Redalyc.Geodesic Deviation Equation for Relativistic Tops and the Detection

Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México

Nieto, J.A.; Saucedo, J.; Villanueva, V.M. Geodesic deviation equation for relativistic tops and the detection of gravitational waves Revista Mexicana de Física, vol. 53, núm. 2, febrero, 2007, pp. 141-145 Sociedad Mexicana de Física A.C. Distrito Federal, México

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How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative REVISTA MEXICANA DE F ISICA´ S 53 (2) 141–145 FEBRERO 2007

Geodesic deviation equation for relativistic tops and the detection of gravitational waves

J.A. Nieto Escuela de Ciencias F ´ısico-Matem aticas,´ Universidad Aut onoma´ de Sinaloa, 80010 Culiac an´ Sinaloa, M exico.´ J. Saucedo Departamento de Investigaci on´ en F ´ısica de la Universidad de Sonora, 83000 Hermosillo Sonora, M exico.´ V.M. Villanueva Instituto de F ´ısica y Matem aticas,´ Universidad Michoacana de San Nicol as´ de Hidalgo, 58040, Morelia, Michoac an,´ M exico.´ Recibido el 18 de julio de 2005; aceptado el 14 de marzo de 2005

In this contribution, we review the derivation of the relativistic top geodesic deviation equation. This equation results in a generalization of the geodesic deviation equation for a pair of nearby point particles. This property is taken into account in investigating the detection of gravitational waves, and we show how our generalized formula for a relativistic top can be used to study the gravitational wave backgrounds. Besides of these facts, we argue that our formulation may be of special interest for detecting the inﬂationary gravitational waves via the polarization of the cosmic background radiation.

Keywords: Geodesic equation; relativistic top; gravitational waves.

En esta contribuci on,´ revisamos la derivaci on´ de la ecuaci on´ de desviaci on´ de geod esicas´ para trompos relativistas. Esta ecuaci on´ es una generalizaci on´ de la ecuaci on´ de geod esicas´ para un par de objetos puntuales cercanos. Esta propiedad es tomada en cuenta para investigar la detecci on´ de ondas gravitacionales, y mostramos c omo´ nuestra f ormula´ generalizada para un trompo relativista puede ser usada para estudiar los fondos de ondas gravitacionales. Adem as´ de estos hechos, argumentamos que nuestra formulaci on´ puede ser de especial inter es´ para detectar las ondas gravitacionales inﬂacionarias a trav es´ de la polarizaci on´ en la radiaci on´ c osmica´ de fondo.

Descriptores: Ecuaci on´ de geod esicas,´ trompo relativista; ondas gravitacionales.

PACS: 04.60.-m; 04.65.+e; 11.15.-q; 11.30.Ly

1. Introduction other things, for distinguishing these two possibilities. Actu- ally, our formulation is so general that the companion of the The importance of the geodesic deviation equation (GDE) for binary pulsar can be any other source of gravitational waves spinless particles is evident when we study gravitational wave such as supernovae or vibrating neutron star systems. phenomena and their detection [1, 2]. In fact, all currently operating projects for the detection of gravitational waves, Another very interesting setting where our proposal of including LIGO [3], VIRGO [4] and LISA [5], have among Ref. 6 may find application is the search for the stochastic their root physical bases such an equation. In this contribu- gravitational wave background (SGWB) as proposed some tion, we review a recently published work [6], which is based time ago by Detweiler [10], who showed that measurements on previous works [7] and [8], where it was proposed that of signal arrival time from the pulsar may be used to inves- the relativistic top equations of motion (RTEM) [9] can be tigate properties of this background. The main strategy for used instead of the GDE for the same purpose of detecting the detection of the SGWB is to consider a number of pulsars gravitational radiation. separated at different parts in the sky. It is clear then that our RTDE formulation may be useful for this proposal. The main contribution of Ref. 6 is the derivation of a relativistic top deviation equation (RTDE), which is reduced to Ultimately, the RTDE may also have an interesting ap- the GDE when the spin tensor associated with the top van- plication in connection with the so called inflationary grav- ishes. Among the main ideas for application of this equation itational waves (see Ref. 11 and references therein). As is is the study of gravitational radiation generated by binary pul- known, the polarization of the cosmic microwave radia- sars and their spin interaction with gravitational waves pro- tion [12] may solve the problem of detecting the gravitational duced by a companion object such as a massive black hole waves produced during the inflationary scenario. Just before (see [6] and references therein). Instead of focusing our at- the universe became transparent to radiation, the plasma mo- tention on the black hole curvature, we think of the black tion caused by the gravitational waves may be generated by hole gravitational waves as being responsible for the timing different sources. In particular, the effect predicted by the effect of the binary pulsars. Our work may be useful, among RTDE may be of physical interest in this scenario. 142 J.A. NIETO, J. SAUCEDO, AND V.M. VILLANUEVA

Our plan for presenting this review is the following. In and DS µν Sec. 2, we brieﬂy review one of the possible mechanisms to = 0 . (8) obtaining the GDE, and we apply similar techniques to ob- Dτ tain the RTDE formulation. In Sec. 3, we explain how the Here Sµν is the internal angular momentum (or the spin RTDE can be applied to the detection of gravitational waves. tensor) per unit mass of the top satisfying the Pirani con- Finally, in Sec. 4, we make some ﬁnal comments. straint [14]

dx 2. The geodesic deviation equation and the rel- Sµν ν = 0 dτ ativistic top (see Refs. 15 to 17). It is worth mentioning that formulae (7) Several methods can be used to obtain the GDE. Some of and (8) can be derived by a number of different methods [9]. them are, in fact, quite brief. For our purpose, however, it An important observation is that (7) can be understood as the turns out to be more convenient to follow the one in Ref. 13. analogue of the geodesic equation (1), and in fact it reduces Consider a point particle whose trajectory is described by to (1) when the spin tensor Sµν vanishes. µ the coordinates x (τ), where τ is the proper time parameter. After comparing equations (6) and (7), we note a great The geodesic equation is similarity. But in fact they are very different in the sense d2xµ dx α dx β that, while equation (6) refers to a pair of nearby point parti- + Γ µ (x) = 0 . (1) cles, (7) is associated with just one physical system: a rela- dτ 2 αβ dτ dτ tivistic top. Nevertheless, this similarity was used as an inspi- µ Here, Γαβ (x) stands for the Christoffel symbols. ration to propose that, just as (6) is used to detect gravitational A nearby point particle must also satisfy a geodesic equa- waves, equation (7) can be used for the same purpose. In or- tion. If we use the coordinates xµ(τ) to describe the position der to further understand the real differences between the two of such a nearby point particle, we have point particle system and the relativistic top, it turns out to be necessary to derive the analogue of (6) for a pair of nearby 2 µ α β d x µ dx dx relativistic tops. For this purpose, we suppose that a nearby + Γ αβ (x ) = 0 . (2) dτ 2 dτ dτ top satisfies the corresponding equations of motion, µ By “nearby” we mean that the coordinates x (τ) can be writ- 2 µ α β α ten as d x µ dx dx 1 µ dx λβ +Γ αβ (x ) = − Rαλβ (x ) S . (9) xµ = xµ + ξµ(x), (3) dτ 2 dτ dτ 2 dτ with ξµ a very small quantity. Consider now a perturbation of the form Then we expand to the first order in ξµ xµ = xµ + ξµ(x) (10) µ µ µ λ Γαβ (x + ξ) = Γ αβ (x) + Γ αβ ,λ ξ , (4) and with µν µν µν α S = S + S ,α ξ (x). (11) µ µ ∂Γαβ Γ ,λ = . Now, after some algebra, (see Ref. 6 for a full explanation αβ ∂x λ and also see Ref. 18), we obtain our master equation: Thus, using (3) and (4), we find that Eq. (2) for ξµ becomes 2 µ α β α D ξ µ dx dx 1 µ Dξ d2ξµ dx α dξ β dx α dx β = −R ξλ − [R Sλβ + 2Γ µ + Γ µ , ξλ = 0 . (5) Dτ 2 αλβ dτ dτ 2 αλβ Dτ dτ 2 αβ dτ dτ αβ λ dτ dτ dx α dx α + Rµ Sλβ ; ξγ + Rµ ; ξσ Sλβ ], (12) Through some algebraic manipulations, this equation can be αλβ dτ γ αλβ σ dτ written in a totally covariant expression (see Ref. 6) which is the covariant form of the relativistic top deviation D2ξµ dx α dx β equation (RTDE). Clearly, (12) reduces to (6) when the spin = −Rµ ξλ , (6) Dτ 2 αλβ dτ dτ tensor Sλβ vanishes. One of the attractive features of (12) is that the spin angular momentum Sλβ of the top appears which is, of course, the famous geodesic deviation equation to be coupled to gravity via the curvature Riemann tensor (GDE) for a pair of nearby freely falling particles in a gravi- Rµ and its gradient. It seems reasonable to think that this tational field background. αλβ characteristic can provide a better description of the proper- Now, we turn to analyzing the equations of motion for ties of the underlying intrinsic curvature of the geometry in a relativistic top moving in a gravitational field background, question. In particular, we shall see in the next section that namely D2xµ 1 dx α the RTDE may be used to study the different properties of a = − Rµ Sλβ , (7) Dτ 2 2 αλβ dτ gravitational wave background.

Rev. Mex. F ´ıs. S 53 (2) (2007) 141–145 GEODESIC DEVIATION EQUATION FOR RELATIVISTIC TOPS AND THE DETECTION OF GRAVITATIONAL WAVES 143

3. The relativistic top deviation equation and Let us now address the problem at hand, namely, the fact gravitational waves that we are interested in applying similar methods as to the above in the case of a system with two nearby relativistic In this section we summarize the consequences of equa- tops. For this purpose let us consider the formula (12) in a tion (12) in the case of gravitational waves. But for com- proper reference frame. Using the fact that S0µ = 0 , due to µν pleteness we shall start by reviewing briefly how formula (6) the Twlczyjew-Pirani constraint S (dx ν /dτ ) = 0 and the is used for this particular case. symmetries of the Riemann curvature tensor we find that the As usual, consider a gravitational wave in a flat back- time component of (12) is ground where the metric gµν can be written as d2ξ0 1 dξ j − 0 kl 2 = Rjkl S , (22) gµν = ηµν + hµν , (13) dt 2 dt where ηµν is the Minkowski metric and | hµν | 1. In the which for the particular case of a gravitational plane wave transverse-traceless gauge the conditions propagating in the z−direction is

µ d2ξ0 dξ a h0µ = 0 , h ij,j = 0 , h µ = 0 , (14) = −R0 Szb , (23) dt 2 azb dt are obeyed, with the indices i, j, . . . , etc running from 1 to where we used the fact that the only nonvanishing compo- 3. The Einstein gravitational ﬁeld equations imply that h ij nents of the Riemann curvature tensor are satisﬁes the wave equation 1 2 Rzazb = R0a0b = −R0azb = − hab, 00 . (24) hij = 0 , (15) 2

2 µ The z component of (12) turns out to be where = ∂ ∂µ is the D’Alambertian. In the gauge given in (14), the space-time components of the Riemann tensor d2ξz dξ a R have the simple form = −Rz Szb , (25) i0j0 dt 2 azb dt 1 R = − h , . (16) where we used (24), and the x and y components are i0j0 2 ij 00 2 a d ξ a b Using (14), one discovers that hij can be written as = −R ξ dt 2 0b0 + × hij = A+e + A×e , (17) dξ 0 ij ij −[Ra Sbz + Ra Sbz , ξ0 + Ra , ξ0Sbz ] 0bz dt 0bz 0 0bz 0 where A+ and A× are two independent dimensionless am- z plitudes and e+ and e× are polarization tensors. For a wave a dξ bz a bz z a z bz ij ij −[Rzbz S + R0bz S ,z ξ + R0bz ,z ξ S ] traveling in the z- direction, the only nonvanishing compo- dt a bz a nents of eij are −R0bz S ,a ξ . (26)

+ + × × ab exx = −eyy and exy = eyx , (18) Note that in (23), (25) and (26) the S component of the spin angular momentum does not appear and only the Szb and in this case A+ and A× turn out to be functions only of component remains . To better understand this, let us deﬁne t − z. In a proper reference frame we have x0 = τ, x i = 0 , so 1 Si = εijk S , (27) that dx 0/dτ = 1 and dx i/dτ = 0 . In this reference frame we 2 jk ﬁnd that (6) becomes where εijk is the Levi-Civita tensor with εxyz =1 . From (27), z d2ξi we note that the component of the intrinsic angular mo- = −Ri ξj. (19) mentum does not play any role in our equations (23), (25) dt 2 0j0 and (26). This means that no effect is expected when the top For a wave propagating in the z- direction, we have is oriented along the direction of propagation of the gravita-

2 z tional wave . d ξ The second interesting observation is that, if Szb is non- 2 = 0 , , (20) dt vanishing, then d2ξ0/dt 2 = 0 and d2ξz/dt 2 = 0 , in contrast as well as to the case of a pair of nearby point particles in which both 2 a d ξ 1 a b of these terms vanish. The third important observation is that = h ,00 ξ , (21) dt 2 2 b the d2ξa/dt 2 equation contains a large number of terms in a b where now, the Latin indices a, b, . . . , etc run from 1 to 2. addition to the usual one, R0b0ξ . Clearly, this means that the The formula (21) tells us that only separations of two nearby solution of (26) will not be as simple as in the case, of nearby point particles in the transverse direction are meaningful. point particles.

Rev. Mex. F ´ıs. S 53 (2) (2007) 141–145 144 J.A. NIETO, J. SAUCEDO, AND V.M. VILLANUEVA

Let us concentrate on the terms in (26) not involving 4. Comments derivatives of Szb and the Riemann tensor, which presum- We have been pursuing the possibility of using relativistic ably represent small order corrections. In this case (26) is tops as detectors of gravitational waves. In two previous reduced to works [7,8], an isolated top was considered, making it diffi- cult to compare our results with the case of DGE. In order to d2ξa dξ 0 dξ z − a b − a bz − a bz overcome this difficulty and to gain further progress towards 2 = R0b0ξ R0bz S Rzbz S . (28) dt dt dt our goal, in this article we have derived the RTDE equation Using (24), we find that (23), (25) and (28) become for a pair of nearby tops. We have shown that the RTDE reduces to the GDE when the spin tensor vanishes. 2 0 a d ξ 1 dξ zb By considering a plane gravitational wave, we find the = − hab, 00 S , (29) dt 2 2 dt solution of the RTDE for two simple cases. In the first case d2ξz 1 dξ a we discover that, if the internal angular momentum of the top = h Szb , (30) dt 2 2 ab, 00 dt is oriented along the vector separation of the two tops, the gravitational wave does not produce any effect on the physi- and cal system. In the second, more general, case we find that the d2ξa 1 1 dξ 0 1 dξ z nearby top will have an effect different from the case of GDE = ha, ξb− ha, Sbz + ha, Sbz , (31) dt 2 2 b 00 2 b 00 dt 2 b 00 dt only to a second order of approximation in the perturbation hab . At first sight, it would appear that these two cases show respectively. From (29) and (30) we discover that we can set that the RTDE formulation, although it may be theoretically 0 z ξ = −ξ + cte. Therefore (31) becomes interesting, does not seem to offer a promising route for ex- 2 a z periments. However, rough estimates can show us that this d ξ 1 a b a dξ bz = h ,00 ξ − h ,00 S . (32) turns out not to be the case (see Ref. 6 for details). dt 2 2 b b dt A possible, interesting extension of the present work Note that if at an initial time the spin of the top Sbz has ori- is to apply to the nongeodesic equations of motion of entation such that spinning particles in a teleparallel gravitational back- dξ z ground (GETGB) [19, 20] a method similar to the one used ξb − 2 Sbz = 0 , (33) to obtain the RTDE. The GETGB equations are an exten- dt sion of the RTEM in the sense that they include torsion in- then there is no transverse motion ξa = const. and there- teractions in addition to the gravity spin interaction. Re- fore (30) allows the solution cently, Garcia [21] has revisited the motion of spinless particles in a teleparallel gravitational wave background and pro- z ξ = const. (34) posed an experimental mechanism for detecting torsion via the GETGB model. The complete picture should be to detect What this means is that, if the spin of the top is oriented along a both gravitational waves, and torsional waves and therefore the vector separation ξ of the two tops, then the gravitational it may be interesting for further research to find the non- wave does not produce any effect on the system. This seems deviation geodesic equations associated with the GETGB to be a new, interesting result since in the ordinary case of model. GDE the wave is always transverse in its physical effects. Finally, we would like to comment on the possibility of Let us now look for a solution to (32) of the form using the RTDE in connection with the inflationary gravi- 1 dξ z tational wave scenario and world brane cosmology. In the ξa = ξa + haξb − ha( | )Sbz , (35) 0 2 b 0 b dt 0 first case, just before the universe becomes transparent, the gravitational waves, produced during inflation, interacts with a where ξ0 = const . We observe that the plasma, producing polarization patterns of the cosmic dξ a 1 dξ z microwave background (CMB). This kind of phenomenon = ha, ξb − ha, ( | )Sbz . dt 2 b 0 0 b 0 dt 0 is especially interesting since recent experiments seems to have measured the variation of the polarization pattern of the Substituting (35) into (30) we find that, to the first order in CMB. The RTDE model can be interesting in this direction h, (30) becomes if the spin tensor Sλβ is identified as the fermi spin of el- d2ξz ≈ 0 , (36) ementary particles. According to the RTDE, a gravitational dt 2 wave should cause two kinds of motions in the constituent Therefore, if initially fermionic particles of such a plasma. The first one is the motion of the particles caused by the forces due to GDE, and the dξ z | = 0 second is the motion in the plasma caused by the spin-gravity dt 0 interaction. Therefore one should expect that the spin-gravity to the first order of approximation, the solution (35) reduces interaction will also leave a “print” in the polarization pattern to the ordinary case of a pair of nearby point particles. of the CMB.

Rev. Mex. F ´ıs. S 53 (2) (2007) 141–145 GEODESIC DEVIATION EQUATION FOR RELATIVISTIC TOPS AND THE DETECTION OF GRAVITATIONAL WAVES 145

In the case of the world brane universes, only gravity Acknowledgments can move in the extra dimensions; all the matter and other forces are confined to the branes. Only gravitational waves J.A. Nieto wishes to thank the DIFUS and the Astronomy (or gravitons) travel from brane to brane carrying some en- Department for their hospitality. VMV wishes to thank the ergy information away from the branes. Presumably, such Universidad Michoacana for financial support through Coor- a gravitational wave affects objects held together by gravity, dinaci on´ de Investigaci on´ Cient ´ıfica CIC 4.14 and CONA- such as stars and galaxies, by distances shorter than millime- CyT project 38293-E. ters. But, according to our above rough estimate, such short- distance changes are also predicted by the RTDE model.

1. S.L. Shapiro and S.A. Teukolsky, Black Holes, White Dwarfs 12 . J. Kovac et al. , Nature 420 (2002) 772, astro-ph/0209478; A. and Neutron Satrs (Wile-Interscience, New York 1983). Kogut et al. , “Wilkinson microwave anisotropy probe (WMAP) 2. K.S. Thorne, in S. Hawking and W. Israel, eds, 300 Year of ﬁrst year observations: polarization, astro-ph/0302213. Gravitation (Cambridge University, Cambridge 1987) p. 330. 13 . S. Weinberg, Gravitation and Cosmology: Principles and Ap- 3. See A. Lazzarini, Ligo Project an Overview, Published in *Les plications of the General Theory of Relativity (John Wiley and Arcs 1996, Dark matter in cosmology, quantum measurements, Sons, Inc. 1972). experimental gravitation* 355-361 Ligo and Ref. there in. 14 . F.A.E. Pirani, Acta Phys. Pol. 15 (1956) 389. 4. See Y. Acker et al. , The Present Status of Virgo Project, Pub- 15 . C.A.P. Galvao and C. Teitelboim, J. Math. Phys. 21 (1980) lished in *Sao Jose dos Campos 1996, Omnidirectional Grav- 1863. itational Radiation Observatory* 241-250 VIRGO and Ref. there in. 16 . S. Hojman and T. Regge, “Spinning tops in external ﬁelds”, Studies in Matematical Physics; Essays in Honor of Valen- 5. See B. Schutz, (ed.), Class. Quantum Grav. 18 (2001) 3965 and tine Bargmann, ed. E.H. Lieb, B. Simon, and A.S. Wightman Ref. there in. (Princeton University Press, Princeton NJ, 1976) 6. J.A. Nieto, J. Saucedo, and V.M. Villanueva, Phys. Lett. A 312 (2003) 175. 17 . W. Tulczyjew, Acta Phys. Pol. 18 (1959) 393. 7. J.A. Nieto and V.M. Villanueva, Nuovo Cim. B 109 (1994) 821. 18 . J.W. Van Holten, Int. J. Mod. Phys. A 17 (2002) 2645, hep- th/0201083 8. J.A. Nieto and M.P. Ryan, Nuovo Cim. A 63 (1981) 71. 9. M. Mathisson, Acta Phys. Pol. 6 (1937) 163; M.H. Pryce, Proc. 19 . J. Nitsch, in Cosmology and Gravitation; Spin, Torsion, Ro- R. Soc. London Ser A 195 (1948) 62; A. Papapetrou, Proc. tation and Supergravity , Eds. P.G. Bergamann and V. Sabbata R. Soc. London Ser. A 201 (1951) 248; W.G. Dixon, Proc. (Plenum Press, 1980). Roy. Soc. London A 314 (1970) 499; S. Hojman, Ph.D. The- 20 . S. Hojman, M. Rosenbaum, and M.P. Ryan, Phys.Rev. D 19 sis, Princeton University (1975). (1979) 430. 10 . S. Detweiler, Astrophys. J. 234 (1979) 1100. 21 . L.C. Garcia de Andrade, “Nongeodesic motion of spinless par- 11 . S. Dodelson, W.H. Kinney, and E.W. Kolb, Phys. Rev. D 56 ticles in the teleparallel gravitational wave background”, gr- (1997) 3207, astro-ph/9702166 qc/0205120.

Rev. Mex. F ´ıs. S 53 (2) (2007) 141–145