Clock effect due to gravitational spin–orbit coupling

Physics Letters A 359 (2006) 252–255 www.elsevier.com/locate/pla

Clock effect due to gravitational spin–orbit coupling S.B. Faruque Department of Physics, Shah Jalal University of Science and Technology, Sylhet 3114, Bangladesh Received 19 May 2006; received in revised form 18 June 2006; accepted 19 June 2006 Available online 27 June 2006 Communicated by P.R. Holland

Abstract In this Letter, we consider a possible gravitational spin–orbit coupling. It is shown that in presence of such a coupling there would appear a clock effect very similar to the gravitomagnetic clock effect. The new clock effect is found to be topological. According to this effect, two counter-orbiting spinning test particles placed on two identical, circular equatorial orbits around a central massive body would take different times to complete a full revolution. We obtain, in this Letter, a qualitative expression for the period difference. © 2006 Elsevier B.V. All rights reserved. PACS: 04.20.Cv Keywords: Gravitational spin–orbit coupling; Clock effect

1. Introduction A novel effect resulting from Einstein’s relativistic theory of gravitation is the gravitomagnetic clock effect (GCE). According to the results found by many authors, see Refs. [1–4], the temporal structure of the space–time around a rotating massive body causes two counter-orbiting test particles placed on two identical, circular equatorial geodesic orbits to take different times to complete a full revolution. The difference in times is termed as the gravitomagnetic clock effect. This effect can be derived either by using geodesic equation of motion in the Kerr field [5], or using gravitomagnetic formalism which is a parallel of electromagnetism [6], or using purely geometric treatment [7]. In this Letter, we calculate a similar clock effect which results due to a possible gravitational spin–orbit coupling. It has been shown in literature [8–10] that the intrinsic spin of a test particle couples with the gravitomagnetic field produced by a rotating massive body. We can consider this effect as a parallel of Zeeman effect [9,10]. We ask, therefore, ourselves that should not then there be a gravitational coupling of spin  of the same particle? Reσ and orbital angular momentum L E-mail address: [email protected] (S.B. Faruque). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.06.046

sponding in the affirmative since in atomic and nuclear domain such a coupling is an established phenomenon, we consider, following Ref. [11], in this Letter a gravitational spin–orbit coupling, and derive a consequence of this, namely, a clock effect. According to this effect, as is going to be shown, two identical test particles, one orbiting counter-clockwise and the other orbiting clockwise, take different times to complete a full revolution around a massive central mass. The period difference is of topological nature having no dependence on position and gravitational constant G. It depends only on σ and m, m being the mass of the orbiting particle. The Letter is organized as follows: in Section 2, we derive the equation of motion and in Section 3 we calculate the main clock effect. Finally, in Section 4, we put a brief summary of the results. 2. Equation of motion It has been shown by Mashhoon [8–10] that the intrinsic spin of a point test particle couples to the gravitational field of a massive rotating body. In the post-Newtonian approximation to general relativity, the Earth’s gravitational field, as for example, can be thought of as consisting of a gravitoelectric field Eg due to its mass together with a gravitomagnetic field Bg due to its angular momentum. For our purposes in this Letter, it is suffi-

S.B. Faruque / Physics Letters A 359 (2006) 252–255

cient to treat the gravitoelectric field at the Newtonian level and neglect post-Newtonian gravitoelectric corrections; therefore,  where Φ is the Newtonian potential. Similarly, Eg = −∇Φ, the gravitomagnetic field, which has no Newtonian analog, can  D , where Ω  D is the be expressed to lowest order as Bg = cΩ dragging frequency of the inertial frame, due to spin angular momentum, and it is given by [8–10] D = Ω

G  c2 r 5

 3(J . r)r − r 2 J ,

(1)

where J is the rotational angular momentum of the massive body of mass M and r denotes radius vector. Let an orbiting test particle of mass m has intrinsic spin σ . According to Mashhoon [8–10], the Hamiltonian associated with the gravitational  D . Now, coupling of σ with the gravitomagnetic field is σ . Ω the non-relativistic Hamiltonian for a particle in a gravitational field Φ is p2 + mΦ, (2) 2m where Φ = −GM/r, to which one must add the lowest order spin contribution,

H0 =

D , Hσ = σ . Ω

(3)

in accordance with the gravitational Larmor theorem [8]. In classical electrodynamics, the magnetic dipole moment q S , for a particle of mass m and charge q is given by μ  = 2mc where S is the orbital angular momentum. The energy associated with the interaction of this magnetic moment with a  Extending this notions to gravimagnetic field B is −μ  . B. toelectromagnetism with qB = −2m, we find that gravitomag netic dipole moment for a gyroscope of spin S is μ  g = −S/c and the energy of interaction with a gravitomagnetic field is  D . A further extension of this result to the in−μ  g . Bg = S . Ω trinsic spin of particles naturally leads to the interaction Hamiltonian given by Eq. (3). Therefore, naturally, we get the total Hamiltonian H = H0 + Hσ = +

G  c2 r 5

p2 + mΦ 2m

 3(J . r)( σ . r) − r 2 ( σ . J) .

(4)

In terms of electrodynamic analog, the Hamiltonian (4) accounts for the Zeeman splitting of the stationary states of a non-relativistic particle of spin σ = h¯ S in a uniform magnetic  In Eq. (4), the B field, the gravitomagnetic field Bg , is field B. produced by the spin of the central rotating mass M. We can extend this notion to the gravitational coupling of the intrinsic  which is an spin σ with its own orbital angular momentum L, established fact in atomic and nuclear physics. Our reasoning gets support from the work of Lee [11]. Lee [11] has shown that for a Dirac particle in the Schwarzschild field, there exists in the Hamiltonian, in addition to the Thomas precession, a gravitational spin–orbit coupling term. The spin–orbit coupling term for a Dirac particle first appeared in the work of Hehl and Ni [12], where they studied the behavior of a Dirac particle in a

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constantly accelerated frame and found a term which they identified as the “new inertial spin–orbit coupling”. Later on, Varju and Ryder [13] compared the non-relativistic behavior of a Dirac particle in the gravitational field of a Schwarzschild metric with that in a constantly accelerated frame. Varju and Ryder found that the non-relativistic Hamiltonians of the two cases show a spin-dependent difference that does not vanish even in an arbitrarily small region. Lee [11] confirmed the result of [13] and showed that, in the Schwarzschild field, there appears not only the term which corresponds to the Thomas precession, which is of kinematical origin, but also a term which they identified as the gravitational spin–orbit coupling term; the strength and form of the two terms being the same. Now, one may argue that the gravitational spin–orbit coupling term arises due to the existence of a gravitational Ampere’s law. For, then, in the rest frame of a Dirac particle, the gravitating mass M moves and it generates the spin-gravity coupling according to the gravitational Ampere’s law. By extending the result of Lee [11], we can consider that a gravitational spin–orbit coupling term arises in the Hamiltonian of a test classical particle which is in orbit around a gravitating body. There also appears a spin–orbit coupling term of kinematical origin, namely, the term corresponding to the Thomas precession. Therefore, we get the total spin–orbit coupling term in the Hamiltonian of the test particle as twice that of the gravitational spin–orbit coupling term alone. Hence, we have the following spin–orbit coupling term: HLσ =

GM  σ . L. mc2 r 3

(5)

Here, we should note that Eq. (5) reflects only the qualitative nature of the gravitational spin–orbit coupling. Now, we consider a simple configuration as follows: the test particle has spin σ that points along the positive z-direction and the particle moves round the central massive body in the equatorial x–y plane. Therefore, the magnitude of the orbital angular momentum of the test particle is L = mr 2 dϕ/dt , where ϕ is the angle measuring the angular position of the particle in the x–y plane. To obtain the force on the particle we use the Hamilton’s equation, Fi = p˙ i = −∂H /∂xi , where H = H0 + Hσ + HLσ . In differentiating the Hamiltonian with respect to position the momentum is kept fixed, as usual. In this way we obtain  3G  GMm σ . r)(J . r) − r 2 ( σ . J) r F = − 3 r + 2 7 5( r c r    2GMσ dϕ 3G    σ . r)J + (J . r) σ + 2 3 − 2 5 ( r. dt c r c r

(6)

Now, in the case where J points along the positive z-direction, and both J and σ are small, we can neglect ( σ . J), and drop  ( σ . r) and (J . r) from Eq. (6). Hence, we are left from Eq. (6) with    2 GMm 2GMσ dϕ dϕ rˆ = − 2 rˆ + 2 2 rˆ . −mr (7) dt dt r c r This is the equation of motion in the limiting and idealized situation we consider further to find an important consequence of in the next section.

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3. Clock effect due to spin–orbit coupling Here, we consider the simple situation, as quoted in the previous section, where the spin of the particle is pointing, say, upward. Then, the equation of motion (7) can be reduced to    2 2GMσ dϕ GM dϕ + − 3 = 0. (8) dt mc2 r 3 dt r The quadratic Eq. (8) for dϕ dt has solutions      2GMσ 2GMσ 2 4GM dϕ 1 ± + = − . dt 2 mc2 r 3 mc2 r 3 r3

(9) 2

Neglecting the first term under the square root since it is ∝ σc4 , which, in realistic cases, is very small, we get  GM GMσ dϕ . =− 2 3 ± (10) dt mc r r3 The second term in this equationis simply the Keplerian angu-

. Now, note that neglectlar speed in circular orbit, ωk = GM r3 ing spin effect, we would have from Eq. (8),   2 GM GM dϕ = 3 , leading to ωk = ± . dt r r3 Here, the positive sign refers to increasing ϕ, i.e., counterclockwise motion with increasing time and the negative sign refers to clockwise motion. Following this notion, we write from Eq. (10), the two angular frequencies   dϕ GMσ (11) = − 2 3 + ωk , dt + mc r   GMσ dϕ = − 2 3 − ωk . (12) dt − mc r The second term in Eqs. (11) and (12) is larger than the first, dt so, in finding dϕ , we use the binomial theorem and obtain the following approximate formulae:   dt 1 σ (13) = + dϕ + ωk mc2 and   1 σ dt =− + . dϕ − ωk mc2

(14)

For the prograde orbit (i.e., Eq. (13)) we obtain the period of circular motion T+ by integrating Eq. (13) from ϕ = 0 to ϕ = 2π . For the retrograde orbit, we obtain the period T− by integrating Eq. (14) from ϕ = 0 to ϕ = −2π . In this way we obtain 2π 2πσ + T+ = ωk mc2 and 2π 2πσ − . T− = ωk mc2 The period difference is 4πσ . T+ − T− = mc2

(15)

(16)

Therefore, we have obtained a clock effect due to gravitational spin–orbit coupling depicted in Eq. (17). Note that Eq. (17) is correct only up to some multiplicative factor that cannot be obtained using the simple qualitative arguments presented in this work. Now, according to Eq. (17), the prograde motion of the test particle is slower than its retrograde motion. This clock effect is similar to the gravitomagnetic clock effect that results from coupling of the spin of the central mass with the orbital motion of the test orbiting particle. In that case one finds [1–4] 4πJ . (18) Mc2 The correspondence between the effect (17), where the coupling is of the spin of the orbiting particle with its own orbital motion, and the effect (18), where the coupling is of the orbital motion of the orbiting particle with the spin of the central mass, is evident. In both cases, the clock effect is determined by the spin and mass of the spinning body.

T+ − T− =

4. Summary In this Letter, we have considered the possibility of gravitational coupling of the spin and orbital angular momentum of a spinning test particle which is in motion round a massive body. This is done in analogy with the spin–orbit interaction of a charged particle in the field of another charged body which is an established phenomenon in atomic and nuclear physics. The interaction potential we have used is exactly the one found by Lee [11]. We have found out the equation of motion of a test particle in circular orbit and calculated the angular speed of prograde and retrograde motion. The periods for these two cases are different. The period difference is found. This is the new clock effect. The clock effect we report in this Letter is due to the gravitational coupling of spin and orbital motion, and it is of topological nature having no dependence on position of the particle and the gravitational constant. The same happens to the usual gravitomagnetic clock effect [1–4]. This is also the case when the spin of the particle interacts with the gravitomagnetic field of the central body, as shown in Refs. [5,14]. In conclusion, we have explored a new aspect of gravitational field and found a new clock effect that we observe to be due to gravitational coupling of the spin and orbital angular momentum of a test particle. This effect needs further investigation. The effect is O(c−2 ) and hence cannot be ruled out of possible observation in future. Acknowledgements I am indebted to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, where this work is done, for inviting me to visit the centre, and providing me with financial and scientific assistances. References

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