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PHYSICAL REVIEW D 15 AUGUST 1996VOLUME 54, NUMBER 4

0

The physics of the Sagnac-Mashhoon effects

Ivano Damiao Soares and Jayme TiomnoCentro Brasileiro de Pesquisas Fı´sicas (CBPF), Rua Dr. Xavier Sigaud, 150 22290-180, Rio de Janeiro, Rio de Janeiro, Bra

~Received 7 February 1996!

We carry out a complete examination of the effects of rotation on the physics of spin-1/2 particles. Theorigin of these effects is connected to the fact that the frame of the rotating experiment is Fermi-Walkertransported, and is related to the inertia rest frame~with respect to which the apparatus rotates with angularvelocityvW ! by an instantaneous Lorentz boost plus a time-dependent spatial rotation, in that order. Two distinctsets of effects are obtained. The first depends on the spin-rotation interaction, and consists of the Mashhooneffect and a mass split effect due to the Lorentz boost mentioned above. In a neutron interferometer the lattereffect produces a phase shift smaller than the Mashhoon phase shift by a factor of orderO(v2/c2), wherev isthe velocity of the particles in the beams of the interferometer. The detection of the Mashhoon effect is crucialto decide if free spin-1/2 particles actually behave as gyroscopes. The second set appears already in the eikonalapproximation and is due to the active boost of the particles by the rotating apparatus; it results in the Sagnaceffect. We also discuss a criterion to fix the frame of the experiment among all mathematically admissibleframes in the Brill-Wheeler formulation, and show that frames used in the literature to derive the Sagnac-Mashhoon effect cannot be physically realized.@S0556-2821~96!01416-6#

PACS number~s!: 03.30.1p, 03.65.Bz, 04.20.Cv

In the realm of special relativity, the laws of physics abasically formulated on inertial frames. However, in masituations the description of a physical system may reqthe use of noninertial frames, as is the case of experimenlaboratories rotating with earth. The rotation of the apparmay have a measurable effect on experiments, such aSagnac effect originally predicted and measured for light@1#and, more recently, measured for neutrons in experimenneutron interferometry@2#. Concerning the Sagnac effect, testablished fact is that the rotation of the experimental aratus relative to the inertia rest frame@3# produces a phasshift in the interference pattern proportional tovW •AW , wherevW is the angular velocity of the apparatus andAW is the ori-ented area enclosed by the interfering beams~this is correctto the first order ofv/v, wherev is the velocity of the neutron or photon!. A few years ago Mashhoon@4# suggestedthe existence of a new effect for spin-1/2 particles, the srotation coupling described by the interaction term2vW •SW ,whereSW is the spin angular momentum operator of the pticle, adding a small contribution to the Sagnac effectneutron interferometry experiments.

With the exception of the paper by Dresden and Yang@5#~where a derivation of the Sagnac effect is made!, all deriva-tions of the Sagnac-Mashhoon effect are incorrect or incplete. The origin of this problem lies in the use of nonphycal frames to describe the motion of the experimeapparatus. We also show that two sets of effects are tdistinguished in experiments with the rotating apparatus.first is a typical wave effect, related to the fact that the fraof the rotating experiment is noninertial, while the secondis due to the dragging of the particles of the experimenthe rotating apparatus. The origin of the effects in both cais due toactiveLorentz transformations realized on the stem, taking it from rest to a state of uniform rotation aboutaxis.

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In the following we adopt a procedure analogous to Dres-den and Yang@5# in their derivation of the Sagnac effect,staying throughout in the inertia rest frame. For simplicity,we consider an experimental apparatus of interferometrywith cylindrical symmetry. The interfering beams have thesemicircular paths as depicted in Fig. 1. The beam is split atS, the semicircular trajectories being produced by a largenumber of mirrors or crystals placed along the pathC. Be-tween two successive mirrors or crystals, the particles of thebeams are assumed to be free.

We introduce inertial cylindrical coordinates (T,R,f,Z).The apparatus rotates about theZ axis with angular velocityv as observed from the inertia rest frame. The coordinates ofeach crystal or mirror of the apparatus are given by (T,R,f1vT,Z) with corresponding four-velocity

u~0!5g~]T1v]f!, ~1!

whereg5~12v2R2!21/2. To complete the description of theapparatus, we introduce three spacelike vector fields, as pre-scribed by Irvine@6#,

FIG. 1. Rotating interferometry apparatus.D is the point ofrecombination of the beams. The incident beam is polarized parallelto the rotation axis.

2808 © 1996 The American Physical Society

y

ns

t

n

u-e-

e of-

rs,

54 2809THE PHYSICS OF THE SAGNAC-MASHHOON EFFECTS

u~1!5]R ,

u~2!5gS vR]T11

R]fD ,

u~3!5]Z , ~2!

connected to the inertia rest frame by the Lorentz boost

LAB5S g

0gvR0

0100

gvR0g0

0001D , A,B50, . . . ,3. ~3!

A general prescription for describing the dynamics ospin-1/2 particles in noninertial frames is given by the BrillWheeler formulation@7#. The equation of motion for thespin-1/2 Dirac particles in the rotating frame~1!, ~2! is thengiven by @7,8#

F ig~]T1v]f!1 ig0g1S ]R11

2RD1 ig0g2S gvR]T1g

R]fD

2S3S C1v

2~g!2D Gc50, ~4!

whereg is the relativistic factor of the Lorentz boost~3! and

C5Mg3g52 ig5]

]Z~5!

is a constant of motion~M is the mass of the particle!. FromEq. ~4! we see thatC corresponds to a trivial symmetry ofthe inertial equation~v50! involving the longitudinalmotion along a given direction, theZ axis. If we choosecinert

to be simultaneously the eigenstate ofp352 i ]Z andC, weobtain

Cc inert5eAM21p32c inert, e561. ~6!

C ceases to be trivial when the system rotates becauseassociated degeneracy is raised by the interaction of the sof the Dirac particle with rotation; the symmetry is preserved, but the degeneracy is raised by a split in the enespectrum due to the split termC1~g!2~v/2! appearing in Eq.~4!. For planar motion~p350!, as is the case of the experi-ment of Fig. 1, and low rotation approximation, this actuallcorresponds to a mass-split termM→M1ev/2. Equation~4! also results from the inertial equation~2igADA1M !cinert50 by the substitution

c inert5Uc,

U5eg0g2b~R!/25coshb/21g0g2sinhb/2, ~7!

whereDA5„]T ,]R11/2R,(1/R)]f ,]z… and tanhb5vR. ThematrixU is a spin-1/2 representation of the Lorentz transfomation ~3!.

In the remainder of the paper, we adopt the low rotatioapproximationv2R2!1 so that in Eqs.~3! and ~4! we maytakeg.1. We choosec in the form

f-

thepin-rgy

y

r-

n

c5x0~R,Z!e2 iET1 ipff, ~8!

whereE andpf are, respectively, eigenvalues of the energand angular momentum operatorsi ]T and2i ]f . Equation~4! reduces then to

~E2vpf!c52 ig5FS1S ]R11

2RD1S2S 2 iEvR1 ipf

R D1S3S C1

v

2 D Gc. ~9!

Now, by two successive Foldy-Wouthuysen transformatio@9,10#, the Hamiltonian equation~9! and the constant of mo-tion ~5! are diagonalized to

~E2vpf!c95F2]R22

1

R]R1

~pf2S3/2!2

R2 22Evpf

2EvS31S C91v

2 D 2G1/2g0c9, ~10!

C95AM21p32g0S3, ~11!

with C9c95AM21p32ec9. If we are now restricted to posi-

tive energy statesg0ce95ce9 , namely,

ce951

2 S 11e12e00D f e

~as in the nonrelativistic limit!, we can interpret the constanof motion C ~or the quantum numbere! as proportional tothe projection of the spin of the particle along the rotatioaxis @11#.

Squaring Eq.~10!, we obtain

~E2vpf!2f e5F2]R22

1

R]R1S pf2e/2

R D 222Evpf

2eEv1S eAM21p321

v

2 D 2G f e . ~12!

The solutions of Eq.~12!, which are regular at the originR50, are Bessel functions of the first kind@13#:

f e5Jpf2e/2„P~v!R…, ~13!

where

P~v!5ASE1ev

2 D 21v2S pf22

1

4D2SAM21p321

ev

2 D 2.~14!

We must now impose boundary conditions on the soltions ~13! connected to the interferometry apparatus dscribed in Fig. 1. Let us consider the inertial casev50,which corresponds to the experiment at rest. The presenccrystals or mirrors atR0, producing semicircular beams, implies that boundary conditions must be imposed atR0. Al-though we do not have a model for the crystals or mirro

r

t

e

-

tns-

n,

s-

q.

2810 54IVANO DAMIA O SOARES AND JAYME TIOMNO

the boundary conditions imposed by them will actually corespond to fixing the momentum of the beams atR0. In fact,from Eq. ~13! we have

P~v50!R05 j pf2e/28 , ~15!

wherej pf2e/28 is a number depending onpf2e/2, associated

with the boundary conditions atR0. On solving Eq.~15!, weobtain, for the inertial energy,

E5A~ j pf2e/28 !2

R02 1M21p3

2, ~16!

leading us to interpretP05( j pf2e/28 /R0) as the azimuthal

momentum of the particles in the beams. For instance, in teikonal approximation@15# for the classical radiusR0 wehave j pf2e/28 ;pf2e/2, which corresponds to imposing tha

uc9u2 have its first maximum approximately atR0. We remarkthat in the circular beams of the experiment the particlhave a fixed polarizatione and, therefore, a fixed energy.The spin polarization dependence of Eq.~16! is an effectanalogous to the spin-orbit interaction in atoms. The azmuthal momentumP0 does not coincide with the eikonalmomentum (pf2e/2)/R0 due to the contribution of the spin-orbit-type interaction to the kinetic energy of the planar motion. For the noninertial case~vÞ0!, noting from Eqs.~7!and ~8! that the energy eigenvalueE remains equal to theinertial energy, we must have, atR0,

P~v!R05 j pf2e/28 1d j pf2e/28 ,

corresponding to a momentum in the beams~p350!,

P~v!5j pf2e/28 1d j pf2e/28

R0.P0A11

ev

E1M

whereP05P(v50)5AE22M2. This gives a momentumvariation

dPmass split>ev

2~E1M !P0.

ev

4vf , ~17!

which depends basically on the mass-split term discussabove, wherevf is the velocity of the particle along thecircular beams@20#. We remark that our treatment of boundary conditions guarantees that onlyactive transformationsare involved in the wave effects derived above.

We now derive the Mashhoon effect, a wave effect of thsame nature of Eq.~17!. On completing the description ofthe apparatus, we introduced in Eq.~2! a triad which isFrenet-Serret transported@6#, but not Fermi-Walker trans-ported. The need for a Fermi-Walker-transported triad comfrom our assumption that free spin-1/2 particles behavegyroscopes and may be used to define the inertia rest fraof our problem. The frame of the apparatus~Fermi-Walkertransported! is obtained from the frame~1!, ~2! by the addi-tional time-dependent rotation

-

he

s

i-

-

ed

e

esasme

RAB5S 1

000

0cosa

2sina0

0sinacosa0

0001D ,

with a5gvT. In other words, we go from the inertia resframe to the frame of the apparatus by the composed traformationRC

A L AB, with L A

B given in Eq.~3!. To avoid cum-bersome calculations of the effects of this additional rotatiowe instead extend the procedure used in obtaining Eq.~17!by noting the following.

A solution c of the Dirac equation in the Fermi-Walkerframe constructed above and a solutionc of Eq. ~4! arerelated by c5S21c, where S5exp igS3vT/2 satisfiesRA

BgB5S21gAS. Now we note that under the Foldy-Wouthuysen transformations leading from Eq.~9! to ~10! theoperatorS3 is transformed into

S5~2gW •pW 1M !

A2~gW •pW !21M2S3,

wheregW •pW 5g1p11g2p252 i (g1DR1g2Df). A straight-forward calculation gives that the Foldy-Wouthuysen tranform of c, which we denote byc9, is

c95e2 i ~ev/2!Tc91 ieS 12M

E D sin vT

2c9

2eP~v!

2E S 00

12e11e

D Jpf2e/221~PR!

N~pf2e/2!e~2 iET1 ipff!

3sinvT

2,

whereN(pf2e/2) is a normalization factor forc9. In thelimit E;M , we haveE[^c9/ i ]Tc9&5E1ev/2, implyingthat c9 behaves approximately asc9 with E→E1ev/2. Anestimate of higher corrections toE may be obtained by cal-culatingE5^c9/ i ]Tc9&, averaging in time, and consideringthat we may approximateN2(pf2e/2)/N2(pf2e/221).1for large quantum numberspf2e/2. We obtain

E5E1ev

21O~vf

4 !.

Hence, in the nonrelativistic limit,c9 will describe par-ticles in the beams with azimuthal momentum given by E~14! with E→E1ev/2: namely,

P~v!.A~E1ev!22SM1ev

2 D 2.P0S 11

ev

2

E

E22M2 1ev

2~E1M ! D , ~18!

whereE is the inertial~v50! energy. The third term in Eq.~18! is exactly Eq.~17!, having its origin from the pure Lor-

54 2811THE PHYSICS OF THE SAGNAC-MASHHOON EFFECTS

entz boost~3!. The second term appears as a direct conquence of the time-dependent rotationRA

B , and gives theMashhoon shift

dPMashhoon5ev

2

E

P0.

ev

2vfS 11

vf2

2 D . ~19!

We note in Eq.~19! a O(v f2 ) correction to the Mashhoon

effect derived in Ref.@4#. Although of distinct origin, it isequal to the mass-split term~17!.

The origin of the Mashhoon effect is thence the additiontime-dependent rotation necessary to produce a FeWalker frame. Therefore the existence of the Mashhoonfect is crucial to decide if free spin-1/2 particles actuabehave as gyroscopes when rotational motion is involvThe phase shift resulting from Eqs.~17! and ~19! will becalculated at the end of the paper.

This exhausts the wave effects due to the motion offrame of the experimental apparatus.

The second set of effects is due to the fact that the rotaapparatus drags the particles in the beams, producing a bin their energy momentum. It results in the Sagnac effeThe dragging of the particles by the apparatus is notscribed in the Brill-Wheeler formulation@14#, but we mayincorporate this effect in the formalism by the following procedure. Let us rewrite Eq.~4! in the form

~2 igADA1M !Uc50, ~20!

where the matrixU defined in Eq.~7! is a solution of

LBAgB5U21gAU ~21!

and gives a spin-1/2 representation of the Lorentz transmation ~3!. The solutionU of Eq. ~21! is defined up to anoperatorV satisfying [U,V]505[V,gA], and we useV todescribe the boost due to the apparatus. If in Eq.~20! wesubstituteU byW5VU, we obtain

$2 i ~U21gAU !V21DAV2 iU21gA~DAU !1M %c50.~22!

To describe the drag of the particle by the apparatus,defineV by

V21DAV5LABDB ~23!

and the wave function of the experiment is now

c5V21U21c inert. ~24!

In the eikonal approximation for the circular rayR0 @15# ~cf.experiment of Fig. 1!, Eq. ~23! has the solution

V5eb~R0!L~R0!.11vR0L~R0!, ~25!

whereL(R)5[(T/R)]f1Rf]T]. The wave function~24! isan inertial solution with boosted four-momentum, and

W215V21U215e2b~R0!@g0g2/21L~R0!#

is the complete boost operator of the rotating experimentfact, under the operator~25! the coordinatesXA transform asXA5V21XAV, and we have

se-

alrmi-ef-llyed.

the

tingoostct.de-

-

for-

we

. In

T5T2vR02f,

f5f2vT,

implying thatV21e2 i (ET2pff)V5e2 i (ET2 pff) where

E5E1vpf ,

pf5pf1vR02E. ~26!

The eikonal wave function of the experiment has thus theform @cf. Eq. ~8!#

c5e2 i ET1 i • pffU21~R0!x~R0 ,Z!,

whereU21x(R0) is the boosted inertial spinor.It is important to note here that the dragging of the par-

ticles by the apparatus is associated with a boost. No drageffect arises associated with the time-dependent rotationRA

B . Also, for exactcylindrical symmetry of the apparatusof Fig. 1, Eq.~26! is the Lorentz boost of the particles by theapparatus atS only.

We remark that thec numberpf is the eigenvalue of theoperatornW •JW , whereJW5LW 1SW /2 is the total angular momen-tum of the particle andnW is the unitary vector of the axis fordefinition off, which coincides with axis of rotation. To seethis we use that the inertial wave function in cylindrical co-ordinates and Cartesian coordinates are related by@16#

c inert Cartes5Pc inert cyl,

whereP5exp[2(if/2)nW•SW ] and, therefore, we have

P~2 i ]f!P215~2 i ]f1S3/2!5nW •S 2 iRW `¹W 1SW

2 Dand

nW •JWc inert Cartes5pfc inert Cartes ~27!

~the domain ofpf are half-integer values!.Now for an observer at the inertia rest frame, the phase

shift due to the rotation of the interferometer is given by thevariation of the difference of wave numbers along the twosemicircular paths of the interfering beamsat a given instantof time: namely,

d f5ECdpW •dxW ,

where the line integral is calculated along the closed pathCof the beams~cf. Fig. 1!. In the approximations consideredwe may assume that the momentum variations arising fromEqs.~17!, ~19!, and~26! are additive. It results then

d f52pvS ~pf2e/2!

vf1

e

2vf12

evf

4 DR0 ,

where we have used that

vf5dE

d~pf /R0!.

~pf2e/2!

R0E

2812 54IVANO DAMIA O SOARES AND JAYME TIOMNO

in the eikonal approximation. From Eq.~27! we may finallyexpress

d f5dSagnac1dMashhoon1dmass split, ~28!

where

dSagnac52pR0

2

vf

^vW •LW &R0

,

dMashhoon52pR0

vfS 11

vf2

2 D K vW •SW

2L ,dmass split5pvfK vW •

SW

2LR0 .

The first effect appears already in the eikonal approximatiand is due to the drag of the particles by the rotating appratus, in agreement with the derivation by Dresden and Ya@5# of the Sagnac effect. The third effect is due to an effetive mass split, being ofO(v f

2 ) smaller than the predictedMashhoon shift. TheO(v f

2 ) correction in the Mashhoonshift adds to the mass-split shift, doubling the latter’s valuFrom the geometry of the experiment of Fig. 1, it is easysee that the Sagnac phase shift can be reexpressed as

dSagnac52kf

vfvW •AW ,

whereAW is the area enclosed by the interfering beams andkfis the wave number of the particles along the beams.

To conclude our analysis we show that other frames usin the literature give a null Sagnac effect, although they a

ona-ngc-

e.to

edre

mathematically admissible and the Hamiltonian in theseframes contains interaction terms analogous to the ones inEq. ~3!. Let us adopt the frame introduced in Ref.@17#,which is frequently used to discuss the interaction of spin-1/2particles with rotation. We restrict ourselves to the case ofthe experiment of Fig. 1. We denote Cartesian inertial coor-dinates byxa5(T,X,Y,Z) and Cartesian rotating coordi-nates byx8a5(T8,X8,Y8,Z8), related by

T85T,

X85X cosvT1Y sinvT,

Y852X sinvT1Y cosvT,

Z85Z. ~29!

It is straightforward to show that a solution of Dirac equationin the Hehl-Ni frame is related to the inertial solution by@18#

cHN~x8!5Uc inert~x!, ~30!

whereU5exp[(i /2)T8vW •SW ]. From Eq. ~30! we can checkthat dpW 85pWHN2pW inert8 50 or, equivalently,pWHN5R21pinert,where R is the three-dimensional time-dependent rotationmatrix defined through Eq.~29!. Thus no interference Sag-nac effect would be measured by the apparatus@19#. In otherwords, the Hehl-Ni frame cannot be physically realized asthe frame of the experiment. A non-null Sagnac effect indi-cates that the experiment is rotating with respect to the iner-tia rest frame, defining a rotating frame connected to theinertia rest frame by an active~instantaneous! Lorentz boost.

ln

@1# M. G. Sagnac, Compte Rendu152, 310 ~1911!; 157, 708~1913!; 157, 1410 ~1916!; A. Silberstein, J. Opt. Soc.5, 291~1921!; A. A. Michelson, H. G. Gale, and F. Pearson, Astrophys. J.61, 140 ~1925!.

@2# S. A. Werner, J. L. Staudenmann, and R. Collela, Phys. RLett. 42, 1103 ~1979!; D. M. Greenberger and A. W. Over-hauser, Rev. Mod. Phys.51, 43 ~1979!.

@3# It is also referred to as the local compass of inertia and maydetermined by an ideal Foulcault pendulum, for instance.

@4# B. Mashhoon, Phys. Rev. Lett.61, 2639~1988!.@5# M. Dresden and C. N. Yang, Phys. Rev. D20, 1846~1979!.@6# W. M. Irvine, Physica30, 1160~1964!. The orthonormal triad

u( i ) is obtained by Irvine under the prescription that they aFrenet-Serret transported along the observer world lineu(0) .

@7# D. R. Brill and J. A. Wheeler, Rev. Mod. Phys.29, 465~1957!.@8# gA are the constant Dirac matrices; we use a representa

such that (gA)15g0gAg0, with (g0)252(gk)251,k51,2,3. Also,g55 ig0g1g2g3 andSk5g5g0gk. Throughoutthe paper we use units such that\5c51.

@9# L. L. Foldy and S. A. Wouthuysen, Phys. Rev.78, 29 ~1950!.@10# J. Tiomno, Physica53, 58 ~1971!.@11# By Cini-Toushek transformations, we can show that, in th

-

ev.

be

re

tion

e

high momentum limit~orM50!, the conserved quantum num-ber e is the projection of the spin of the Dirac field along thedirection of the momentum of the field~cf. Ref. @12#!.

@12# B. D. B. Figueiredo, I. Damia˜o Soares, and J. Tiomno, Class.Quantum Grav.9, 1593~1992!.

@13# Up to a normalization factorNpf2e/2 .@14# The Brill-Wheeler formalism describes only the spinorial

boost of the inertial spinor wave function, according to Eq.~7!.@15# In the eikonal approximation for the classical radiusR0, Eq.

~2! assumes the form

Hg0]T1g2]f

R01g1

1

2R01g3]z1iMJc50.

@16# We remark thatP is the operator which transforms the Diracinertial Hamiltonian in cylindrical coordinates into the Diracinertial Hamiltonian in Cartesian coordinates.

@17# F. W. Hehl and W. T. Ni, Phys. Rev. D42, 2045~1990!.@18# The operatorU transforms the Dirac Hamiltonian in the iner-

tial rest frame~in Cartesian rotating coordinates! into the DiracHamiltonian in the HN frame~in Cartesian rotating coordi-nates!. The difference between the HN tetrad and the inertiatetrad, in Cartesian rotating coordinates, is a spatial rotatiodepending on the anglevT8.

@19# We note that the Hehl-Ni frame isnot Fermi-Walker trans-

y-

l

54 2813THE PHYSICS OF THE SAGNAC-MASHHOON EFFECTS

ported. To transform it into a Fermi-Walker frame, we need torealize a time-dependent rotation, resulting in the inertiaframe expressed in Cartesian rotating coordinates.

@20# The smallness of this term results from the fact that it is actu

l

-

ally the difference between the mass-split term and the FoldWouthuysen term2 ivW •SW ]T , which is a rotomagnetic mo-ment interaction originating from the rotomagnetic potentia2 iEvR produced by the Lorentz boost@cf. Eq. ~9!#.