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arXiv:gr-qc/0111069v1 21 Nov 2001

Null Result for the Violation of Equivalence Principle with

Free-Fall Rotating Gyroscopes

J. LUO

a *

, Y. X. Nie

b

, Y. Z. Zhang

c,d

, Z. B. Zhou

a

a

Department of Physics, Huazhong Univeristy of Science and

Technology, Wuhan 430074, China

b

Institute of Physics, Chinese Academy of Sciences, Beijing

100080, China

c

CCAST(World Lab.), P. O. Box 8730, Beijing 100080

d

Institute of Theoretical Physics, Chinese Academy of Sciences,

Beijing 100080, China**

(June 20, 2001)

Abstract

The diﬀerential acceleration between a rotating mechanical gyroscope and a

non-rotating one is directly measured by using a double fr ee-fall interferome-

ter, and no ap parent diﬀerential acceleration has been obs erved at the relative

level of 2×10

-6

. It means that the equivalence principle is still valid for rotat-

ing extended bodies, i.e., the spin-gravity interaction between the extended

bodies h as not been observed at this level. Also, to the limit of our experi-

mental sensitivity, there is no observed asymmetrical eﬀect or anti-gravity of

the rotating gyroscopes as reported by hayasaka et al.

PACS number(s): 04.80.Cc, 04.90.+e

Typeset using REVT

E

X

1

I. INTRODUCTION

It is well known that spin-interactions of elementary particles, spin-orbit coupling and

spin-spin coupling, have been studied in both theory and experiment for long time. Fur-

thermore, gravitational couplings (i.e. the spin-gravitoelectric coupling [1,2] and the spin-

gravitomagnetic coupling [3,4]) and spin-rotation coupling [5–7] between intrinsic spins and

rotating bodies have been also investigated for long time (see, e.g., [8]).

However, the status of research for rotation (spin)-coupling between macroscopic ro-

tating bodies is greatly diﬀerent . The spin-orbit coupling for motion of mechanical gy-

roscope has been already well known in Newton’s mechanics. With the exception of the

spin-orbit coupling, on the other hand, Einstein theory of general relativity also predicts the

spin-gravitational coupling of mecha nical g yroscope, which has been investigated by many

authors, e.g. see Ref. 8. In particular, the Stanford Gravity Probe B (GPB)group has

theoretically studied for long time on t hese types of gravitomagnetic eﬀects and planed to

perform a satellite orbital experiment in order to seek the couplings of rotor spin to Earth

spin and rotor spin to rotor orbit [9]. As pointed out by Zhang et al. [10], however, the me-

chanical gyroscope spin is essentially diﬀerent from the intrinsic spin of elementary particle.

In fact, an extended body could have two diﬀerent types of motion, i.e. orbit motion (the

motion of the center-of-mass) and rotation. Thus a extra force (or force moment), which

could come from t he spin-spin (i.e. rotation-rotation) coupling between rotating macroscopic

bodies, might change the three types of motion for the rotating bodies: (i) spin precession

(i.e. a change of spin direction), ( ii) a change of the rotation rate, and (iii) a change of

the motion of the center-of-mass. It is known that general relativity (GR) only predicts

(i), i.e. spin precession. While any possible connections of GR with (ii) and (iii) are now

still open problems. Thus the Stanford GPB project simply includes a measurement of the

spin precession rather than the (ii) and (iii). In addition, although other gravitational the-

ories, such as the gauge theories of gravitation with torsion [11], seem to include spin-spin

coupling of ﬂuid, it is diﬃcult to discuss the spin-interaction between rotating rigid balls

2

within the framework of these theories. For this reason, Zhang et al. recently developed

a phenomenological model for the rotation-rota tion interaction between the rotating rigid

balls [10], which predicts (iii), i.e. the eﬀect of the coupling, gyroscope spin to Earth spin,

on the orbital acceleration of the gyroscope free-falling in Earth’ s gr avitational ﬁeld. In

this sense this type of spin-spin coupling would violate the equivalence principle (EP) f or

the free-fall gyroscopes.

EP, as one of the fundamental hypo t heses of Einstein’s general relativity, has been tested

by many experiments [12 –18]. Recently, some diﬀerent t ests of EP for gravitational self-

energy [19] and spin-polarized macroscopic objects [20,21] have been reported. However,

in all of the experiments as well as the Satellite Test of the Equivalence Principle (STEP)

and the G alileo Galilei (GG) space projects as well as the MICROSCOPE space mission

[22–24], it is non-rotating bodies that are used. In addition, as pointed out a bove, a lthough

a gyroscope is used in the Stanford Gravity Probe B project, only the precession of the

gyroscopic spin is to be observed, which is irrelevant to the orbital motion.

Some relevant experiments have been perfo r med by use of mechanical gyroscopes a nd

give contradictory results [25–30 ]. In particular, the observations in these experiments were

made by means of beam balance, and so only the gr avity a nd its reacting for ce were working,

which is irrelevant to inertial fo rce. Therefore, this type of experiment is simply a test of

statics indep endent of EP.

Recently, Hayasaka et al. investigated the eﬀect of a rotating gyroscope on the fall-

acceleration by comparing the fall-times of the gyroscope with diﬀerential rot ating sense

using the time-counter combined with three couples of the laser-emitters and receivers [31].

Their experimental data show that the gravity acceleration of the right-rotating roto r at

18000 rpm is smaller than that of non-rotating one at the relative level of 10

-4

, and t he

gravity acceleration of the left-rotating rotor almost identical with that of the non-rotating

(i.e. an asymmetric coupling). But the phenomenological theory for rotating r ig id balls in

Ref. 10 predicts a symmetric spin-spin coupling which is in the order of 10

-14

much less than

the observation in Ref. 31. As pointed out above, this type of free-fall experiment is a test

3

of dynamics, which is closely related to EP. And hence, it is necessary to do a new dynamic

test of EP by use of free-fall gyroscopes.

In this article, we shall report a new dynamic test of the spin-spin coupling between

a gyroscope and the Earth. Based on the theoretical model in Ref. 10, a dimensionless

parameter representing the strength of violation of EP can be deﬁned a s follows:

η

s

=

∆g

g

= κ

⇀

S

1

·

⇀

S

e

Gm

1

M

e

R

1

−

⇀

S

2

·

⇀

S

e

Gm

2

M

e

R

2

, (1)

where G is the Newtonian gravitational constant, m

1

, m

2

and M

e

are the masses of the two

gyroscopes and the Earth, respectively, and

⇀

S

1

,

⇀

S

2

, and

⇀

S

e

are the spin angular momentums

of them correspondingly, R

1

and R

2

are the distances between the centers of the two gyro-

scopes and the Earth, respectively, and the parameter κ represents the universal coupling

factor f or the spin-spin interaction. Therefore, in a double free-fall (D FF) experiment, in

which two gyroscopes with diﬀerent ia l rotating senses drop freely, an observed non-zero value

of η

s

would imply violation of the EP or existence of spin-spin force between the gyroscope

and the Earth.

II. EXPERIMENTAL DESCRIPTION

The schematic diagram of the DFF experiment is shown in Fig. 1. A frequency-stabilized

He-Ne laser beam (633 nm) with the relative length standard of 1.3×10

-8

is split by two beam

splitters and sent vertically to the two corner-cube-retroreﬂectors (CCRs) ﬁxed on the bot-

toms of the test masses, respectively, and then combined again and forms interference fringes

on a 12 ns-response-time photodiode (RS Ltd., OS D15-5T). The diﬀerential vertical displace-

ment of both test masses, the gyroscopes with diﬀerential rotating senses, is continuously

monitored by the interferometer and sampled by means o f a 10 MHz data-acquisition-card

(Gage Ltd., Cs1250) combined with an external rubidium atomic clock (SRS Inc., SR620),

which provides a relative time standard of 10

-10

, and then stored in a computer. The test

masses are freely dropped in two 12 m-high vacuum tubes of about 20 ∼ 50 mPa. Compared

4

with the SFF experiment employed by Hayasaka et a l., the DFF scheme can minimize the

environmental noises such a s the tides, gravity gradient, seismic noise, and air damping and

so on, because the diﬀerential mode design can suppress some common errors of bo th falling

objects.

As we known, the sensitivity of such a G alilean experiment in which both dropping

objects are put side-by-side is limited by the alignment of the beam propagation away

from the vertical line [17]. For example, an error in the verticality of 5

′′

will contribute

an uncertain diﬀerential acceleration of 0.3 µGal (1 Gal = 1 cm/s

2

). A proposed method

to reduce this error is to locate the dropping masses directly one above the other, but the

design and operation would be very complicate. However, in order to test the asymmetrical

gravity acceleration eﬀect of 10

-4

as reported by Hayasaka et al, the side-by-side setup is

employed here, and t he two test masses are separated horizontally (south-north) by 480 mm.

This design is very convenient for us to drive the gyroscopes and release them.

Each of the two test masses consists of a steel gyroscope with a mass of 420.0±2.5 g,

a diameter of about 55 mm and a height of about 32 mm, a CCR of 76.4±0.4 g and a

diameter of 40 mm as well as an outer aluminum fra me of 159.4±0.9 g . Tinned copper

wires with a diameter of 0.25 mm are used to suspend the test masses and melt by an

instantaneous large current ( >1 50 A) provided by a capacitor array, and then the test

masses are r eleased and drop freely [32]. A DC three-phase motor is used to drive one of

the gyroscopes, and the other is in non-rotating status. The rotating speed of the gyroscope

can be adjusted by changing the input voltage of the motor. Simultaneous measurement of

the driving frequency of the motor and the rotating rate o f the gyroscope rotor in a vacuum

container of about 3 Pa showed that the rota t ing frequency of the rotor is equal to that of

the motor with a n uncertainty of 1%. It is useful for recording the rotating speed of the

gyroscope without adding an external measurement system in the vacuum chamber. The

rotating speed of the gyroscope is kept at (17000±200) rpm. A mechanical claw is used to

grasp t he test mass during the speedup progress of the gyroscope, and it is then loosed when

the gyroscope runs normally. The free-fall test masses are captured by two 1.2 m-length

5

tubes with an assembly of thin rubber- rings and aluminum foils, respectively. Because of

the lack of a return mechanism, which could reset the dropping objects under the vacuum

condition, we have to open the vacuum tubes a fter each free-fall measurement.

The diameter of the laser beam is kept in a range of 3.0 ∼ 3.2 mm by a two-lens

collimation assembly during 20 m optical length so that the beam wavefront eﬀect can be

neglected here. The diﬀerential radiation pressure on the test masses is less than 3×10

-4

µGal for 0.5 mW laser power used here. The angles o f the beam aligned with the local

vertical are monitored by a telescope combined with two horizontal oil references, and then

fed back to align the beam splitters by four ﬁne screws. The aligned verticality is kept within

50

′′

for each laser beam, the maximum uncertainty of the diﬀerential acceleration due to the

aligned verticality is 30 µGal.

The test mass with a non-rotation rotor is released about 3 ms before the other with a

left- or right-rotating rotor in order to obtain a n interference fringe rate of about 100 kHz

by mean of two diﬀerential relay switches. The amplitude spectrum of the seismic noise in

our laboratory is abo ut 10

−9

/(f/Hz)

2

m/

√

Hz [3 3], which will contribute an uncertainty of

about 1 µGal to the ﬁnal experiment result.

The sample data in each free-fall are pro cessed as following steps. First, the DC-oﬀset

and the amplitude of each interference fringe are determined from the original time-voltage

data {t

i

, V

i

}. Second, by calculating an inverse function of the fringe using the DC-oﬀset

and the amplitude determined, we can t ransform the data {t

i

, V

i

} into the time-diﬀerential

displacement data {t

i

, ∆z

i

}. Finally, the data {t

i

, ∆z

i

} are ﬁtted by a parabolic trajectory

perturbed with a linear vertical gravity gradient γ. The diﬀerential displacement between

both test masses is given by the equation as follows

∆z = ∆z

0

+ ∆v

0

t + (∆g + γ∆z

0

)t

2

/2 + ∆v

0

γt

3

/6 , (2)

where unknown para meters ∆z

0

, ∆v

0

= g∆t

0

, and ∆g are the initial diﬀerential vertical

displacement, velocity, and acceleration at the same height, respectively. It is evident that

the initial diﬀerential displacement, which includes their original suspending diﬀerence and

6

descent height due to the release time-delay ∆t

0

, has to be measured accurately. Here the

suspending height diﬀerence of bot h test masses is less than 1 mm, and their descent height

due to 3 ms delay is abo ut 50 µm. In this case, the vertical gravity gradient eﬀect is about

0.3 µGal. In addition, it is noted that the ﬁtting initial time diﬀerence, which is here deﬁned

as the time diﬀerence of the ﬁtting initial data away from the real release t ime of the latter

test mass, will contribute an uncertain acceleration diﬀerence due to the coupling between

the initial diﬀerential velocity and the vertical gravity gradient. In general, the ﬁtting initial

time diﬀerence should be kept below 0.1 s for 1 µG al uncertainty.

A known systematic error due to the ﬁnite speed of light is given by [34]

∆g/g ≃ 3∆v

0

/C , (3)

and the correction is about 0.3 µGal in our experiment. Another systematic error due to

residual gas drag could be calculated as follows [35]

∆g/g = A∆v

0

p

q

8µ/(πRT )/(4mg) , (4)

where A ( ≈ 1 70 cm

2

) is the total surface area of the test mass, µ and R are the molecular

weight of residual gas a nd the gas constant, m is the mass of the falling object, T is the

temperature, and p is the residual pressure. The uncertain a cceleration due to the drag

eﬀect is less than 5 µGal at p = 50 mPa and T = 300 K.

Variation of the magnetic ﬂux density is within 0.1 Gauss near the right-, left-, or non-

rotating rotor, and the geomagnetic ﬂux density is about 0.4 Gauss here. The estimation

shows that the eﬀect of the geomagnetic ﬁeld on the steel rotor is at the level of 10

-10

Gal.

An acceleration diﬀerence due to interaction between a possible horizontal velocity dif-

ference ∆v

h

and rotation of the Earth is given by

∆g/g = 2∆

⇀

v

h

×

⇀

Ω

≤ 2∆v

h

Ω cos λ , (5)

where Ω is the angular frequency of the Earth’s rotating, and λ ( ≃ 30 degree) is the latitude

of o ur laboratory. The ∆v

h

is estimated smaller than 4.3 mm/s according to interference

7

intensity of the two interference beams reﬂected from the CCRs versus the falling length

(6 mm deviation fo r 10 m-fall height). Therefore, the uncertain acceleration due to the

procession eﬀect is less than 54 µGal. It means that the hor izontal velocity diﬀerence would

have to be monitored in the further exp eriment with a higher precision.

A possible lifting force for a rotating rotor due to the residual gas ﬂow’s circulation can

be calculated based o n the Zhukovskii’s theorem a s follows [36]

⇀

F

lift

= m

⇀

a

lift

= −2ρ

gas

⇀

V

×

⇀

ω

, (6)

where

⇀

V

is the velocity of the rotating rotor,

⇀

ω

( ∼ 1700 0 rpm) is the angular velocity

of the rotating rotor, and ρ

gas

is the residual ga s density in the vacuum tube. Because the

interferometric measurement here is nearly insensitive to the horizontal motions of the two

test masses, the lifting eﬀect on the vertical acceleration diﬀerence would be zero if

⇀

ω

was

exactly along the vertical axis. The maximum uncertain rotation directio n of the rotating

rotor away the vertical axis is estimated within 2.4 mrad, thus a possible vertical acceleration

diﬀerence between the rotors due to the gas ﬂow’s lifting is at the level of 10

-10

Gal, which

can be neglected here.

III. EXPERIMENTAL RESULTS

A typical voltage output from the photodiode is shown in Fig. 2. Figure 2( a) is the

intensity curve of the interference fringe as the ﬁrst dropping object (non-rotating here) is

released, and the rate of the fringe increases with the falling of the non-rotating test mass

until the other is also released. As both test masses drop f r eely, the rate o f the fringe is

modulated by t heir acceleration diﬀerence or the noises, as shown in Fig. 2(b).

Figure 3 lists 3 sets of 5 measurements each of N-L, N-R, and N-N, where L, R, and N

represent left-, right-, and non-rotating, respectively. The uncertain diﬀerential acceleration

of each free-fall comes to the level of 1000 µGal, while the ﬁtting standard deviation (±1σ) is

only a few µGal, but it is noted that the uncer tainty is independent of their rotating senses.

8

Statistical result shows that relative uncertainty of the diﬀerential acceleration between the

non- and left-rotating test masses is ∆g

N-L

/g = (0.90 ± 0.94) × 10

−6

, and that between the

non- and right-rotating is ∆g

N-R

/g = (0.67 ±1.92) ×10

−6

. They are almost the same as the

background limit of ∆g

N-N

/g = (0.56 ± 1.44) × 10

−6

.

Summarizing the data obtained in Ref. 25, the weight loss, resulting from the mass reduc-

tion or the acceleration decrease, for right-rotating around the vertical a xis is approximately

formulated by Hayasaka and Takeuchi, in units of dynes, as follows

∆W (ω) = αmr

eq

ω , (7)

where m is the mass of rotor (in g), ω is the angular frequency of rotat io n (in rad/s), and

r

eq

is the equivalence radius (in cm), deﬁned as follows

mr

eq

=

Z Z

ρ(r, z)2πr

2

drdz , (8)

where ρ(r, z) is the density of the ro t or materials. Their experimental result shows that the

factor α is about 2×10

−5

/s. Considering the generalization of the possible anomalous weight

change of the rotating gyroscopes, the possible weight loss of the two rotating directions of

the gyroscope could be given as follows [29,37]

∆W (ω) = βIω , (9)

where I is t he inertia moment of the rotating rotor, β could be considered as a factor de-

pendent upon the anomalous eﬀect. Based on the above formulas, all reported experimental

tests of the anomalous eﬀect are tabulated in Table I as suggested by Newman [38]. It is

noted that some unknown parameters are calculated according to a uniform composition

rotor assumption.

¿From the results of our DFF experiment, there is no apparent diﬀerential acceler ation

between the rotating and non-rotating test masses within our experimental limits. Therefore,

we can conclude that the diﬀerential acceleration between the r otating and non-rotating

gyroscopes is almost 2 orders of magnitude smaller than reported in Ref. 31, and the

9

diﬀerential acceleration eﬀect between the right- and left- ver sus the non-rotating has not

observed in our experiment at the relative level of 2×10

-6

. It means that EP is still valid

for extended rotating bodies, and t he spin-spin interaction between the rotating extended

bodies has not been o bserved at this level. And then, a ccording to the Eq. (1) and the

approximately uniform sphere mode of the Earth, it can be concluded that κ ≤2×10

-18

kg

-1

,

which sets an upper limit for the spin-spin interaction between a rotating extended body

and the Earth.

IV. DISCUSSION

A large limitation in our experiment comes from the friction coupling between the ro-

tating rotor and the f r ame of the test mass. The friction coupling not only causes a high-

frequency mechanical vibration o f the CCR at the frequency of the rotating rotor, but also

results in a slowly rotating motio n of the frame, which frequency is about 1 Hz. Another

main limitation had been proved to come fro m the outgassing eﬀect of the vacuum pump

with a full rated pumping speed 1500 L/s due to the asymmetrical outgassing for the two

tubes here. It is hoped that the sensitivity of our DFF experiment could be improved by

one or two orders in the near future, and the upper limit o f the dependent fa ctors α or β

could be improved t o 10

-9

. Therefore, the new EP f or the rota ting extended bodies could

be tested at the same level correspondingly.

Acknowledgments We are grateful to Prof. W. R. Hu and Prof. R. D. Newman for

their discussion and useful suggestion. This work was supp o r ted by the Ministry of Science

and Technology of China under Grant No: 95-Yu-34 and the Na t ional Natural Science

Foundation of China under Grant No: 19835040.

* Email address: junluo@public.wh.hb.cn.

** Mailing address of Y. Z. Zhang (e-mail: yzhang@itp.ac.cn)

10

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13

FIGURES

FIG. 1. Schematical diagram of a free-fall interferomater used to measure the diﬀerential ac-

celeration between two gyroscopes with diﬀerential rotating senses.

FIG. 2. Interference fringe intensity as the ﬁrst test mass is released. The fringe rate increases

with its falling; (b) Interference fringe intensity as both the test masses d rop freely.

FIG. 3. Statistical result of the relative diﬀerential acceleration between two test masses with

diﬀerent rotating senses. L, R, and N represent left, right-, and non-ratating, respectively, L

sta

,

R

sta

, and N

sta

represent the statistical values of the corresponding diﬀerential acceleration, and

the error bars denote ±1σ.

FIG. 4. Summary of test experiments of anomalous weight change of the rotating rotors.

Experiment Method M D r

eq

I ω

max

∆W α β

(g) (cm) (cm) (g·cm

2

) (rpm) (dyn) (s

-1

) (cm

-1

s

-1

)

Hayasaka & Takeuchi BB 140 5.2 1.85

a

473

b

13000 7.6 2.14×10

-5

1.17×10

-5

175 5.8 2.26

a

736

b

11.7 2.17×10

-5

1.16×10

-5

Faller et al. BB 451 5.1 1.70

b

1466

b

6000 <0.39 ≤ 8.14×10

-7

≤ 4.25×10

-7

Quinn & Picard BB 330 4.0 1.33

b

660

b

8000 <0.06 ≤ 1.60×10

-7

≤ 1.06×10

-7

Nitschke &Wilmarth BB 142 3.84 1.28

b

328

a

22000 <0.07 ≤ 1.64×10

-7

≤ 0.91×10

-7

Imanishi et al. BB 129 5.0 1.94

a

551

a

11000 <0.32 ≤ 1.12×10

-6

≤ 5.10×10

-7

Hayasaka et al. SFF 175 5.8 1.93

b

970

a

18000 24.9 3.90×10

-5

1.36×10

-5

Luo et al. DFF 420 5.49 1.83

b

1582

b

17000 <0.80 ≤ 5.89×10

-7

≤ 2.86×10

-7

a

data provided by the corresponding reference.

b

data calculated a ccording to the assumption of a unifrom composition rotor.

14

High-speed

AD

Laser

∆

z

=

∆

z

0

+

∆

v

0

t

+

∆

gt

2

/2

t

PD

Computer

Rub. clock

Amplifier

Rotor

CCR

Beam splitter

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.5

2.0

2.5

3.0

3.5

Interference intensity (V)

Time (ms)

52.00

52.05

52.10

52.15

52.20

1.5

2.0

2.5

3.0

3.5

Interference intensity (V)

Time (ms)

-8

-6

-4

-2

0

2

4

6

8

N-L

N-N

sta

sta

N-R

N1 N3 N5

L1 L3 L5

sta

R1 R3 R5

N-L

N-R

N-N

∆

g/g (10

-6

)

Number of Experiment