We show that the whole of the gravitomagnetic clock effect in the orbit of a spinning test particle which is revolving round a spinning central massive astrophysical body can be calculated using a gravitational spin-orbit coupling potential which involves the two spins and the orbital angular momentum of the test particle.Pokazujemo da se ukupni gravitomagnetski satni učinak za stazu ispitne čestice, koja se vrti i kruži oko središnjeg masivnog astrofizičkog tijela koje se vrti, može izračunati primjenom gravitacijskog potencijala vezanja spina i staze dviju vrtnji i staznog momenta impulsa ispitne čestice

Faruque, S. B.

Shahrear, Pabel

English

Hrčak - Portal of scientific journals of Croatia

Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7LETTER TO THE EDITORON THE GRAVITOMAGNETIC CLOCK EFFECTS. B. FARUQUEa,1 and PABEL SHAHREARbaDepartment of Physics, Shahjalal University of Science and Technology, Sylhet 3114,BangladeshbDepartment of Mathematics, Shahjalal University of Science and Technology, Sylhet3114, BangladeshReceived 8 May 2008; Accepted 15 October 2008Online 30 October 2008We show that the whole of the gravitomagnetic clock effect in the orbit of a spin-ning test particle which is revolving round a spinning central massive astrophysicalbody can be calculated using a gravitational spin-orbit coupling potential whichinvolves the two spins and the orbital angular momentum of the test particle.PACS number: 04.20.Cv UDC 530.12Keywords: clock effect, spinning particle, spin-orbit couplingGeneral relativity predicts that two freely counter-revolving test particles in theexterior field of a central rotating mass take different periods of time to completethe same full orbit; the time difference is termed as the gravitomagnetic clock effect(GCE). Let us consider circular geodesics in the equatorial plane of a Kerr blackhole of mass M and spin angular momentum J . Let t+ denote the period of co-rotating (i.e. motion in the same sense of the rotation of the central body) motionand t−denote that of counter-rotating motion along such an orbit according toasymptotically static inertial observers. Thent+ − t− =4πJMc2. (1)This result was first derived by Cohen and Mashhoon [1]. Subsequently, varioustheoretical aspects of this effect have been investigated [2 – 7]. On the observational1Corresponding author. E-mail address: awsbf62@yahoo.comFIZIKA B (Zagreb) 17 (2008) 3, 429–434 429faruque and shahrear: on the gravitomagnetic clock effectside, the possibility of its detection has been considered by a number of authors[8 – 13].A remarkable addition to the scenario of GCE came through the work of Faruque[14], namely, when the orbiting particle possesses spin, it contributes to the gravit-omagnetic clock effect an additional amount given by 6πS/mc2, where S is the spinof the orbiting test body whose mass is m. Subsequently, Bini et al. [15] have shownthat the clock effect can be classified in three simple situations: (1) co-rotating or-bit with test particle spin up against counter-rotating orbit with test particle spinup, (2) co-rotating orbit with spin down against counter-rotating orbit with spindown, and (3) co-rotating orbit with spin up against counter-rotating orbit withspin down and vice versa. If we picture these situations, we see that actually in thefirst case the two spins ~J and ~S are parallel and in the second case they are anti-parallel. In the third case an orbit of the first case is differentiated with an orbitof the second case. In this short article, we shall show, in a very concise way, howthe clock effects in all these cases can be obtained using gravitational spin-orbitcoupling. First of all we quote the results from Refs. [14, 15]:Case I:t+ − t− =4πJMc2+6πSmc2, (2)Case II:t+ − t− =4πJMc2−6πSmc2, (3)Case III:t+ − t− =4πJMc2. (4)The derivation of these results was done by solving Mathisson-Papapetrou equa-tions in Ref. [15], and the result in Eq. (3) was found in Ref. [14] by approximationprocedures. However, it is shown in Refs. [16] and [17] that in the simple situation ofcircular motion of a spinning particle in the equatorial plane of a non-spinning cen-tral body, the second term in Eqs. (2) and (3) can be obtained using gravitationalspin-orbit coupling. Subsequently, this line of derivation of clock effect is elaboratedby Mashhoon and Singh [18]. As noted in Ref. [19], it is not possible to decide bythe distant static observer whether he is measuring a time delay of spinning clocksin a non-rotating space-time or a time delay of non-spinning clocks moving ongeodesics in a rotating space-time. That is, there is an equivalence. Hence, we canassume that the set of Eqs. (2 – 4) should follow from a single Hamiltonian con-taining spin-orbit coupling potentials where both spins are involved. As shown byBarker and O’Connel [20], the spin of the central source couples with the orbitalangular momentum of the orbiting particle, and there is coupling between the spinand orbital angular momentum of the orbiter, too. Hence, in a scenario where thegravito-electric part of the field is approximated by the Newtonian potential, the430 FIZIKA B (Zagreb) 17 (2008) 3, 429–434faruque and shahrear: on the gravitomagnetic clock effecttotal Hamiltonian can be written asH =p22m+ mΦ + VLJ + VLS , (5)whereVLJ = ξ1Gc2r3~J · ~L , VLS = ξ2GMmc2r3~S · ~L , (6)where ξ1 and ξ2 are constants of the order of unity. Their exact value can not beascertained as yet. However, the results of Eqs.(2 – 4) can be generated by takingξ1 = 1 and ξ2 = 3/2. Hence, in what follows we will use these values. In a simplesituation where ~J , ~S and ~L all point either in the +ẑ or −ẑ direction, we canproceed to calculate the clock effects using the procedure followed in Ref. [16]. Inthe equatorial circular orbit, if ~J and ~S point along the +ẑ direction and in thefirst consideration we take ~L to point also in the +ẑ direction, then the equationof motion reads−mr(dϕdt)2r̂ = −GMmr2r̂ +2Gmc2r2(J +3M2mS)(dϕdt)r̂ . (7)When ~J points along the +ẑ direction and ~S points along the −ẑ direction, theequation of motion reads−mr(dϕdt)2r̂ = −GMmr2r̂ +2Gmc2r2(J −3M2mS)(dϕdt)r̂ . (8)Equation (7) corresponds to the first case we discussed above and Eq. (8) corre-sponds to the second case. The third case is generated with Eq. (7) and (8). Weshall briefly sketch how Eq. (7) leads to the clock effect in Eq. (2).Case I: ~J and ~S point along the +ẑ directionThis case is described by Eq. (7) which is reduced to the following form(dϕdt)2+2Gc2r3(J +3M2mS)(dϕdt)−GMr3= 0 , (9)which is a quadratic equation having solutions, neglecting smaller term issuingunder the square root, given bydϕdt= −Gc2r3(J +3M2mS)±√GMr3. (10)Here, the positive sign refers to the co-rotating orbit and the negative sign refers tothe counter-rotating orbit with reference to the direction of ~J . The second term inFIZIKA B (Zagreb) 17 (2008) 3, 429–434 431faruque and shahrear: on the gravitomagnetic clock effectEq. (10) is larger than the first, so, in finding dt/dϕ, we use the binomial theoremand obtain the following approximate formulae(dtdϕ)+=1ωK+Gc2r3ω2K(J +3M2mS)=1ωK+JMc2+32Smc2, (11)(dtdϕ)−= −1ωK+JMc2+32Smc2, (12)where ωK =√GM/r3 is the Keplerian angular frequency. Now, for the co-rotatingorbit, we obtain the period of circular motion by integrating Eq. (11) from ϕ = 0to ϕ = 2π. For the counter-rotating orbit, we obtain the period by integratingEq. (12) from ϕ = 0 to ϕ = −2π. In this way, we obtaint+ =2πωK+2πJMc2+3πSmc2, (13)andt−=2πωK−2πJMc2−3πSmc2. (14)The period difference ist+ − t− =4πJMc2+6πSmc2. (15)Thus, we have found the result depicted in Eq. (2) for clock effect in co-rotatingorbit with spin up against counter-rotating orbit with spin up.Case II: ~J points along the +ẑ direction and ~S points along the −ẑdirectionThis case is described by Eq. (8). Following the same procedure just describedfor the Case I, we obtain the time periodst+ =2πωK+2πJMc2−3πSmc2, (16)t−=2πωK−2πJMc2+3πSmc2. (17)The period difference in this case ist+ − t− =4πJMc2−6πSmc2. (18)Thus we have found the result depicted in Eq. (3) for the clock effect in co-rotatingorbit with spin down against counter-rotating orbit with spin down.432 FIZIKA B (Zagreb) 17 (2008) 3, 429–434faruque and shahrear: on the gravitomagnetic clock effectCase III: Co-rotating with spin up against counter-rotating with spindown or co-rotating with spin down against counter-rotating with spinupIn this case the time period in Eq.(17) is subtracted from the time period inEq.(13) to get the clock effect. Or the time period in Eq.(14) is subtracted fromthe time period in Eq.(16). In both cases we obtaint+ − t− =4πJMc2. (19)Thus we have found Eq. (4) for the clock effect in the co-rotating with spin upagainst the counter-rotating with spin down or the co-rotating with spin downagainst the counter-rotating with spin up.We now summarize: The gravitomagnetic clock effect in the orbit of a spinningparticle which is revolving round a spinning central body contains two terms in-volving the two spins. In previous studies, the results were calculated by solvingMathisson-Papapetrou equations. Here, we have shown that whole set of the resultscan be derived using spin-orbit coupling potential. The signs of the terms came outhere exactly as those in the previous studies. Hence, not only the contribution ofthe spin of the orbiting test particle, but also the contribution of the spin of thecentral body can be thought of as due to gravitational spin-orbit coupling. Thisgives us a complete view for the whole picture of the gravitomagnetic clock effect;this unified view is, as far as know, not shown elsewhere in literature.References[1] J. M. Cohen and B. Mashhoon, Phys. Lett. A 181 (1993) 353.[2] O. Semerak, Class. Quantum Grav. 16 (1999) 3769.[3] B. Mashhoon and N. O. Santos, Ann. Phys. 9 (2000) 49.[4] D. Bini, R. T. Jantzen and B. Mashhoon, Class. Quantum Grav. 18 (2001) 653.[5] W. B. Bonnor and B. R. Steadman, Class. Quantum Grav. 16 (1999) 1853.[6] A. Tartaglia, Gen. Rel. Grav. 32 (2000) 1745.[7] L. Iorio, H. Lichtenegger and B. Mashhoon, Class. Quantum Grav. 19 (2002) 39.[8] L. Iorio, Int. J. Mod. Phys. D 10 (2001) 465.[9] L. Iorio, Class. Quantum Grav. 18 (2001) 4303.[10] H. Lichtenegger, F. Gronwald and B. Mashhoon, Adv. Space Res. 25 (2000) 1255.[11] A. Tartaglia, Class. Quantum Grav. 17 (2000) 783.[12] A. Tartaglia, Class. Quantum Grav. 17 (2000) 2381.[13] H. Lichtenegger, L. Iorio and B. Mashhoon, Ann. Phys. 15 (2006) 868.[14] ] S. B. Faruque, Phys. Lett. A 327 (2004) 95.[15] D. Bini, F. de Felice and A. Geralico, Class Quantum Grav. 21 (2004) 5441.[16] S. B. Faruque, Phys. Lett. A 359 (2006) 252.FIZIKA B (Zagreb) 17 (2008) 3, 429–434 433faruque and shahrear: on the gravitomagnetic clock effect[17] P. Shahrear and S. B. Faruque, Int. J. Mod. Phys. D 16 (2007) 1863.[18] B. Mashhoon and D. Singh, Phys. Rev. D 74 (2006) 124006.[19] D. Bini, F. de Felice and A. Geralico, Class. Quantum Grav. 21 (2004) 5427.[20] B. M. Barker and R. F. O’Connel, Phys. Rev. D 12 (1975) 329.AcknowledgementsS. B. Faruque gratefully acknowledges the receipt of a financial grant from theAdvanced Studies Board of Shahjalal University of Science and Technology, Sylhet.O GRAVITOMAGNETSKOM SATNOM UČINKUPokazujemo da se ukupni gravitomagnetski satni učinak za stazu ispitne čestice,koja se vrti i kruži oko sredǐsnjeg masivnog astrofizičkog tijela koje se vrti, možeizračunati primjenom gravitacijskog potencijala vezanja spina i staze dviju vrtnji istaznog momenta impulsa ispitne čestice.434 FIZIKA B (Zagreb) 17 (2008) 3, 429–434

O gravitomagnetskom satnom č

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia