# Gravitomagnetic Stern–Gerlach Force

Bahram Mashhoon1,2<EMAIL_ADDRESS>1Department of Physics and
Astronomy, University of Missouri, Columbia, Missouri 65211, USA
2School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.
O. Box 19395-5531, Tehran, Iran

###### Abstract

A heuristic description of the spin-rotation-gravity coupling is presented and
the implications of the corresponding gravitomagnetic Stern–Gerlach force are
briefly mentioned. It is shown, within the framework of linearized general
relativity, that the gravitomagnetic Stern–Gerlach force reduces in the
appropriate correspondence limit to the classical Mathisson spin-curvature
force.

spin-vorticity coupling, spin-gravity coupling

###### pacs:

03.30+p, 04.20.Cv

##  I Introduction

Consider a free test particle of mass $m$ moving with velocity $\mathbf{V}$ in
an inertial frame of reference in Minkowski spacetime. The free particle moves
on a straight line with constant velocity forever. Here, the Minkowski metric
is

$-ds^{2}=\eta_{\mu\nu}\,dX^{\mu}dX^{\nu}\,,\qquad X^{\mu}=(ct,X,Y,Z)\,,$ (1)

where Greek indices run from 0 to 3, while Latin indices run from 1 to 3. The
Minkowski metric tensor $\eta_{\mu\nu}$ is given by diag$(-1,1,1,1)$.
Throughout this paper, we use the convention that c = 1, unless specified
otherwise. The equation of motion of the particle is obtained via the
variational principle of stationary action $\delta\mathcal{S}=0$, where

$\mathcal{S}=\int-m\,ds=\int\mathcal{L}\,dt\,,\qquad\mathcal{L}=-m(1-V^{2})^{1/2}\,.$
(2)

The corresponding Hamiltonian is $\mathcal{H}_{0}=\gamma\,mc^{2}$, where
$\gamma$ is the Lorentz factor.

Let us now imagine that the static inertial observer at the origin of the
spatial coordinates in Minkowski spacetime decides to refer the motion of the
free particle to axes that rotate with angular velocity $\Omega(t)$ about the
$Z$ axis. This static observer thus becomes noninertial and its new reference
frame has coordinates $(ct,\mathbf{r})$, where $\mathbf{r}=(x,y,z)$. Then,
$\mathbf{V}=\mathbf{v}+\bm{\Omega}(t)\times\mathbf{r}$, where
$\mathbf{v}=d\mathbf{r}/dt$ is the velocity of the particle with respect to
the new rotating axes. From

$\mathcal{L}=-m\left[1-(\mathbf{v}+\bm{\Omega}\times\mathbf{r})^{2}\right]^{1/2}\,,$
(3)

we find the canonical momentum

$\mathbf{p}=\frac{\partial\mathcal{L}}{\partial\mathbf{v}}=\gamma
m(\mathbf{v}+\bm{\Omega}\times\mathbf{r})\,$ (4)

and the Hamiltonian LL

$\mathcal{H}=\mathcal{H}_{0}-\bm{\Omega}(t)\cdot\mathbf{L}\,,$ (5)

where $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ is the orbital angular momentum
of the free point particle.

If the particle carries with it an “intrinsic” spin vector $\mathbf{S}$, then
$\mathbf{S}$ remains constant along the straight trajectory of the particle in
the inertial frame. However, with respect to the rotating coordinate system,
$\mathbf{S}$ appears to precess with angular velocity $-\bm{\Omega}(t)$. Let
$s_{i}$, $i=1,2,3$, be the components of $\mathbf{S}$ with respect to the
rotating axes; then,

$\frac{ds_{i}}{dt}+\epsilon_{ijk}\Omega^{j}s^{k}=0\,.$ (6)

On the other hand, for a true intrinsic quantum spin vector with the
commutation relations

$[s_{p},s_{q}]=i\hbar\epsilon_{pqn}s^{n}\,$ (7)

that is invariant under the rotation of coordinates, the Heisenberg equation
of motion for such a quantum observable,

$\hbar\frac{ds_{k}}{dt}=i[\mathcal{H}_{SR},s_{k}]\,,$ (8)

results in Equation (6) if the spin-rotation Hamiltonian is of the form

$\mathcal{H}_{SR}=-\mathbf{S}\cdot\bm{\Omega}(t)\,.$ (9)

This is the Hamiltonian that accounts for the precessional motion of the spin
in the quantum theory. It follows that in the quantum case there is an
additional contribution to the classical Hamiltonian (5) such that the total
Hamiltonian of the particle in the rotating frame is given by
$\mathcal{H}+\mathcal{H}_{SR}$. Hence,

$\mathcal{H}_{Total}=\mathcal{H}_{0}-\bm{\Omega}(t)\cdot\mathcal{J}\,,$ (10)

where $\mathcal{J}:=\mathbf{L}+\mathbf{S}$ is the total angular momentum of
the free particle. This is a natural result, since $\mathcal{J}$ is the
generator of rotations in the quantum theory. The energy eigenvalues as
measured by the noninertial static observer include the spin-rotation
coupling, which is a quantum inertial effect that is independent of mass of
the particle. In the classical limit, $\hbar\to 0$ and we recover Equation
(5).

Spin-rotation coupling is a general phenomenon that is due to the inertia of
intrinsic spin. Physical states in quantum theory are described by mass and
spin, which characterize the irreducible unitary representations of the
inhomogeneous Lorentz group. The inertial properties of mass are well known.
Phenomena associated with the spin-rotation coupling reveal the inertial
properties of intrinsic spin. Spin-rotation coupling has extensive
observational support BMB ; BM3 ; BM5 ; Mash ; HN ; BM5a ; SoTi ; LR ; PJA ;
MNHS ; BM4 ; Ash ; HaMa ; AnMa ; BMash ; Bliokh:2015yhi ; SP ; Pa ; ShHe ;
LaPa ; Pan:2011zza ; Arminjon:2013kxa . It has recently been observed directly
in neutron interferometry Mashhoon:2005fe ; Werner ; HRW ; DSH ; MK ;
Mashhoon:2015nea ; DDSH ; DDKWLSH . Furthermore, it has significant
applications in spintronics MISM ; MISM2 ; MIM ; CB ; PAP ; IMM ; NaTa ; Hama
; MIHSM ; Taka ; KYM ; OMOS ; MOM ; KMN ; KMMN ; Kaz . For further discussion
and references, see Mashhoon:1997qc ; Mashhoon:2003ax .

###  I.1 Spin-Vorticity Coupling

Consider a laboratory experiment involving a rotating system, which creates a
congruence in spacetime. As a body rotates, we expect that the intrinsic spins
of the constituent particles all remain fixed with respect to the local
inertial frame; therefore, the intrinsic spins all appear to precess with
respect to the body-fixed frame. In the continuum limit, it may be that the
local angular velocity of motion becomes dependent on position, in which case
the spin-rotation coupling naturally goes over to the spin-vorticity coupling
MOM ; OMOS

$\mathcal{H}_{SV}=-\frac{1}{2}\,\mathbf{S}\cdot\bm{\omega}\,,$ (11)

where $\bm{\omega}$ is the vorticity

$\bm{\omega}=\nabla\times\mathbf{V}\,$ (12)

and $\mathbf{V}$ is the velocity field of the congruence. If the angular
velocity is spatially uniform such that
$\mathbf{V}=\bm{\Omega}\times\mathbf{r}$, then $\bm{\omega}=2\,\bm{\Omega}$
and $\mathcal{H}_{SV}$ reduces to $\mathcal{H}_{SR}$. For a description of
moving macroscopic matter in continuum mechanics, see section E.4.1 of Ref.
HeOb . For recent work on spin-vorticity coupling, see KMN ; KMMN .

###  I.2 Stern–Gerlach Force due to Spin-Vorticity Coupling

In general, vorticity depends on position and we might then expect the
appearance of an attendant Stern–Gerlach force as well; that is,

$f_{\mu}=-\partial_{\mu}(\mathcal{H}_{SV})=\frac{1}{2}\,\mathbf{S}\cdot\nabla_{\mu}\,\bm{\omega}\,.$
(13)

Such a spin-dependent force could then lead to the generation of a spin
current. This idea was apparently first proposed in Ref. MIHSM  and received
experimental confirmation in Taka ; KYM ; OMOS . For the extension of these
ideas to fluid spintronics, see MOM  and the references cited therein.
Moreover, Ref. Kaz  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
approximation scheme known as gravitoelectromagnetism (GEM) that is based on
the well-known analogy with Maxwell’s electrodynamics. We are interested in
the weak exterior field of a compact rotating astronomical source with mass
$M$ and proper angular momentum $\mathbf{J}$. The spacetime metric,
$-ds^{2}=g_{\mu\nu}\,dx^{\mu}dx^{\nu}$, is given in a Cartesian system of
coordinates $x^{\alpha}=(ct,\mathbf{x})$ by Mashhoon:2003ax

$-ds^{2}=-c^{2}\left(1-2\frac{\Phi}{c^{2}}\right)dt^{2}-\frac{4}{c}({\bf
A}\cdot d{\bf
x})dt+\left(1+2\frac{\Phi}{c^{2}}\right)\delta_{ij}dx^{i}dx^{j}\,,$ (14)

which represents Minkowski spacetime plus a linear perturbation due to the
source. That is, $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. We neglect all metric
perturbation terms of $O(c^{-4})$ in this weak-field and slow-motion
approximation method. In Equation (14), $\Phi(t,\mathbf{x})$ is the
gravitoelectric potential and $\mathbf{A}(t,\mathbf{x})$ is the
gravitomagnetic vector potential. For the exterior field of a rotating
astronomical mass, for instance, $-\Phi$ is the Newtonian gravitational
potential and $\mathbf{A}$ is due to mass current and vanishes in the
Newtonian limit ($c\to\infty$). Very far from the rotating source,

$\Phi\sim\frac{GM}{r}\,,\qquad\mathbf{A}\sim\frac{G}{c}\frac{\mathbf{J}\times\mathbf{x}}{r^{3}}\,,$
(15)

where $r=|\mathbf{x}|$. The GEM potentials satisfy the transverse gauge
condition

$\frac{1}{c}\frac{\partial\Phi}{\partial
t}+\nabla\cdot\left(\frac{1}{2}\mathbf{A}\right)=0\,.$ (16)

Moreover, in analogy with electrodynamics, the GEM fields are defined by

$\mathbf{E}=-\nabla\Phi-\frac{1}{c}\frac{\partial}{\partial
t}\left(\frac{1}{2}\mathbf{A}\right)\,,\qquad\mathbf{B}=\nabla\times\mathbf{A}\,,$
(17)

in terms of which Einstein’s field equations in this case become formally
similar to Maxwell’s equations AE . For discussions of the non-Newtonian
gravitomagnetic effects, see Mashhoon:2003ax ; RuTa .

We are interested in the motion of a classical spinning point particle in the
GEM field. The relevant equations in this case are the Mathisson–Papapetrou
(“pole-dipole”) equations Math ; Papa , namely,

$\frac{DP^{\mu}}{d\tau}=F^{\mu}\,,\qquad
F^{\mu}=-\frac{1}{2}R^{\mu}{}_{\nu\alpha\beta}\,u^{\nu}S^{\alpha\beta}\,,$
(18) $\frac{DS^{\mu\nu}}{d\tau}=P^{\mu}u^{\nu}-P^{\nu}u^{\mu}\,.$ (19)

In these equations, $F^{\mu}$, $F^{\mu}u_{\mu}=0$, is the Mathisson spin-
curvature force Mashhoon:2008si , $u^{\mu}=dx^{\mu}/d\tau$ is the 4-velocity
of the pole-dipole particle and $\tau$ is its proper time. Moreover, $P^{\mu}$
is the 4-momentum of the particle and $S^{\mu\nu}$ is its spin tensor that
satisfies the Frenkel–Pirani supplementary condition Frenkel ; Pirani

$S^{\mu\nu}\,u_{\nu}=0\,.$ (20)

It follows from these equations that

$P^{\mu}=m\,u^{\mu}+S^{\mu\nu}\,\frac{Du_{\nu}}{d\tau}\,,$ (21)

where $m:=-P^{\mu}u_{\mu}$ is the mass of the spinning particle and is a
constant of the motion. That is, differentiating $m=-P\cdot u$ and using
$F\cdot u=0$ together with Equation (21), we find

$\frac{dm}{d\tau}=-\left(m\,u^{\mu}+S^{\mu\nu}\,\frac{Du_{\nu}}{d\tau}\right)\frac{Du_{\mu}}{d\tau}=0\,,$
(22)

since $u\cdot u=-1$ and $S^{\mu\nu}$ is antisymmetric. In the massless limit
($m\to 0$), the Mathisson–Papapetrou equations together with the
Frenkel–Pirani supplementary condition behave properly; indeed, the spinning
massless test particle follows a null geodesic with spin parallel or
antiparallel to its world line Mashhoon . The Frenkel–Pirani supplementary
condition is therefore appropriate for a point particle.

Let us define the spin vector of the particle $S^{\mu}$ via

$S_{\mu}=-\frac{1}{2}\,\eta_{\mu\nu\rho\sigma}\,u^{\nu}S^{\rho\sigma}\,,\qquad
S^{\alpha\beta}=\eta^{\alpha\beta\gamma\delta}\,u_{\gamma}S_{\delta}\,,$ (23)

where
$\eta_{\alpha\beta\gamma\delta}=(-g)^{1/2}\epsilon_{\alpha\beta\gamma\delta}$
is the Levi-Civita tensor and $\epsilon_{\alpha\beta\gamma\delta}$ is the
alternating symbol with $\epsilon_{0123}=1$. The Mathisson spin-curvature
force now takes the form

$F_{\mu}=~{}^{*}R_{\mu\nu\rho\sigma}\,u^{\nu}\,S^{\rho}\,u^{\sigma}\,,\qquad^{*}R_{\mu\nu\rho\sigma}=\frac{1}{2}\,\eta_{\mu\nu\alpha\beta}\,R^{\alpha\beta}{}_{\rho\sigma}\,,$
(24)

in terms of the dual Riemann tensor, and the spin dynamics is represented by

$(g_{\mu\nu}+u_{\mu}\,u_{\nu})\,\frac{DS^{\nu}}{d\tau}=0\,,$ (25)

so that $S^{\mu}$, $S^{\mu}\,u_{\mu}=0$, is Fermi–Walker transported along the
world line of the spinning particle Pirani .

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 as the attraction of gravity of the source in order to
keep the particle fixed in space. The 4-velocity vector of the particle is
then given by $u^{\mu}=(1+\Phi/c^{2})\delta^{\mu}_{0}$. A natural orthonormal
tetrad frame $\lambda^{\mu}{}_{(\alpha)}$ adapted to the static test pole-
dipole particle with $u^{\mu}=\lambda^{\mu}{}_{(0)}$ is given in the
$(ct,x,y,z)$ coordinate system by

$\lambda^{\mu}{}_{(0)}=(1+\Phi/c^{2},0,0,0)\,,$ (26)
$\lambda^{\mu}{}_{(1)}=(-2A_{1}/c^{2},1-\Phi/c^{2},0,0)\,,$ (27)
$\lambda^{\mu}{}_{(2)}=(-2A_{2}/c^{2},0,1-\Phi/c^{2},0)\,,$ (28)
$\lambda^{\mu}{}_{(3)}=(-2A_{3}/c^{2},0,0,1-\Phi/c^{2})\,,$ (29)

where the tetrad axes are primarily along the local GEM coordinate axes. The
projection of the spin vector on the adapted tetrad frame is given by

$S_{(\alpha)}=S_{\mu}\,\lambda^{\mu}{}_{(\alpha)}\,,$ (30)

which implies that $S_{(0)}=0$ and

$\frac{dS_{(i)}}{d\tau}=\left[\frac{D\lambda^{\mu}{}_{(i)}}{d\tau}\,\lambda_{\mu(j)}\right]\,S^{(j)}\,.$
(31)

A straightforward calculation reveals that to linear order in the perturbation

$\frac{D\lambda^{\mu}{}_{(i)}}{d\tau}\,\lambda_{\mu(j)}=\partial_{j}\,A_{i}-\partial_{i}\,A_{j}\,.$
(32)

Therefore,

$\frac{dS_{(i)}}{d\tau}=\epsilon_{ijk}B^{j}\,S^{(k)}\,$ (33)

and the spin vector precesses with an angular velocity given by the local
gravitomagnetic field. 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 combined into a 4-vector in analogy with
electrodynamics and the corresponding GEM field tensor is then projected on
the tetrad frame $\lambda^{\mu}{}_{(\alpha)}$ to obtain the measured
gravitoelectric and gravitomagnetic fields at the location of the spinning
particle. However, $\lambda^{\mu}{}_{(\alpha)}$ differs from
$\delta^{\mu}_{\alpha}$ by terms that are linear in the spacetime
perturbation; therefore, in our approximation method $\mathbf{E}$ and
$\mathbf{B}$ 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 with its precession such that

$\mathcal{H}_{SG}=\frac{1}{c}\,\mathbf{S}\cdot\mathbf{B}\,.$ (34)

We assume here that a particle with intrinsic spin behaves in the
correspondence limit like an ideal gyroscope. For instance, in connection with
experiments in an Earth-based laboratory, to every Hamiltonian we must add the
spin-rotation-gravity contribution

$\delta\mathcal{H}\approx-\bm{\Omega}_{\oplus}\cdot\mathbf{S}+\bm{\Omega}_{P}\cdot\mathbf{S}\,,$
(35)

where $\bm{\Omega}_{\oplus}$ and $\bm{\Omega}_{P}=\mathbf{B}_{\oplus}/c$ refer
to the Earth’s rotation frequency and the corresponding gravitomagnetic
precession frequency, respectively. In fact, we have approximately

$\bm{\Omega}_{P}=\frac{G}{c^{2}r^{5}}[3({\bf J}\cdot{\bf r})\,{\bf r}-{\bf
J}r^{2}]\,.$ (36)

In the recent GP-B experiment Francis1 ; Francis2 , the non-Newtonian
gravitomagnetic field of the Earth has been directly measured and the
prediction of general relativity has been verified at about the 19% level.

In Equation (35), the difference in the energy of a spin-$1/2$ particle
polarized vertically up and down relative to the surface of the Earth is given
by $\hbar\Omega_{\oplus}\approx 10^{-19}\text{eV}$ and $\hbar\Omega_{P}\approx
10^{-29}\text{eV}$. For recent attempts to measure the spin-gravity term, see
Tarallo:2014oaa ; GPS . Furthermore, the gravitomagnetic field depends upon
position; therefore, there exists a gravitomagnetic Stern–Gerlach force
$-\nabla(\bm{\Omega}_{P}\cdot\mathbf{S})$ on a spinning particle that is
independent of its mass and hence violates the principle of equivalence and
the universality of free fall. This force naturally leads to a differential
deflection of polarized beams. For various implications of the spin-gravity
coupling, see BMc ; BMd ; Bah1 ; Bah2 ; Ior1 ; Bini:2004kz ; CMP ; Ran ; Ior2
; Mashhoon:2013jaa ; Plyatsko:2017kux ; Yuri ; Obukhov:2017avp .

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. Projecting the gravitomagnetic Stern–Gerlach force,
$f_{\mu}=-\partial_{\mu}\mathcal{H}_{SG}$ on the orthonormal tetrad frame
$\lambda^{\mu}{}_{(\alpha)}$, we have

$f_{(0)}=0\,,\qquad f_{(i)}=-\frac{1}{c}\partial_{i}\,B_{j}\,S^{(j)}\,.$ (37)

Here, $f_{(0)}$ vanishes because the gravitomagnetic field is assumed to be
independent of time. On the other hand, the Mathisson spin-curvature force
projected on the tetrad frame $\lambda^{\mu}{}_{(\alpha)}$ 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 orthonormal tetrad frame $\Theta^{\mu}{}_{\hat{\alpha}}$ adapted to an
observer; the measured components of the curvature are then

$R_{\mu\nu\rho\sigma}\,\Theta^{\mu}{}_{\hat{\alpha}}\,\Theta^{\nu}{}_{\hat{\beta}}\,\Theta^{\rho}{}_{\hat{\gamma}}\,\Theta^{\sigma}{}_{\hat{\delta}}\,.$
(39)

Taking the symmetries of the Riemann tensor into account, one can express
Equation (39) in the standard manner as a $6\times 6$ matrix with indices that
range over the set $\\{01,02,03,23,31,12\\}$. The general form of this matrix
is

$\left[\begin{array}[]{cc}\mathbb{E}&\mathbb{H}\cr\mathbb{H}^{T}&\mathbb{S}\cr\end{array}\right]\,,$
(40)

where $\mathbb{E}$ and $\mathbb{S}$ are symmetric $3\times 3$ matrices and
$\mathbb{H}$ is traceless. Here, $\mathbb{E}$, $\mathbb{H}$ and $\mathbb{S}$
represent the measured gravitoelectric, gravitomagnetic and spatial components
of the Riemann curvature tensor, respectively. If the spacetime is Ricci flat,
then Equation (40) takes the form

$\left[\begin{array}[]{cc}\mathbb{E}&\mathbb{H}\cr\mathbb{H}&-\mathbb{E}\cr\end{array}\right]\,,$
(41)

where $\mathbb{E}$ and $\mathbb{H}$ are now symmetric and traceless. That is,
in the Ricci-flat case, the Riemann curvature tensor degenerates into the Weyl
conformal curvature tensor whose gravitoelectric and gravitomagnetic
components are then

$\mathbb{E}_{\hat{a}\hat{b}}=C_{\alpha\beta\gamma\delta}\,\Theta^{\alpha}{}_{\hat{0}}\,\Theta^{\beta}{}_{\hat{a}}\,\Theta^{\gamma}{}_{\hat{0}}\,\Theta^{\delta}{}_{\hat{b}}\,,\qquad\mathbb{H}_{\hat{a}\hat{b}}=C^{*}_{\alpha\beta\gamma\delta}\,\Theta^{\alpha}{}_{\hat{0}}\,\Theta^{\beta}{}_{\hat{a}}\,\Theta^{\gamma}{}_{\hat{0}}\,\Theta^{\delta}{}_{\hat{b}}\,,$
(42)

where $C^{*}_{\alpha\beta\gamma\delta}$ is the unique dual of the Weyl tensor
given by

$C^{*}_{\alpha\beta\gamma\delta}=\frac{1}{2}\,\eta_{\alpha\beta}{}^{\mu\nu}\,C_{\mu\nu\gamma\delta}\,,$
(43)

since the right and left duals of the Weyl tensor coincide.

In our GEM scheme, $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and the gauge-
invariant curvature tensor is given by

$R_{\mu\nu\rho\sigma}=\frac{1}{2}(h_{\mu\sigma,\,\nu\rho}+h_{\nu\rho,\,\mu\sigma}-h_{\nu\sigma,\,\mu\rho}-h_{\mu\rho,\,\nu\sigma})\,.$
(44)

We recall that to lowest order
$\lambda^{\mu}{}_{(\alpha)}\approx\delta^{\mu}_{\alpha}$ and hence in the
exterior of a GEM source, we have the Weyl tensor in the form (41) with
symmetric and traceless matrices given by

$\mathbb{E}_{ij}=-\frac{1}{c^{2}}\Phi_{,ij}+O(c^{-4})=\frac{1}{c^{2}}E_{j,i}+O(c^{-4})\,$
(45)

and

$\mathbb{H}_{ij}=-\frac{1}{c^{2}}\partial_{i}B_{j}+\frac{1}{c^{3}}\,\epsilon_{ijk}\,\frac{\partial
E_{k}}{\partial t}+O(c^{-4})\,.$ (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 the Mathisson spin-curvature force.

The spin interactions discussed in this paper all involve Hamiltonians that
are similar to that of the traditional Zeeman effect. Moreover, the
gravitational Larmor theorem can be invoked to connect spin-gravity coupling
with the spin-rotation coupling. The local equivalence between magnetism and
rotational inertia was first established via Larmor’s original theorem Larmor
. The gravitational Larmor theorem is an expression of Einstein’s local
principle of equivalence of gravitation and inertia Mashhoon:2003ax ; Bahram .

Consider a steady-state configuration with exterior metric (14). In this
stationary gravitational field, the temporal coordinate can be subjected to a
simple scale transformation of the form $t\mapsto(1+\Phi_{0}/c^{2})t$, where
$\Phi_{0}$ is a constant such that $|\Phi_{0}|\ll c^{2}$. The only consequence
of this transformation is that $-g_{00}=1-2\Phi_{0}\mapsto
1-2(\Phi-\Phi_{0})/c^{2}$, while the other terms in the metric remain
unchanged since we neglect all terms of $O(c^{-4})$. In a sufficiently small
neighborhood around any event in the exterior GEM spacetime, we can replace
the metric by that of an accelerated system in Minkowski spacetime. The
resulting metric is to linear order of the form
$(\eta_{\mu\nu}+\ell_{\mu\nu})dX^{\mu}dX^{\nu}$, where BMB ; Mashhoon:2003ax

$\ell_{00}=-2\,\mathbf{a}_{L}\cdot\mathbf{X}\,,\qquad\ell_{0i}=(\bm{\Omega}_{L}\times\mathbf{X})_{i}\,,\qquad\ell_{ij}=0\,.$
(47)

This has the form of a first-order perturbation where $\mathbf{a}_{L}$ is the
constant translational acceleration and $\bm{\Omega}_{L}$ is the constant
rotational frequency of the accelerated system. A comparison with the GEM
metric reveals that the corresponding gravitoelectric and gravitomagnetic
potentials are given by

$\Phi-\Phi_{0}=-\mathbf{a}_{L}\cdot\mathbf{X}\,,\qquad\mathbf{A}=-\frac{1}{2}\,\bm{\Omega}_{L}\times\mathbf{X}\,.$
(48)

We neglect the spatial curvature of the GEM metric in this analogy. Moreover,
$\mathbf{E}=-\nabla\Phi=\mathbf{a}_{L}$ and
$\mathbf{B}=\nabla\times\mathbf{A}=-\bm{\Omega}_{L}$ are the corresponding
fields. It is clear that the spin-rotation Hamiltonian
$\mathcal{H}_{SR}=-\mathbf{S}\cdot\bm{\Omega}_{L}$ corresponds to the spin-
gravity Hamiltonian $\mathcal{H}_{SG}=\mathbf{S}\cdot\mathbf{B}$ via the
gravitational Larmor theorem.

##  III Linear Gravitational Waves

The general linear approximation of general relativity involves GEM fields of
massive systems as well as linearized gravitational waves. The purpose of this
section is to discuss spin-gravity coupling for linearized gravitational
waves; in particular, we are interested in the corresponding Stern–Gerlach
force. For related studies, see Shen:2004uw ; Ramos:2006sb ; Hojman:2016mox ;
Papini:2019qxn ; Papini:2020fgo ; Ruggiero:2020nef ; Ruggiero:2020oxo ;
Ruggiero:2021qnu  and the references cited therein.

Consider a free linear gravitational radiation field, which can be expressed
as a Fourier sum of plane monochromatic components each with frequency
$\omega_{g}$ and wave vector $\mathbf{k}_{g}$, $\omega_{g}=c|\mathbf{k}_{g}|$.
The gravitational potential of the radiation is given by the symmetric tensor
$\bar{h}_{\mu\nu}$, which is a perturbation on the background Minkowski
spacetime; that is, $g_{\mu\nu}=\eta_{\mu\nu}+\bar{h}_{\mu\nu}(x)$, where
$x^{\alpha}=(ct,x,y,z)$. In the transverse-traceless (TT) gauge,
$\bar{h}^{\mu\nu}{}_{,\nu}=0$, $\bar{h}_{0\mu}=0$ and
$\bar{h}^{\mu}{}_{\mu}=0$. In this gauge, the gravitational potentials
$\bar{h}_{ij}(x)$ each satisfies the standard wave equation.

For the sake of definiteness, let the incident radiation be a monochromatic
plane wave propagating along the $x$ direction. Then, $\bar{h}_{ij}$ can be
written as

$(\bar{h}_{ij})=\begin{bmatrix}0&0&0\\\ 0&h_{+}&h_{\times}\\\
0&h_{\times}&-h_{+}\end{bmatrix}\,,$ (49)

where

$h_{+}=\tilde{h}_{+}\,\cos[\omega_{g}(t-x)+\varphi_{+}]\,,\qquad
h_{\times}=\tilde{h}_{\times}\,\cos[\omega_{g}(t-x)+\varphi_{\times}]\,$ (50)

represent the $\oplus$ (“plus”) and $\otimes$ (“cross”) linear polarization
states of the radiation. Here, $(\tilde{h}_{+},\varphi_{+})$ and
$(\tilde{h}_{\times},\varphi_{\times})$ are constants associated with the
independent states of the radiation field.

It is a general result that in a spacetime with a metric of the form
$-dt^{2}+g_{ij}(x)dx^{i}dx^{j}$, observers that remain permanently at rest in
space follow geodesic world lines. Thus imagine this class of geodesic
observers each at rest in space with a 4-velocity vector
$e^{\mu}{}_{\hat{0}}=\delta^{\mu}_{0}$ in the spacetime under consideration
here. To each such observer, we associate an adapted orthonormal tetrad frame
$e^{\mu}{}_{\hat{\alpha}}$ that is parallel propagated along its world line.
It is straightforward to show that

$e^{\mu}{}_{\hat{\alpha}}=\delta^{\mu}_{\alpha}-\frac{1}{2}\,\bar{h}^{\mu}{}_{\alpha}\,.$
(51)

The projection of the curvature tensor (44) in the case of the incident
gravitational wave on the tetrad frame (51) results in
$R_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}=R_{\mu\nu\rho\sigma}\,e^{\mu}{}_{\hat{\alpha}}\,e^{\nu}{}_{\hat{\beta}}\,e^{\rho}{}_{\hat{\gamma}}\,e^{\sigma}{}_{\hat{\delta}}$.
Here, $e^{\mu}{}_{\hat{\alpha}}$ is in effect $\delta^{\mu}_{\alpha}$ in our
linear approximation scheme and the GEM components of curvature can be
represented as in Equation (41) with

$\mathbb{E}_{ij}=\frac{1}{2}\,\omega_{g}^{2}\,\bar{h}_{ij}\,,\qquad\mathbb{H}_{ij}=\frac{1}{2}\,\omega_{g}^{2}\begin{bmatrix}0&0&0\\\
0&h_{\times}&-h_{+}\\\ 0&-h_{+}&-h_{\times}\end{bmatrix}\,.$ (52)

For measurement purposes, it proves interesting to set up a quasi-inertial
Fermi normal coordinate system with coordinates
$X^{\hat{\mu}}=(cT,\widehat{X},\widehat{Y},\widehat{Z})$ based on the
nonrotating tetrad frame (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 $(x,y,z)=(x_{0},y_{0},z_{0})$. The spacetime metric in the
Fermi frame is given by

$-ds^{2}=g_{\hat{\mu}\hat{\nu}}\,dX^{\hat{\mu}}dX^{\hat{\nu}}\,$ (53)

where

$g_{\hat{0}\hat{0}}=-1-R_{\hat{0}\hat{i}\hat{0}\hat{j}}\,X^{\hat{i}}X^{\hat{j}}\,,$
(54)
$g_{\hat{0}\hat{i}}=-\frac{2}{3}\,R_{\hat{0}\hat{j}\hat{i}\hat{k}}\,X^{\hat{j}}X^{\hat{k}}\,$
(55)

and

$g_{\hat{i}\hat{j}}=\delta_{ij}-\frac{1}{3}\,R_{\hat{i}\hat{k}\hat{j}\hat{l}}\,X^{\hat{k}}X^{\hat{l}}\,.$
(56)

In these expansions, we have neglected third and higher-order terms. In close
analogy with the GEM case, we can define the gravitoelectric potential
$\hat{\Phi}$ and gravitomagnetic vector potential $\hat{\mathbf{A}}$ via
$g_{\hat{0}\hat{0}}=-1+2\hat{\Phi}$ and $g_{\hat{0}\hat{i}}=-2\hat{A}_{i}$;
that is,

$\hat{\Phi}=-\frac{1}{2}\,R_{\hat{0}\hat{i}\hat{0}\hat{j}}\,X^{\hat{i}}X^{\hat{j}}\,,\qquad\hat{A}_{i}=\frac{1}{3}\,R_{\hat{0}\hat{j}\hat{i}\hat{k}}\,X^{\hat{j}}X^{\hat{k}}\,.$
(57)

Similarly, the corresponding fields can be defined as in Equation (17); in
fact, to lowest order we find

$\hat{E}_{i}=R_{\hat{0}\hat{i}\hat{0}\hat{j}}\,X^{\hat{j}}\,,\qquad\hat{B}_{i}=-\frac{1}{2}\,\epsilon_{ijk}\,R^{\hat{j}\hat{k}}{}_{\hat{0}\hat{l}}\,X^{\hat{l}}\,.$
(58)

Concentrating on the incident gravitational wave under consideration in this
section, Equation (52) implies

$\hat{\Phi}=-\frac{1}{4}\,\omega_{g}^{2}\,[h_{+}(\widehat{Y}^{2}-\widehat{Z}^{2})+2h_{\times}\widehat{Y}\widehat{Z}]\,$
(59)

and

$\hat{A}_{1}=\frac{2}{3}\,\hat{\Phi}\,,\quad\hat{A}_{2}=\frac{1}{6}\,\omega_{g}^{2}\,\widehat{X}\,(h_{+}\widehat{Y}+h_{\times}\widehat{Z})\,,\quad\hat{A}_{3}=\frac{1}{6}\,\omega_{g}^{2}\,\widehat{X}\,(h_{\times}\widehat{Y}-h_{+}\widehat{Z})\,.$
(60)

Moreover, the relevant GEM fields are

$\hat{E}_{1}=0\,,\qquad\hat{E}_{2}=\frac{1}{2}\,\omega_{g}^{2}\,(h_{+}\widehat{Y}+h_{\times}\widehat{Z})\,,\qquad\hat{E}_{3}=\frac{1}{2}\,\omega_{g}^{2}\,(h_{\times}\widehat{Y}-h_{+}\widehat{Z})\,,$
(61)
$\hat{B}_{1}=0\,,\qquad\hat{B}_{2}=-\hat{E}_{3}\,,\qquad\hat{B}_{3}=\hat{E}_{2}\,,$
(62)

which are clearly transverse to the direction of wave propagation,
$|\hat{\mathbf{E}}|=|\hat{\mathbf{B}}|$ and
$\hat{\mathbf{E}}\cdot\hat{\mathbf{B}}=0$. For the incident wave, the
gravitoelectric and gravitomagnetic potentials are defined via the
$g_{\hat{0}\hat{\mu}}$ components of the Fermi metric and the remaining
spatial components can be expressed as

$(g_{\hat{i}\hat{j}})=\begin{bmatrix}1+\hat{A}_{1}&\hat{A}_{2}&\hat{A}_{3}\\\
\hat{A}_{2}&1-\xi_{+}&-\xi_{\times}\\\
\hat{A}_{3}&-\xi_{\times}&1+\xi_{+}\end{bmatrix}\,,$ (63)

where

$\xi_{+}=\frac{1}{6}\,\omega_{g}^{2}\,h_{+}\,\widehat{X}^{2}\,,\qquad\xi_{\times}=\frac{1}{6}\,\omega_{g}^{2}\,h_{\times}\,\widehat{X}^{2}\,.$
(64)

Within this Fermi coordinate system, let us imagine the class of observers
that stay spatially at rest. It is straightforward to show that a proper
orthonormal tetrad frame $\Lambda^{\hat{\mu}}{}_{\tilde{\alpha}}$ adapted to
this class of observers is given in $(t,\widehat{X},\widehat{Y},\widehat{Z})$
coordinates by

$\Lambda^{\hat{\mu}}{}_{\tilde{0}}=(1+\hat{\Phi},0,0,0)\,,$ (65)
$\Lambda^{\hat{\mu}}{}_{\tilde{1}}=(-2\hat{A}_{1},1-\tfrac{1}{3}\hat{\Phi},0,0)\,,$
(66)
$\Lambda^{\hat{\mu}}{}_{\tilde{2}}=(-2\hat{A}_{2},-\hat{A}_{2},1+\tfrac{1}{2}\xi_{+},0)\,,$
(67)
$\Lambda^{\hat{\mu}}{}_{\tilde{3}}=(-2\hat{A}_{3},-\hat{A}_{3},\xi_{\times},1-\tfrac{1}{2}\xi_{+})\,,$
(68)

where the tetrad axes are primarily along the Fermi coordinate axes.

Consider now a spinning test particle held fixed in space at
$(\widehat{X},\widehat{Y},\widehat{Z})$ by a reference observer in the Fermi
frame. Projecting the spin vector $S^{\hat{\mu}}$ in the Fermi frame on the
tetrad frame $\Lambda^{\hat{\mu}}{}_{\tilde{\alpha}}$ of the local reference
observer,
$S_{\tilde{\alpha}}=S_{\hat{\mu}}\Lambda^{\hat{\mu}}{}_{\tilde{\alpha}}$, we
find $S_{\tilde{0}}=0$, as before, and

$\frac{dS_{\tilde{i}}}{d\tilde{t}}=\left[\frac{D\Lambda^{\hat{\mu}}{}_{\tilde{i}}}{d\tilde{t}}\,\Lambda_{\hat{\mu}\tilde{j}}\right]\,S^{\tilde{j}}\,,$
(69)

where $\tilde{t}$ is the proper time of the reference observer and
$dt=(1+\hat{\Phi})d\tilde{t}$. A detailed calculation reveals that to lowest
order in $\widehat{X}$, $\widehat{Y}$ and $\widehat{Z}$ within the Fermi
coordinate system

$\frac{D\Lambda^{\hat{\mu}}{}_{\tilde{i}}}{d\tilde{t}}\,\Lambda_{\hat{\mu}\tilde{j}}=\partial_{j}\,\hat{A}_{i}-\partial_{i}\,\hat{A}_{j}\,;$
(70)

hence,

$\frac{dS_{\tilde{i}}}{d\tilde{t}}=\epsilon_{ijk}\hat{B}^{j}\,S^{\tilde{k}}\,.$
(71)

Thus, as before, the dominant effect is that the spin vector precesses with an
angular velocity given by the local gravitomagnetic field. We note that
$\Lambda^{\hat{\mu}}{}_{\tilde{\alpha}}$ differs from $\delta^{\mu}_{\alpha}$
by terms linear in the perturbation; hence, the gravitomagnetic field in
Equation (71) is in effect the field measured by the reference observer. The
corresponding Stern–Gerlach force,
$f_{\hat{\mu}}=-\partial_{\hat{\mu}}(\mathbf{S}\cdot\hat{\mathbf{B}})$, to
lowest order in $\widehat{X}$, $\widehat{Y}$ and $\widehat{Z}$ as measured by
the reference observer, is $f_{\tilde{0}}=0$ and

$f_{\tilde{1}}=0\,,\qquad
f_{\tilde{2}}=\tfrac{1}{2}\omega_{g}^{2}(h_{\times}S^{\tilde{2}}-h_{+}S^{\tilde{3}})\,,\qquad
f_{\tilde{3}}=-\tfrac{1}{2}\omega_{g}^{2}(h_{+}S^{\tilde{2}}+h_{\times}S^{\tilde{3}})\,.$
(72)

On the other hand, the Mathisson force (24) as measured by the reference
observer is given by $F_{\tilde{0}}=0$ and

$F_{\tilde{i}}=\mathbb{H}_{\tilde{i}\tilde{j}}\,S^{\tilde{j}}\,,$ (73)

where $\mathbb{H}_{\tilde{i}\tilde{j}}$ is given to lowest order by Equation
(52). This is a consequence of the fact that in our approximation scheme
$\Lambda^{\hat{\mu}}{}_{\tilde{\alpha}}$ is in effect given by
$\delta^{\mu}_{\alpha}$ for the calculation of the measured components of the
curvature tensor. It is then evident that the resulting components of the
Mathisson force for the gravitational wave field under consideration in this
section coincide with 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
the Frenkel–Pirani supplementary condition imply that the spin vector of a
test pole-dipole particle is Fermi–Walker transported along its world line
Pirani . For a spinning test particle held spatially at rest by a fiducial
observer in the Ricci-flat region of an arbitrary gravitational field within
the framework of linearized general relativity, the Fermi–Walker equation for
the spin vector indicates that its measured components undergo a precessional
motion with an angular velocity that is given by the locally measured
gravitomagnetic field. For an intrinsic quantum spin, there is therefore a
spin-gravitomagnetic field coupling Hamiltonian associated with such
precessional motion that can be obtained from Heisenberg’s equation of motion.
The gravitomagnetic field generally depends upon position; therefore, there is
an accompanying Stern–Gerlach force connected with such a spin-gravity
coupling. We show that under appropriate conditions, this Stern–Gerlach force
reduces in the correspondence limit to Mathisson’s classical spin-curvature
force.

## Acknowledgments

I am grateful to Friedrich Hehl and Yuri Obukhov for valuable discussions.

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