aa r X i v : . [ g r- q c ] J a n Spin-Gravity Coupling
Bahram Mashhoon
Department of Physics and AstronomyUniversity of Missouri-ColumbiaColumbia, Missouri 65211, USAMathisson’s spin-gravity coupling and its Larmor-equivalent interac-tion, namely, the spin-rotation coupling are discussed. The study of thelatter leads to a critical examination of the basic role of locality in rela-tivistic physics. The nonlocal theory of accelerated systems is outlined andsome of its implications are described.PACS numbers: 04.20.Cv, 03.30.+p
1. Introduction
The proper theory of the motion of a spinning mass in a gravitationalfield is due to Mathisson [1], Papapetrou [2], and Dixon [3]. A main aspectof this theory, which already appears in the work of Mathisson [1], is theexistence of a spin-curvature force F α = − c R αβµν u β S µν . (1)Here u µ is the unit four-velocity vector of the spinning mass; that is, u µ = dx µ /dτ , where x µ = ( ct, x, y, z ) and τ /c is the proper time. The signatureof the metric is +2 throughout this paper.In the linear approximation of general relativity, with the spinning massheld at rest in the stationary exterior field of a rotating central source andkeeping only first-order terms in spin, F α = (0 , F ), where [4] F = −∇ ( S · Ω P ) . (2)Here Ω P is the precession frequency of a test gyroscope held at rest outsidethe central source of angular momentum J ; far from the source, Ω P = Gc r [3( J · r ) r − J r ] , (3) (1) spin-ms printed on October 30, 2018 so that c Ω P = B g is the familiar dipolar gravitomagnetic field of the source.It follows from Eq. (2) that one can define the Hamiltonian for the spin-gravity coupling as H = S · Ω P . (4)This Mathisson Hamiltonian is a direct analogue of − µ · B coupling inelectrodynamics [5]. Imagine now the test gyroscope that is held at rest butprecesses with frequency Ω P as before. If the gravitational interaction isturned off, the gyro keeps its direction fixed with respect to the backgroundglobal inertial frame by the principle of inertia. The former precessionalmotion is recovered, however, from the viewpoint of a local observer thatis at rest in a frame of reference rotating with frequency Ω = − Ω P . Thisis an instance of the gravitational Larmor theorem [5], which follows fromEinstein’s principle of equivalence. To this latter motion in the rotatingframe, one can associate a new Hamiltonian H ′ , which can be obtained from H by replacing Ω P with − Ω . Thus the Hamiltonian due to the coupling ofspin with rotation is given by H ′ = − S · Ω . (5)The classical couplings (4) and (5) are expected to be valid for intrinsic spinas well. This is mainly based on the study of relativistic wave equations ingravitational fields and accelerated frames of reference, see [6] for someexamples; a more complete discussion as well as list of references is given in[7]. It follows from the inertia of intrinsic spin that to every spin Hamiltonianin a laboratory fixed on the Earth, one must add δ H ≈ − S · Ω ⊕ + S · Ω P ⊕ . (6)For a spin- particle, the spin-rotation part of Eq. (6) implies that themaximum energy difference between spin-up and spin-down states is ¯ h Ω ⊕ ≈ − eV. As pointed out in [8], the experimental results of [9] constitute anindirect measurement of this coupling. Further evidence in this direction,based on an analysis of the muon g − h Ω P ⊕ ≈ − eV. As discussed in [7], even in a space-borne laboratory inorbit around Jupiter, this Mathisson coupling would still be too small to bemeasurable at present by several orders of magnitude. An interesting recentdiscussion of the theoretical as well as observational aspects of spin-gravitycoupling is contained in [11].Finally, a fundamental aspect of the Mathisson coupling should be notedhere: For a classical gyro, its spin is proportional to its mass and the grav-itational force (2) is then proportional to the mass of the gyro, as it should pin-ms printed on October 30, 2018 be; however, for a spin- particle, the magnitude of spin is ¯ h/
2. Photon helicity-rotation coupling
Consider a thought experiment in which an observer rotates uniformlywith frequency Ω about the direction of propagation of an incident planemonochromatic electromagnetic wave of frequency ω . The object of theexperiment is to measure ω ′ , the wave frequency according to the rotatingobserver. Specifically, we assume that the wave propagates along the z direction and the observer follows a circle of radius r about the origin ofspatial coordinates in the ( x, y ) plane. The natural orthonormal tetradframe associated with the observer is given by λ µ (0) = γ (1 , − β sin ϕ, β cos ϕ, , (7) λ µ (1) = (0 , cos ϕ, sin ϕ, , (8) λ µ (2) = γ ( β, − sin ϕ, cos ϕ, , (9) λ µ (3) = (0 , , , . (10)Here ϕ = Ω t = γ Ω τ /c , β = r Ω /c , and γ = (1 − β ) − / . The observer’slocal temporal axis is along its four-velocity λ µ (0) and its spatial frame λ µ ( i ) , i = 1 , ,
3, is such that its axes point along the radial, tangential, and z directions, respectively.According to the standard Doppler effect, the frequency of the wavemeasured by the observer is ω ′ D = − k µ λ µ (0) = γω , where the Lorentz factoraccounts for time dilation. In this general approach, the rotating observeris assumed to be pointwise inertial and hence at rest in a comoving inertialframe (“hypothesis of locality”) and the Doppler effect follows from theinvariance of the phase of the wave under Lorentz transformations betweenthe global background inertial frame and the instantaneous inertial framesof the observer. spin-ms printed on October 30, 2018 There is, however, another way to measure frequency based on the factthat at least a few periods of the wave must be registered before the observercan determine ω ′ . To this end, we suppose that the observer can makepointwise determinations of the incident field. The result can be expressedin terms of instantaneous Lorentz transformations or equivalently as F ( α )( β ) ( τ ) = F µν λ µ ( α ) λ ν ( β ) . (11)This quantity, upon Fourier analysis, yields [12] ω ′ = γ ( ω ∓ Ω) . (12)The upper (lower) sign refers to an incident positive (negative) helicity wave.For the photon energy, we find that E ′ = γ ( E ∓ ¯ h Ω) , (13)where ± ¯ h is the photon helicity. Thus Eqs. (12) and (13) contain, inaddition to the transverse Doppler effect, the influence of the spin-rotationcoupling. Eq. (12) can be written as ω ′ = ω ′ D (1 ∓ Ω /ω ), where Ω /ω is theratio of the reduced wavelength of the radiation λ/ (2 π ) to the accelerationlength L of the observer, L = c/ Ω. The Doppler effect is recovered whenthis ratio vanishes in the JWKB limit.For oblique incidence, the analogue of Eq. (13) is E ′ = γ ( E − ¯ hM Ω) , (14)where ¯ hM is the total angular momentum of the radiation along the axis ofrotation. Thus ω ′ = γ ( ω − M Ω), where M = 0 , ± , ± , . . . , for a scalar ora vector field, while M ∓ = 0 , ± , ± , . . . , for a Dirac field. In the JWKBapproximation, Eq. (14) may be expressed as E ′ = γ ( E − J · Ω ); hence, E ′ = γ ( E − v · p ) − γ S · Ω , where J = r × p + S and v = Ω × r . It isimportant to note that ω ′ vanishes for ω = M Ω, while ω ′ can be negativefor ω < M Ω. The former circumstance poses a basic difficulty, while thelatter is a consequence of the absolute character of accelerated motion [12].It is useful to provide an intuitive explanation for the appearance of thespin-rotation term in Eq. (12). In an incident positive (negative) helicitywave, the electric and magnetic fields rotate with frequency ω in the posi-tive (negative) sense about the direction of propagation of the wave. Theobserver rotates about this direction with frequency Ω; therefore, relativeto the observer, the electric and magnetic fields of the incident wave rotatewith frequency ω − Ω ( ω + Ω) in the positive (negative) helicity case. Whilethe relative circular motion accounts for the subtraction (addition) of fre-quencies, the Lorentz factor in Eq. (12) takes care of time dilation. This pin-ms printed on October 30, 2018 factor is unity for the rotating observer at r = 0, hence ω ′ = ω ∓ Ω in thiscase; the fact that only the Lorentz factor distinguishes rotating observersat different radii in Eq. (12) follows intuitively from the circumstance thateach such observer is locally equivalent to the one at r = 0, since each islocally a center of rotation of frequency Ω.The existence of spin-rotation coupling in Eq. (12) can be observationallydemonstrated by various means including the GPS, where it accounts for thephenomenon of phase wrap-up. That is, for γ ≪ ≪ ω , ω ′ ≈ ω ∓ Ωhas been verified with ω/ (2 π ) ∼ / (2 π ) ∼ ω ′ = γ ( ω − Ω) for incident positive-helicity radiationhas a fundamental consequence that must now be addressed. This relationimplies that ω ′ = 0 for ω = Ω. The incident radiation stands completelystill with respect to all observers that uniformly rotate with frequency ω about the direction of propagation of the wave. That by a mere rotationan observer can stand still with an electromagnetic wave is analogous tothe pre-relativistic formula for the Doppler effect where an observer movingwith speed c along a beam of light would see an electromagnetic field thatis spatially oscillatory but at rest. This paradoxical circumstance played arole in Einstein’s path to relativity theory (see p. 53 of [15], which containsEinstein’s autobiographical notes). The origin of this defect in Eq. (12)must be sought in Eq. (11), namely the assumption that the field measuredby the rotating observer is pointwise the same as that measured by themomentarily comoving inertial observer (“hypothesis of locality”); a briefcritique of this notion of locality is contained in the next section. The othernonlocal assumption, involving the Fourier analysis of the measured field, isreasonable, since a number of periods of the wave must be received by theaccelerated observer before ω ′ could be adequately measured.
3. Hypothesis of locality
According to the standard theory of relativity, Lorentz invariance is ex-tended to accelerated observers in Minkowski spacetime via the hypothesisof locality, namely, the assumption that an accelerated observer, at eachinstant along its worldline, is momentarily equivalent to an otherwise iden-tical hypothetical comoving inertial observer. For time determination, thisassumption reduces to the clock hypothesis. Thus an accelerated observeris pointwise inertial and this supposition provides operational significancefor Einstein’s principle of equivalence [16].Regarding the source of this important postulate of relativity theory,it must be noted that Lorentz introduced it as an approximation in hisdiscussion of the Lorentz-Fitzgerald contraction of electrons in curvilinear spin-ms printed on October 30, 2018 motion (see section 183 of [17]). Einstein mentioned it in his discussion ofaccelerated systems (see p. 60 of [18]). Weyl likened it to the assumptionof adiabaticity in thermodynamics (see pp. 176-177 of [19]).The locality assumption originates from Newtonian mechanics, wherethe state of a particle is determined by its position and velocity. The accel-erated observer shares the same state with the comoving inertial observer;hence, locality is exact and no new physical assumption is needed if allphysical phenomena could be reduced to pointlike coincidences of classicalparticles and null rays. However, when wave phenomena are taken into con-sideration, the locality hypothesis would be approximately valid whenever λ ≪ L . Here λ is the characteristic wavelength of the phenomena underobservation and L , the acceleration length, is the characteristic length scalefor the variation of the state of the observer. In practice, deviations fromlocality are expected to be of order λ/ L and are generally very small, since L is quite long; for instance, c /g ⊕ ≈ c/ Ω ⊕ ≈
28 AU for an ob-server in a laboratory fixed on the Earth. The consistency of these ideascan be illustrated by two examples of general interest.Imagine a classical charged particle of mass m and charge q that issubject to an external force F ext . The accelerated charge radiates electro-magnetic radiation with characteristic wavelength λ ∼ L . The hypothesisof locality is thus violated since λ/ L ∼
1. This means that the state of thecharged particle cannot be given at each instant by its position and velocityalone. This is consistent with the equation of motion of the particle, whichreduces to the Abraham-Lorentz equation m d v dt − q c d v dt + · · · = F ext (15)in the nonrelativistic approximation.Consider next muon decay in a storage ring [20]. This experiment hasverified with good accuracy relativistic time dilation τ µ = γτ µ , where τ µ isthe lifetime of the muon at rest. To mimic the circular acceleration of amuon in a storage ring and take the quantum nature of this particle intoaccount, one can suppose that the muon decays from a high-energy Landaulevel in a constant magnetic field. Based on the detailed calculation reportedin [21], τ µ ≈ γτ µ " (cid:18) λ C L (cid:19) , (16)where λ C is the Compton wavelength of the muon and L = c /a , where a ∼ g ⊕ is the effective acceleration of the muon. The correction to thestandard formula in Eq. (16) is very small ( ∼ − ), but nonzero. pin-ms printed on October 30, 2018
4. Nonlocality
To go beyond the hypothesis of locality, let us return to Eq. (11) andconsider its generalization. Let F ( α )( β ) ( τ ) be the field that is actually mea-sured by the accelerated observer. Here τ is measured by the backgroundinertial observers using dτ = cdt/γ . The most general linear relationshipbetween F ( α )( β ) ( τ ) and the field measured by the infinite sequence of comov-ing inertial observers, given by Eq. (11), that preserves causality is givenby [22] F ( α )( β ) ( τ ) = F ( α )( β ) ( τ ) + Z ττ K ( γ )( δ )( α )( β ) ( τ, τ ′ ) F ( γ )( δ ) ( τ ′ ) dτ ′ . (17)Here τ is the instant at which the acceleration is turned on and the kernel K is such that it vanishes in the absence of acceleration. The integral inEq. (17) has the form of an average over the past worldline of the acceleratedobserver; moreover, it is expected to vanish in the JWKB limit ( λ/ L → F ( α )( β ) and F ( α )( β ) is unique.How should the kernel be determined? This involves various complica-tions [22], but a key idea is that the kernel should be so chosen as to preventthe circumstance encountered in section 2. That is, we introduce the fun-damental postulate that a basic radiation field can never stand completelystill with respect to an arbitrary observer. A detailed treatment of the non-local theory of accelerated systems is contained in [24] and the referencescited therein. This theory is in agreement with available observational data;moreover, it forbids the existence of a fundamental scalar (or pseudoscalar)field.What are the implications of nonlocality for the photon helicity-rotationcoupling in the thought experiment of section 2? There are basically twoaspects of the problem that are altered by nonlocality:(i) As determined by the rotating observer, for ω > Ω the amplitude ofthe positive-helicity incident wave is enhanced, while the amplitude of thenegative-helicity wave is diminished.(ii) For ω = Ω, the field is not static in the positive helicity case; instead,it varies like t as in the case of resonance.It is important to verify these purely nonlocal effects experimentally.The task here is complicated by the fact that the behavior of rotating mea-suring devices must be known. An interesting discussion of such issues ofprinciple is contained in [25]. We therefore turn to a different approachbased on the correspondence principle in nonrelativistic quantum mechan-ics. The study of electrons in rotational motion within the framework of spin-ms printed on October 30, 2018 quantum theory could shed light on the question of the correct classicaltheory of accelerated systems.In connection with (i), the cross section σ for the photoionization ofhydrogen atom has been studied with the electron in a circular state withrespect to the incident radiation that would correspond to the motion of theobserver in section 2. A detailed investigation reveals that σ + > σ − , where σ + ( σ − ) is the cross section in the case that the electron rotates in the same(opposite) sense as the helicity of the incident radiation [26].The situation in (ii) can be mimicked by the transition of an electron ina circular “orbit” about a uniform magnetic field to the next energy state asa result of absorption of a photon of frequency Ω c and definite helicity thatis incident along the direction of the magnetic field. Here Ω c is the electroncyclotron frequency. Let P be the probability of transition to the nextenergy state. A detailed investigation reveals that in the correspondenceregime, P + ∝ t , while P − = 0, corresponding to the positive and negativehelicity cases, respectively [26].It appears from these studies that the nonlocal theory is in better agree-ment with quantum theory than the standard theory of relativity that isbased on the hypothesis of locality [26].
5. Discussion
Mathisson’s spin-gravity Hamiltonian leads, via the gravitational Lar-mor theorem, to the spin-rotation Hamiltonian. For the photon, helicity-rotation coupling has the consequence that a rotating observer can in prin-ciple be comoving with an electromagnetic wave such that the wave is os-cillatory in space but stands completely still with respect to the observer.The source of this difficulty is the hypothesis of locality that is the basis forthe extension of Lorentz invariance to accelerated observers and the sub-sequent transition to general relativity. The nonlocal theory of acceleratedsystems is briefly described; in this theory, instead of the locality assump-tion, where a curved worldline is in effect replaced at each instant by thestraight tangent worldline, one considers in addition an average over thepast worldline of the observer. The consequences of this nonlocal specialrelativity are briefly described. The nonlocal theory is in agreement withavailable observational data. It remains to extend this theory to a nonlocaltheory of gravitation.
Acknowledgement
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