arXiv:1705.02794v1 [gr-qc] 8 May 2017 ntecs fafml fgoeis h omri o so not is former the ) (especially of literature family the a lat- of in case the studied the While well in is enough. to problem is equation curves ter (only) deviation of and families needed the accelerated not of are of generalization laws position the transport relative such the two, namely line other, in- the world the only one to is of respect one deviation with if the Differently, understanding other. in on the terested defined onto operation quantities line This world of two a law of congruence. transport the some and of necessitates that momentum lines Imagine compare world point. to labeled different base any like each of would curves, space one Mathisson- of in position congruence the collection the a by The to forms particles according [1–3]. of model 4-momentum (MPD) of spin Papapetrou-Dixon terms and in described vector are which particles, well. as general- equation associated deviation the line enter world will ized coupling back- This the of and ground. curvature spin geodesics, the the with more between moments multipole coupling no higher the are to due lines quadrupole accelerated world but (spin, their the structure etc.), If with moment, two tensor. endowed Riemann the rate are the connecting to The particles vector proportional is deviation lines equation. de- the world deviation of is variation geodesic of background the given by a scribed in along moving orbits particles geodesic test of lines world neighboring two ‡ † ∗ lcrncades [email protected] address: Electronic [email protected] address: Electronic lcrncades [email protected] address: Electronic nti ae ecnie uc fsinn test spinning of bunch a consider we paper this In between acceleration relative the relativity general In eito n rcsineet ntefil fawa gravita weak a of field the in effects precession and Deviation ytmatce oarfrnewrdln.The too. line. derived world is reference set coordinates Fermi a been to has attached “laboratory” spinnin system the a off quantities model MPD physical the measurable of extension suitable neighboring two a between by vector solved ori propaga spin initial an of vector the deviation direction around the the oscillations to undergoing vector gravitat orthogonal spin the 2-plane with the interaction on The moving associated with magnitude. rest Mathisson-Papapetr at constant are the with particles the to wave the according of studied passage are wave plane NN-Ntoa aoaoiso enr,Vaedell’Univer Viale Legnaro, of Laboratories National - INFN eito n rcsineet fabnho pnigparti spinning of bunch a of effects precession and Deviation .INTRODUCTION I. siuoprl plczoidlCloo“.Pcn, N I- CNR Picone,” “M. Calcolo del Applicazioni le per Istituto CAe,Paz el eubia1,I612Psaa Ital Pescara, I-65122 10, Repubblica della Piazza ICRANet, oaoBini Donato Dtd pi 0 2021) 20, April (Dated: noel Ortolan Antonello ∗ n nraGeralico Andrea and exact oua,ee nfmla pctmso hsclinterest. physical of familiar in even popular, edscwrdlns[–1.Ide,ti a eachieved be can this for Indeed, works [9–11]. de- lines previous coordinates to world generalizing TT approximation geodesic coordinates, between field Fermi weak transformation and the exact of the as rive well space- background of as the advantage of time take character as We symmetrical acts high which system.” the about line, reference world system “laboratory particle’s a coordinate spinning Fermi physically reference a a a construct get relative we of to measurement motion order the nearby spin of in two interpretation the Finally, meaningful between of effect points. equation precession spacetime transport the the giving and vector neighbor- lines as We two world congruence between components. equation ing vector the deviation spin the of the solve of determining particle then evolution wave, single the as the the a well of with such motion interaction of the response the the to investigate constant system first with We direction associated given with a magnitude. wave along the aligned of vector passage spin the before rest at are the instead is analysis. which present the there, of all spin focus at the main discussed for general- equation not transport a is in the vector However, to developed been 8]. leading [7, has extension equation Refs. an deviation defined line Such world are ized line, vector it. world spin along particle and only the 4-momentum off both model because MPD an en- requires the also Studying particles of such extension bodies [6]. of two extended Ref. between motion in small relative structure of quadrupolar GPW gen- case with later a dowed the 5], in [4, to spin Refs. eralized in with investigated particles been test of has spinning spacetime motion of The that con- is the lines particles. and world (GPW) accelerated wave of plane that gruence is gravitational spacetime weak background a the which of in case the in lem eew osdrabnho pnigprilswhich particles spinning of bunch a consider we Here prob- a such study to is paper present the of aim The rnfrainbtenT oriae and coordinates TT between transformation nain h rnpr qain o both for equations transport The entation. pnvco lge ln ie direction given a along aligned vector spin ‡ ol ie fsc oguneaethen are congruence a such of lines world pb osrcigaFricoordinate Fermi a constructing by up keep to particles the causes wave ional lsi h edo ekgravitational weak a of field the in cles i` ,I300Lgao(D,Italy (PD), Legnaro I-35020 sit`a 2, atceswrdln.I re obtain order In line. world particle’s g ino h ae ihtetransverse the with wave, the of tion uDxn(P)mdl eoethe Before model. (MPD) ou-Dixon † 08 oe tl and Italy Rome, 00185 y inlwave tional 2

α α α α 1 in closed form only for a few spacetimes around very equations x = x (τU ) so that U = dx /dτU . special world lines [12, 13], while most applications use a For convenience, let us introduce the unit timelike vec- series expansion approximation from the very beginning toru ¯ associated with p, i.e., [14–19]. 2 The physical problem underlying this study concerns p = mu,¯ p · p = −m , (2) eventually new effects related to the interaction between gravitational waves and spinning test particles which are which is generally distinct from U (but having support poorly (or even not at all) investigated up to now, and only along it) and related to it by a boost, i.e., may lead to measurable effects in the near future. In fact, u¯ = γ(¯u,U)[U + ν(¯u,U)] . (3) deviation and precession effects are the goal of a novel type of gravitational wave detectors, based on magne- Here m is an “effective mass” of the spinning particle, tized samples placed at a some given distance within a which is in general different from the “bare mass” of a common superconducting shield [20, 21]. We will show non-spinning particle and not a conserved quantity along that measuring the different orientation of the spin vec- the path, due to the evolution of the spin. The relative tors of two neighboring such samples due to the pas- velocity ν(¯u,U), following [24], is given by sage of the wave may allow to estimate the product be- tween the amplitude and the frequency of the gravita- β αµ β P (U) µS Fα(spin) tional wave. ν(¯u,U) = 2 αµ , (4) m + S Fα Uµ The paper is organized as follows. In Sec. II we in- (spin) troduce the MPD model for spinning particles and the where P (U)= g + U ⊗ U projects orthogonally to U. generalized deviation equation for accelerated world lines It is also useful to introduce the spin vector in a curved background. These concepts are then ap- plied in Sec. III to the case of a weak gravitational 1 Sα = η(¯u)αβγS , (5) wave spacetime, associated with a monochromatic grav- 2 βγ itational wave traveling along a fixed spatial direction. αβγ σαβγ We discuss the motion as well as the spin deviation ef- where η(¯u) =u ¯ση is obtained from the 1-volume fects in such a case, completing the section with some 4-form ησαβγ . The vector S is orthogonal tou ¯ by defini- indication of possible measurements of the deviation ef- tion and its magnitude fects. In Appendix A we shortly recall the geodesics of 2 α 1 αβ this metric and in Appendix B we give the new result s = S Sα = SαβS (6) concerning the exact map between spacetime and Fermi 2 coordinates adapted to the world line of a spinning body. is a constant of motion, as it can be easily checked by Notation and conventions follow Ref. [23]. The metric differentiating both sides along U and using Eqs. (1). signature is chosen as − +++; Greek indices run from 0 Further constants of motion are associated with Killing to 3, whereas Latin ones run from 1 to 3. vectors of the background spacetime. More precisely, if ξ is a Killing vector, then

α 1 αβ II. SPINNING PARTICLES AND WORLD LINE Jξ = ξαp + S ∇βξα (7) 2 DEVIATION is a constant of the motion. Let us consider a family of spinning test particles in a given gravitational field (to be specified later). The particles form a congruence of accelerated world lines, A. World line deviation equation whose evolution follows the MPD We are interested here in studying deviation effects α Dpα 1 among the particles of the congruence. Let x = = − Rα U βSγδ ≡ F α , α dτ 2 βγδ (spin) x (τU , σ) denote the family of associated world lines, U each curve labelled by σ and parametrized by a proper DSαβ = p[αU β] , time parameter τU . As a consequence, the unit time- dτU like tangent vector to the lines of the congruence is αβ S pβ = 0 . (1)

As standard, U denotes the unit timelike vector tangent 1 If the particles have additional structure besides spin, e.g., to the particle’s world line CU , p (a timelike vector, not µν quadrupolar, octupolar structure, Eqs. (1) are modified by corre- unitary) is the generalized momentum and S is the spin sponding additional terms. It is therefore quite natural to study tensor. Both p and S have support on the world line CU these equations only at the linear order in spin. When working parametrized by the proper time τU , i.e., with parametric within this approximation we will explicitly specify it. 3

U α = ∂xα/∂τ , whereas the deviation vector is Y α = along the reference curve and defined only along it) and σ U (τU ) ∂xα/∂σ. The deviation vector Y is Lie-transported along solve the parallel transport equations along these curves. α α U Their parametric equations x = y (σ, τ(ref)) in terms of the affine arclength parameter σ are the solutions of the £U Y = ∇U Y − ∇Y U =0 . (8) following equations

2 α β β In addition, Y is not necessary orthogonal to U and not d y µ α dy dy + Γ(y (σ)) βγ =0 , (14) unitary; therefore, in the Local Rest Space of U (hereafter dσ2 dσ dσ LRSU ) the “effective” deviation vector or the position with initial conditions vector of nearby particles is its spatial projection β α α dy β ⊥ y | = x (τ ) , = Yˆ (τ ) . (15) Y ≡ P (U)Y. (9) σ=0 (ref) dσ (ref) σ=0

From Eq. (8) it follows that Once the curve (the family of curves) CYˆ with tangent ˆ β vector Y (τ(ref), σ) ≡ dy /dσ is explicitly obtained, the ∇U Y = ∇Y U, (10) solution of the parallel transport equations along it, so that differentiating (covariantly) again along U this αβ... ∇Yˆ T =0 , (16) relation (as well as adding and subtracting terms prop- erly), leads to leads to the corresponding extended quantities off the ref- αβ... erence world line, i.e., T (σ, τ(ref)). Subsequently, for ∇U ∇U Y = ∇U ∇Y U − ∇Y ∇U U + ∇Y ∇U U fixed values of σ, one can evolve these along U, i.e., along any other curve of the congruence U, and com- = [∇U , ∇Y ]U + ∇Y a(U) pare with the corresponding quantities already existing = R(U, Y )U + ∇ a(U) Y there. = −K(U) Y, (11) The Riemann tensor identities for the commuting vec- tor fields U and Y , i.e., where we have used the Lie transport condition (8) in µ µ ν α β the Riemann tensor definition and we have introduced a [∇U , ∇Y ]C = R ναβC U Y , (17) “strain tensor” [25] imply γ δ Kαβ(U)= −RαγβδU U − ∇βa(U)α , (12) αβ... α µβ... γ δ 0 = ∇Y ∇U T + R µγδT U Y β αµ... γ δ built up with the tidal-electric Riemann tensor field, i.e., +R µγδT U Y + ..., (18)

γ δ E(U)αβ = RαγβδU U , (13) which give in turn compatibility conditions for the cor- responding extended fields. and the of the acceleration spatial In this paper, we will assume that the particles of the vector field a(U) = ∇U U of the congruence U. In the congruence U are spinning test particles, while the gen- special case of U geodesic we recognize the evolution eral formulas derived above may involve any kind of ac- equation for Y as the geodesic deviation equation (11), celeration. whose 3 + 1 version was discussed in Refs. [25, 26]. The above considerations are general enough, in the sense that they hold for any family of world lines U. Moreover, III. DEVIATION EFFECTS IN THE FIELD OF the geodesic deviation equation and its generalization to A WEAK GRAVITATIONAL WAVE any accelerated family of world lines, Eq. (11), allow to define the deviation vector Y (and hence its spatial pro- Let us consider as a background spacetime that of a jection P (U)Y ) only along U. In different words, taken a single gravitational wave travelling along the x axis, with reference world line within the congruence U, the vector line element

Y is defined as an applied vector to any point along it. 2 2 2 2 2 Note that Lie-transporting along U an initially spatial ds = −dt + dx + (1 − h+)dy +(1+ h+)dz vector with respect to U does not lead in general to a −2h×dydz , (19) transported vector which is still orthogonal to U. If one is interested in propagating the main tensors with h+,× depending on t − x only. In terms of the null (momentum and spin) off the reference world line of the coordinates congruence U the deviation vector plays an essential role. u = t − x , v = t + x , (20) Indeed, one should consider the set of all spatial geodesics ˆ ⊥ ⊥ CYˆ with unit tangent vector Y = Y /||Y || emanat- such that α ing from a generic point x (τ(ref)) of the reference world 1 1 ˆ ˆ ∂u = (∂t − ∂x) , ∂v = (∂t + ∂x) , (21) line where Y = Y (τ(ref)) (function of the proper time 2 2 4 the metric becomes A. Spinning particle motion

ds2 = −dudv + (1 − h )dy2 +(1+ h )dz2 + + The motion of an extended body in the spacetime of −2h×dydz a weak gravitational wave (19) has been studied in Refs. A B = −dudv + gABdx dx , (22) [4–6]. In the case of a spinning test particle and working at linear order in spin large simplifications arise. For A where x , A = 2, 3 denote the coordinates on the wave instance, in this case p = mU, with m constant along U. front. The analysis of any kind of motion in this space- We recall below the main results corresponding to the time is simplified because of the existence of the three case of a spinning particle initially (before the passage of Killing vectors ∂x, ∂y, ∂v. the wave) at rest at the origin of the coordinates, with Hereafter we will consider a monochromatic wave with constant spin components. The solution for the orbit α α U = dx /dτU is the following h+(u)= A+ sin ωu, h×(u)= A× cos ωu , (23)

t(τU ) = τU , x(τU )=0 and limit our analysis to the linear order in h+, h×. Lin- 1 ear polarization corresponds to either A+ =0or A× = 0, y(τ ) = A S0[cos(ωτ ) − 1] whereas circular polarization is assured by the condition U 2 × 2 U A+ = ±A×. In general, one uses the “polarization angle” 1 0 −1 − A+S3 [sin(ωτU ) − ωτU ] ψ = tan (A×/A+). 2 Let us fix a family of fiducial observers at rest with re- 1 z(τ ) = − A S0[cos(ωτ ) − 1] spect to the (t,x,y,z) coordinate grid, with four-velocity U 2 × 3 U 1 0 n = ∂ . (24) − A S [sin(ωτU ) − ωτU ] , (31) t 2 + 2

An adapted triad to n is given by 0 where Si (i = 1, 2, 3) are the values of the coordinate exˆ = ∂x components of the particle’s spin at τU = 0. In compact 1 1 form eyˆ = 1+ h+ ∂y + h×∂z  2  2 1 xA(τ ) = A [cos(ωτ ) − 1](S0 )A 1 1 U 2 × U + ezˆ = h×∂y + 1 − h+ ∂z . (25) 1 2  2  − A [sin(ωτ ) − ωτ ](S0 )A . (32) 2 + U U × This triad deals “symmetrically” with eyˆ and ezˆ and has a special geometrical meaning, since each leg of the triad Similarly undergoes a Fermi-Walker transport along n, namely 1 0 U = ∂t − ωA×S+ sin(ωτU ) P (n)∇neaˆ =0 , (26) 2 1 0 with P (n) = g + n ⊗ n projecting orthogonally to n. − ωA+S×(cos(ωτU ) − 1) , (33) 2 Moreover, the dependence on the coordinates of these frame vectors is only through t−x, implying a trivial Lie where we have chosen the location of the particle to be transport along any direction on the wave front. the origin at τU = 0. It is customary to define the two polarization tensors The coordinate components of the spin vector are given by e+ = ∂y ⊗ ∂y − ∂z ⊗ ∂z t x 0 e× = ∂y ⊗ ∂z + ∂z ⊗ ∂y , (27) S (τU ) = 0 , S (τU )= S1 , 1 with associated notation Sy(τ ) = S0 + S0A sin(ωτ ) U 2 2 2 + U 1 e+ X = X+ , e× X = X× , (28) + A S0[cos(ωτ ) − 1] , 2 × 3 U where the symbol denotes right contraction between z 0 1 0 a tensor and a vector. In components, for example, S (τU ) = S3 − S3 A+ sin(ωτU ) j 2 [e+ X]i = (X+)i = (e+)ij X , that is 1 0 + A×S2 [cos(ωτU ) − 1] , (34) 2 3 2 (X+)i = δi X2 − δi X3 , (29) so that or

0 1 0 1 0 X = X ∂ − X ∂ S = S + A+S sin(ωτU )+ A×S (cos(ωτU ) − 1) . + 2 y 3 z 2 + 2 × X× = X3∂y + X2∂z . (30) (35) 5

0 α 0 B. Deviation effects where Y = Y ∂α is the constant solution correspond- ing to the geodesic case and initial conditions have been 0 From these results, a straightforward computation chosen so that Y (τU = 0) = Y . When expressed with gives the acceleration of U (i.e., the spin force per unit respect to the frame (25) it reads mass)

1 Y = Y(geo) + Ys , (41) a(U) = − ω2 −S0A sin(ωτ )+ S0A cos(ωτ ) ∂ 2 3 + U 2 × U y 0  0  − S2 A+ sin(ωτU )+ S3 A× cos(ωτU ) ∂z with (see, e.g., Ref. [23])

1 2 0   0 = − ω −A+ sin(ωτU )S× + A× cos(ωτU )S+ . 2 α 0   Y(geo) = Y eαˆ (36) 1 − A Y z 0 cos(ωτ )+ A Y y 0 sin(ωτ ) e 2 × U + U yˆ as well as the associated derivative   1 y 0 z 0 1 − A×Y cos(ωτU ) − A+Y sin(ωτU ) ezˆ , 3 0 u 0 2 ∇Y a(U) = ω Y S3 A+ cos(ωτU )   2 (42) 0  +S2 A× sin(ωτU ) ∂y whereas the spinning part is given by 0  0 − −S2 A+ cos(ωτU )+ S3 A× sin(ωτU ) ∂z

1 3 0 u 0  0 1 0 u 0 0 = ω Y A+ cos(ωτU )S + A× sin(ωτU )S , Y = − ωY A S (cos(ωτ ) − 1)+ A S sin(ωτ ) e 2 × + s 2 + 3 U × 2 U yˆ   (37) 1   − ωY 0 u A S0(cos(ωτ ) − 1) − A S0 sin(ωτ ) e . 2 + 2 U × 3 U zˆ 0 u 0 t 0 x   where Y = Y − Y , and the only nonvanishing (43) components of the (symmetric and tracefree) electric part of the Riemann tensor are Similarly 1 1 E(U) = − ω2A sin(ωτ )= − ω2h = −E(U) , xx 2 + U 2 + yy 1 1 Yˆ = Yˆ + Yˆ , (44) E(U) = − ω2A cos(ωτ )= − ω2h . (38) (geo) s xy 2 × U 2 × In tensorial form where

1 2 E(U)= − ω hpep , (39) 1 Z a 0 2 Yˆ = 1+ Y ∂a , (45) p=+X,× (geo) W  2W 2  where here and below hp is meant to be evaluated along the orbit. 2 a 0 b 0 with W = δabY Y and Finally, the solution for the deviation vector is given by y 0 z 0 Z = 2Y Y A× cos(ωτU ) 0 Y = Y +[(Y y 0)2 − (Y z 0)2]A sin(ωτ ) , (46) 1 + U − ωY u 0 S0 A (cos(ωτ ) − 1)+ S0 A sin(ωτ ) , 2 × + U + × U   (40) and

x 0 2 x 0 ω 0 (Y ) Y y 0 z 0 Yˆ = − (T· Y ) ∂t + ∂x − Y ∂y + Y ∂z s 2W  W 2  (Y y 0)2 + (Y z 0)2   ω Y x 0 − (K· Y 0) Y z 0∂ − Y y 0∂ , (47) 2W (Y y 0)2 + (Y z 0)2 y z  6 with the two vectors T and K given by

0 0 T = A+S×[cos(ωτU ) − 1]+ A×S+ sin(ωτU ) , 0 0 K = A+S+[cos(ωτU ) − 1] − A×S× sin(ωτU ) . (48)

Let us consider now the spatial geodesics emanating with α, β and E given by Eq. (49). Their behavior as from a generic point on the world line U. These are functions of σ is shown in Fig. 1 (a) for a fixed value of 2 given in Appendix A (for µ = −1 and λ = σ) and the the proper time parameter τU . Fig. 2 (b) shows, instead, values of the constants there correspond to their behavior as functions of τU for a given displacement σ. Y 0 y Y 0 z Y 0 x α = , β = , E = − . (49) W W W The transport equations (16) can then be solved and the solution for the transported spin vector is given by C. Measuring the spin deviation α α α α S (τU , σ) = S (τU )+ S˜ (τU , σ), with S (τU ) given by Eq. (34) and A physical measurement in a gravitational field by a given observer necessitates a continuous locally inertial ˜1 0 0 0 0 S (τU , σ) = (βS2 + αS3 )C + (αS2 − βS3 )S , system along the world line of the observer to get a cor- ˜2 0 0 0 0 rect interpretation. This can be realized by setting up S (τU , σ) = (−βS1 + ES3 )C + (−αS1 + ES2 )S , a Fermi in the neighborhood of the S˜3(τ , σ) = (−αS0 + ES0)C + (βS0 − ES0)S , (50) U 1 2 1 3 observer’s world line. The transformation between the background coordinates (t,x,y,z) and Fermi coordinates with S˜0(τ , σ)= S˜1(τ , σ), where U U (T,X,Y,Z) is explicitly derived in Appendix B. Here we simply use the results to rewrite the deviation compo- A× C = [cos(ω(σE + τU )) − cos(ωτU )] , nents of the transported spin vector in a form which is 2E ˜T ˜X A suitable for a measurement process. We find S = S S = + [sin(ω(σE + τ )) − sin(ωτ )] , (51) and 2E U U

1 S˜X = − (S0Y − S0Z)A [sin(ω(T − X)) − sin(ωT )] + (S0Z + S0Y )A [cos(ω(T − X)) − cos(ωT )] , 2X 2 3 + 2 3 × 1  S˜Y = (S0Y + S0X)A [sin(ω(T − X)) − sin(ωT )] + (S0Z + S0X)A [cos(ω(T − X)) − cos(ωT )] , 2X 1 2 + 1 3 × 1 S˜Z = − (S0Z + S0X)A [sin(ω(T − X)) − sin(ωT )] − (S0Y + S0X)A [cos(ω(T − X)) − cos(ωT )] , (52) 2X 1 3 + 1 2 ×  which imply

S0 S˜X X + S˜Y Y + S˜Z Z = 1 {2Y ZA [cos(ω(T − X)) − cos(ωT )] + (Y 2 − Z2)A [sin(ω(T − X)) − sin(ωT )]} . (53) 2X × +

Let us prepare our device in such a way that the test simplifies as body is spinning around the Z-axis before the passage 0 0 of the wave with constant spin, i.e., S1 = 0 = S2 X s0 0 S˜ = {ZA+[sin(ω(T − X)) − sin(ωT )] and S3 ≡ s0. The interaction with the gravitational 2X wave causes the spin vector to acquire nonvanishing spa- −Y A×[cos(ω(T − X)) − cos(ωT )]} , tial components all varying with time and displacements s S˜Y = 0 A [cos(ω(T − X)) − cos(ωT )] , along the three spatial directions as from Eq. (52), which 2 × s S˜Z = − 0 A [sin(ω(T − X)) − sin(ωT )] . (54) 2 +

It is easy to check then the following (exact) properties 7

(a) (b)

i FIG. 1: The behavior of the spatial components S˜ (τU ,σ) of the transported spin vector as functions of σ is shown in panel (a) for a fixed value τU = π/2 of the proper time parameter along the world line of the spinning particle. Panel (b) shows instead 0 their behavior as functions of τU for a given displacement σ = 1. The choice of parameters is as follows: S1 = s0 sin θs cos φs, 0 0 a 0 S2 = s0 sin θs sin φs, S3 = s0 cos θs, with s0 = 1, θs = π/4= φs, Y = 0.1, ω = 1, and A× = A+ tan ψ, with A+ = 1, ψ = π/6, as an example.

(a) (b)

3 2 FIG. 2: The parametric curve S˜ (τU ,σ) versus S˜ (τU ,σ) is shown for different values of (a) τU and (b) σ for the same parameter values as in Fig. 1. satisfied by the spin components for S˜Y , S˜Z corresponds actually to a circle, recalling the mechanical properties of the ellipsoid of inertia of a rigid ˜X ˜Y ˜Z S X + S Y + S Z =0 , body. Vice versa, in the configuration space X,Y,Z, with ˜Y 2 ˜Z 2 (S ) (S ) 2 2 ωX the spin components taken as fixed, the above equation 2 + 2 = s0 sin , (55) implies a fixed value for the X and straight line in the A× A+  2  Y -Z plane, leading to an unexpected simple geometrical in which the dependence on T has disappeared. A nice characterization of an otherwise complicated situation. geometrical interpretation of Eqs. (55) can be given ei- ther in the spin space, i.e., at fixed X,Y,Z or in the configuration space, i.e., at fixed S˜X , S˜Y , S˜Z . In fact, in the spin space, the transverse components of the spin Sufficiently close to the the reference spinning parti- “belong” to an ellipse which intersects a plane. Using cle’s world line it is enough to take the expansion of the component S˜X as a parameter, the explicit solution the above expressions up to the first order in the spa- 8 tial Fermi coordinates, which gives should be able to detect the magnetic-like part of a grav- itational wave by comparing two distant samples of el- 1 S˜X = − ωs [Y A sin(ωT )+ ZA cos(ωT )] + O(2) , ementary spins, differently (and complementary) to the 2 0 × + LIGO/VIRGO interferometers which succeeded in mea- 1 suring the electric-like part of a gravitational wave using S˜Y = ωs XA sin(ωT )+ O(2) , 2 0 × free falling masses. Our present study provides the un- 1 derlying theoretical framework for the prediction of ob- S˜Z = ωs XA cos(ωT )+ O(2) . (56) 2 0 + served signals by this new kind of detectors. For instance, for the gravitational wave event If we consider a second spinning particle located at coor- GW150914 recently observed by LIGO [27], with max- dinates (X, 0, 0) with the same spin vector of the particle imum amplitude h ∼ 10−21 at a frequency ω = 150 Hz, ⊥ −21 in the origin, the second and third equations in (56) de- we find S˜ /s0 ∼ 2.5 × 10 , if the detector consists of scribe a -induced relative precession, as the direc- two magnetometers separated by a distance of L = 104 tions of the two spins define a generalized cone in space. Km [20]. Let us focus on the component of the spin vector on the wave front, with magnitude IV. CONCLUDING REMARKS S˜⊥ = (S˜Y )2 + (S˜Z )2 . (57) q The observation of gravitational wave signals It is worth noticing that for a circularly polarized GPW GW150914 [27] and GW151226 [28] by LIGO, be- S˜⊥ is the amplitude of the relative precession cone in- sides opening new horizons in gravitational wave duced by a GPW on the two spinning particles. It turns astronomy and astrophysics, provides important tests of out that to the first order in the distance (omitting the the general theory of relativity in the strong field regime. 2 O(2) notation too) and taking A+ = ±A× = h However, up to now, the two LIGO interferometers have observed only the electric-like part of the waves, 1 measuring the induced displacements of their free falling S˜⊥(X) = s hωX . (58) 2 0 mirrors. The observation of the magnetic-like part of a gravitational wave, i.e., the measurement of the induced In Earth based laboratories separated by a distance X = effects on mass current or spinning particles is still L we have missing. We have shown that by using spinning test particles one can identify observable effects associated ⊥ 1 S˜ (L)= s0hωL , (59) with the variation of the polarization. 2 We have considered a bunch of spinning test particles where O(2) expansion requires ωL/c . 1 (here the speed initially at rest before the passage of the wave (plane, monochromatic, transverse in the present analysis, for of light c is made explicit). Therefore, measuring S˜⊥(L) simplicity), with associated spin vector aligned along a at different distances L leads to a direct information given direction with constant magnitude. The interac- about the gravitational wave concerning the product of tion with the gravitational wave causes the particles to amplitude and frequency. keep moving on the 2-plane orthogonal to the direction of The present result is interesting especially because it is propagation of the wave. The transverse components of under current consideration a novel experimental scheme the spin vector undergo oscillations around their initial enabling the investigation of spin couplings [20], based orientation, whereas the component parallel to the direc- on synchronous measurements of optical magnetometer tion of the wave propagation does not change. By solving signals from several devices operating in magnetically the transport equations for both the deviation vector an shielded environments placed at distant locations [21, 22]. spin vector between two neighboring world lines of such a The primary interest is to probe couplings between spins congruence we have shown that the relative precession of and “exotic” fields beyond the Standard Model, e.g., two spins, assumed to be along the transverse direction of axion-like fields, by analyzing the correlation between a Fermi frame, which is our “laboratory” system and sep- signals from multiple, geographically separated magne- arated by a distance L along the direction of propagation tometers. However, we argue that the same scheme can of the GPW) is given by 1 hωL. Even if the correspond- be equally applied to the kind of interaction under in- 2 ing amplitudes of the GPW detected by LIGO seem not vestigation here, the gravitational wave signal inducing very promising, it is true that the technology to measure a torque on atomic spins. In this way, the apparatus spin precessions on a network of optical magneometers already exists [20] and will be refined in the next years. As a final remark, we observe that S˜⊥(L) could be also 2 Clearly, extending the present calculation to higher orders in the measured by comparing the magnetizations due to elec- distance is a trivial task, and does not require special considera- tron spins in two ferromagnetic samples, separated by a tion. distance L, and magnetized by parallel external magnetic 9

fields.

Appendix A: Geodesics

The geodesic of the metric (19) are given by [29, 30]

1 U = [(µ2 + f + E2)∂ + (µ2 + f − E2)∂ ] (geo) 2E t x +[α(1 + h+)+ βh×]∂y + [β(1 − h+)+ αh×]∂z , (A1) where α, β and E are conserved Killing quantities, µ2 =1, 0, −1 corresponding to timelike, null and spacelike geodesics respectively, and

2 2 f = α (1 + h+)+ β (1 − h+)+2αβh× . (A2)

The corresponding parametric equations of the geodesic orbits are then easily obtained t(λ) = t0 + Eλ + x(λ) − x0 , λ x(λ) = x + (µ2 + α2 + β2 − E2) 0 2E 1 1 2 2 − (α − β )A [cos ω(Eλ + t − x ) − cos ω(t − x )] − αβA×[sin ω(Eλ + t − x ) − sin ω(t − x )] , ωE2 2 + 0 0 0 0 0 0 0 0  1 y(λ) = y + αλ − αA [cos ω(Eλ + t − x ) − cos ω(t − x )] − βA×[sin ω(Eλ + t − x ) − sin ω(t − x )] , 0 ωE  + 0 0 0 0 0 0 0 0  1 z(λ) = z + βλ + βA [cos ω(Eλ + t − x ) − cos ω(t − x )] + αA×[sin ω(Eλ + t − x ) − sin ω(t − x )] ,(A3) 0 ωE  + 0 0 0 0 0 0 0 0 

α α where λ is an affine parameter and x0 = x (λ = 0).

Appendix B: Fermi coordinates

Let us construct a Fermi coordinate system (T,X,Y,Z) = (T,X1,X2,X3) in a neighborhood of the (accelerated) world line U of the spinning particle. They are defined by

i T = τU , X = σ(ξ · Fi)|Q , (B1)

dxµ(σ) where ξ = dσ ∂µ is the unit vector tangent to the unique spacelike geodesic segment of proper length σ connecting a generic point Q on the particle’s world line with a generic spacetime point P near Q and satisfying the condition (ξ · U)|Q = 0, and {Fi} is a Fermi-Walker spatial triad given by

yˆ zˆ F1 = exˆ , F2 = ν ∂t + eyˆ , F3 = ν ∂t + ezˆ . (B2)

yˆ zˆ The velocity components can be identified directly from Eq. (33), by re-writing U = ∂t + ν eyˆ + ν ezˆ. By using the solution (A3) for the spatial geodesics, the spatial Fermi coordinates around the point Q turn out to be given by

σ 1 2 2 X = 1 − α(0)α(1) − β(0)β(1) − (α(0) − β(0))A+ sin(ωτU ) − α(0)β(0)A× cos(ωτU ) , 2 2  2  1 − α(0) − β(0) q 1 1 Y = σ α + α + α A sin(ωτU )+ β A× cos(ωτU ) ,  (0) (1) 2 (0) + 2 (0)  1 1 Z = σ β + β + α A× cos(ωτU ) − β A sin(ωτU ) , (B3)  (0) (1) 2 (0) 2 (0) +  10 to first order in the gravitational wave amplitudes, where we have used the orthogonality condition between ξ and U at Q to express the constant of motion E = E(0) + E(1) in terms of α = α(0) + α(1) and β = β(0) + β(1). The arclength parameter σ as well as the constants α and β must then be expressed in terms of the background coordinates (t,x,y,z). We are interested here in the inverse transformation. Therefore, solving Eq. (B3) for σ, α and β yields

σ = X2 + Y 2 + Z2 , p 1 1 σα = Y − ZA cos(ωT ) − Y A sin(ωT ) , 2 × 2 + 1 1 σβ = Z − Y A cos(ωT )+ ZA sin(ωT ) , (B4) 2 × 2 + with in addition

1 1 σE = −X − ωA (S0Z + S0Y )[cos(ωT ) − 1]+ ωA (S0Y − S0Z) sin(ωT ) , (B5) 2 + 2 3 2 × 2 3

α α which, once inserted in Eq. (A3) with x0 = x (τU ), finally gives the exact coordinate transformations

1 Y 2 − Z2 cos(ω(T − X)) − cos(ωT ) t = T − A − sin(ωT ) + ω(S0Z + S0Y )[cos(ωT ) − 1] 2 +  X  ωX  2 3 

YZ sin(ω(T − X)) − sin(ωT ) 1 0 0 +A× + cos(ωT ) − ω(S Y − S Z) sin(ωT ) ,  X  ωX  2 2 3  1 Y 2 − Z2 cos(ω(T − X)) − cos(ωT ) x = X − A − sin(ωT ) 2 + X  ωX  YZ sin(ω(T − X)) − sin(ωT ) +A× + cos(ωT ) , X  ωX  cos(ω(T − X)) − cos(ωT ) 1 1 y = Y + A Y − sin(ωT ) − S0[sin(ωT ) − ωT ] +   ωX 2  2 3 

sin(ω(T − X)) − sin(ωT ) 1 1 0 −A× Z + cos(ωT ) − S [cos(ωT ) − 1] ,   ωX 2  2 2  cos(ω(T − X)) − cos(ωT ) 1 1 z = Z − A Z − sin(ωT ) + S0[sin(ωT ) − ωT ] +   ωX 2  2 2 

sin(ω(T − X)) − sin(ωT ) 1 1 0 −A× Y + cos(ωT ) + S [cos(ωT ) − 1] . (B6)   ωX 2  2 3 

This transformation contains the new (spinning) terms additively, in the sense that it is formally of the type

α α α µ α µ x = X + f(geo)(X )+ f(spin)(X ) . (B7)

The spacetime metric in Fermi coordinates is then simply evaluated by

∂xα ∂xβ g (X)= g (x(X)) . (B8) AB ∂XA ∂XB αβ 11

Up to the second order in the spatial Fermi coordinates its nonvanishing components are

0 0 1 2 2 2 0 0 2 gT T = −1 − S Z + S Y − (Y − Z ) ω A sin(ωT )+ S Y − S Z + YZ ω A× cos(ωT )+ O(3) ,  2 3 2  + 2 3   1 g = − ω2 (Y 2 − Z2)A sin(ωT )+2Y ZA cos(ωT ) + O(3) , T X 3 + × 1   g = ω2X [Y A sin(ωT )+ ZA cos(ωT )] + O(3) , TY 3 + × 1 g = ω2X [−ZA sin(ωT )+ Y A cos(ωT )] + O(3) , T Z 3 + × 1 g = 1+ ω2 (Y 2 − Z2)A sin(ωT )+2Y ZA cos(ωT ) + O(3) , XX 6 + × 1  1  g = − g + O(3) , g = − g + O(3) , XY 2 TY XZ 2 T Z 1 1 g = 1+ ω2X2A sin(ωT )+ O(3) , g = ω2X2A cos(ωT )+ O(3) , YY 6 + Y Z 6 × 1 g = 1 − ω2X2A sin(ωT )+ O(3) , (B9) ZZ 6 + in agreement with the series-expanded metric components in Fermi coordinates. Of course here we also have the exact (non-expanded) form of the metric, having identified the exact coordinate transformation from one coordinate system to the other.

Acknowledgments

D.B. thanks ICRANet for partial support. A.O. thanks Dr. M. L. Ruggiero for useful discussion.

[1] M. Mathisson, “Neue mechanik materieller systemes,” nate system and electromagnetic detectors of grav- Acta Phys. Polon. 6, 163 (1937). itational waves,” Nuovo Cim. B 71, 37 (1982). [2] A. Papapetrou, “Spinning test particles in general rel- doi:10.1007/BF02721692 ativity. 1.,” Proc. Roy. Soc. Lond. A 209, 248 (1951). [10] A. I. Nesterov, “Riemann normal coordinates, Fermi doi:10.1098/rspa.1951.0200 reference system and the geodesic deviation equation,” [3] W. G. Dixon, “Dynamics of extended bodies in gen- Class. Quant. Grav. 16, 465 (1999) doi:10.1088/0264- eral relativity. I. Momentum and angular momen- 9381/16/2/011 [gr-qc/0010096]. tum,” Proc. Roy. Soc. Lond. A 314, 499 (1970). [11] M. Rakhmanov, “Fermi-normal, optical, and wave- doi:10.1098/rspa.1970.0020 synchronous coordinates for spacetime with a plane grav- [4] M. Mohseni and H. R. Sepangi, “Gravitational itational wave,” Class. Quant. Grav. 31, 085006 (2014) waves and spinning test particles,” Class. Quant. doi:10.1088/0264-9381/31/8/085006 [arXiv:1409.4648 Grav. 17, 4615 (2000) doi:10.1088/0264-9381/17/22/302 [gr-qc]]. [gr-qc/0009070]. [12] C. Chicone and B. Mashhoon, “Explicit Fermi co- [5] M. Mohseni, R. W. Tucker and C. Wang, “On the motion ordinates and tidal dynamics in de Sitter and of spinning test particles in plane gravitational waves,” Godel spacetimes,” Phys. Rev. D 74, 064019 (2006) Class. Quant. Grav. 18, 3007 (2001) doi:10.1088/0264- doi:10.1103/PhysRevD.74.064019 [gr-qc/0511129]. 9381/18/15/314 [gr-qc/0308042]. [13] D. Klein and P. Collas, “Exact Fermi coordinates for a [6] D. Bini, C. Cherubini, A. Geralico and A. Ortolan, class of spacetimes,” J. Math. Phys. 51, 022501 (2010) “Dixon’s extended bodies and weak gravitational waves,” doi:10.1063/1.3298684 [arXiv:0912.2779 [math-ph]]. Gen. Rel. Grav. 41, 105 (2009) doi:10.1007/s10714-008- [14] B. Leaute and B. Linet, “Principle of equivalence and 0657-x [arXiv:0910.2843 [gr-qc]]. ,” Int. J. Theor. Phys. 22, 67 (1983). [7] M. Mohseni, “Spinning particles in gravitational doi:10.1007/BF02086898 wave space-time,” Phys. Lett. A 301, 382 (2002) [15] D. Bini, A. Geralico and R. T. Jantzen, “Kerr metric, doi:10.1016/S0375-9601(02)00988-X [gr-qc/0208072]. static observers and Fermi coordinates,” Class. Quant. [8] M. Mohseni, “Worldline deviation and spin- Grav. 22, 4729 (2005) doi:10.1088/0264-9381/22/22/006 ning particles,” Phys. Lett. B 587, 133 (2004) [gr-qc/0510023]. doi:10.1016/j.physletb.2004.02.059 [gr-qc/0403055]. [16] M. Ishii, M. Shibata and Y. Mino, “Black hole tidal [9] P. Fortini and C. Gualdi, “Fermi normal coordi- problem in the Fermi normal coordinates,” Phys. Rev. 12

D 71, 044017 (2005) doi:10.1103/PhysRevD.71.044017 [24] K. P. Tod, F. de Felice and M. Calvani, “Spinning test [gr-qc/0501084]. particles in the field of a black hole,” Nuovo Cim. B 34, [17] D. Bini, A. Geralico, M. L. Ruggiero and A. Tartaglia, 365 (1976). doi:10.1007/BF02728614 “Emission versus Fermi coordinates: Applications to [25] D. Bini, F. de Felice and A. Geralico, “Strains in Gen- relativistic positioning systems,” Class. Quant. Grav. eral Relativity,” Class. Quant. Grav. 23, 7603 (2006) 25, 205011 (2008) doi:10.1088/0264-9381/25/20/205011 doi:10.1088/0264-9381/23/24/028 [arXiv:1408.4283 [gr- [arXiv:0809.0998 [gr-qc]]. qc]]. [18] D. Bini, C. Cherubini, S. Filippi and A. Geralico, “C [26] D. Bini, F. de Felice and A. Geralico, “Strains and axial metric: The equatorial plane and Fermi coordinates,” outflows in the field of a rotating black hole,” Phys. Rev. Class. Quant. Grav. 22, 5157 (2005) doi:10.1088/0264- D 76, 047502 (2007) doi:10.1103/PhysRevD.76.047502 9381/22/23/015 [arXiv:1408.4277 [gr-qc]]. [arXiv:1408.4592 [gr-qc]]. [19] D. Bini, A. Geralico and R. T. Jantzen, “Fermi [27] B. P. Abbott et al. [LIGO Scientific and Virgo coordinates in Schwarzschild spacetime: closed form Collaborations], “Observation of Gravitational Waves expressions,” Gen. Rel. Grav. 43, 1837 (2011) from a Binary Black Hole Merger,” Phys. Rev. Lett. doi:10.1007/s10714-011-1163-0 [arXiv:1408.4947 [gr-qc]]. 116, 061102 (2016) doi:10.1103/PhysRevLett.116.061102 [20] S. Pustelny et al., “Global Network of Optical Magne- [arXiv:1602.03837 [gr-qc]]. tometers for Exotic (GNOME): Novel scheme for [28] B. P. Abbott et al. [LIGO Scientific and Virgo Col- exotic physics searches,” physics.atom-ph/1303.5524 laborations], “GW151226: Observation of Gravita- [21] T.W. Kornack, R. K. Ghosh, and M.V. Romalis, tional Waves from a 22-Solar-Mass Binary Black Hole “Nuclear Spin Gyroscope Based on an Atomic Co- Coalescence,” Phys. Rev. Lett. 116, 241103 (2016) magnetometer,” Phys. Rev. Lett. 95, 230801 (2005) doi:10.1103/PhysRevLett.116.241103 [arXiv:1606.04855 doi:10.1103/PhysRevLett.95.230801 [gr-qc]]. [22] D. F. Jackson Kimball, J. Dudley, Y. Li, S. Thulasi, [29] Bini, D., de Felice, F., “Gyroscopes and gravita- S. Pustelny, D. Budker and M. Zolotorev, “Magnetic tional waves,” Class. Quantum Grav. 17, 4627 (2000) shielding and exotic spin-dependent interactions,” Phys. doi:10.1088/0264-9381/17/22/303 Rev. D 94, 082005 (2016) [arXiv:1606.00696 [physics.ins- [30] F. Sorge, D. Bini and F. de Felice, “Gravitational waves, det]]. gyroscopes and frame dragging,” Class. Quant. Grav. 18, [23] C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravi- 2945 (2001). doi:10.1088/0264-9381/18/15/309 tation,” San Francisco 1973, 1279p