Geodesic deviation in a nonlinear gravitational wave spacetime
aa r X i v : . [ g r- q c ] J un Geodesic deviation in a nonlinear gravitationalwave spacetime
Hristu Culetu,Ovidius University, Dept.of Physics and Electronics,B-dul Mamaia 124, 900527 Constanta, Romania,e-mail : [email protected] 21, 2016
Abstract
The tidal effects generated by a nonlinear gravitational wave are inves-tigated in double-null v - u coordinates, as an exact solution of Einstein’sfield equations. The components ξ v and ξ u of the separation vector be-have as in flat space but the transversal components ξ x and ξ y dependnonlinearly on v through the Bessel and Neumann functions, far from thenull surface v = 0. We show that the same results are obtained by meansof the tetrad formalism. In General Relativity (GR), the equation of motion for a test particle is given bythe geodesic equation which describe the worldline, i.e., the path through spaceand time of the particle. A geodesic is a curve of extremal length between anytwo points: its length remains unchanged to first order in small changes in thecurve [1]. In flat space the freely falling particles maintain their separation butthey do not in curved space. That leads to the geodesic deviation equation andshow mathematically how the tidal forces of a gravitational field (which causetrajectories of neighboring particles to diverge) can be represented by curvatureof spacetime. The affine parameter along a geodesic can be interpreted as propertime and is, thus, related to the reading of a clock that is transported with it[2]. To study the geodesic deviation phenomenon we have to find the variationof the separation vector between two neighboring geodesics, one of them beingthe reference-geodesic. The Fermi-normal coordinates are the best candidatefor to investigate the second order effect of geodesic deviation. Manasse andMisner [3] computed the Schwarzschild (KS) metric in Fermi normal coordinatessurrounding a radial geodesic, choosing a suitable orthonormal frame along thegeodesic. They showed that the components of the Riemann tensor have the1implest form in those coordinates and all of them have the right dimension.Thus, normal coordinates provide a nice way by which a freely falling observercan report local experiments.Crispino et al. [4] analyzed the tidal forces produced in the Reissner-Nordstrom spacetime, finding some mismatches w.r.t. the KS geometry, de-pending of the mass-to-charge ratio. The radial components of the separationvector start decreasing inside the event horizon, unlike the KS case. Fleury [5]states that tidal forces experienced by ultrarelativistic particles in the directionof their motion are much smaller than those experienced orthogonally to theirmotion. Philipp and Puetzfeld [6] reconsidered the geodesic deviation in New-tonian gravitation and then determined relativistic effects within the theory ofGR. Studying the first order deviation equation for orbits around the Earth,they uncovered artificial effects that are due to the linearized framework. Theso-called Shirokov effect, rather than being a new feature of GR, they identifiedit as being the relict of the approximated description of a well-known perigeeprecession. Aldrovandi et al. [7] assert that a physical gravitational wave can-not be represented by a solution to a linear wave equation. In their view, theeffects produced by the 2nd order solution on free particles consist of nonlinearoscillations along the direction of propagation.Our aim in this paper is to investigate the tidal effect produced on a testparticle by a gravitational wave which is an exact solution of Einstein’s fieldequations. We use the line-element from [8] in double-null coordinates, for towork with more simple form of the components of the Riemann tensor, com-pared to the Cartesian coordinates. Section 2 deals with the calculation of thecomponents of the separation vector ξ a between two nearby geodesics , far awayfrom the null surface t − z = 0, for a wave traveling along the z - axis. Theonly components with a nonlinear variation w.r.t. the affine parameter alongthe geodesic are the transversal components ξ x and ξ y , expressed in terms ofthe Neumann and modified Bessel functions. A more convenient tetrad basedformalism is applied in Section 3, for to obtain th deviation vector. Finally, inSection 4 we analize the implications of our results. Let us consider two freely falling particles which are very close to each other.Their worldlines are infinitesimally separated timelike geodesics [5]. We usethe proper time τ to parameterize them and introduce the separation vector ξ a ( τ ) = x a ( τ ) − y a ( τ ), which is orthogonal to the geodesics, i.e. u a ξ a = 0,where u a (with u a u a = −
1) denotes the velocity 4-vector of one of the particles,tangent to the trajectory (here the index a runs from 0 to 3). It is clear that ξ a represents their separation vector in the rest frame. The Jacobi equation D ξ a dτ + R abcd u b ξ c u d = 0 (2.1)2ives the second covariant derivative of the deviation vector in terms of theRiemann curvature tensor, up to the linear order in the deviation and its firstderivative (along the reference curve [6]). From (2.1) we infer that ξ a will havea nonlinear time dependence only if the geometry is curved, namely R abcd = 0.The capital D in (2.1) signifies the covariant derivative and R abcd = ∂ c Γ abd − ... .We shall work in the local inertial frame at the point of the first geodesic where ξ a originates. In this coordinate system the coordinate distances are properdistances [1]. What is more, in that frame the covariant derivative acquires asimple form: in the local inertial frame the Christoffel symbols all vanish at thatpoint, so the second derivative is just an ordinary one, w.r.t. the proper time.Therefore, Eq. (2.1) may now be written as [1] d ξ a dτ + R abcd u b ξ c u d = 0 , (2.2)where u a = dx a /dτ is the velocity 4 - vector of the two particles.We write down our proposed line-element corresponding to a nonlinear planegravitational wave (GW) in double-null coordinates [8], propagating along thez - coordinate ds = − dvdu + e − b √ v b dx + e b √ v b dy , (2.3)where v = t − z is the retarded null coordinate, u = t + z is the advanced nullcoordinate and b is a constant length, taken here as the Planck length ( t, x, y, z denote the Cartesian coordinates). We take v to play the role of time and thecoordinates are x = v, x = u, x = x, x = y . The geodesics in the spacetime(2.3) were investigated in [8]. We found there that ˙ v = dv/dτ = 1, so that onecould take v = τ representing the proper time along the geodesics. We willspecialize to the case ˙ x = u x = 0 , ˙ y = u y = 0, whence u a = (1 , , , Keeping in mind that u a ξ a = 0, one finds that ξ a = ( ξ v , − ξ v , ξ x , ξ y ), ξ a ξ a = 1.The nonzero components of the curvature tensor for (2.3) are given by [8] R uxvx = b bv − v − b ) √ v + b ( v + b ) e − b √ v b R uyvy = b bv + 2(2 v − b ) √ v + b ( v + b ) e b √ v b R xvvx = b bv − v − b ) √ v + b ( v + b ) R yvvy = b bv + 2(2 v − b ) √ v + b ( v + b ) , (2.4)(the others are obtained by symmetry operations). By means of (2.4) it is aneasy task to find that the components ξ v and ξ u have a linear dependence on v . Hence d ξ v dv = 0 , d ξ u dv = 0 , (2.5) The fact that ˙ u = du/dv = 1 comes from Eq. (4.6) from [8], with ˙ x = ˙ y = 0.
3s far as the other two components of ξ a are concerned, it is too complicatemathematically to work directly with the exact expression (2.4) of the curvaturetensor, for to find the functional dependence of ξ x ( v ) , ξ y ( v ). Therefore, werestrict ourselves to a more simple approach and consider that v >> b , i.e. theregion far from the surface v = b , which is very close to the null surface v = 0when b is taken of the order of the Planck length. That restriction leads to thefollowing equations for the other two components of ξ a d ξ x dv + bv ξ x = 0 , (2.6)and d ξ y dv − bv ξ y = 0 . (2.7)Let us deal firstly with Eq. (2.6). It is a linear, second order differential equa-tion. Its solution may be expressed in terms of the Bessel functions , oneobtains, using Mathematica ξ x ( v ) = C r vb J r bv ! + C r vb Y r bv ! , (2.8)where C , C are constants of integration, J is the Bessel function of the firstkind and index one and argument 2 p b/v and Y is the Neumann function ofindex one. [9]. The first term gives us r vb J r bv ! = 1 − b v + b v − ... , v >> b (2.9)But Y is divergent when its argument tends to zero ( v >> b ) so that ξ x ( v )becomes infinite when the retarded time v → ∞ . Now the two freely fallingparticles move away each other indefinitely.Let us look now for the solution of (2.7). That means to replace b with − b in (2.8). Therefore, we have ξ y ( v ) = C r vb I r bv ! + C r vb K r bv ! , (2.10)where C , C are constant of integration, I is the modified Bessel function ofindex one and K is the Macdonald function of index one. The first term of ther.h.s. of (2.10) can be written as a series expansion r vb I r bv ! = 1 + b v + b v − ... , v >> b (2.11) Thanks go to Viviana Ene and Luminita Cosma for helpful suggestions on the Besselfunctions. K diverges when b/v →
0. Hence, | ξ y ( v ) | increases indefinitely in this limit, having a similar behaviour with ξ x ( v ). Thatconclusion is also valid in flat space because, if we neglect the second term in(2.6) and (2.7), the ”no gravitation” situation is recovered.We turn now our attention to the opposite case, namely v << b . Notingthat in Eqs. (2.4) the above approximation leads to R xvvx = − R yvvy = 12 b , (2.12)if we neglect v /b w.r.t. unity. One obtains d ξ x dv − b ξ x = 0 , (2.13)and d ξ y dv + 12 b ξ y = 0 . (2.14)The solution for ξ x ( v ) is given by ξ x ( v ) = C e − v √ b + C e v √ b , (2.15)with C and C - constants of integration. It is clear that, for v << b , theexponentials become unity and ξ x tends to a constant. In other words, there isno geodesic deviation along the x - direction. The solution for ξ y ( v ) appears as ξ y ( v ) = C sin v √ b + C cos v √ b , (2.16)with C and C - constants of integration. We have again v << b , whence sin ( v/b √ ≈ , cos ( v/b √ ≈ Let us now compare the results obtained in Section 2 with those given by thetetrad formalism, applied for the metric (2.3). We use in this section the indices a, b = 0 , , , µ, ν = 0 , , , η ab e aµ e bν = g µν , η ab e µa e νb = g µν , (3.1)where η ab = ( − , , ,
1) is the Minkowski metric and e aµ are the vierbeins. Theonly nonzero components of g µν are g uv = g vu = − , g xx = exp ( b/ √ v + b ) , g yy = exp ( − b/ √ v + b ). At each point we may define a local Lorentz frame using theorthonormal basis vectors e a which are not derived from any coordinate frame[10], with the scalar product e a e b = η ab and e a = e µa e µ ( e µ is the coordinatebasis vector). 5he local Lorentz frame at each point defines a family of ideal observerswhose worldlines are the integral curves of the unit vector field e . The spacialunit vectors e i ( i = 1 , ,
3) define the orthogonal space coordinates axes of alocal laboratory frame, valid close to observer’s trajectory [10]. In a freelyfalling frame with u µ = ( ˙ v, ˙ u, ˙ x, ˙ y ), we will choose ˙ x = ˙ y = 0, so that ˙ u = ˙ v in this case (see [8]) and, therefore, u µ = ( ˙ v, ˙ v, , e µ = (cid:18) ˙ v ,
12 ˙ v , , (cid:19) , e µ = (cid:18) − ˙ v ,