粒子自旋---引力耦合理论 On the Spin-Rotation Coupling and an Application to the Rotation of Earth
JianQi Shen1, 2 HongYi Zhu1, 2 ShenLei Shi2
(1.State Key Laboratory of Modern Optical Instrumentation, HangZhou, 310027; 2. ZheJiang Institute of Modern Physics and Physical Department of ZheJiang University, HangZhou, 310027)
ABSTRACT:
The Kerr metric of spherically symmetric gravitational field is analyzed by considering the coordinate transformation from the rotating system to the fixing system, and consequently that the inertial field (with the exception of the centrifugal force field) of the rotating system is one part of its gravitomagnetic field is verified. We investigate the spin-rotation coupling and the geometric phase factor in the time-dependent gravitomagnetic field. A possible application of this geometric phase factor to the investigation of the fluctuation of the angular velocity of Earth’s rotation is also discussed in this present paper.
Keywords:
spin-rotation coupling, gravitomagnetic field, invariant theory, geometric phase factor
It is easily verified that the form of the weak field approximation of the equation of the gravitational field is in analogy with Maxwell’s equation of electromagnetic field. It is the most outstanding point that the former also possesses both the gravitoelectric potential written as and the gravitomagnetic potential as , and the corresponding gravitomagnetic field strength is of the form . A particle with intrinsic spin possesses the gravitomagnetic moment of such magnitude that it equals the spin of the particle. Thus the interaction between the gravitomagnetic moment and the gravitomagnetic field is also called the coupling of the spin with the gravitomagnetic field, of which the Hamiltonian is given by
. (1)
The strongest gravitomagnetic field, which we can find on the Earth, is the one caused by the rotation of the Earth. Since the Earth is a noninertial reference frame due to its rotation, the particle with spin is coupled to a more strong gravitomagnetic field which embodies the interaction between the spin and the noninertial frame in addition to the interaction expressed by Eq.(1). It is apparently seen that the spin-noninertial frame interaction is in connection with the Coriolis force. These two gravitomagnetic fields have different origins and properties. The gravitomagnetic field from the motion of material, expressed by , is analogous to the way the electric current leads to the magnetic field. And its strength is dependent on the Newton’s gravitational constant G. The gravitomagnetic field associated with the Coriolis force is caused by the choice of the coordinate. That is, the coordinate transformation from the rotating frame to the fixing frame results in the inertial force, observed by the observer fixed in the rotating frame in accordance with Newton’s Law.
Obviously, the latter magnetic field is independent of the Newton’s gravitational constant G. Due to the smallness of G, the coupling of the latter gravitomagnetic field with spin is stronger than that of Eq. (1). In the present paper, the interaction between this inertial force field and spin is just investigated. According to the equivalence principle, the inertial force is gravitational force by nature. Consequently, these two gravitomagnetic forces can be derived from the equation of gravitational field, which has been proved in Sec.1. Thus we obtain the Hamiltonian of the coupling of spin-rotation. It is known that Mashhoon’s approach to deriving the intrinsic spin-rotation coupling is given by analyzing the Doppler’s effect of the rotating frame with respect to the fixing frame. In this paper, however, the transformation of the gravitomagnetic potential is studied through the coordinate transformation, the Hmiltonian of the interaction between the intrinsic spin and the noninertial frame thus obtained.
The reason why the coupling of the spin (or gravitomagnetic moment) with the noninertial frame and the gravitomagnetic field is of great importance, is that the development of laser technology and applications of the precise instrumentation make possible the investigation of quantum mechanics in weak-gravitation field. The use of these weak effects of the relativistic quantum gravity enables people to test the basic principles of general relativity. Although the equivalence principle is still accurate, there are some interesting phenomena, such as the violation of Galileo’s principle of free falling body for the particle with spin in the Kerr space-time.
Since the analogy can be drawn between the gravity and electromagnetism in some aspects, Aharonov, Carmi and Anandan, Dresden, Sakurai et al. proposed the geometric effect of the vector potential of inertial force and the effect of the quantum interfere associated with gravity, respectively.
On the rotating reference frame, a particle was acted on by the inertial centrifugal force and Coriolis force, which are analogous to the electric force and magnetic force. Thus the matter wave on the rotating frame will possess an integral phase factor which has been called Aharonov-Carmi (A-C) effect, or the gravitational A-B effect. Overhauser and Colella, Werner, Standenmann et al. have proved the existence of A-C effect by making use of the neutron interferometry experiment. In fact, A-C effect is due to the interaction of the momentum and the noninertial frame. Although the spin of a particle such as neutron interacting with the noninertial frame has the same origin of the A-C effect, namely, both arise from the presence of the Coriolis force, the A-C effect mentioned above doesn’t contain the spin-rotation interaction. Berry’s theory of the gemometric phase proposed in 1984 is applicable only to the case of adiabatic approximation. In 1991, on the basis of the Lerris-Riesenfeld(L-R), invariant theory X、C、Gao et al. put forward to th invariant-related unitary transformation formulation which is appropriate to treat the cases of non-adiabatic and non-cyclic process. Hence, L-R theory developed into this generalized invariant theory which is a powerful tool to investigate the geometric phase factor. In Sec. 2 the spin-rotation coupling is taken into consideration by using this method.
1. the interaction between spin and gravitomagnetic field
The Kerr Metric of the exterior gravitational field of the rotating spherically symmetric body is of the form
(2)
where are the displacement of spherical coordinate; denotes the angular momentum of unit mass and possesses the dimension of length. Since the space-time coordinate of Kerr metric (2) is on the rotating reference frame, we transform it into that the fixing reference frame. Due to the smallness of Earth’s rotating velocity, , one can use the Galileo transformation
, (3)
with being the radial velocity, the space-time coordinate of the fixing frame, the space-time coordinate of the rotating frame, then . Substitution of Eq. (3) into the expression (2) leads to
(4) where in leads to the inertial centrifugal force . Neglecting the terms corresponding to in , we can obtain
, (5)
thus the gravitomagnetic potential can be written as
. (6)
It can be easily seen that, , the first term of , is closely analogous to the , which corresponds to the magnetic potential of the rotating charged spherical shell. Then we can calculate the exterior gravitomagnetic strength of the Kerr space-time and the result is with being the angular momentum of unit mass of the rotating spherical body.
In accordance with the form of post-Newtonian approximation of the equation of the geodesic equation, the gravitomagnetic strength can be defined by with . Set , then the gravitomagnetic strength, which arises from the choice of the reference frame, is
, (7)
which is related to the rotation and a part of the gravitomagnetic field. From the point of view of Newtonian mechanics, it is the innertial force field in essence, not resulted from the mass-current denisty. Since we have assumed that the velocity of a particle is parallel to namely, the Lorentz force acted on the particle in the gravitomagnetic force is given by
, (8)
hence, we conclude that the Lorentz force in the noninertial reference frame is the well-known Coriolis force. The gravitational field is in analoy with the electromagnetic field, by reason of the analogous forms of the gravitational field equation of post-Newtonian approximation and the Maxwell electromagnetic equation. Then, the interaction Hamiltonian of the coupling of a spinning particle with the gravitomagnetic field is of the form
, (9)
The two parts of the gravitomagnetic field have different origins: the former, , arises from the motion of mass and is in connection with the gravitational constant G; the latter, , originated in the transformation of coordinate from the rotating frame into the fixing frame. And it is independent of the gravitational constant G. The latter is stronger than the former by 1020 times for the Earth, thus, is neglected and (9) is rewritten as
, (11)
For the neutron, then we have
. (12)
2. The exact solution of the time-dependent Schrdinger equation describing the spin-rotation coupling of the neutron
The variation of the Earth’s rotating angular velocity may be caused by the motion of interior matter, tidal force, “YangLiu”, atmosphere etc. Once we obtain the information on the fluctuation of the rotating velocity of the Earth, it is possible to investigate the motion of matter on the Earth. For the sake of detecting the fluctuation of the rotating velocity, we propose an approach to measure the geometric phase factor arising from the interaction of neutron spin with the Earth’s rotation, by making use of the neutron – gravity interferometry experiment. The time-dependent Schrdinger equation which describes the neutron spin-rotation coupling is as follows:
, (13)
Set , and , then the expression (12) can be written as
. (14)
According to L-R invariant theory, an invariant is easily constructed which satisfies the following invariant equation
. (15)
Since the Set forms a complete algebra, and the is the linear combiation of this algebraic generator, the invariant is surely of the form
. (16)
Where the time-dependent parameters and satisfy two auxiliary equations
, (17)
which are obtained by inserting (14) and (16) into Eq.(15). In Eq.(17), the dot denotes the time derivative.
According to the L-R invariant theory, the particular solution of the time-dependent Schrdinger equation differs from the eigenstate of the invariant only by a time-dependent phase factor which is expressed as , where
, (18)
Thus the particular solution of the time-dependent Schrdinger is given by
. (19)
In order to obtain the eigenstate of the invariant and the expression for the phase, we transform the time-dependent invariant into a time-independent invariant , namely,
, (20)
Where
, . (21)by using the invariant – related unitary transformation formulation.
Complicated and lengthy calculation yields
, , (22)
then , which may be written as
. (23)
is time-independent.
Since the eigenstates of corresponding to the eigenvalue are and respectively, the particular solution of the Schrdinger equation (13) can be written as
, (24)
and
(25)
where the phase comprise the dynamical phase and geometric phase which are defined as
. (26)
Using the Baber-Campbell-Hausdortt formulation, we get the expression
, (27)
for the dynamical phase and
. (28)
for the geometric phase in the neutron spin-rotation couplings.
From (24)-(28) we obtain the exact solutions of the time-dependent Schrdinger equation (13). It can be further verified that in the expression (27) vanishes in terms of the auxiliary equation (17). Hence, the solutions of Eq.(13) contain only the geometric phase. Since the geometric phase factor possesses the physical meanings, it can be measured in the neutron-gravity interferometry experiment. The signs of the geometric phase of the neutron corresponding to the spin-up and spin-down are opposite and thus this effect due to the spin- rotation coupling will be detected by the neutron interferometer. Once the time-dependent expression for the geometric phase, is obtained, we can investigated the fluctuation of the Earth’s rotating angular velocity in terms of the auxiliary equations(17) and Eq.(18).
3.Conclusion and remarks
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