Riemann Normal Coordinates, Fermi Reference System and the Geodesic

Home , Normal coordinates

View metadata, citationRiemann and similar normal papers at coordinates,core.ac.uk Fermi reference system and the geodesic deviationbrought to you by CORE equation provided by CERN Document Server

Alexander I. Nesterov ∗ We obtain the integral formulae for computing the tetrads and metric components in Riemann normal coordinates and Fermi coordinate system of an observer in arbitrary motion. Our approach admits essential enlarging the range of validity of these coordinates. The results obtained are applied to the geodesic deviation in the ﬁeld of a weak plane gravitational wave and the computation of plane-wave metric in Fermi normal coordinates.

KEY WORDS: Riemann coordinates, Fermi reference system

∗Departamento de F´ısica, Universidad de Guadalajara, Guadalajara, Jalisco, M´exico; e-mail: [email protected]

1 I. INTRODUCTION

µ The well known Riemann normal coordinates satisfy the conditions gµν p = ηµν and Γνλ p = 0 at a point p,the origin of the coordinate system. In these cordinates the metric is presented| in powers of the| canonical parameter of the geodesics out from p. In 1922 Fermi [1] showed that it is possible to generalize the Riemann coordinates in such a way that the Christoffell symbols vanish along given any curve in a Riemannian manifold, leaving the metric there rectangular. For a geodesic curve Misner and Manasse [2] introduced a Fermi normal coordinates, which satisfy Fermi conditions along a given geodesic and calculated the first-order expansion of the connection coefficients and the second-order expansion of the metric in powers of proper distance normal to the geodesic. Their construction is a special case of Fermi coordinates defined by Synge [3]. The last ones form a natural coordinate system for a nonrotating accelerated observer. In 1973 Misner, Thorn and Wheeler (MTW) [4] defined a Fermi coordinates for an accelerated and rotating observer and calculated the first-order expansion of the metric. Extending MTW work Mitskievich and Nesterov [5] and later Ni and Zimmerman [6] obtained the second-order expansion of the metric and the first-order expansions of the connection coefficients. Li and Ni [7,8] derived the third-order expansion of the metric and the second-order expansions of the connection coefficients. The size of neighbourhood V , where Riemann and Fermi normal coordinates do not involve singularaties, is determined by conditions

0 1 R u min , | αγδβ | ,  0 0  R 1/2 R | αγδβ | | αγδβ,µ |   0 where u is a canonical parameter along the geodesics from the origin of coordinates. The ﬁrst condition u Rαγδβ 1/2 | − determines the size of V where the curvature has not yet caused geodesics to cross each other. The second condition| determines the domain where the curvature does not change essentially. For instance, for gravitational waves with wavelength λ the Riemann tensor is A exp(ikx)/λ2,whereA is the dimensionless amplitude. This yields ∼

u min λ/√A, λ . { } 18 Generally it is assumed A 10− . This means that the size of V is restricted by u0 λ. So the application of Riemann (or Fermi) coordinates≤ to the modern experiments may be very restrictive since λ is often supposed being in the order of 300 km. Thus for enlarging the range of validity by a factor 1/√A (which is about 109 in our example) it is necesary to take into account derivatives of any order of the Riemann tensor. Some years ago Marzlin [9] has considered Fermi coordinates for weak gravitational ﬁeld gµν = ηµν + hµν , hµν 1 and dervived the metric as a Taylor expansion valid to all orders in the geodesic distance from the world line| of| Fermi observer. Here we develop a new approach to Riemann and Fermi coordinates, which admits enlarging the range of validity 0 1/2 of these coordinates up to u Rαγδβ − . The point is to use the integral formulae. Recently this approach has been used to calculate a geometric| phase| shift for a light beam propagating in the ﬁeld of a weak gravitational wave [10,11]. The paper is organized as follows. In Section 2 we obtain the integral formulae for computing the tetrad and metric components in Riemann normal and Fermi coordinates. In Section 3 we compute the plane-wave metric in Fermi normal coordinates and analyse the geodesic deviation equation for this case. In this paper we use the space-time signature (+, , , ); Greek indices run from 0 to 3, Latin a, b, c from1to2 and i, j, k from 1 to 3. − − −

II. GENERAL RESULTS

A. Riemann normal coordinates

For arbitrary point p0 in some neighbourhood V (p0) there is the unique geodesic γ(u) connecting p0 and p which, α using exponential mapping, we may write as p =expγ(u)(uξ), where ξ = ∂/∂u = ξ e(α), u being the canonical parameter while basis is parallely propagated along γ(u). The Riemann normal coordinates with the origin at the α α 1 point p0, are deﬁned as X = uξ ,whereuξ =expγ−(u)(p). The following basic equations summarize the most important properties of the Riemannian normal coordinates:

2 α β β β β α e(α)(p0 )=(∂/∂X )p0 =(Λα ∂/∂x )p0 , Λα =(∂x /∂X )p0 (1) 0 2 0 Γα =0, Γα XβXγ =0, Γα = Rα Xδ + , (2) βγ βγ βγ −3 (βγ)δ ··· 1 0 1 0 g = η + R XγXδ + R XγXδXµ αβ αβ 3 αγδβ 3! αγδβ,µ 1 0 16 0 0 + 6 R + R Rρ XγXδXλXµ + (3) 5! αγδβ,λ,µ 3 αγδρ λµβ ··· The size of neighbourhood V , where these coordinates do not involve singularaties, is determined by conditions 0 1 R u min , | αγδβ | . (4)  0 0  R 1/2 R | αγδβ | | αγδβ,µ | 0 1/2 The first condition u Rαγδβ − determines the size of V where the curvature has not yet caused geodesics to cross each other. The| second condition| determines the domain where the curvature does not change essentially. For studing Riemann and Fermi normal coordinates one useful technique is the geodesic deviation equation [2]. Further we apply it for our computation. Let us consider a point p and a frame e (p ) in this point. For 0 { (α) 0 } arbitrary geodesic γ(u) out from p0 the equation of geodesic deviation has the following form 2η + R(ξ,η)ξ =0, (5) ∇ξ where ξ = ∂/∂u is tangent vector to γ(u), η is vector of deviation and R is operator of curvature. We suppose that the basis e(α) is parallely propagated along γ(u). In tetrad components the equation (5) takes the form d2η(α) = R(α) ξ(β)ξ(γ)η(δ) (6) du2 (β)(γ)(δ) (α) β (α) where η = η eβ ,etc. β β In Riemann normal coordinates there are natural deviation vectors ηα = uδα∂/∂X satisfying Eq.(5) [2]. Then (5) yields (α) (α) d2e 2 de β + β = e(α)Rλ ξγξδ. (7) du2 u du λ γδβ We solve this equation by succesive approximations [12]. Let us write (7) in the equivalent integral form 1 u τ e(α) = δα + dτ dτ τ e(α)Rλ ξγ ξδ, (8) β β u 0 0 λ γδβ Z0 Z0 (α) α whereweassumeeβ = δβ at the origin of Riemann coordinates. Now we start with e(α) e(α) = δα. β ≈ 0β β (α) To improve this first approximation we feed back e0β into the integral (8) getting 1 u τ e(α) = δα + dτ dτ τ Rα ξγ ξδ, (9) 1β β u 0 0 γδβ Z0 Z0 (α) (α) Repeteang this process by substituting the new eiβ back into Eq. (8), we find our solution eβ as infinite series

(α) ∞ (n)(α) eβ = eβ , (10) n=0 X (0)(α) α eβ = δβ u τ (1)(α) 1 (0)(α) λ γ δ eβ = dτ dτ 0τ 0 eλ R γδβξ ξ , u 0 0 Z u Z τ (2)(α) 1 (1)(α) λ γ δ eβ = dτ dτ 0τ 0 eλ R γδβξ ξ , u 0 0 Z u Z τ (n+1)(α) 1 (n)(α) e = dτ dτ τ e Rλ ξγξδ. β u 0 0 λ γδβ Z0 Z0

3 (n)(α) Noting that the order of magnitude of functions eβ is given in the neighbourhood V by

γ δ n (n)(α) R X X e | αγδβ |max , β ≤ (2n +1)!! we conclude that the radius of convergence of series (10) is determined by

1 u . 0 1/2 ' ( Rαγδβ ) | |max Now using (7-10) one obtains for the components of orthonormal tetrad in Riemann coordinates the following integral formula:

u τ (α) α 1 α γ δ 2 e = δ + dτ dτ 0τ 0R ξ ξ + (R ). (11) β β u γδβ O Z0 Z0 Remark 2.1. Integrating by parts one can write (11) as

u τ u τ τ 0 (α) α α γ δ 2 α γ δ 2 e = δ + dτ dτ 0R ξ ξ dτ dτ 0 dτ 00R ξ ξ + (R ). (12) β β γδβ − u γδβ O Z0 Z0 Z0 Z0 Z0 This form is more convenient for the comparison with the results obtained in Fermi coordinates (see Subsection B). For the covariant derivative of the tetrad we obtain the following useful formula (see Appendix A)

1 u e µ = Rµ ξρτdτ + (R2). (13) ∇∂λ (ν) −u νλρ O Z0 (µ) (ν) With the aid of (11), it is straightforward to derive the metric in Riemann coordinates by using gαβ = ηµν eα eβ . The result is u τ 2 γ δ 2 g = η + dτ dτ 0τ 0R ξ ξ + (R ), (14) αβ αβ u αγδβ O Z0 Z0 and can be rewritten by using Taylor expansion as

1 0 1 0 6 0 g = η + R XγXδ + R XγXδXµ + R XγXδXλXµ + , αβ αβ 3 αγδβ 3! αγδβ,µ 5! αγδβ,λ,µ ··· that agrees with Eq.(3). In fact, we are interested in expressions which should involve (in the right-hand sides) the arbitrary chosen initial non-Riemann coordinates only. Then

u τ α σ γ δ (α) α 1 λ ∂X ∂x ∂x ∂x κ ρ 2 e = δ + dτ dτ 0τ 0R ξ ξ + (R ). (15) β β u γδσ ∂xλ ∂Xβ ∂Xκ ∂Xρ O Z0 Z0 µ µ µ ν The transformation from arbitrary coordinate system to Riemann one takes the form x = x (p0)+Λν X + (Γ) (or xµ = xµ(p )+Λµ ξν u + (Γ) ) and the equations (14), (15) can be written as O 0 ν O u τ (α) α 1 λ 1α γ δ σ κ ρ e = δ + dτ dτ 0τ 0R Λ− Λ Λ Λ ξ ξ + (R), (16) β β γδσ λ κ ρ β O u 0 0 Z u Z τ 2 λ γ δ σ κ ρ g = η + dτ dτ 0τ 0R Λ Λ Λ Λ ξ ξ + (R), (17) αβ αβ u λγδσ α κ ρ β O Z0 Z0 λ λ µ µ ν where R γδβ = R γδσ x (p0)+Λν ξ τ 0 while the integration is performed.

4 B. Fermi coordinates

Let us consider the proper system of reference e(α) of a single observer which moves along a worldline γ with an acceleration and rotation [4] De (α) = Ω e , (18) ds · (α) where Ωµν = Gµτ ν Gν τ µ + Eµνγδτ ω − γ δ where Eµνγδ is the axial tensor of Levi-Civita and τ µ is the unit tangent vector to the line γ; Gµ being the four- δ µ µ µ i acceleration of observer, ω the four-rotation of the tetrad. Fermi coordinates are deﬁned as X = δ0 s + δi ξ u (see e.g. [2,4]). Here s is proper time along the observer’s world line γ, ξ is a unit vector on γ tangent to the spacelike geodesic starting from γ, ortogonal to it and parametrized by proper length u.Infact,ξ describes the direction in which such a space-like geodesic goes, and it possesses only spatial components diﬀerent from zero (the temporal coordinate is directed along γ). In Fermi coordinates, the covariant components of the corresponding orthonormal (ν) tetrad eµ parallel transported along the spacelike geodesic, are represented as expansions (see Appendix B)

1 0 1 0 1 0 e(µ) = δµ +Ωµ Xi + Rµ XiXj + R Ωµ XiXjXk + Rµ XiXjXk + , 0 0 i 2 ij0 6 0ij0 k 6 ij0,k ··· 1 0 1 0 1 0 e(µ) = δµ + Rµ XiXj + R0 Ωµ XiXjXk + Rµ XiXjXk + , (19) p p 6 ijp 12 ijp k 12 ijp,k ··· 0 where quantities of the type of Q are taken on γ. The expansion up to 3d order of the metric and up to 1st order of the connection coeﬃcients (Christoﬀel symbols) is given by [5–8]

0 0 p α k p k p α i j k g00 =1+2Ω0pX +ΩαkΩ pX X + R0kp0 X X + Rαij0 Ω kX X X 1 0 1 0 + R Ω XiXjXk + R XiXjXk + , 3 0ij0 0k 3 0ij0,k ··· 2 0 1 0 1 0 g =Ω Xp + R XkXl + R Ω XkXlXm + R Ω XkXlXm 0i ip 3 ikl0 4 ikl0 0m 6 0kl0 im 1 0 1 0 + R Ωm XkXlXn + R XkXlXm + , 6 iklm n 4 ikl0,m ··· 1 0 1 0 0 g = η + R XkXp + (R Ω XkXlXm+ R Ω XkXlXm) ij ij 3 ikpj 12 ikl0 jm jkl0 im 1 0 + R XkXlXm + , 6 iklj,m ··· 0 Γµ =Ωµ +(δ0Ω˙ µ + δ0Ωµ Ωλ Ω0 Ωµ + Rµ )Xi + , 0ν ν ν i ν λ i − ν i νi0 ··· 2 0 Γµ = Rµ Xk + . (20) ij −3 (ij)k ··· For applications it is convenient to write the expansion for the metric tensor in the following form

0 0 g =(1+GX)2 (ω X)2+ R XkXp+ Rl ωpXiXjXk 00 − × 0kp0 ij0 lkp 4 0 1 0 + R G XiXjXk + R XiXjXk + , 3 0ij0 k 3 0ij0,k ··· 2 0 1 0 1 0 g = (ω X) + R XkXp + R G XkXlXm + R ωnXkXlXm 0i − × i 3 0kpi 4 ikl0 m 6 0kl0 imn 1 0 1 0 + Rm ωpXkXlXn + R XkXlXm + , 6 kli mnp 4 ikl0,m ··· 1 0 1 0 0 g = η + R XkXp + R ωnXkXlXm+ R ωnXkXlXm ij ij 3 ikpj 12 ikl0 jmn jkl0 imn 1 0 + R XkXlXm + , 6 iklj,m ···

5 j k j where the expression (ω X)i denotes ijkω X and GX := GiX . The radius of convergence× of series (20) is determined by (see− Introduction)

∂ ∂ η = δµ ,η= uδµ . 0 0 ∂Xµ i i ∂Xµ The straightforward calculation leads to

1 u τ e(α) = δα + dτ dτ τ e(α)Rλ ξiξj, (21) p p u 0 0 λ ijp Z0 Z0 0(α) d e u τ e(α) = δα + 0 u + dτ dτ e(α)Rλ ξiξj. (22) 0 0 du 0 λ ij0 Z0 Z0 where

0(α) 0(α) d e e = δα, µ = δ0 Ωα ξi. µ µ du µ i The iterative scheme (10) yields

(α) ∞ (n)(α) (0)(α) α 0 α i eβ = eβ , eβ = δβ + δβΩ iξ u, (23) n=0 X u τ (n+1)(α) 1 (n)(α) e = dτ dτ τ e Rλ ξγ ξδ, (24) β u 0 0 λ γδβ Z0 Z0 and the radius of convergence of series (23) is determined by conditions

u τ (α) α 1 α 0 α i λ j k 2 ep = δp + dτ dτ 0τ 0(δλ + δλΩ iξ τ 0)R jkpξ ξ + (R ), (25) u 0 0 O Z Zu τ (α) α α i α 0 α i λ j k 2 e = δ +Ω ξ u + dτ dτ 0(δ + δ Ω ξ τ 0)R ξ ξ + (R ). (26) 0 0 i λ λ i jk0 O Z0 Z0 Integrating by parts we combine (25), (26) into one expression

u τ (α) α 0 α i α 0 α i λ j k eβ = δβ + δβΩ iξ u + dτ dτ 0(δλ + δλΩ iξ τ 0)R jkβξ ξ Z0 Z0 u τ τ 0 2 p α 0 α i λ j k 2 dτ dτ 0 dτ 00δ (δ + δ Ω ξ τ 00)R ξ ξ + (R ), (27) −u β λ λ i jkp O Z0 Z0 Z0 and the componets of metric tensor are given by

0 i 0 i 0 0 γ i j 2 gαβ = ηαβ + δβΩαiξ u + δαΩβiξ u + δαδβΩγiΩj ξ ξ u u τ µ 0 µ i 0 i λ j k +(δα + δαΩ iξ u) dτ dτ 0(ηµλ + δλΩµiξ τ 0)R jkβξ ξ Z0 Z0

6 u τ τ 0 2 p 0 i λ j k dτ dτ 0 dτ 00δ (η + δ Ω ξ τ 00)R ξ ξ − u µ βλ λ βi jkp Z0 Z0 Z0 # u τ µ 0 µ i 0 i λ j k +(δβ + δβΩ iξ u) dτ dτ 0(ηµλ + δλΩµiξ τ 0)R jkαξ ξ Z0 Z0 u τ τ 0 2 p 0 i λ j k 2 dτ dτ 0 dτ 00δ (η + δ Ω ξ τ 00)R ξ ξ + (R ), (28) − u µ αλ λ αi jkp O Z0 Z0 Z0 # This integral formula agrees with the series (20). The transformation from arbitrary coordinate system xµ to Fermi one takes the form { } xµ = xµ(X0)+Λµ(X0)Xi + , i ··· and in the coordinates xµ one can write Eq.(27) as { } u τ (α) α 0 γ 1α i 1α 0 1α γ i λ δ κ j k eβ = δβ + δβΩ iΛ−γ ξ u + dτ dτ 0(Λ−γ +Λλ Λ−γ Ω iξ τ 0)R δκβΛj Λk ξ ξ Z0 Z0 u τ τ 0 2 p 1α 0 1α γ i λ δ κ j k dτ dτ 0 dτ 00δ (Λ− +Λ Λ− Ω ξ τ 00)R Λ Λ ξ ξ + (R), −u β γ λ γ i δκp j k O Z0 Z0 Z0 α µ where it is used the same notation R βγδ for the curvature tensor in the coordinate system x and

α α β 0 β 0 i R µνλ = R µνλ x (X )+Λi (X )ξ τ 00 while the integration is performed.

III. WEAK PLANE-WAVE METRIC IN FERMI NORMAL COORDINATES

We compute the plane-wave metric in Fermi normal coordinates for a geodesic observer in the weak gravitational- wave ﬁeld, whose metric tensor is usually written in synchronous coordinates,

2 µ ν a b ds = ηµν dx dx + habdx dx , where a and b run from 1 to 2 while

hab = hab(t z),h22 = h11 = h+,h12 = h . − − × Using the deﬁnition of the Riemann tensor 1 R = (h + h h h ), µνλσ 2 νλ,µ,σ µσ,ν lambda − µλ,ν,σ − νσ,µ,λ one ﬁnds that in the linearized theory non-zero components of Riemann tensor are 1 R = R = R = h¨ , 3ab3 0ab0 − 3ab0 2 ab R = R ,R = R , µ22ν − µ11ν µ12ν µ21ν dot being derivative with respect to t. The next step is to choose an orthonormal frame along the world line γ of observer. Taking into account the µ µ transformation to the Fermi normal coordinates x = X + (hab) and that the timelike base vector must be the tangent ∂/∂t, one obtains O

e = ∂/∂T = ∂/∂t, e = ∂/∂X = ∂/∂x, (0)|γ (1)|γ e = ∂/∂Y = ∂/∂y, e = ∂/∂Z = ∂/∂z, (2)|γ (3)|γ where X0 = T, X1 = X, X2 = Y, X3 = Z are setted. Using (28) we compute the metric tensor in the Fermi normal coordinates. The result is

7 a b g00 =1+HabX X , (29) 2 gab = ηab + FabZ , (30) 1 g = (H + F ) XbZ, (31) 0a 2 ab ab 1 g = (H + F ) XaXb, (32) 03 −2 ab ab g = 1+F XaXb, (33) 33 − ab g = F XbZ, (34) a3 − ab where 1 H = h (T Z) h (T )+Zh˙ (T ) , ab Z2 ab − − ab ab 2 Z F = H H (T,Y)Y 2dY. ab ab − Z3 ab Z0 ˙ ¨ It is easy to see that H0 = F , H (T,0) = 3F (T,0) = (1/2)h (T )and ab − ab ab ab ab 1 F = ((H F )Z)0, ab 2 ab − ab here and below dot being derivative with respect to T and prime derivative with respect to Z. The nonvanishing components of Christoﬀel symbols are found to be

a 1 2 a b 1 b Γ = H0 Z , Γ = H X (H˙ H0 )X Z, (35) 0b 2 ab 00 ab − 2 ab − ab a b b a 1 2 Γ =2F X + F 0 X Z, Γ = F Z F 0 Z (36) 33 ab ab 3b − ab − 2 ab a b b 0 3 b Γ = H X H0 ZX , Γ =Γ = H X , (37) 03 − ab − ab 0a 0a ab 0 3 1 2 3 b Γ =Γ =2F Z + F 0 Z , Γ = F X , (38) ab ab ab 2 ab a3 − ab 0 3 1 a b 0 1 a b Γ =Γ = H˙ X X , Γ = F 0 X X , (39) 00 00 2 ab 33 −2 ab 0 3 1 a b 3 1 a b Γ =Γ = H0 X X , Γ = F 0 X X , (40) 03 03 2 ab 33 −2 ab and the computation of the non-zero components of Riemann tensors in Fermi normal coordinates yields

R3ab3 = R0ab0 = R3ab0 1 − = F¨ Z2 +2H (H˙ + F˙ )Z . (41) 2{ ab ab − ab ab } This leads to 1 R = R = R = h¨ (T Z). (42) 3ab3 0ab0 − 3ab0 2 ab − Thus in Fermi coordinates the curvature is the same function (1/2)h¨ , but depending on (T Z). ab − Plane monochromaric gravitational wave. The only nonzero components hab = Aab sin[k(t z)] of wave amplitudes are − h = h = A sin[k(t z)], 11 − 22 + − h12 = h21 = A sin[k(t z)]. × − Computation yields kZ sin(kZ) cos(kZ) 1 H = k2A cos(kT) − + sin(kT) − , (43) ab ab (kZ)2 (kZ)2 kZ sin(kZ) cos(kZ) 1+(kZ)2/2 F = cos(kT) − 2 − ab (kZ)2 − (kZ)3 cos(kZ) 1 kZ sin(kZ) +sin(kT) − +2 − (44) (kZ)2 (kZ)3

8 This leads to kZ sin(kZ) cos(kZ) 1 g =1+k2A XaXb cos(kT) − + sin(kT) − , (45) 00 ab (kZ)2 (kZ)2 sin(kZ) kZ cos(kZ) 1+(kZ)2/2 g = k2A XaXb cos(kT) − + − 03 ab (kZ)2 (kZ)3 sin(kZ) kZ cos(kZ) 1 +sin(kT) − − , (46) (kZ)2 − (kZ)2 sin(kZ) kZ cos(kZ) 1+(kZ)2/2 g = k2A XbZ cos(kT) − + − 0a − ab (kZ)2 (kZ)3 sin(kZ) kZ cos(kZ) 1 +sin(kT) − − , (47) (kZ)2 − (kZ)2 kZ sin(kZ) cos(kZ) 1+(kZ)2/2 g = η + k2A Z2 cos(kT) − 2 − ab ab ab (kZ)2 − (kZ)3 cos(kZ) 1 kZ sin(kZ) +sin(kT) − +2 − (48) (kZ)2 (kZ)3 kZ sin(kZ) cos(kZ) 1+(kZ)2/2 g = k2A XbZ cos(kT) − 2 − a3 − ab (kZ)2 − (kZ)3 cos(kZ) 1 kZ sin(kZ) +sin(kT) − +2 − (49) (kZ)2 (kZ)3 kZ sin(kZ) cos(kZ) 1+(kZ)2/2 g = 1+k2A XaXb cos(kT) − 2 − 33 − ab (kZ)2 − (kZ)3 cos(kZ) 1 kZ sin(kZ) +sin(kT) − +2 − , (50) (kZ)2 (kZ)3 whereweset

b 1 2 1 2 AabX = A+(δaX δaY )+A (δaY + δaX), × a b − 2 2 AabX X = A+(X Y )+2A XY. − × Our results agree with the metric obtained in [13]. Geodesic deviation. Let us consider the geodesic deviation of two neighbouring geodesics in the gravitational ﬁeld of a plane weak wave. The separation vector η satisﬁes a geodesic deviation equation (6), which reads

d2η(α) = R(α) ζ(β)ζ(γ)η(δ), (51) dλ2 (β)(γ)(δ) λ being a canonical parameter along the geodesic and ζ a tangent vector to it. The solution of this equation is given by

λ τ 0 0 (α) α α β γ δ 2 η = η + dτ dτ 0R ζ ζ η , + (R ) (52) 0 βγδ 0 O Z0 Z0 α α where we set η0 = η (0) and integration being performed along the basic geodesic line canonically parametrized by λ The separation magnitude is found to be

λ τ 0 0 β γ δ α l = l dτ dτ 0R ζ ζ η η , (53) 0 − αβγδ 0 0 Z0 Z0 Now let us consider the geodesic deviation of two neighbouring particles A and B assuming that the the origin of Fermi reference frame attached to A0s geodesic and for T = 0 the particles are in the plane Z = 0. In this case 0 α α β β the components of the separation vector are nothing but the coordinates of particle B: η = X , ζ = δ0 and the canonical parameter λ beingapropertimeτ along the geodesic line of the observer. Then (53) reads

9 λ τ 1 a b l = l dτ dτ 0R X X . (54) 0 − l 0ab0 0 Z0 Z0 This yields

1 a b l = l0 hab(T )X X (55) − l0 and for the plane monochromatic gravitational wave one obtains the well known result [4]

l0 l = l0 (A+ cos 2ϕ + A sin 2ϕ) sin(kT). (56) − 2 ×

ACKNOWLEDGEMENTS

It is a pleasure to acknowledge numerous conversations with N. V. Mitskievich. This work was supported by CONACyT grant 1626 P-E.

APPENDIX A

Here we shall obtain the formula (Eq.(13) in the text)

1 u 0 e µ = τdτRµ ξδ + (R). ∇∂λ (ν) −u νλδ O Z0 Let us start with the Taylor expansion for tetrad in the neighbourhood V (p0 )

0 µ 0 µ 0 µ d e 1 d2 e 1 d3 e e µ = δµ + (ν) u + (ν) u2 + (ν) u3 + , (57) (ν) ν du 2! du2 3! du3 ··· u being the canonical parameter along geodesics starting in the point p0. Using in the Riemann normal coordinates the equation of parallel propagation de µ (ν) +Γµ e σξλ =0, (58) du σλ (ν) we rewrite (57) as 1 0 1 0 e µ = δµ Γµ XσXγ Γµ XσXγXδ + + (R2). (59) (ν) ν − 2! νσ,γ − 3! νσ,γ,δ ··· O The expansion of the connection coeﬃcients is given by 0 1 0 Γµ =Γµ Xi + Γµ XiXl + . (60) νλ νλ,i 2! νλ,i,l ··· Applying Eq. (59), (60) we obtain the following series for the covariant derivatives of tetrad 0 0 1 0 0 e µ =Γµ Xσ Γµ Xσ + (Γµ Γµ )XσXγ ∇∂λ (ν) νλ,σ − ν(λ,σ) 2! νλ,σ,γ − ν(λ,σ,γ) 1 0 0 + (Γµ Γµ )XσXγXδ + (R2). (61) 3! νλ,σ,γ,δ − ν(λ,σ,γ,δ) O Now using the relations 1 0 0 0 Rµ Xσ =(Γµ Γµ )Xσ, 2 νσλ νλ,σ − ν(λ,σ) 2 0 0 0 Rµ XσXγ =(Γµ Γµ )XσXγ, 3 νσλ,γ νλ,σ,γ − ν(λ,σ,γ) 3 0 0 0 Rµ XσXγXδ =(Γµ Γµ )XσXγXδ, etc 4 νσλ,γδ νλ,σ,γ,δ − ν(λ,σ,γ,δ)

10 one can write the expansion (61) as

1 0 2 0 3 0 e µ = Rµ Xσ + Rµ XσXγ + Rµ XσXγXδ + ∇∂λ (ν) 2 νσλ 3! νσλ,γ 4! νσλ,γ ··· n 0 µ γ1 γ2 γn 2 + R νγ λ,γ , ,γ X X X + + (R ). (n +1)! 1 2 ··· n ··· ··· O It is convenient to present this series in the form

0 ∞ dn (Rµ ξσ) un+1 e µ = νσλ + (R2) (62) ∇∂λ (ν) dun (n +2)(n!) O n=0 X Straightforward calculation yields the following integral representation of Eq.(61):

1 u e µ = Rµ ξστdτ + (R2). (63) ∇∂λ (ν) −u νλσ O Z0 Integrating by parts it found to be

u u τ µ µ σ 1 µ σ 2 e = R ξ dτ + dτ dτ 0R ξ + (R ). (64) ∇∂λ (ν) − νλσ u νλσ O Z0 Z0 Z0

APPENDIX B

Here we shall obtain the series (Eq.(19) in the text)

1 0 1 0 1 0 e(µ) = δµ +Ωµ Xi + Rµ XiXj + R Ωµ XiXjXk + Rµ XiXjXk + , 0 0 i 2 ij0 6 0ij0 k 6 ij0,k ··· 1 0 1 0 1 0 e(µ) = δµ + Rµ XiXj + R0 Ωµ XiXjXk + Rµ XiXjXk + . (65) p p 6 ijp 12 ijp k 12 ijp,k ··· We start with the Taylor expansion for tetrad in a world tube surrounding the world line γ of Fermi observer:

(ν) 2 (ν) 3 (ν) de (0) 1 d e (0) 1 d e (0) e(ν) = δµ + µ u + µ u2 + µ u3 + , (66) µ ν du 2! du2 3! du3 ··· u being the canonical parameter along spacelike geodesics ortogonal to γ. Using in Fermi coordinates the equation of parallel propagation

de(ν) µ Γσ e(ν)ξi =0, (67) du − µi σ we rewrite Eq.(66) as

0 0 0 0 (ν) ν ν i 1 ν λ ν i l eµ = δµ+ Γ µi X + Γ µi,l + Γ µiΓ λl X X 2! !

0 0 0 0 0 0 0 0 1 ν λ ν λ ν λ σ ν i l p + Γ µi,l,p + Γ µi,lΓ λp + Γ µiΓ λl,p + Γ µiΓ λlΓ σp X X X + . (68) 3! ! · For computing of the connection coeﬃcients and its derivatives one needs the spacelike geodesic deviation equation

D2ηµ Rµ ξiξjην =0. du2 − ijν It is convenient to write it as

11 d2ηµ dΓµ dην + νi ξiην +2Γµ ξi +Γλ Γµ ξiξjην = Rµ ξiξjην . (69) du2 du νi du νi λj ijν µ µ µ µ Using deviation vectors η(i) = uδi and η0 = δ0 , we write (69) as follows:

dΓµ pi ξiu +2Γµ ξi +Γλ Γµ ξiξju = Rµ ξiξju, (70) du pi pi λj ijp dΓµ 0i ξi +Γλ Γµ ξiξj = Rµ ξiξj. (71) du 0i λj ij0

0 λ 0 λ From Eq.(18) one obtains Γ µν = δµΩ ν . These conditions together with (70), (71) yield

0 0 d Γµ 1 0 Γλ = δ0 Ωλ , ij ξj = Rµ ξj ξk, (72) µν µ ν du 3 jki 0 d Γµ 0 0i ξi = Rµ Ω0 Ωµ ξiξk, (73) du ik0 − i k 0 2 µ 0 0 d Γ oi 0 ξi = Rµ R Ωµ Rµ Ω du2 ij0,k− 0ij0 k− ij0 0k 1 0 Rµ Ωp +2Ωµ Ω Ω ξiξj ξk, (74) −3 ijp k i 0j 0k 0 2 µ 0 d Γ ij 1 1 0 ξj = Rµ R Ωµ ξjξkξp. (75) du2 2 jki,p − 6 0jki p Finally, applying (72 - 75) and (68) we compute

1 0 e(µ) = δµ +Ωµ Xi + Rµ XiXj 0 0 i 2 ij0 1 0 1 0 + R Ωµ XiXjXk + Rµ XiXjXk + , 6 0ij0 k 6 ij0,k ··· 1 0 1 0 e(µ) = δµ + Rµ XiXj + R0 Ωµ XiXjXk p p 6 ijp 12 ijp k 1 0 + Rµ XiXjXk + . 12 ijp,k ···

12 [1] Fermi E 1922 Atti Acad. Naz. Lincei Rend. Cl. Sci. Fiz. Mat. Nat. 31 21, 51 [2] Manasse F K and Misner C W 1963 J. Math. Phys. 4, 735 [3] Synge J L 1960 Relativity: The General Theory (North-Holland, Amsterdam) [4] Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (Freeman, New York) [5] Mitskievich N V and Nesterov A I 1976 Izv. Vuzov. (Fisica).No.9 92 [6] Ni W -T and Zimmermann M 1978 Phys. Rev. D 17 1473 [7] Li W -Q and Ni W -T 1979 J. Math. Phys. 20 1473 [8] Li W -Q and Ni W -T 1979 J. Math. Phys. 20 1925 [9] Marzlin K -P 1994 Phys. Rev. D 50 888 [10] Mitskievich N V and Nesterov A I 1995 Gen. Relat. Grav. 27 361 [11] Mitskievich N V and Nesterov A I 1996 Int. J. Theor. Phys. 35 2677 [12] Arfken J 1968 Mathematical Methods for Physicists (Academic Press, New York) [13] Faraoni V 1992 Nuovo Cim. B 43 631