Quantification of GR effects in muon g-2, EDM and other spin precession experiments
Recently, Morishima, Futamase and Shimizu published a series of manuscripts, putting forward arguments, based on a post-Newtonian approximative calculation, that there can be a sizable general relativistic (GR) correction in the experimental determination of the muon magnetic moment based on spin precession, i.e., in muon g-2 experiments. In response, other authors argued that the effect must be much smaller than claimed. Further authors argued that the effect exactly cancels. Also, the known formulae for de Sitter and Lense-Thirring effect do not apply due to the non-geodesic motion. All this indicates that it is difficult to estimate from first principles the influence of GR corrections in the problem of spin propagation. Therefore, in this paper we present a full general relativistic calculation in order to quantify this effect. The main methodology is the purely differential geometrical tool of Fermi-Walker transport over a Schwarzschild background. Also the Larmor precession due to the propagation in the electromagnetic field of the experimental apparatus is included. For the muon g-2 experiments the GR correction turns out to be very small, well below the present sensitivity. However, in other similar storage ring experimental settings, such as electric dipole moment (EDM) search experiments, where the so-called frozen spin method is used, GR gives a well detectable effect, and should be corrected for. All frozen spin scenarios are affected which intend to reach a sensitivity of 0.1 microradians/second for the spin precession in the vertical plane.
Keywords: Thomas precession, Larmor precession, spin precession, muon g-2, anomalous magnetic moment, electric dipole moment, EDM
In a recent series of papers [1, 2, 3], it was claimed that, in the muon anomalous magnetic moment experiments [4, 5, 6, 7], there can be a general relativistic (GR) correction to the precession effect of the muon spin direction vector when orbiting in the magnetic storage ring sitting on the Earth’s surface in a Schwarzschild metric. These calculations were based on a post-Newtonian approximation, and the authors claimed that the pertinent effect may cause an unaccounted systematic error in the measurement of the muon’s anomalous magnetic moment, often referred to as g-2. Other papers [8, 9] responded that the effect is much smaller. Further papers  responded that the effect exactly cancels. Moreover, the usual formulae of de Sitter and Lense-Thirring precession  do not apply, since the pertinent orbit is non-geodesic. All this suggests that it is relatively difficult to say something from first principles on the magnitude of GR corrections for spin transport in a gyroscopic motion along a forced orbit. Motivated by these, in the present paper, we intend to quantify the pertinent effect in the context of GR. We use the differential geometrical tool of Fermi-Walker transport of vectors along trajectories in spacetime. In this way, the kinematic precession, called the Thomas precession, can be quantified over the Schwarzschild background field of the Earth. This is then compared to the Minkowski limit, i.e., when GR is neglected. The effect of the Larmor precession in the electromagnetic field of the experimental setting is also quantified, and its corresponding GR correction is also evaluated. The calculations show that the GR corrections for the actual g-2 experimental setting [4, 5, 6, 7] is very small, well below the experimental sensitivity. There are, however, other spin precession experiments, such as the electric dipole moment (EDM) search experiments [12, 13, 14], where it turns out that GR gives a rather large signal. Since these experiments are intended as sensitive probes for Beyond Standard Model (BSM) scenarios, their experimental data should be corrected for the GR effect. In particular, the EDM experiments [12, 13, 14] could be thought of also as sensitive GR experiments on spin propagation of elementary particles, kind of microscopic versions of Gravity Probe B [15, 16] gyroscope experiment. During the past years there have been a few papers warning about the possibility of such an effect [17, 18, 19, 20, 21]. These estimations, however, are not fully covariant Lorentz geometric GR calculations, but are mostly special relativistic or semi-general relativistic, or applying other kind of approximations such as not fully taking into account GR for the electrodynamic part. As a result, the estimations [17, 18, 19, 20] differ from our geometric GR calculation in the details of the particle velocity dependence, and with some factors. The post-Newtonian estimation of , where the GR effect for the special case of a purely electric frozen spin storage ring is quantified, is confirmed by our covariant calculations. Since the EDM signal is expected to sit on this large GR background of the order of , the pertinent factors matter a lot for discrimination from a BSM signature with the planned precision of . The GR signal, however, can also be disentangled from a true EDM signal due to their opposite space reflection behavior, i.e., by switching beam direction.
The structure of the paper is as follows. Section 2 outlines the kinematic setting of our model of the experimental situation. In Section 3 and 4, we discuss, from a geometrical point of view, the Fermi-Walker transport (gyroscopic equation) and the general relativistic Thomas precession, respectively. The GR corrections to the Thomas precession is evaluated in Section 5. In Section 6, the idealized model of electromagnetic fields in an electromagnetic storage ring over a Schwarzschild background is outlined, and their Fermi-Walker-Larmor spin transport is evaluated in Section 7. The analytical formulae are derived for a combined Thomas and Larmor precession over the Schwarzschild background in Section 8. Finally, in Section 9 the total GR corrections are evaluated, which is followed by our concluding remarks in Section 10.
2 The kinematic setting
The kinematic setting of the experiment is outlined in Fig. 1. The gravitational field of the Earth is modelled by a Schwarzschild metric with being the corresponding Schwarzschild radius, . The non-sphericity of the Earth as well as its rotation is neglected. We use the standard Schwarzschild coordinates , and thus the components of the Schwarzschild metric read as:
In such coordinates, the Earth’s surface is at an level-surface, we denote this radius by . By convention, the North pole of the spherical coordinates is adjusted such that it corresponds to the central axis of the storage ring, i.e., this axis is at . The entire storage ring is located at an , surface, where the corresponding coordinate value is denoted by . The radius of the storage ring is then . Throughout the paper, the coordinate indices are denoted by fonts like and take their value from the index set . Occasionally, the alternative notation is used as equivalent symbols for the indices . Moreover, we will also use the Penrose abstract indices , with index symbol fonts like in order to aid the notation of various tensorial trace expressions in a coordinate independent way.
The trajectory of the orbiting particle inside the storage ring on the Earth’s surface is described by a worldline with coordinate components
where for convenience the worldline is parameterized by the Killing time and not with its proper time. Here, denotes the circular frequency of the orbiting particle trajectory, in terms of the proper time of the laboratory system. It is seen that the particles are assumed to be orbiting on a closed circular trajectory, i.e., a beam balanced against falling towards the Earth is assumed. This is justified by the fact that according to  an electrostatic beam focusing optics is used in the g-2 experimental setup, which is resting on the surface of the Earth, together with the storage ring. The initial spin direction vector at is a unit pseudolength spacelike vector, orthogonal to the curve . The amount of precession can be quantified via also evolving the initial spin direction vector along the worldline of the beam injection point of the storage ring (laboratory observer), described by the curve having coordinate components
The worldlines and intersect at each full revolution, i.e., at each , with being any non-negative integer. In these intersection points the propagated spin direction vectors can be eventually compared. The unit tangent vector fields, i.e., the four velocity fields of these curves are and , with and , respectively.
In order to evaluate the spin direction vector along any point of the worldline or , it needs to be transported along the pertinent trajectories. This is described by the Fermi-Walker transport, i.e., by the relativistic gyroscopic transport . Let be a future directed unit timelike vector field, then the Fermi-Walker derivative of a vector field along is defined as:
where denotes the Levi-Civita covariant derivation associated to the metric . The Fermi-Walker derivative is distinguished by the fact that holds, as well as the property that for any two vector field and satisfying and , the identity holds. In particular, whenever one has , then also and hold. A vector field is said to be Fermi-Walker transported along the integral curves of a future directed timelike unit vector field , whenever the equation
is satisfied, which is just the relativistic gyroscope equation [15, 16]. The rationale behind considering the Fermi-Walker transport as a relativistic model of the gyroscope evolution is that for the transport of a vector field along a timelike curve with future directed unit tangent vector field , the initial constraints
are conserved during evolution, and no artificial vorticity is added. Note, that physically the spin vector has constant pseudolength and is always perpendicular to the worldline of the particle, and this constraint needs to be preserved throughout the evolution. Also note, that intuitively the Fermi-Walker transport can be regarded as the parallel transport of a rigid orthonormal frame along a unit timelike vector field, the timelike element of the frame coinciding to that of the transporting vector field.
Whenever an electromagnetic field is also present, the charged particles with spin are governed by the equations of motion
where the first equation is the relativistic Newton equation with the electromagnetic force, and the second equation is the Bargmann-Michel-Telegdi (BMT) equation [24, 25]. Here, denotes the particle mass, denotes the particle charge, denotes the magnetic moment of the particle, and denotes the spin magnitude ( for particles), while is the four velocity of the particle and is the spin direction vector of the particle.
Let us note that for charged spinning objects with non-trivial internal structure the BMT equation cannot directly be applied. Instead it can be regarded as a special limiting case of the so-called electromagnetically extended Mathisson-Papapetrou-Dixon (MPD) equations [26, 27, 28, 29, 30], which describe in the pole-dipole approximation the motion of a charged spinning body on a curved spacetime in the presence of electromagnetic field. When the electromagnetic dipole moment tensor is taken to be proportional to the spin tensor, and the curvature effects and the second order spin effects are all neglected, and after introducing a spin supplementary condition111It is necessary to introduce spin supplementary conditions (SSCs) as the MPD equations are not closed. One can use, e.g., the Mathisson-Pirani [26, 31] or the Tulczyjew-Dixon [28, 32] SSCs., the BMT equation is obtained. In the present context the treated objects are charged particles with spin without relevant internal structure, and therefore the BMT equation can be safely considered to be enough to describe the spin propagation, and is indeed used for the engineering design of accelerator facilities for spin-polarized particles.
In Sections 3, 4, 5 merely the Fermi-Walker transport , i.e., the gyroscopic kinematics of the spin direction vector along the worldlines Eq.(2) and Eq.(3) will be studied in order to extract the Thomas precession over a Schwarzschild background. These results apply to any forced circular motion over a Schwarzschild background. Following that, in Sections 6, 7, 8, 9 the GR modifications to the electromagnetic (Larmor) precession is quantified, which contributes in addition when the forced circular orbit is achieved via an electromagnetic field acting on a charged particle.
3 The absolute Fermi-Walker transport of four vectors
The Fermi-Walker transport differential equation of a vector field along a curve reads in components as
where denotes the Christoffel symbols in the used coordinates, and
is the pseudolength function of the tangent vector field , and denotes derivative against the curve parameter . In our calculations, for convenience reasons, we use the Killing time as the parameter of the worldline curves.
In order to calculate Fermi-Walker transported vector fields and along the curves and , we introduce the vector valued functions and . One should note that the coordinate components of the tangent vectors
These linear differential equations need to be solved for the vector valued functions and .
In order to solve the transport equations Eq.(25), one needs the expressions of the Christoffel symbols over Schwarzschild spacetime in our coordinate conventions. The only non-vanishing components at a point are:
where the index symmetry property also needs to be taken into account. Observe, that due to the time translational and spherical symmetry of the Schwarzschild spacetime, the Christoffel symbols in our adapted coordinates only have dependence on surfaces, i.e., also on the surface of the Earth. Since the curves and evolve on the Earth’s surface, i.e., on the surface, the Christoffel symbol coefficients in Eq.(25) can merely have dependence along these curves. But since the pertinent curves are also , or more precisely curves, the Christoffel symbol coefficients in Eq.(25) are completely constant along these. Similarly, the metric tensor components are also constants along these world lines. Moreover, also the vector valued functions and are constant. All these imply that the homogeneous linear differential equations Eq.(25) have constant coefficients, and therefore they can be eventually solved relatively easily, by a matrix exponentiation.
In the following, we denote by the symbol the particular constant value of the Schwarzschild Christoffel symbols along the curves or , in our coordinates. Similarly, will denote the particular constant value of the metric tensor components along these world lines. These are obtained by simply substituting the values , and any value of and into Eq.(34) and Eq.(1). Similarly, the symbol and will denote the constant value of the constant vector valued functions Eq.(19). Also, their pseudolengths are constant, and . With these notations, we are left with homogeneous linear differential equations with constant coefficients:
Direct evaluation shows that holds, and thus any Fermi-Walker transported vector field along the curve satisfies for all . It also means that the Fermi-Walker derivative along is proportional to the Lie derivative against the Killing time translation vector field . Taking this into account, our pair of differential equations simplify as
being the Fermi-Walker transport tensor. The index pulled up version of the Fermi-Walker transport tensor can be shown to be antisymmetric by direct substitution. Therefore, it describes a Lorentz transformation generator. Moreover, holds by construction. Therefore, the Fermi-Walker transport tensor describes a pure rotation in the space of -orthogonal vectors, called to be the Thomas rotation, and describes an absolute, i.e., observer independent rotation effect of the spin direction four vector. The concrete formula for the Fermi-Walker transport tensor is
in our coordinate conventions.
4 The relative Fermi-Walker transport as seen by the laboratory observer
As shown in the previous section, the Fermi-Walker transport of four vectors along is relatively simple notion described by the tensor . This needs to be translated to the transport of spatial vectors orthogonal to the laboratory observer , known to be the Thomas precession, which is a phenomenon also including effects relative to an observer. The procedure for quantifying this effect is rather well known already in the special relativistic scenario [33, 34].
Recall that the worldline of the beam injection point in the laboratory is the curve with a four velocity vector , described by Eq.(3). Let us consider such a curve in each point of the storage ring. In other words: take the initial vector, and extend it via requiring to all , defining the four velocity field of the curve . It will obey the Fermi-Walker transport equation along itself. Then, extend it via the Lie transport to any point of the storage ring world sheet. This vector field will have a family of integral curves
indexed by . These will be the worldlines of the laboratory observer. Similarly, to Eq.(3), these will have the tangent vector field
and will have corresponding unit tangent vector field, i.e., four velocity with . By this construction, the observer vector field is present at each point of the storage ring world sheet as shown in the top panel of Fig. 2, with the property , . Actually, any vector at the initial spacetime point can be spread as a reference vector to any point of the storage ring worldsheet, using this “Lie extension” , . Since this spread vector field is vorticity-free, by means of Frobenius theorem it can be Einstein synchronized with orthogonal surfaces. These happen to coincide with the Killing time surfaces. The Einstein synchronized observer observes -time evolution of vector fields along the curve via first spreading the initial vector using the above Lie extension as a reference, and then comparing the parallel transport evolution of the vector field along to the evolution of the Lie extended spread reference vector field along the parallel transport and subsequent Lie transport, in order to match the comparison spacetime point. This is illustrated in the bottom panel of Fig. 2. As a consequence, the covariant -time derivative of vector fields along formally can be written as , where denotes the Lie extended vector field of the vector at the given point of the curve, in order to make sense of the formula. In terms of coordinate components, this is described by
for a vector field along the curve , in our coordinate choice.
It is important to recall that the evolving Fermi-Walker transported spin direction vector field is always orthogonal to . Let us denote by at a point of the subspace of -orthogonal vectors (-space vectors). Also, let denote the orthogonal vectors to at a point of the laboratory observer world sheet.
Take a solution of the Fermi-Walker transport equation , where the vector is initially (and thus also eternally) -space vector, i.e., resides in . The Einstein synchronized laboratory observer , at a corresponding spacetime point, observes it via Lorentz boosting it back to . That shall be denoted by , being an -space vector at the same spacetime point. The Lorentz boost at a spacetime point from a future directed unit timelike vector to an other one is given by the formula:
It is uniquely characterized by the following properties: it is the -isometry taking to (and thus to ) that acts as the identity on the subspace . With this notation, one has that the original Fermi-Walker transported four vector field is described by . Since that was required to satisfy the Fermi-Walker transport equation, it must satisfy
along the curve . Applying now inverse boost , i.e., boost from to , and using subsequently the Leibniz rule for covariant derivation, one infers that
must be satisfied. Using this and Eq.(51), the -time derivative of the observed Fermi-Walker transported vector field can be given:
with the -Fermi-Walker transport tensor
Using now the fact that we took special coordinates such that the coordinate components of , and are constant, we get an explicit form for the coordinate components
By direct substitution it is seen that the index pulled up version is antisymmetric, and therefore corresponds to a Lorentz transformation generator. Also, it is seen that , and therefore it is an -rotation generator, called to be the Thomas precession, which includes the relative observer effects as well. The concrete coordinate components of the Thomas precession tensor is
where the notation is used. It is remarkable that only the components and depend on .
In order to extract the angular velocity vector of the -rotation generator , one needs to take the spatial Hodge dual in the space of . This is given by the formula
where denotes the determinant of the matrix of the metric in our coordinates, and is the Levi-Civita symbol. The concrete coordinate components of the Thomas precession angular velocity vector is
Let us introduce the vector fields , , , which are by construction an orthonormal basis in the space at each point, in the direction of , , . The metric projections of onto this orthonormal basis is
It is remarkable that only the