ptep Progress of Theoretical and Experimental Physics 2050-3911 Oxford University Press 10.1093/ptep/pty066 pty066 arXiv:1801.10244 Papers Theoretical Particle Physics PTEP/B30 PTEP/B31 PTEP/E00 PTEP/E01 The general relativistic effects on the magnetic moment in Earth’s gravity MorishimaTakahiro moris@nagoya-u.jp FutamaseToshifumi ShimizuHirohiko M Laboratory for Particle Properties, Department of Physics, Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan Department of Astrophysics and Atmospheric Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita, Kyoto 603-8555, Japan E-mail: moris@nagoya-u.jp

Toshifumi Futamase, tof@cc.kyoto-su.ac.jp; Hirohiko M Shimizu, hirohiko.shimizu@nagoya-u.jp

 06 2018 01 06 2018 28 06 2018 2018 6 063B07 31 01 2018 27 04 2018 11 05 2018 © The Author(s) 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. 2018 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Funded by SCOAP3 Abstract

The magnetic moment of free fermions in the Earth’s gravitational field has been studied on the basis of general relativity. Adopting the Schwarzschild metric for the background spacetime, the dipole coupling between the magnetic moment and the magnetic field has been found to be dependent on the gravity in the calculation up to the post-Newtonian order . The gravity dependence can be formulated by employing the effective value of the magnetic moment as a gravity-dependent quantity for the cases of minimal coupling, non-minimal coupling, and a mixture of the two. The gravitationally induced anomaly is found to be canceled in the experimental values of the anomalous magnetic moment measured in the Penning trap and storage ring methods.

 B30 B31 E00 E01 1. Introduction

The magnetic moment is regarded as one of the fundamental properties of elementary particles. The magnetic moment of fermions with a rest mass of , a charge of , and a spin of is defined as where is the constant factor specific to each elementary particle, referred to as the gyromagnetic ratio or Landé g-factor1. Here we consider particles with a spin of 1/2 such as electrons, protons, and atoms placed in the Earth’s gravitational field and consider if their magnetic moments, defined by Eq. (1), are independent of gravity in the same way as the mass or the electric charge. We focus on the influence of the spacetime distortion on the basis of general relativity, which is not a quantum mechanical effect mediated by the graviton at the Planck scale but a classical mechanical effect remaining even in low-energy regions.

Below, we consider observables measured using instruments fixed on the Earth’s surface. General relativity requires that the translational motion of a charged particle with an electric charge of obeys the geodesic equation where is the proper time, is the four velocity vector, is the electromagnetic tensor, and is the Christoffel symbol. The Earth’s gravitational field can be described by the Schwarzschild metric, approximated up to the post-Newtonian order in the expansion series of , as where is the Earth’s gravitational potential at a distance of from the center of a uniformly distributed spherical mass with a radius of (see Appendix B). Equation (2) becomes where , , and

Here we suggest that the and represent the electric and magnetic fields to be measured by the observer on the Earth’s surface, and correspond to the field vectors in the flat spacetime.

At the limit of , Eq. (2) results in which is the well-known equation of motion of charged particles in the flat spacetime (Minkowski spacetime) consistently with special relativity. Additionally, in the region of , it results in the Lorentz equation of motion

For simplicity, we consider the case where the electric field is zero and the motion is limited on the horizontal plane and obtain

According to the correspondence in classical mechanics, the magnetic moment can be related to the angular momentum of a charged particle moving on a circle under a constant magnetic field, which leads to . Equation (8) can be written as

The factor represents the effect of the spacetime curvature due to the Earth’s gravitational field. Here we note that is defined in the flat spacetime, while the magnetic field is measured by the observer on the Earth’s surface. Therefore, in this paper, we define the effective magnetic moment as to include the gravitational effect. The above discussion remains within the classical picture since the lepton is not treated as a fermion with quantized spin and it is not trivial to extend the influence of gravity for quantum mechanical particles. Therefore the effective magnetic moment should be examined based on the formalism compatible with both quantum mechanics and general relativity.

In quantum field theory, the magnetic moment of a fermion is related to the form factors and defined in the vertex function (see Refs. [1,2]) and the g-factor in the flat spacetime is defined as

The origin of the magnetic moment varies according to the type of fermions as , for Dirac particles, i.e., fermions with , the internal structure and radiative corrections of which can be ignored; , for neutrons and neutral atoms, i.e., fermions with and ; and , for protons and ions, i.e., fermions with and . We applied quantum mechanics in the curved spacetime in general relativity for the three types:

Type A: Dirac particles with and (, ),

Type B: fermions with and (, 0),

Type C: fermions with and (, 0).

The coupling between the fermions and the electromagnetic field corresponds to minimal coupling for Type A, non-minimal coupling for Type B, and a mixture of minimal and non-minimal coupling for Type C.

In this paper, we generalize the Dirac equation in curved spacetime by employing the covariant formulation. For simplicity, we employ the Schwarzschild metric as the background spacetime, derive the Hamiltonian of free fermions to be observed using instruments fixed on the Earth’s surface up to the post-Newtonian order for the above three types and evaluate the effective value of the magnetic moment. We introduce the effective magnetic moment of fermions in the Earth’s gravitational field and discuss the influence on precision measurements of the anomalous magnetic moment of electrons and muons.

 2. Effective magnetic moment in curved spacetime

The influence of general relativity in the Hamiltonian of fermions has been considered and reported in several papers [3–5]. Wajima et al. [5] employed the covariant Dirac equation and regarded its time component as the Hamiltonian and calculated its expectation value up to the post-Newtonian order , where is the spinor function, the covariant derivative, the spin connection, the -matrices in the curved spacetime (Appendix A), and a small quantity to be used as the variable for the power series expansion to identify the post-Newtonian approximation. Since the electric charge and the magnetic moment were not involved in their method, we develop and generalize their method to derive the Hamiltonian containing gravitational effects in general relativity on the basis of the covariant Dirac equation with the electric charge and the magnetic moment. Here we adopt the Heisenberg equation for the magnetic moment as the definition of the magnetic moment in the flat spacetime. We generalize Eq. (15) using the general relativistic Hamiltonian to evaluate the effective magnetic moment .

2.1. Dirac particles with <inline-formula><tex-math notation="LaTeX" id="ImEquation69"><![CDATA[$\mathrm{g}=2$]]></tex-math></inline-formula> (Type A)

In this section, we consider Type A, for which the form factors are and , corresponding to point-like fermions, i.e., electrons and muons without radiative corrections. The covariant Dirac equation for this case is given with minimal coupling with the four potential as which leads to the Hamiltonian which is the general form of the Hamiltonian of Dirac particles with with the electric charge and the magnetic moment. It is uniquely determined for a given metric of the spacetime. Adopting the Schwarzschild metric (3), we can rewrite Eq. (17), after some algebra to rearrange it (see Appendix D), as

This Hamiltonian is a matrix. We can obtain a even matrix by applying the Foldy–Wouthuysen–Tani transformation [6–8]. Its left-up matrix (the large component) given as corresponds to the Hamiltonian of the Dirac particle in curved spacetime. Substituting , Eq. (19) leads to

Terms of the order of and are well known, while some of terms of the order of are newly introduced coupling terms between the gravitational potential , the magnetic moment , and the magnetic field . The effective magnetic moment is given as for sufficiently slow particles moving in a static spacetime. The result is consistent with the gravitationally modified magnetic moment obtained in the formalism [9,10]2. The above discussions are restricted to Dirac fermions minimally coupled to electromagnetic fields. Non-minimal coupling cases such as neutrons and spatially spread cases such as fermions dressed with renormalization and quantum radiative corrections are not involved.

 2.2. Fermions with <inline-formula><tex-math notation="LaTeX" id="ImEquation86"><![CDATA[$\mathrm{g}\ne2, e=0$]]></tex-math></inline-formula> (Type B)

Neutrons and neutral atoms correspond to the case of form factors of and since . Since they cannot minimally couple to the electromagnetic field, their covariant Dirac equation contains the following non-minimal coupling between the magnetic moment and the electromagnetic tensor : where and are second-rank covariant tensors (see Appendix A). The Hamiltonian is calculated as

In the same manner as described in the previous section, the large component in the Schwarzschild spacetime is obtained as

Accordingly, the effective magnetic moment is obtained as

This result shows that the effective magnetic moment with the general relativistic influence up to the order of has a common form for both Type A and Type B, in which the minimal and non-minimal coupling terms are employed in the Dirac equations (16) and (22), respectively.

 2.3. Fermions with <inline-formula><tex-math notation="LaTeX" id="ImEquation96"><![CDATA[$g\ne2$]]></tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX" id="ImEquation97"><![CDATA[$e\ne0$]]></tex-math></inline-formula> (Type C)

Spread charged fermions have the form factors of and . This case corresponds to electrons and muons dressed with quantum radiative corrections and also composite particles such as protons and atoms containing internal structure. The corresponding covariant Dirac equation contains both minimal coupling and non-minimal coupling terms as

The influence of radiative corrections and internal structure is contained in the last term in Eq. (26), which can be confirmed to observe that it returns to Eq. (16) to eliminate the last term corresponding to the spread of fermions. The Hamiltonian in -matrix format is calculated as and the large component in the -matrix format Hamiltonian as where

Accordingly, the effective magnetic moment is

Equation (30) can be understood as the general expression of the effective magnetic moment of fermions including leptons and hadrons. Although the g-factor of proton or atoms largely deviates from , the effective magnetic moment is given in the same form as Type A: Dirac particles with minimal coupling (16) and Type B: neutral fermions with non-minimal coupling (22).

 2.4. Effective anomalous magnetic moment in curved spacetime

The effective magnetic moment to be observed with the instrument fixed on the Earth’s surface has been obtained as Eq. (30) up to the post-Newtonian order adopting the Schwarzschild metric as the background spacetime, common to leptons and hadrons of Type A, Type B, and Type C. The effective values of the gyromagnetic ratio (g-factor) and the anomalous magnetic moment , to be measured in ground-based experiments, can be defined as which implies that the -factor and the anomalous magnetic moment depend on the gravitational field as a result of the influence of the curved spacetime in general relativity. For an example, the anomalous magnetic moment of fermions with is evaluated as on the Earth’s surface.

This implies that the Earth’s gravitational field induces an anomalous magnetic moment of for electrons, muons, etc. regardless of the type of fermions, in addition to quantum radiative corrections.

 2.5. Parametric post-Newtonian (PPN) approximation

Here we consider the case in which the parametric post-Newtonian (PPN) approximation is adopted as the background spacetime:

This is a primitive version of one of the possible extensions of general relativity [11], in which the parameter choice of and corresponds to the scalar–tensor gravity model and the choice of , corresponds to standard general relativity. In the same manner as in the case of general relativity, the large-component Hamiltonian can be obtained via the covariant Dirac equation (26) using the PPN metric as

Accordingly, the effective magnetic moment can be derived, in the same manner, as

We note that the parameter of the theoretical model of gravity explicitly appears in the gravitationally induced correction term. This implies that the precise determination of the magnetic moment in a specific local circumstance may determine the model parameter .

 3. Penning trap experiment to measure electron <inline-formula><tex-math notation="LaTeX" id="ImEquation116"><![CDATA[$\mathrm{g}-2$]]></tex-math></inline-formula> in the Earth’s gravity

The g-factor of leptons exactly equals 2 according to the Dirac theory (=2, for ), which is one of the important consequences of the Dirac equation for free fermions with an electric charge of and a spin of minimally coupling to an electromagnetic field. However, the real value of deviates from 2 by a fraction of about as a result of the quantum radiative corrections. This deviation is referred to as the anomalous magnetic moment, defined as where is the magnetic moment and is the Bohr magneton. The Bohr magneton is defined as = for the lepton mass and spin . For electrons, a comparison between the theoretical value, which was obtained with higher orders of quantum radiative corrections based on the standard model of elementary particles (SM) [12,15], and the experimental value, which was obtained in the precision measurement [13–16], shows the extremely precise agreement up to the 12th digit: as shown in Table 1, which clearly validates the accuracy of the quantum electrodynamics (QED) employed in the standard model of elementary particles [12,15,16].

Comparison of the theoretical value and the experimental value of the anomalous magnetic moment of the electron.

  Anomalous magnetic moment () Ref.
Experiment [13–16]
Theory [12,15]
Difference  
(Deviation) (1.1 )  

3.1. Interpretation of the experimental result of electron <inline-formula><tex-math notation="LaTeX" id="ImEquation134"><![CDATA[$\mathrm{g}-2$]]></tex-math></inline-formula>

Below, we examine the interpretation of the experimental result of the Penning trap experiment [13–15].

The cyclotron frequency and the spin precession frequency of an electron moving in a uniform magnetic field are measured in the Penning trap experiment [13–15]. The anomalous magnetic moment of electrons is regarded as being equal to the ratio of the anomalous spin precession frequency to the cyclotron frequency as

The has been treated as the electron anomalous magnetic moment in the conventional analysis without consideration of the gravitational effect. However, this experiment was carried out in the Earth’s gravitational field. Thus, the experimentally measured frequencies in Eq. (37) are the effective frequencies in a curved spacetime. Hereafter, we derive the effective values of the cyclotron frequency and the spin precession frequency in a curved spacetime.

General relativity requires that the translational motion of a free particle with an electric charge of is described by the covariant equation (2). Assuming that the Earth’s gravitational field can be described with the Schwarzschild metric shown in Eq. (3), the equation of translational motion of a free electron can be written as Eq. (4). In general, the solution of this equation gives a spiral motion that is a combination of translational acceleration and rotational motion. The cyclotron frequency is a measure of the rotational motion. In other words, the cyclotron frequency corresponds to the frequency of the rotational motion of the constant-norm vector parallel to the velocity. We define the unit vector parallel to the 3D velocity as

Here we use and assume that the effects of the gravitational gradient are negligibly small; the rotational component of the translational motion is given as

Using the cyclotron frequency in the flat spacetime the effective cyclotron frequency can be written as which represents the cyclotron frequency in the Earth’s gravitational field based on general relativity.

In the measurement of the electron experiment (i.e., the Penning trap experiment), the electron velocity is sufficiently small as and the effective value of the electron’s cyclotron frequency is given as

We can interpret that the is used as the experimentally measured cyclotron frequency to be observed with the ground-based instrument.

Here we analyze the behavior of an electron spin in the Earth’s gravitational field. The time evolution of an electron spin in the flat spacetime is described by the equation where . Rewriting this equation with the four vector, we obtain which is known as the Bargmann–Michel–Telegdi (BMT) equation [22]. In the same manner as in Eq. (2), employing the covariant derivative in the BMT equation, we obtain the covariant equation of spin motion in general relativity as

Here we substitute the Schwarzschild metric defined by Eq. (3) into the covariant BMT equation. In the same way as we have derived the cyclotron frequency based on the equation of translational motion of electrons, we consider the case where the gradient of the gravitational potential is negligibly small, such as the case where the motion is limited on a horizontal plane. We define the four spin vector as ; the spatial component of the generalized BMT equation (44) up to the post-Newtonian order can be written as

We can take in the measurement of electron (i.e., the Penning trap experiment), which implies that the effective value of the electron spin precession frequency to be observed with the ground-based instrument is given as

The experimental value of the anomalous magnetic moment obtained in the Penning trap experiment, denoted as below, should be compared with the ratio of the kinematically calculated effective values and . Substituting the effective values and into Eq. (37), we obtain

Although both effective values and are dependent on the Earth’s gravitational field, the gravitational effects are canceled in their ratio and coincides with . It is reasonable to regard the high-precision agreement between the experimental and theoretical values as confirming the cancellation of gravitational effects. It also suggests that, if we take the ratio of the conventionally defined in Eq. (35) to the Bohr magneton, the “true” anomalous magnetic moment deviates from the “conventional” value as

Since the Earth’s gravity induces an anomaly even for additionally to radiative corrections, the comparison between the experimental value measured in the curved spacetime and the theoretical value calculated in the flat spacetime results in a difference of

Here we introduce the redefinition of the effective Bohr magneton including the general relativistic effects of the Earth’s gravitational field as

In this case, we can treat as an invariant quantity independent of the gravitational field if we cancel the general relativistic effects by redefining the anomalous magnetic moment using the ratio of the effective magnetic moment to the effective Bohr magneton as instead of its ratio to the Bohr magneton in the flat spacetime. This redefinition restores the validity of the test of the standard model of elementary particles via the comparison of the theoretical value and the experimental value .

 4. Storage ring experiment to measure muon <inline-formula><tex-math notation="LaTeX" id="ImEquation172"><![CDATA[$\mathrm{g}-2$]]></tex-math></inline-formula> in the Earth’s gravity

In the storage ring experiments for such as CERN [17,18] and BNL E821 [19–21], the muon anomalous magnetic moment was determined from the relation where is the ratio of the anomalous spin precession frequency of muons to the proton spin precession frequency measured using the nuclear magnetic resonance, and is the ratio of the magnetic moment of muons to that of protons. This can be understood as being determined from the relation where is the Larmor precession frequency of muons. This interpretation can be obtained using the classical kinematical framework and is the basis of the design of the experiment to measure the muon [15–21]. We reconsider this interpretation including the spacetime curvature in the Earth’s gravitational field on the basis of general relativity.

4.1. Spin motion in flat spacetime

Here we give an overview of the procedure to evaluate the experimental value of the anomalous magnetic moment using the equation of motion of spin in the flat spacetime (see, e.g., Chap. 11 in Ref. [23]). The BMT equation (43) described in the laboratory frame (inertial frame) can be decomposed into a set of equations for time and spatial components as

The 3D spin vector in the inertial frame and the 3D spin vector are related through the Lorentz transformation

The equation of motion of the spin in the rest frame can be derived as

Rewriting the proper time with the time in the inertial frame , we can reproduce the familiar equation

We note that the last term in the equation of motion of spin in the rest frame (57) represents the relativistic term known as Thomas precession, which does not appear in the non-relativistic equation of motion of spin (42). The electric and magnetic fields in the inertial frame and are related to those in the rest frame and through the Lorentz transformation as

The equation of motion of spin in the rest frame can be rewritten with the quantities in the inertial frame by substituting Eqs. (6) and (58), which results in

By using the spin precession frequency , the anomalous spin precession frequency can be written as with . We assume that the motion is confined in the horizontal plane and the magnetic field is applied vertically. In addition, we assume that the electric term (the second term) can be canceled as by selecting a muon momentum of 29.3, which is referred to as the magic momentum. The anomalous magnetic moment of the muon in the flat spacetime, which is defined as , can be written as by rewriting the magnetic field using the Larmor precession frequency . Equation (61) represents the experimental value in the conventional interpretation. The experimental values of the anomalous magnetic moment using Eqs. (60) and (61) are evaluated within the classical kinematical framework. The relations were applied to achieve high precision by employing the magic momentum to suppress the contribution of electric fields.

 4.2. Spin motion in curved spacetime

In this section, we apply the procedure of the evaluation of the experimental value of the anomalous magnetic moment in the flat spacetime to that in the curved spacetime.

The BMT equation in the curved spacetime in the inertial frame (44) can be decomposed into the time and spatial components as

Considering that the contribution arising from the gradient of the gravitational potential is sufficiently small to be ignored in the post-Newtonian order (see Appendix E), Eq. (62) can be simplified as

Following the procedure in the flat spacetime, we derive the relation between the spins in the inertial frame and the rest frame using the Lorentz transformation. The Lorentz transformation can be naturally defined in the flat spacetime since the Lorentz invariance is satisfied at any point. However, the Lorentz transformation cannot be defined in the general coordinate system (curved spacetime), which disturbs the applicability of the same procedure. We avoid the difficulty by employing the local inertial frame to satisfy the local Lorentz invariance (see Appendix C). The four vector in the general coordinate system can be related to the four vector in the local inertial frame , using the tetrad , as

Using the explicit expression of the tetrad up to the post-Newtonian order in the Schwarzschild metric (shown in Eq. (B.7)), the transformation from the general coordinate system into the local inertial frame can be simplified as

We rewrite the BMT equation in the curved spacetime (63) using the local inertial frame and obtain which is an analogue to the relation in the flat spacetime. The local Lorentz transformation is obtained as

In the same manner as the flat spacetime, using the BMT equation in the curved spacetime (66) and the local Lorentz transformation (67), the equation of motion of spin in the rest frame can be written as which is similar to the case of the flat spacetime. This relation gives a relation similar to the case of the flat spacetime: by replacing the proper time with the time in the inertial frame . Here we notice that the electric and magnetic fields in the rest frame and are given through a local Lorentz transformation, using the electric ands magnetic fields in the inertial frame and , as

Thus, the equation of motion of spin in the rest frame can be written using the electric and magnetic fields in the inertial frame and as

Equation (71) has the same form as the equation of spin motion (59). We convert the electric and magnetic fields and in the local inertial frame to those in the general coordinate system to consider the experimental values with the ground-based instrument.

The equation of motion of spin in the rest frame represented in the general coordinate system is obtained as

Putting and using the effective value of the spin precession frequency in the curved spacetime (72) and the effective value of the cyclotron frequency in the curved spacetime (40), the effective value of the anomalous spin precession frequency is obtained as

When the motion is confined on the horizontal plane, a magnetic field is applied vertically, and the momentum is chosen to cancel the contribution of the electric field, the effective value of the anomalous magnetic moment of the muon is interpreted as

The Larmor precession frequency in the curved spacetime can be interpreted as a special solution of the BMT equation with = and =, which corresponds to the null translational motion, as

Therefore, we obtain which show that the effective value of the anomalous magnetic moment in the curved spacetime is equal to that in the flat spacetime up to the post-Newtonian order of general relativity. Thus the effective values of cyclotron frequency, spin precession frequency, and Larmor precession frequency have been derived based on the primitive consideration based on a kinematical framework. However those effective values in curved spacetime are, respectively, different from those values in the flat spacetime, the gravitational contribution is canceled in the ratio (76) and the anomalous magnetic moment in the curved spacetime coincides with the case in the flat spacetime.

 4.3. Gravitational influence on the storage ring experiment to measure muon <inline-formula><tex-math notation="LaTeX" id="ImEquation224"><![CDATA[$\mathrm{g}-2$]]></tex-math></inline-formula>

We have found that the gravitational influence on the magnetic moment of muons in the storage ring experiment is canceled in the ratio based on the kinematical consideration up to the post-Newtonian order using the Schwarzschild metric if the muon momentum is chosen to satisfy where is the experimental value of the anomalous magnetic moment. Consequently, the comparison between the experimental and theoretical values remains valid as before. However, Eq. (73) implies that general relativity modifies the contribution of the term. The perfect cancellation of the term corresponds to which introduces an offset to the experimental value of the anomalous magnetic moment as large as . Therefore, if the term is perfectly canceled in the experiment, is measured instead of and the offset may be misidentified as a discrepancy between the theory and the experiment. The general relativistic offset may be clarified in future experiments with improved accuracy.

 5. Conclusion

The magnetic moment of free fermions to be observed using instruments fixed on the Earth’s surface has been examined by evaluating the effective magnetic moment including the influence of the gravitational field up to the post-Newtonian order , adopting the Schwarzschild metric as the background spacetime. We found that the magnetic moments of fermions are influenced by the spacetime curvature as for the cases of minimal coupling, non-minimal coupling, and a mixture of the two. In the same manner, the effective values of the gyromagnetic ratio and the anomalous magnetic moment depend on the Earth’s gravitational field as , , which seems inconsistent with the precise agreement between the experimental values and the theoretical values calculated in the flat spacetime.

However, experimental values of the anomalous magnetic moment were obtained as a ratio of the spin precession frequency and the cyclotron frequency in the Penning trap and the storage ring experiments, and the gravitational influence is canceled to recover the consistency between the experiment and the theory. Proper treatment of the experimental offset of the anomalous magnetic moment measured in the storage ring method would be necessary to compare the theoretical and experimental values, taking into consideration the change of the cancellation condition of the electric field contribution.

 Acknowledgements

This work is supported in part by a Grant-in-Aid for Science Research from Japan Society for the Promotion of Science 501100001691 (JSPS) (No. 17K05453 to T.F).

 Funding

Open Access funding: SCOAP.

 Appendix A. <inline-formula><tex-math notation="LaTeX" id="ImEquation238"><![CDATA[$\boldsymbol{\gamma}$]]></tex-math></inline-formula>-matrix <p>The <inline-formula><tex-math notation="LaTeX" id="ImEquation239"><![CDATA[$\gamma$]]></tex-math></inline-formula>-matrices in a curved spacetime satisfy <disp-formula id="pty066-MA-1"><label>(A.1)</label><tex-math notation="LaTeX" id="Equation79"><![CDATA[ \begin{eqnarray} \{\gamma^\mu, \gamma^\nu\} &=& \gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2 g^{\mu\nu} I_4 , \nonumber\\ \{\gamma_\mu, \gamma_\nu\} &=& \gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=2 g_{\mu\nu} I_4 , \end{eqnarray}]]></tex-math></disp-formula> where <inline-formula><tex-math notation="LaTeX" id="ImEquation240"><![CDATA[$I_4$]]></tex-math></inline-formula> is the <inline-formula><tex-math notation="LaTeX" id="ImEquation241"><![CDATA[$4\times4$]]></tex-math></inline-formula> unit matrix, <inline-formula><tex-math notation="LaTeX" id="ImEquation242"><![CDATA[$g_{\mu\nu}$]]></tex-math></inline-formula> the metric tensor of the curved spacetime. In the flat spacetime, the above relation leads to <disp-formula id="pty066-MA-2"><label>(A.2)</label><tex-math notation="LaTeX" id="Equation80"><![CDATA[ \begin{eqnarray} \{\gamma^{(\alpha)}, \gamma^{(\beta)}\} &=& \gamma^{(\alpha)}\gamma^{(\beta)}+\gamma^{(\beta)}\gamma^{(\alpha)}=2 \eta^{\alpha\beta} I_4, \nonumber\\ \{\gamma_{(\alpha)}, \gamma_{(\beta)}\} &=& \gamma_{(\alpha)}\gamma_{(\beta)}+\gamma_{(\beta)}\gamma_{(\alpha)}=2 \eta_{\alpha\beta} I_4, \end{eqnarray}]]></tex-math></disp-formula> where <inline-formula><tex-math notation="LaTeX" id="ImEquation243"><![CDATA[$\eta_{\alpha\beta}$]]></tex-math></inline-formula> is the metric tensor of the flat spacetime (Minkowski metric). We define the tetrad <inline-formula><tex-math notation="LaTeX" id="ImEquation244"><![CDATA[$e^{\mu}_{(\alpha)}$]]></tex-math></inline-formula> satisfying the relation <disp-formula id="pty066-MA-3"><label>(A.3)</label><tex-math notation="LaTeX" id="Equation81"><![CDATA[ \begin{equation} g_{\mu\nu} \,\,e^\mu_{(\alpha)} e^\nu_{(\beta)} = \eta_{\alpha\beta} = \left( \begin{array}{cccc} \epsilon^{-2} & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right)\!.\,\,\,\,\,\, \end{equation}]]></tex-math></disp-formula></p> <p>The relation between the <inline-formula><tex-math notation="LaTeX" id="ImEquation245"><![CDATA[$\gamma$]]></tex-math></inline-formula>-matrices in the curved spacetime and the flat spacetime can be written as <disp-formula id="pty066-MA-4"><label>(A.4)</label><tex-math notation="LaTeX" id="Equation82"><![CDATA[ \begin{equation} \gamma^{\mu}=\gamma^{(\alpha)} \,\,e^\mu_{(\alpha)} ,\,\,\,\,\, \gamma_{\mu}=\gamma_{(\alpha)} \,\,e_\mu^{(\alpha)}. \end{equation}]]></tex-math></disp-formula></p> <p>Using the tetrad <inline-formula><tex-math notation="LaTeX" id="ImEquation246"><![CDATA[$e^{\mu}_{(\alpha)}$]]></tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX" id="ImEquation247"><![CDATA[$\gamma$]]></tex-math></inline-formula>-matrices, the spin connection <inline-formula><tex-math notation="LaTeX" id="ImEquation248"><![CDATA[$\Gamma_\mu$]]></tex-math></inline-formula> can be written as <disp-formula id="pty066-MA-5"><label>(A.5)</label><tex-math notation="LaTeX" id="Equation83"><![CDATA[ \begin{eqnarray} {\Gamma}_\mu &=& \frac{1}{4}\, g^{\lambda\nu} \, e_\lambda^{(\alpha)} \, \nabla_\mu \, e_\nu^{(\beta)} \,\,\frac{1}{2}\, \Big[ \gamma_{(\alpha)}, \gamma_{(\beta)} \Big] . \end{eqnarray}]]></tex-math></disp-formula></p> <p>Here we show the explicit representation of <inline-formula><tex-math notation="LaTeX" id="ImEquation249"><![CDATA[$\gamma$]]></tex-math></inline-formula>-matrices in the flat spacetime using the standard Dirac representation as <disp-formula id="pty066-MA-6"><label>(A.6)</label><tex-math notation="LaTeX" id="Equation84"><![CDATA[ \begin{eqnarray} && \gamma^{(0)} = {\epsilon} \beta = \epsilon \left( \begin{array}{cc} I & 0 \\ 0 & -I \end{array} \right)\!,\,\,\, \gamma^{(i)} = \left( \begin{array}{cc} 0 & \sigma_i \\ - \sigma_i & 0 \end{array} \right)\!, \nonumber\\ && \gamma_{(0)} = \frac{1}{\epsilon} \left( \begin{array}{cc} I & 0 \\ 0 & -I \end{array} \right)\!,\,\,\,\,\, \gamma_{(i)} = \left( \begin{array}{cc} 0 & -\sigma_i \\ \sigma_i & 0 \end{array} \right)\!, \end{eqnarray}]]></tex-math></disp-formula> where <inline-formula><tex-math notation="LaTeX" id="ImEquation250"><![CDATA[$\sigma_i$]]></tex-math></inline-formula> is the Pauli matrix satisfying <disp-formula id="pty066-MA-7"><label>(A.7)</label><tex-math notation="LaTeX" id="Equation85"><![CDATA[ \begin{equation} \sigma_i \, \sigma_j = \delta_{ij} + i\, \epsilon_{ijk} \,\sigma_k \end{equation}]]></tex-math></disp-formula> and their explicit representation is <disp-formula id="pty066-MA-8"><label>(A.8)</label><tex-math notation="LaTeX" id="Equation86"><![CDATA[ \begin{equation} \sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\!,\,\,\,\, \sigma_2 = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right)\!,\,\,\,\, \sigma_3 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)\!.\,\,\,\,\,\,\, \end{equation}]]></tex-math></disp-formula></p> <p>The <inline-formula><tex-math notation="LaTeX" id="ImEquation251"><![CDATA[$4\times4$]]></tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX" id="ImEquation252"><![CDATA[$\rho_i$]]></tex-math></inline-formula> matrices are Dirac matrices satisfying <disp-formula id="pty066-MA-9"><label>(A.9)</label><tex-math notation="LaTeX" id="Equation87"><![CDATA[ \begin{equation} \rho_i \, \rho_j = \delta_{ij} + i\, \epsilon_{ijk} \,\rho_k \end{equation}]]></tex-math></disp-formula> and their explicit representation is <disp-formula id="pty066-MA-10"><label>(A.10)</label><tex-math notation="LaTeX" id="Equation88"><![CDATA[ \begin{equation} \rho_1 = \left( \begin{array}{cc} 0 & I \\ I & 0 \end{array} \right)\!,\,\,\,\, \rho_2 = \left( \begin{array}{cc} 0 & -iI \\ iI & 0 \end{array} \right)\!,\,\,\,\, \rho_3 = \left( \begin{array}{cc} I & 0 \\ 0 & -I \end{array} \right)\!,\,\,\,\,\,\,\, \end{equation}]]></tex-math></disp-formula> which leads to <disp-formula id="pty066-MA-11"><label>(A.11)</label><tex-math notation="LaTeX" id="Equation89"><![CDATA[ \begin{equation} {\bf \alpha}_i = \rho_1 \sigma_i = \left( \begin{array}{cc} 0 & \sigma_i \\ \sigma_i & 0 \end{array} \right)\!, \,\,\,\,\,\,\,\, {\bf \beta} = \rho_3 = \left( \begin{array}{cc} I & 0 \\ 0 & -I \end{array} \right)\!. \end{equation}]]></tex-math></disp-formula></p> <p>Here we define the second-rank tensor <inline-formula><tex-math notation="LaTeX" id="ImEquation253"><![CDATA[$\sigma^{(\alpha)(\beta)}$]]></tex-math></inline-formula> as <disp-formula id="pty066-MA-12"><label>(A.12)</label><tex-math notation="LaTeX" id="Equation90"><![CDATA[ \begin{gather} \sigma^{(\alpha)(\beta)} = \frac{i}{2}\,\big[\gamma^{(\alpha)}, \gamma^{(\beta)}\big] ,\,\,\,\,\,\,\,\, \gamma^{(\alpha)} \gamma^{(\beta)} = \eta^{\alpha\beta}- i \,\sigma^{(\alpha)(\beta)} \\ \end{gather}]]></tex-math></disp-formula> <disp-formula id="pty066-MA-13"><label>(A.13)</label><tex-math notation="LaTeX" id="Equation91"><![CDATA[ \begin{gather} \sigma^{(0)(k)} = \frac{i}{2}\,\big[\gamma^{(0)}, \gamma^{(k)}\big] = i\epsilon\rho_1\sigma_k = i\epsilon\alpha_k \nonumber\\ \sigma^{(i)(j)} = \frac{i}{2}\,\big[\gamma^{(i)}, \gamma^{(j)}\big] = \epsilon^{ijk} \sigma_k\\ \end{gather}]]></tex-math></disp-formula> <disp-formula id="pty066-MA-14"><label>(A.14)</label><tex-math notation="LaTeX" id="Equation92"><![CDATA[ \begin{gather} \sigma_{(0)(k)} = \frac{i}{2}\,\big[\gamma_{(0)}, \gamma_{(k)}\big] = -i\epsilon^{-1}\rho_1\sigma_k = -i\epsilon^{-1}\alpha_k \nonumber\\ \sigma_{(i)(j)} = \frac{i}{2}\,\big[\gamma_{(i)}, \gamma_{(j)}\big] = \epsilon_{ijk} \sigma_k . \end{gather}]]></tex-math></disp-formula></p> <p>Using the tetrad <inline-formula><tex-math notation="LaTeX" id="ImEquation254"><![CDATA[$e^{(\alpha)}_{\mu}$]]></tex-math></inline-formula>, the second-rank tensors <inline-formula><tex-math notation="LaTeX" id="ImEquation255"><![CDATA[$\sigma^{\mu\nu}$]]></tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX" id="ImEquation256"><![CDATA[$F_{\mu\nu}$]]></tex-math></inline-formula> can be written as <disp-formula id="pty066-MA-15"><label>(A.15)</label><tex-math notation="LaTeX" id="Equation93"><![CDATA[ \begin{eqnarray} \sigma^{\mu\nu} &=& e^{\mu}_{(\alpha)} \, e^{\nu}_{(\beta)} \sigma^{(\alpha)(\beta)} \nonumber\\ F_{\mu\nu} &=& e_{\mu}^{(\alpha)} \, e_{\nu}^{(\beta)} F_{(\alpha)(\beta)} \end{eqnarray}]]></tex-math></disp-formula> and <disp-formula id="pty066-MA-16"><label>(A.16)</label><tex-math notation="LaTeX" id="Equation94"><![CDATA[ \begin{eqnarray} \sigma^{\mu\nu} F_{\mu\nu} &=& \sigma^{(\alpha)(\beta)}\,F_{(\alpha)(\beta)} . \end{eqnarray}]]></tex-math></disp-formula></p> </sec></app> <app id="APP2"><title>Appendix B. Geometrical values in the Schwarzschild metric <p>The Schwarzschild metric can be written as <disp-formula id="pty066-MB-1"><label>(B.1)</label><tex-math notation="LaTeX" id="Equation95"><![CDATA[ \begin{eqnarray} ds^2 &=& \left(1 - \epsilon^2\frac{2GM}{r}\right) c^2 dt^2 - \frac{dr^2 }{(1 - \epsilon^2\frac{2GM}{r})} - r^2(d\theta^2+\sin^2 \theta\, d{\varphi}^2) \end{eqnarray}]]></tex-math></disp-formula> in spherical coordinates. This can be rewritten, using isotropic coordinates and expanding up to the post-Newtonian order <inline-formula><tex-math notation="LaTeX" id="ImEquation257"><![CDATA[$O(\epsilon^2)$]]></tex-math></inline-formula>, as <disp-formula id="pty066-MB-2"><label>(B.2)</label><tex-math notation="LaTeX" id="Equation96"><![CDATA[ \begin{eqnarray} ds^2 &=& \epsilon^{-2} (1 + \epsilon^2 2\phi+\epsilon^4 2\phi^2) \, dt^2 - (1 - \epsilon^2 2\phi) (dx^2 + dy^2 + dz^2) \nonumber\\ & & \hspace{1cm} + O(\epsilon^4) , \end{eqnarray}]]></tex-math></disp-formula> where <inline-formula><tex-math notation="LaTeX" id="ImEquation258"><![CDATA[$\phi = -GM/r$]]></tex-math></inline-formula> is the Earth’s gravitational potential. When the metric tensor <inline-formula><tex-math notation="LaTeX" id="ImEquation259"><![CDATA[$g_{\mu\nu}$]]></tex-math></inline-formula> is uniquely defined, the Christoffel symbol <inline-formula><tex-math notation="LaTeX" id="ImEquation260"><![CDATA[$\Gamma^\lambda_{\mu\nu}$]]></tex-math></inline-formula> and the tetrad <inline-formula><tex-math notation="LaTeX" id="ImEquation261"><![CDATA[$e^{\mu}_{(\alpha)}$]]></tex-math></inline-formula> are also uniquely determined in the following.</p> <sec id="SECB.1"><title>B.1. Metric tensor <inline-formula><tex-math notation="LaTeX" id="ImEquation262"><![CDATA[$g_{\mu\nu}$]]></tex-math></inline-formula>

Using the definition and Eq. (B.2), the explicit expression of each component of the four metric tensor and is given up to the post-Newtonian order as

B.2. Christoffel symbol <inline-formula><tex-math notation="LaTeX" id="ImEquation266"><![CDATA[$\Gamma^\lambda_{\mu\nu}$]]></tex-math></inline-formula>

The Christoffel symbols are defined as

Substituting Eq. (B.3), we obtain explicit expressions for the Christoffel symbols up to the post-Newtonian order as where .

 B.3. Tetrad <inline-formula><tex-math notation="LaTeX" id="ImEquation269"><![CDATA[$e^\mu_{(\alpha)}$]]></tex-math></inline-formula>

Using Eqs. (B.3) and (B.5), the tetrad satisfying can be obtained up to the post-Newtonian order as

Appendix C. Local inertial frame <p>The tetrad enables us to define the local inertial frame in which the local Lorentz invariance (LLI) is satisfied. The four vectors in the local inertial frame <inline-formula><tex-math notation="LaTeX" id="ImEquation272"><![CDATA[$x^{(\mu)}$]]></tex-math></inline-formula> and in the general coordinate system <inline-formula><tex-math notation="LaTeX" id="ImEquation273"><![CDATA[$x^{\mu}$]]></tex-math></inline-formula> are related to each other using the tetrad as <disp-formula id="pty066-MC-1"><label>(C.1)</label><tex-math notation="LaTeX" id="Equation102"><![CDATA[ \begin{eqnarray} x^{(a)} &\equiv& e^{(a)}_{\mu} x^{\mu} = (\tilde{x}^{0}, \tilde{\boldsymbol{x}}) . \end{eqnarray}]]></tex-math></disp-formula></p> <p>Using the representation of the tetrad up to the post-Newton order <inline-formula><tex-math notation="LaTeX" id="ImEquation274"><![CDATA[$O(\epsilon^2)$]]></tex-math></inline-formula> shown in Eq. (<xref ref-type="disp-formula" rid="pty066-MB-7">B.7</xref>), the physical quantities relevant in this paper can be expressed as <disp-formula id="pty066-MC-2"><label>(C.2)</label><tex-math notation="LaTeX" id="Equation103"><![CDATA[ \begin{eqnarray} \tilde{S}^0 &=& (1+\epsilon^2\phi) \, {S}^0 \nonumber\\ \tilde{\boldsymbol{S}} &=& (1-\epsilon^2\phi) \, \boldsymbol{S} \nonumber\\ \tilde{\boldsymbol{\beta}} &=& \frac{ \tilde{\boldsymbol{u}} }{ \tilde{u}^0 } = (1-2\epsilon^2\phi) \,{\boldsymbol{\beta }} \nonumber\\ \tilde{ t} &=& (1+\epsilon^2\phi) \,\,t \nonumber\\ \tilde{\boldsymbol{x}} &=& (1-\epsilon^2\phi) \,\,\boldsymbol{x} \nonumber\\ \tilde{ \gamma } &=& \big( 1-2\epsilon^2\phi\, (\gamma^2-1) \big) \,{{ \gamma}} . \end{eqnarray}]]></tex-math></disp-formula></p> <p>The conversion of a tensor of rank 2 (electromagnetic tensor) is given as <disp-formula id="pty066-MC-3"><label>(C.3)</label><tex-math notation="LaTeX" id="Equation104"><![CDATA[ \begin{eqnarray} F_{(a)(b)} &=& e_{(a)}^{\mu} e_{(a)}^{\nu} F_{\mu\nu} , \end{eqnarray}]]></tex-math></disp-formula> which results in <disp-formula id="pty066-MC-4"><label>(C.4)</label><tex-math notation="LaTeX" id="Equation105"><![CDATA[ \begin{eqnarray} \tilde{\boldsymbol{E}} &=& F_{(0)(\alpha)} = e_{(0)}^{0} e_{(\alpha)}^{\alpha} F_{0\alpha} = \boldsymbol{E} \nonumber\\ \tilde{\boldsymbol{B}} &=& F_{(\alpha)(\beta)} = e_{(\alpha)}^{\alpha} e_{(\beta)}^{\beta} F_{\mu\nu} = (1+2\epsilon^2\phi) \boldsymbol{B} . \end{eqnarray}]]></tex-math></disp-formula></p> </sec> </app> <app id="APP4"><title>Appendix D. Normalization of the Hamiltonian <p>The scalar product of a spinor function <inline-formula><tex-math notation="LaTeX" id="ImEquation275"><![CDATA[$\Psi$]]></tex-math></inline-formula> is defined as <disp-formula id="pty066-MD-1"><label>(D.1)</label><tex-math notation="LaTeX" id="Equation106"><![CDATA[ \begin{equation} \left\langle \Psi \,{|}\, \Psi \right\rangle = \int \Psi^{\dagger} \Psi\, \sqrt{h}\, d^3x , \end{equation}]]></tex-math></disp-formula> where <inline-formula><tex-math notation="LaTeX" id="ImEquation276"><![CDATA[$h_{ij}$]]></tex-math></inline-formula> are the spatial components of the 4D tensor <inline-formula><tex-math notation="LaTeX" id="ImEquation277"><![CDATA[$g_{\mu\nu}$]]></tex-math></inline-formula> : <inline-formula><tex-math notation="LaTeX" id="ImEquation278"><![CDATA[$h_{ij}=-g_{ij}$]]></tex-math></inline-formula>. The expectation value of the Hamiltonian is given as <disp-formula id="pty066-MD-2"><label>(D.2)</label><tex-math notation="LaTeX" id="Equation107"><![CDATA[ \begin{equation} \left\langle {\cal H} \right\rangle = \left\langle \Psi \,{|}\, {\cal H} \,{|}\, \Psi \right\rangle = \int \Psi^{\dagger}\, {\cal H} \,\Psi\, \sqrt{h}\, d^3x . \end{equation}]]></tex-math></disp-formula></p> <p>Here we put <disp-formula id="pty066-MD-3"><label>(D.3)</label><tex-math notation="LaTeX" id="Equation108"><![CDATA[ \begin{eqnarray} \Psi' = h^{\frac{1}{4}}\, \Psi ,\,\,\,\,\, \,\,\,\,\, {\cal H}' = h^{\frac{1}{4}}\, {\cal H} \, h^{-\frac{1}{4}}\, \end{eqnarray}]]></tex-math></disp-formula> to obtain <disp-formula id="pty066-MD-4"><label>(D.4)</label><tex-math notation="LaTeX" id="Equation109"><![CDATA[ \begin{eqnarray} \left\langle \Psi \,{|}\, {\cal H} \,{|}\, \Psi \right\rangle &=& \int \Psi^{\dagger}\, {\cal H} \,\Psi\, \sqrt{h}\, d^3x \nonumber\\ &=& \int {\Psi'}^{\dagger}\, {\cal H}' \,{\Psi'}\, d^3x \nonumber\\ &=& \left\langle \Psi' \,{|}\, {\cal H}' \,{|}\, \Psi' \right\rangle . \end{eqnarray}]]></tex-math></disp-formula></p> <p>This corresponds to the normalization of the expectation value of the Hamiltonian into the expectation value in the local flat spacetime as <inline-formula><tex-math notation="LaTeX" id="ImEquation279"><![CDATA[${\cal H}'$]]></tex-math></inline-formula>, which justifies the description of <inline-formula><tex-math notation="LaTeX" id="ImEquation280"><![CDATA[$\epsilon$]]></tex-math></inline-formula>-dependent terms such as the post-Newtonian terms as a perturbation to the terms in the flat spacetime [<xref ref-type="bibr" rid="B5">5</xref>].</p> </sec> </app> <app id="APP5"><title>Appendix E. Typical scale of the gradient terms of the gravitational field <p>In general the equation of translational motion of a charged particle in curved spacetime involving general relativistic effects can be written as in Eq. (<xref ref-type="disp-formula" rid="pty066-M-2">2</xref>). Using the Schwarzschild metric and considering up to the post-Newtonian order <inline-formula><tex-math notation="LaTeX" id="ImEquation281"><![CDATA[$O(\epsilon^2)$]]></tex-math></inline-formula>, the equation can be written as in Eq. (<xref ref-type="disp-formula" rid="pty066-M-4">4</xref>). Here we estimate the magnitude of the contribution of the terms containing the gradient of the gravitational potential <inline-formula><tex-math notation="LaTeX" id="ImEquation282"><![CDATA[$\nabla\phi$]]></tex-math></inline-formula> (last two terms in Eq. (<xref ref-type="disp-formula" rid="pty066-M-4">4</xref>)) in the precise measurement of <inline-formula><tex-math notation="LaTeX" id="ImEquation283"><![CDATA[${\rm g}_{\mu} - 2$]]></tex-math></inline-formula> in the storage ring apparatus.</p> <p>The BNL E821 experiment [<xref ref-type="bibr" rid="B19">19</xref>–<xref ref-type="bibr" rid="B21">21</xref>] has apparatus parameters of <inline-formula><tex-math notation="LaTeX" id="ImEquation284"><![CDATA[$\gamma$]]></tex-math></inline-formula> = 29.3, <inline-formula><tex-math notation="LaTeX" id="ImEquation285"><![CDATA[$B$]]></tex-math></inline-formula> = 1.45 [T], <inline-formula><tex-math notation="LaTeX" id="ImEquation286"><![CDATA[$r$]]></tex-math></inline-formula> = 7112 [mm] and <inline-formula><tex-math notation="LaTeX" id="ImEquation287"><![CDATA[$\Omega_{\rm c} = 42 $]]></tex-math></inline-formula> [MHz], and the magnitudes of the electromagnetic interaction terms are estimated as <disp-formula id="pty066-ME-1"><label>(E.1)</label><tex-math notation="LaTeX" id="Equation110"><![CDATA[ \begin{eqnarray} f_{\rm EM} &=& \Bigg|\, \big( 1 + (2\gamma^2 + 1) \epsilon^2\phi \big) \,\frac{e}{\gamma\,m} \,\boldsymbol{\beta} \times \boldsymbol{B} \,\Bigg| \nonumber\\ &=& \Big|\, \big( 1 + (2\gamma^2 + 1) \epsilon^2\phi \big) \,\, \Omega_{\rm c} \times \beta \,\Big| \nonumber\\ &\simeq& 4.2 \times 10^{7} \,\,\,{\rm [s^{-1}] } . \end{eqnarray}]]></tex-math></disp-formula></p> <p>On the other hand, the magnitudes of terms containing the gradient of the gravitational potential are <disp-formula id="pty066-ME-2"><label>(E.2)</label><tex-math notation="LaTeX" id="Equation111"><![CDATA[ \begin{eqnarray} f_{\nabla\phi} &=& \Big|\, \epsilon (1+\beta^2)\nabla\phi \Big|\, \nonumber\\ &\simeq& 2 \times \frac{GM/R^2 }{c} \nonumber\\ &\simeq& 6.5 \times 10^{-8} \,\,{\rm [s^{-1}] } . \end{eqnarray}]]></tex-math></disp-formula></p> <p>Therefore, the relative magnitudes of the contributions of the gravity gradient and the electromagnetic interaction are <disp-formula id="pty066-ME-3"><label>(E.3)</label><tex-math notation="LaTeX" id="Equation112"><![CDATA[ \begin{eqnarray} \frac{f_{\nabla\phi}}{f_{\rm EM}} &\simeq& 1.5\times 10^{-15} , \end{eqnarray}]]></tex-math></disp-formula> which shows that the gravity gradient contribution is <inline-formula><tex-math notation="LaTeX" id="ImEquation288"><![CDATA[$10^{-15}$]]></tex-math></inline-formula> times smaller than the electromagnetic interaction and is even <inline-formula><tex-math notation="LaTeX" id="ImEquation289"><![CDATA[$10^{-5}$]]></tex-math></inline-formula> times smaller than the post-Newtonian effects. Consequently, the contribution of the gravity gradient is negligible as long as the magnitude of the relevant contribution is <inline-formula><tex-math notation="LaTeX" id="ImEquation290"><![CDATA[$10^{-10}$]]></tex-math></inline-formula> times that of the main contribution.</p> </sec></app></app-group> <fn-group> <title>Footnotes

1 In this paper we use units .

2 The formalism with the Schwarzschild metric corresponds to and , which leads to the gravitational dependence of for the magnetic moment [10].

 References ItzyksonC. and ZuberJ. B., Quantum Field Theory (McGraw-Hill, New York, 1985). PeskinM. E. and SchroederD. V., An Introduction to Quantum Field Theory (Addison-Wesley, New York, 1997). AnandanJ., Phys. Rev. D 30, 1615 (1984). (http://dx.doi.org/10.1103/PhysRevD.30.1615) ParkerL., Phys. Rev. D 22, 1922 (1980). (http://dx.doi.org/10.1103/PhysRevD.22.1922) WajimaS., KasaiM., and FutamaseT., Phys. Rev. D 55, 1964 (1997). (http://dx.doi.org/10.1103/PhysRevD.55.1964) FoldyL. L. and WouthuysenS. A., Phys. Rev. 78, 29 (1950). (http://dx.doi.org/10.1103/PhysRev.78.29) TaniS., Prog. Theor. Phys. 6, 267 (1951). (http://dx.doi.org/10.1143/ptp/6.3.267) CaseK. M., Phys. Rev. 95, 1323 (1954). (http://dx.doi.org/10.1103/PhysRev.95.1323) LightmanA. P. and LeeD. L., Phys. Rev. D 8, 364 (1973). (http://dx.doi.org/10.1103/PhysRevD.8.364) WillC. M., Phys. Rev. D 10, 2330 (1974). (http://dx.doi.org/10.1103/PhysRevD.10.2330) WillC. M., Theory and Experiment in Gravitational Physics (Cambridge University Press, Cambridge, UK, 1993). AoyamaT., HayakawaM., KinoshitaT., and NioM., Phys. Rev. D 91, 033006 (2015). (https://doi.org/10.1103/PhysRevD.91.033006) HannekeD., FogwellS., and GabrielseG., Phys. Rev. Lett. 100, 120801 (2008). (http://dx.doi.org/10.1103/PhysRevLett.100.120801) HannekeD., Fogwell HoogerheideS., and GabrielseG., Phys. Rev. A 83, 052122 (2011). (http://dx.doi.org/10.1103/PhysRevA.83.052122) MohrP. J., NewellD. B., and TaylorB. N., Rev. Mod. Phys. 88, 035009 (2016). (http://dx.doi.org/10.1103/RevModPhys.88.035009) PatrignaniC. et al [Particle Data Group], Chin. Phys. C 40, 100001 (2016). (https://doi.org/10.1088/1674-1137/40/10/100001) BaileyJ. et al, Nucl. Phys. B 150, 1 (1979). (https://doi.org/10.1016/0550-3213(79)90292-X) BaileyJ. et al, Phys. Lett. B 68, 191 (1977). (http://dx.doi.org/10.1016/0370-2693(77)90199-X) BennettG. W. et al, Phys. Rev. D 73, 072003 (2006). (https://doi.org/10.1103/PhysRevD.73.072003) MillerJ. P., de RafaelE. and RobertsB. L., Rep. Prog. Phys. 70, 795 (2007). (https://doi.org/10.1088/0034-4885/70/5/R03) PaleyJ. M., Measurement of the anomalous magnetic moment of the negative muon to 0.7 parts per million, Ph.D. Thesis, Boston University (2004). BargmannV., MichelL., and TelegdiV. L., Phys. Rev. Lett. 2, 435 (1959). (http://dx.doi.org/10.1103/PhysRevLett.2.435) JacksonJ. D., Classical Electrodynamics (Wiley, New York, 1998), 3rd ed.