b\displaystyle-e^{i}_{1}e^{j}_{2}e^{a}_{2}e^{b}_{2}+e^{i}_{1}e^{j}_{2}e^{a}_{1}e^{b}_{1}-e^{i}_{2}e^{j}_{1}e^{a}_{2}e^{b}_{2}+e^{i}_{2}e^{j}_{1}e^{a}_{1}e^{b}_{1}
=\displaystyle= ϵiamk^mδjb+ϵiamk^mk^jk^bδiaϵjbnk^n+k^ik^aϵjbnk^n\displaystyle-\epsilon_{iam}\hat{k}_{m}\delta_{jb}+\epsilon_{iam}\hat{k}_{m}\hat{k}_{j}\hat{k}_{b}-\delta_{ia}\epsilon_{jbn}\hat{k}_{n}+\hat{k}_{i}\hat{k}_{a}\epsilon_{jbn}\hat{k}_{n}
=\displaystyle= 𝒢ijabEM(k),\displaystyle-\mathcal{G}^{\mathrm{E-M}}_{ijab}(\textbf{k}),

which can be derived using the methods shown in Eqs. (57)-(59). Therefore, we have

GijklEM(r,r,t,t)=GijklME(r,r,t,t).\displaystyle G^{\mathrm{E-M}}_{ijkl}(\textbf{r},{\textbf{r}}^{\prime},t,t^{\prime})=-G^{\mathrm{M-E}}_{ijkl}(\textbf{r},{\textbf{r}}^{\prime},t,t^{\prime}). (62)

Appendix C Comparison of the magnitudes of the repulsive quantum gravitoelectric-gravitomagnetic interaction and the attractive gravitoelectric-gravitoelectric and gravitomagnetic-gravitomagnetic interactions

To compare the magnitude of the repulsive quantum gravitoelectric-gravitomagnetic interaction with those of the attractive gravitoelectric-gravitoelectric and gravitomagnetic-gravitomagnetic interactions, we start from the original integral expressions for the interaction potentials, as presented in Eq. (39) of this paper, Eq. (28) in Ref. Ford2016 , and Eq. (39) in Ref. hao2024 . For two isotropically polarizable objects which are both gravitoelectrically and gravitomagnetically polarizable, we assume, for convenience, that their gravitoelectric and gravitomagnetic polarizabilities are equal, and their energy level spacings are the same, i.e., α^A=α^B=α\hat{\alpha}_{A}=\hat{\alpha}_{B}=\alpha, χ^A=χ^B=χ\hat{\chi}_{A}=\hat{\chi}_{B}=\chi, and ωA=ωB=ω0\omega_{A}=\omega_{B}=\omega_{0}. Then, with the help of Eqs. (41) and (42), the original integral expressions for the three types of potentials can be reorganized as

ΔEABcross(r)=G2ω02r4πr110+𝑑x49αχT1(x)[1(ω0r)2+x2]2e2x,\Delta E_{AB}^{\mathrm{cross}}\left(r\right)=\frac{G^{2}\omega_{0}^{2}r^{4}}{\pi r^{11}}\int_{0}^{+\infty}{dx\,\frac{4}{9}\alpha\chi T_{1}\left(x\right)\left[\frac{1}{\left(\omega_{0}r\right)^{2}+x^{2}}\right]^{2}e^{-2x}}, (63)
ΔEABGEGE(r)=G2ω02r4πr110+𝑑xα2T2(x)[1(ω0r)2+x2]2e2x,\Delta E_{AB}^{\mathrm{GE}-\mathrm{GE}}\left(r\right)=-\frac{G^{2}\omega_{0}^{2}r^{4}}{\pi r^{11}}\int_{0}^{+\infty}{dx\,\alpha^{2}T_{2}\left(x\right)\left[\frac{1}{\left(\omega_{0}r\right)^{2}+x^{2}}\right]^{2}e^{-2x}}, (64)

and

ΔEABGMGM(r)=G2ω02r4πr110+𝑑x1681χ2T2(x)[1(ω0r)2+x2]2e2x,\Delta E_{AB}^{\mathrm{GM}-\mathrm{GM}}\left(r\right)=-\frac{G^{2}\omega_{0}^{2}r^{4}}{\pi r^{11}}\int_{0}^{+\infty}{dx\,\frac{16}{81}\chi^{2}T_{2}\left(x\right)\left[\frac{1}{\left(\omega_{0}r\right)^{2}+x^{2}}\right]^{2}e^{-2x}}, (65)

respectively. Note that the results above are shown in natural units, with x=urx=ur, and T1(x)T_{1}(x) and T2(x)T_{2}(x) take the forms

T1(x)=x2(45+90x+81x2+42x3+14x4+4x5+x6),T_{1}\left(x\right)=x^{2}\left(45+90x+81x^{2}+42x^{3}+14x^{4}+4x^{5}+x^{6}\right), (66)

and

T2(x)=315+630x+585x2+330x3+129x4+42x5+14x6+4x7+x8,T_{2}\left(x\right)=315+630x+585x^{2}+330x^{3}+129x^{4}+42x^{5}+14x^{6}+4x^{7}+x^{8}, (67)

respectively. Then, the magnitudes of the attractive and repulsive quantum gravitational interactions are given by |ΔEABGEGE(r)|+|ΔEABGMGM(r)||\Delta E_{AB}^{\mathrm{GE}-\mathrm{GE}}\left(r\right)|+|\Delta E_{AB}^{\mathrm{GM}-\mathrm{GM}}\left(r\right)| and |ΔEABcross(r)||\Delta E_{AB}^{\mathrm{cross}}\left(r\right)|, respectively, where “|||\,\,|” denotes the absolute value. In the following, we prove that their difference,

ΔE=|ΔEABGEGE(r)|+|ΔEABGMGM(r)||ΔEABcross(r)|,\Delta E=|\Delta E_{AB}^{\mathrm{GE}-\mathrm{GE}}\left(r\right)|+|\Delta E_{AB}^{\mathrm{GM}-\mathrm{GM}}\left(r\right)|-|\Delta E_{AB}^{\mathrm{cross}}\left(r\right)|, (68)

is always positive, indicating that the repulsive quantum gravitoelectric-gravitomagnetic interaction cannot dominate.

Substituting Eqs. (63)-(65) into Eq. (68), then it can be shown that ΔE>0\Delta E>0 implies

0+𝑑x{[α2T2(x)+1681χ2T2(x)][49αχT1(x)]}[1(ω0r)2+x2]2e2x>0.\int_{0}^{+\infty}{dx\left\{\left[\alpha^{2}T_{2}\left(x\right)+\frac{16}{81}\chi^{2}T_{2}\left(x\right)\right]-\left[\frac{4}{9}\alpha\chi T_{1}\left(x\right)\right]\right\}\left[\frac{1}{\left(\omega_{0}r\right)^{2}+x^{2}}\right]^{2}e^{-2x}}>0. (69)

According to Eqs. (23), (24), and (67), since α\alpha, χ\chi, and T2(x)T_{2}(x) are all positive, it follows from the inequality of arithmetic and geometric means that

α2T2(x)+1681χ2T2(x)89αχT2(x).\alpha^{2}T_{2}\left(x\right)+\frac{16}{81}\chi^{2}T_{2}\left(x\right)\geq\frac{8}{9}\alpha\chi T_{2}\left(x\right). (70)

Therefore, a sufficient condition for inequality (69) to hold is

δ0+𝑑x[89T2(x)49T1(x)][1(ω0r)2+x2]2e2x>0,\delta\equiv\int_{0}^{+\infty}{dx\left[\frac{8}{9}T_{2}\left(x\right)-\frac{4}{9}T_{1}\left(x\right)\right]\left[\frac{1}{\left(\omega_{0}r\right)^{2}+x^{2}}\right]^{2}e^{-2x}}>0, (71)

which means that the condition ΔE>0\Delta E>0 describing the magnitude of the repulsive interaction energy as being smaller than the attractive interaction energy translates into δ>0\delta>0.

Now, let us discuss whether δ>0\delta>0 is valid in both the nonretarded and retarded regimes, as well as in the transition regime where the interobject distance is comparable to the transition wavelength.

Refer to caption
Figure 1: The value of δ\delta as a function of ω0r\omega_{0}r in the transition regimes between the nonretarded and retarded regimes.

(1) In the nonretarded regime, performing the integral in Eq. (71) and then expanding the result in a Taylor series in the region where ω0r1\omega_{0}r\ll 1, we obtain

δ=70π(ω0r)3+𝒪[1(ω0r)],\delta=\frac{70\pi}{\left(\omega_{0}r\right)^{3}}+\mathcal{O}\left[\frac{1}{\left(\omega_{0}r\right)}\right], (72)

which is positive. Here, 𝒪(xn)\mathcal{O}(x^{n}) denotes terms on the order of xnx^{n} or higher, which are considered negligibly small and thus omitted.

(2) In the retarded regime, we perform the integral in Eq. (71) and then expand the result in a Taylor series in the region where ω0r1\omega_{0}r\gg 1, yielding

δ=699(ω0r)4+𝒪[1(ω0r)6],\delta=\frac{699}{\left(\omega_{0}r\right)^{4}}+\mathcal{O}\left[\frac{1}{\left(\omega_{0}r\right)^{6}}\right], (73)

which is also positive.

(3) In the transition regime between the nonretarded and retarded regimes, we show numerically the value of δ\delta as a function of ω0r\omega_{0}r in these transition regimes in Fig. 1, which clearly shows that δ>0\delta>0 always holds.

Considering the analysis above, for two isotropically polarizable objects which are both gravitoelectrically and gravitomagnetically polarizable and have the same gravitoelectric and gravitomagnetic polarizabilities and energy level spacing, the repulsive interaction energy cannot dominate over the attractive one, irrespective of the gravitoelectric and gravitomagnetic polarizabilities and the interobject distance.

References