aa r X i v : . [ g r- q c ] F e b Gravitomagnetic Stern–Gerlach Force
Bahram Mashhoon , ∗ Department of Physics and Astronomy,University of Missouri,Columbia, Missouri 65211, USA School of Astronomy,Institute for Research in Fundamental Sciences (IPM),P. O. Box 19395-5531, Tehran, Iran (Dated: March 2, 2021)A heuristic description of the spin-rotation-gravity coupling is presented and theimplications of the corresponding gravitomagnetic Stern–Gerlach force are brieflymentioned. It is shown, within the framework of linearized general relativity, thatthe gravitomagnetic Stern–Gerlach force reduces in the appropriate correspondencelimit to the classical Mathisson spin-curvature force.
PACS numbers: 03.30+p, 04.20.CvKeywords: spin-vorticity coupling, spin-gravity coupling
I. INTRODUCTION
Consider a free test particle of mass m moving with velocity V in an inertial frame ofreference in Minkowski spacetime. The free particle moves on a straight line with constantvelocity forever. Here, the Minkowski metric is − ds = η µν dX µ dX ν , X µ = ( ct, X, Y, Z ) , (1)where Greek indices run from 0 to 3, while Latin indices run from 1 to 3. The Minkowskimetric tensor η µν is given by diag( − , , , ∗ Electronic address: [email protected] that c = 1, unless specified otherwise. The equation of motion of the particle is obtainedvia the variational principle of stationary action δ S = 0, where S = Z − m ds = Z L dt , L = − m (1 − V ) / . (2)The corresponding Hamiltonian is H = γ mc , where γ is the Lorentz factor.Let us now imagine that the static inertial observer at the origin of the spatial coordinatesin Minkowski spacetime decides to refer the motion of the free particle to axes that rotatewith angular velocity Ω( t ) about the Z axis. This static observer thus becomes noninertialand its new reference frame has coordinates ( ct, r ), where r = ( x, y, z ). Then, V = v + Ω ( t ) × r , where v = d r /dt is the velocity of the particle with respect to the new rotatingaxes. From L = − m (cid:2) − ( v + Ω × r ) (cid:3) / , (3)we find the canonical momentum p = ∂ L ∂ v = γm ( v + Ω × r ) (4)and the Hamiltonian [1] H = H − Ω ( t ) · L , (5)where L = r × p is the orbital angular momentum of the free point particle.If the particle carries with it an “intrinsic” spin vector S , then S remains constant alongthe straight trajectory of the particle in the inertial frame. However, with respect to therotating coordinate system, S appears to precess with angular velocity − Ω ( t ). Let s i , i = 1 , ,
3, be the components of S with respect to the rotating axes; then, ds i dt + ǫ ijk Ω j s k = 0 . (6)On the other hand, for a true intrinsic quantum spin vector with the commutation relations[ s p , s q ] = i ~ ǫ pqn s n (7)that is invariant under the rotation of coordinates, the Heisenberg equation of motion forsuch a quantum observable, ~ ds k dt = i [ H SR , s k ] , (8)results in Equation (6) if the spin-rotation Hamiltonian is of the form H SR = − S · Ω ( t ) . (9)This is the Hamiltonian that accounts for the precessional motion of the spin in the quantumtheory. It follows that in the quantum case there is an additional contribution to the classicalHamiltonian (5) such that the total Hamiltonian of the particle in the rotating frame is givenby H + H SR . Hence, H T otal = H − Ω ( t ) · J , (10)where J := L + S is the total angular momentum of the free particle. This is a naturalresult, since J is the generator of rotations in the quantum theory. The energy eigenvaluesas measured by the noninertial static observer include the spin-rotation coupling, which isa quantum inertial effect that is independent of mass of the particle. In the classical limit, ~ → A. Spin-Vorticity Coupling
Consider a laboratory experiment involving a rotating system, which creates a congruencein spacetime. As a body rotates, we expect that the intrinsic spins of the constituent particlesall remain fixed with respect to the local inertial frame; therefore, the intrinsic spins allappear to precess with respect to the body-fixed frame. In the continuum limit, it may bethat the local angular velocity of motion becomes dependent on position, in which case thespin-rotation coupling naturally goes over to the spin-vorticity coupling [43, 44] H SV = − S · ω , (11)where ω is the vorticity ω = ∇ × V (12)and V is the velocity field of the congruence. If the angular velocity is spatially uniformsuch that V = Ω × r , then ω = 2 Ω and H SV reduces to H SR . For a description of movingmacroscopic matter in continuum mechanics, see section E.4.1 of Ref. [50]. For recent workon spin-vorticity coupling, see [45, 46]. B. Stern–Gerlach Force due to Spin-Vorticity Coupling
In general, vorticity depends on position and we might then expect the appearance of anattendant Stern–Gerlach force as well; that is, f µ = − ∂ µ ( H SV ) = 12 S · ∇ µ ω . (13)Such a spin-dependent force could then lead to the generation of a spin current. This ideawas apparently first proposed in Ref. [40] and received experimental confirmation in [41–43].For the extension of these ideas to fluid spintronics, see [44] and the references cited therein.Moreover, Ref. [47] deals with the application of spin-vorticity coupling in fluid dynamics. II. SPIN-GRAVITY COUPLING
Within the framework of linearized general relativity, we use here the approximationscheme known as gravitoelectromagnetism (GEM) that is based on the well-known analogywith Maxwell’s electrodynamics. We are interested in the weak exterior field of a compactrotating astronomical source with mass M and proper angular momentum J . The spacetimemetric, − ds = g µν dx µ dx ν , is given in a Cartesian system of coordinates x α = ( ct, x ) by [49] − ds = − c (cid:18) − c (cid:19) dt − c ( A · d x ) dt + (cid:18) c (cid:19) δ ij dx i dx j , (14)which represents Minkowski spacetime plus a linear perturbation due to the source. Thatis, g µν = η µν + h µν . We neglect all metric perturbation terms of O ( c − ) in this weak-fieldand slow-motion approximation method. In Equation (14), Φ( t, x ) is the gravitoelectricpotential and A ( t, x ) is the gravitomagnetic vector potential. For the exterior field of arotating astronomical mass, for instance, − Φ is the Newtonian gravitational potential and A is due to mass current and vanishes in the Newtonian limit ( c → ∞ ). Very far from therotating source, Φ ∼ GMr , A ∼ Gc J × x r , (15)where r = | x | . The GEM potentials satisfy the transverse gauge condition1 c ∂ Φ ∂t + ∇ · (cid:18) A (cid:19) = 0 . (16)Moreover, in analogy with electrodynamics, the GEM fields are defined by E = −∇ Φ − c ∂∂t (cid:18) A (cid:19) , B = ∇ × A , (17)in terms of which Einstein’s field equations in this case become formally similar to Maxwell’sequations [51]. For discussions of the non-Newtonian gravitomagnetic effects, see [49, 52].We are interested in the motion of a classical spinning point particle in the GEM field. Therelevant equations in this case are the Mathisson–Papapetrou (“pole-dipole”) equations [53,54], namely, DP µ dτ = F µ , F µ = − R µναβ u ν S αβ , (18) DS µν dτ = P µ u ν − P ν u µ . (19)In these equations, F µ , F µ u µ = 0, is the Mathisson spin-curvature force [55], u µ = dx µ /dτ is the 4-velocity of the pole-dipole particle and τ is its proper time. Moreover, P µ is the4-momentum of the particle and S µν is its spin tensor that satisfies the Frenkel–Piranisupplementary condition [56, 57] S µν u ν = 0 . (20)It follows from these equations that P µ = m u µ + S µν Du ν dτ , (21)where m := − P µ u µ is the mass of the spinning particle and is a constant of the motion.That is, differentiating m = − P · u and using F · u = 0 together with Equation (21), we find dmdτ = − (cid:18) m u µ + S µν Du ν dτ (cid:19) Du µ dτ = 0 , (22)since u · u = − S µν is antisymmetric. In the massless limit ( m → S µ via S µ = − η µνρσ u ν S ρσ , S αβ = η αβγδ u γ S δ , (23)where η αβγδ = ( − g ) / ǫ αβγδ is the Levi-Civita tensor and ǫ αβγδ is the alternating symbolwith ǫ = 1. The Mathisson spin-curvature force now takes the form F µ = ∗ R µνρσ u ν S ρ u σ , ∗ R µνρσ = 12 η µναβ R αβ ρσ , (24)in terms of the dual Riemann tensor, and the spin dynamics is represented by( g µν + u µ u ν ) DS ν dτ = 0 , (25)so that S µ , S µ u µ = 0, is Fermi–Walker transported along the world line of the spinningparticle [57].Consider now a pole-dipole particle held at rest in space in the exterior GEM field.Nongravitational torque-free forces are necessary to counter the Mathisson force as well asthe attraction of gravity of the source in order to keep the particle fixed in space. The4-velocity vector of the particle is then given by u µ = (1 + Φ /c ) δ µ . A natural orthonormaltetrad frame λ µ ( α ) adapted to the static test pole-dipole particle with u µ = λ µ (0) is given inthe ( ct, x, y, z ) coordinate system by λ µ (0) = (1 + Φ /c , , , , (26) λ µ (1) = ( − A /c , − Φ /c , , , (27) λ µ (2) = ( − A /c , , − Φ /c , , (28) λ µ (3) = ( − A /c , , , − Φ /c ) , (29)where the tetrad axes are primarily along the local GEM coordinate axes. The projectionof the spin vector on the adapted tetrad frame is given by S ( α ) = S µ λ µ ( α ) , (30)which implies that S (0) = 0 and dS ( i ) dτ = (cid:20) Dλ µ ( i ) dτ λ µ ( j ) (cid:21) S ( j ) . (31)A straightforward calculation reveals that to linear order in the perturbation Dλ µ ( i ) dτ λ µ ( j ) = ∂ j A i − ∂ i A j . (32)Therefore, dS ( i ) dτ = ǫ ijk B j S ( k ) (33)and the spin vector precesses with an angular velocity given by the local gravitomagneticfield. We can regard the gravitomagnetic field in Equation (33) as the locally measured field within our approximation scheme. That is, the GEM potentials can be combinedinto a 4-vector in analogy with electrodynamics and the corresponding GEM field tensoris then projected on the tetrad frame λ µ ( α ) to obtain the measured gravitoelectric andgravitomagnetic fields at the location of the spinning particle. However, λ µ ( α ) differs from δ µα by terms that are linear in the spacetime perturbation; therefore, in our approximationmethod E and B are indeed the same as the measured fields.If the spin vector is of quantum origin and represents the intrinsic spin of the “point”particle, then a spin-gravity Hamiltonian in terms of measured quantities is associated withits precession such that H SG = 1 c S · B . (34)We assume here that a particle with intrinsic spin behaves in the correspondence limit like anideal gyroscope. For instance, in connection with experiments in an Earth-based laboratory,to every Hamiltonian we must add the spin-rotation-gravity contribution δ H ≈ − Ω ⊕ · S + Ω P · S , (35)where Ω ⊕ and Ω P = B ⊕ /c refer to the Earth’s rotation frequency and the correspondinggravitomagnetic precession frequency, respectively. In fact, we have approximately Ω P = Gc r [3( J · r ) r − J r ] . (36)In the recent GP-B experiment [59, 60], the non-Newtonian gravitomagnetic field of theEarth has been directly measured and the prediction of general relativity has been verifiedat about the 19% level.In Equation (35), the difference in the energy of a spin-1 / ~ Ω ⊕ ≈ − eV and ~ Ω P ≈ − eV. For recent attempts to measure the spin-gravity term, see [61, 62]. Furthermore,the gravitomagnetic field depends upon position; therefore, there exists a gravitomagneticStern–Gerlach force −∇ ( Ω P · S ) on a spinning particle that is independent of its massand hence violates the principle of equivalence and the universality of free fall. This forcenaturally leads to a differential deflection of polarized beams. For various implications ofthe spin-gravity coupling, see [63–72].We now wish to establish a general correspondence between the gravitomagnetic Stern–Gerlach force and the Mathisson spin-curvature force for a steady-state configuration. Pro-jecting the gravitomagnetic Stern–Gerlach force, f µ = − ∂ µ H SG on the orthonormal tetradframe λ µ ( α ) , we have f (0) = 0 , f ( i ) = − c ∂ i B j S ( j ) . (37)On the other hand, the Mathisson spin-curvature force projected on the tetrad frame λ µ ( α ) is given by F (0) = 0 and F ( i ) = c ∗ R ( i )(0)( j )(0) S ( j ) . (38)We want to show that f ( i ) reduces to F ( i ) in the correspondence limit.In an arbitrary gravitational field, one can project the Riemann tensor onto an orthonor-mal tetrad frame Θ µ ˆ α adapted to an observer; the measured components of the curvatureare then R µνρσ Θ µ ˆ α Θ ν ˆ β Θ ρ ˆ γ Θ σ ˆ δ . (39)Taking the symmetries of the Riemann tensor into account, one can express Equa-tion (39) in the standard manner as a 6 × { , , , , , } . The general form of this matrix is E HH T S , (40)where E and S are symmetric 3 × H is traceless. Here, E , H and S represent themeasured gravitoelectric, gravitomagnetic and spatial components of the Riemann curvaturetensor, respectively. If the spacetime is Ricci flat, then Equation (40) takes the form E HH − E , (41)where E and H are now symmetric and traceless. That is, in the Ricci-flat case, the Riemanncurvature tensor degenerates into the Weyl conformal curvature tensor whose gravitoelectricand gravitomagnetic components are then E ˆ a ˆ b = C αβγδ Θ α ˆ0 Θ β ˆ a Θ γ ˆ0 Θ δ ˆ b , H ˆ a ˆ b = C ∗ αβγδ Θ α ˆ0 Θ β ˆ a Θ γ ˆ0 Θ δ ˆ b , (42)where C ∗ αβγδ is the unique dual of the Weyl tensor given by C ∗ αβγδ = 12 η αβ µν C µνγδ , (43)since the right and left duals of the Weyl tensor coincide.In our GEM scheme, g µν = η µν + h µν and the gauge-invariant curvature tensor is givenby R µνρσ = 12 ( h µσ, νρ + h νρ, µσ − h νσ, µρ − h µρ, νσ ) . (44)We recall that to lowest order λ µ ( α ) ≈ δ µα and hence in the exterior of a GEM source, wehave the Weyl tensor in the form (41) with symmetric and traceless matrices given by E ij = − c Φ ,ij + O ( c − ) = 1 c E j,i + O ( c − ) (45)and H ij = − c ∂ i B j + 1 c ǫ ijk ∂E k ∂t + O ( c − ) . (46)It follows from these results and Equations (37)–(38) that for a stationary GEM field,the gravitomagnetic Stern–Gerlach force in the correspondence limit is the same as theMathisson spin-curvature force.The spin interactions discussed in this paper all involve Hamiltonians that are similarto that of the traditional Zeeman effect. Moreover, the gravitational Larmor theorem canbe invoked to connect spin-gravity coupling with the spin-rotation coupling. The localequivalence between magnetism and rotational inertia was first established via Larmor’soriginal theorem [73]. The gravitational Larmor theorem is an expression of Einstein’s localprinciple of equivalence of gravitation and inertia [49, 74].Consider a steady-state configuration with exterior metric (14). In this stationary gravi-tational field, the temporal coordinate can be subjected to a simple scale transformation ofthe form t (1+Φ /c ) t , where Φ is a constant such that | Φ | ≪ c . The only consequenceof this transformation is that − g = 1 − − − Φ ) /c , while the other terms0in the metric remain unchanged since we neglect all terms of O ( c − ). In a sufficiently smallneighborhood around any event in the exterior GEM spacetime, we can replace the metricby that of an accelerated system in Minkowski spacetime. The resulting metric is to linearorder of the form ( η µν + ℓ µν ) dX µ dX ν , where [2, 49] ℓ = − a L · X , ℓ i = ( Ω L × X ) i , ℓ ij = 0 . (47)This has the form of a first-order perturbation where a L is the constant translational acceler-ation and Ω L is the constant rotational frequency of the accelerated system. A comparisonwith the GEM metric reveals that the corresponding gravitoelectric and gravitomagneticpotentials are given by Φ − Φ = − a L · X , A = − Ω L × X . (48)We neglect the spatial curvature of the GEM metric in this analogy. Moreover, E = −∇ Φ = a L and B = ∇ × A = − Ω L are the corresponding fields. It is clear that the spin-rotationHamiltonian H SR = − S · Ω L corresponds to the spin-gravity Hamiltonian H SG = S · B viathe gravitational Larmor theorem. III. LINEAR GRAVITATIONAL WAVES
The general linear approximation of general relativity involves GEM fields of massivesystems as well as linearized gravitational waves. The purpose of this section is to discussspin-gravity coupling for linearized gravitational waves; in particular, we are interested inthe corresponding Stern–Gerlach force. For related studies, see [75–82] and the referencescited therein.Consider a free linear gravitational radiation field, which can be expressed as a Fouriersum of plane monochromatic components each with frequency ω g and wave vector k g , ω g = c | k g | . The gravitational potential of the radiation is given by the symmetric tensor ¯ h µν , whichis a perturbation on the background Minkowski spacetime; that is, g µν = η µν + ¯ h µν ( x ), where x α = ( ct, x, y, z ). In the transverse-traceless (TT) gauge, ¯ h µν ,ν = 0, ¯ h µ = 0 and ¯ h µµ = 0.In this gauge, the gravitational potentials ¯ h ij ( x ) each satisfies the standard wave equation.For the sake of definiteness, let the incident radiation be a monochromatic plane wave1propagating along the x direction. Then, ¯ h ij can be written as(¯ h ij ) = h + h × h × − h + , (49)where h + = ˜ h + cos[ ω g ( t − x ) + ϕ + ] , h × = ˜ h × cos[ ω g ( t − x ) + ϕ × ] (50)represent the ⊕ (“plus”) and ⊗ (“cross”) linear polarization states of the radiation. Here,(˜ h + , ϕ + ) and (˜ h × , ϕ × ) are constants associated with the independent states of the radiationfield.It is a general result that in a spacetime with a metric of the form − dt + g ij ( x ) dx i dx j ,observers that remain permanently at rest in space follow geodesic world lines. Thus imag-ine this class of geodesic observers each at rest in space with a 4-velocity vector e µ ˆ0 = δ µ in the spacetime under consideration here. To each such observer, we associate an adaptedorthonormal tetrad frame e µ ˆ α that is parallel propagated along its world line. It is straight-forward to show that e µ ˆ α = δ µα −
12 ¯ h µα . (51)The projection of the curvature tensor (44) in the case of the incident gravitational wave onthe tetrad frame (51) results in R ˆ α ˆ β ˆ γ ˆ δ = R µνρσ e µ ˆ α e ν ˆ β e ρ ˆ γ e σ ˆ δ . Here, e µ ˆ α is in effect δ µα in ourlinear approximation scheme and the GEM components of curvature can be represented asin Equation (41) with E ij = 12 ω g ¯ h ij , H ij = 12 ω g h × − h + − h + − h × . (52)For measurement purposes, it proves interesting to set up a quasi-inertial Fermi normalcoordinate system with coordinates X ˆ µ = ( cT, b X, b Y , b Z ) based on the nonrotating tetradframe (51) along the world line of an arbitrary fiducial static geodesic observer. Here, T = t is the proper time of the reference observer fixed at ( x, y, z ) = ( x , y , z ). The spacetimemetric in the Fermi frame is given by − ds = g ˆ µ ˆ ν dX ˆ µ dX ˆ ν (53)2where g ˆ0ˆ0 = − − R ˆ0ˆ i ˆ0ˆ j X ˆ i X ˆ j , (54) g ˆ0ˆ i = − R ˆ0ˆ j ˆ i ˆ k X ˆ j X ˆ k (55)and g ˆ i ˆ j = δ ij − R ˆ i ˆ k ˆ j ˆ l X ˆ k X ˆ l . (56)In these expansions, we have neglected third and higher-order terms. In close analogy withthe GEM case, we can define the gravitoelectric potential ˆΦ and gravitomagnetic vectorpotential ˆ A via g ˆ0ˆ0 = − g ˆ0ˆ i = − A i ; that is,ˆΦ = − R ˆ0ˆ i ˆ0ˆ j X ˆ i X ˆ j , ˆ A i = 13 R ˆ0ˆ j ˆ i ˆ k X ˆ j X ˆ k . (57)Similarly, the corresponding fields can be defined as in Equation (17); in fact, to lowest orderwe find ˆ E i = R ˆ0ˆ i ˆ0ˆ j X ˆ j , ˆ B i = − ǫ ijk R ˆ j ˆ k ˆ0ˆ l X ˆ l . (58)Concentrating on the incident gravitational wave under consideration in this section,Equation (52) implies ˆΦ = − ω g [ h + ( b Y − b Z ) + 2 h × b Y b Z ] (59)and ˆ A = 23 ˆΦ , ˆ A = 16 ω g b X ( h + b Y + h × b Z ) , ˆ A = 16 ω g b X ( h × b Y − h + b Z ) . (60)Moreover, the relevant GEM fields areˆ E = 0 , ˆ E = 12 ω g ( h + b Y + h × b Z ) , ˆ E = 12 ω g ( h × b Y − h + b Z ) , (61)ˆ B = 0 , ˆ B = − ˆ E , ˆ B = ˆ E , (62)which are clearly transverse to the direction of wave propagation, | ˆ E | = | ˆ B | and ˆ E · ˆ B = 0.For the incident wave, the gravitoelectric and gravitomagnetic potentials are defined via the g ˆ0ˆ µ components of the Fermi metric and the remaining spatial components can be expressedas ( g ˆ i ˆ j ) = A ˆ A ˆ A ˆ A − ξ + − ξ × ˆ A − ξ × ξ + , (63)3where ξ + = 16 ω g h + b X , ξ × = 16 ω g h × b X . (64)Within this Fermi coordinate system, let us imagine the class of observers that stayspatially at rest. It is straightforward to show that a proper orthonormal tetrad frame Λ ˆ µ ˜ α adapted to this class of observers is given in ( t, b X, b Y , b Z ) coordinates byΛ ˆ µ ˜0 = (1 + ˆΦ , , , , (65)Λ ˆ µ ˜1 = ( − A , − ˆΦ , , , (66)Λ ˆ µ ˜2 = ( − A , − ˆ A , ξ + , , (67)Λ ˆ µ ˜3 = ( − A , − ˆ A , ξ × , − ξ + ) , (68)where the tetrad axes are primarily along the Fermi coordinate axes.Consider now a spinning test particle held fixed in space at ( b X, b Y , b Z ) by a referenceobserver in the Fermi frame. Projecting the spin vector S ˆ µ in the Fermi frame on the tetradframe Λ ˆ µ ˜ α of the local reference observer, S ˜ α = S ˆ µ Λ ˆ µ ˜ α , we find S ˜0 = 0, as before, and dS ˜ i d ˜ t = (cid:20) D Λ ˆ µ ˜ i d ˜ t Λ ˆ µ ˜ j (cid:21) S ˜ j , (69)where ˜ t is the proper time of the reference observer and dt = (1+ ˆΦ) d ˜ t . A detailed calculationreveals that to lowest order in b X , b Y and b Z within the Fermi coordinate system D Λ ˆ µ ˜ i d ˜ t Λ ˆ µ ˜ j = ∂ j ˆ A i − ∂ i ˆ A j ; (70)hence, dS ˜ i d ˜ t = ǫ ijk ˆ B j S ˜ k . (71)Thus, as before, the dominant effect is that the spin vector precesses with an angular velocitygiven by the local gravitomagnetic field. We note that Λ ˆ µ ˜ α differs from δ µα by terms linearin the perturbation; hence, the gravitomagnetic field in Equation (71) is in effect the fieldmeasured by the reference observer. The corresponding Stern–Gerlach force, f ˆ µ = − ∂ ˆ µ ( S · ˆ B ),to lowest order in b X , b Y and b Z as measured by the reference observer, is f ˜0 = 0 and f ˜1 = 0 , f ˜2 = ω g ( h × S ˜2 − h + S ˜3 ) , f ˜3 = − ω g ( h + S ˜2 + h × S ˜3 ) . (72)4On the other hand, the Mathisson force (24) as measured by the reference observer is givenby F ˜0 = 0 and F ˜ i = H ˜ i ˜ j S ˜ j , (73)where H ˜ i ˜ j is given to lowest order by Equation (52). This is a consequence of the fact thatin our approximation scheme Λ ˆ µ ˜ α is in effect given by δ µα for the calculation of the measuredcomponents of the curvature tensor. It is then evident that the resulting components of theMathisson force for the gravitational wave field under consideration in this section coincidewith those of the Stern–Gerlach force given by Equation (72) in the correspondence limit. IV. DISCUSSION
The Mathisson–Papapetrou equations for a spinning test particle together with theFrenkel–Pirani supplementary condition imply that the spin vector of a test pole-dipoleparticle is Fermi–Walker transported along its world line [57]. For a spinning test particleheld spatially at rest by a fiducial observer in the Ricci-flat region of an arbitrary gravita-tional field within the framework of linearized general relativity, the Fermi–Walker equationfor the spin vector indicates that its measured components undergo a precessional motionwith an angular velocity that is given by the locally measured gravitomagnetic field. For anintrinsic quantum spin, there is therefore a spin-gravitomagnetic field coupling Hamiltonianassociated with such precessional motion that can be obtained from Heisenberg’s equationof motion. The gravitomagnetic field generally depends upon position; therefore, there is anaccompanying Stern–Gerlach force connected with such a spin-gravity coupling. We showthat under appropriate conditions, this Stern–Gerlach force reduces in the correspondencelimit to Mathisson’s classical spin-curvature force.
Acknowledgments
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