Uniform circular motion in General Relativity: Existence and extendibility of the trajectories

Uniform circular motion in General Relativity: Existence and extendibility of the trajectories Daniel de la Fuente?, Alfonso Romero†and Pedro J. Torres?∗ ?Departamento de Matem´atica Aplicada, Universidad de Granada, 18071 Granada, Spain E-mail: [email protected] E-mail: [email protected] †Departamento de Geometr´ıa y Topolog´ıa, Universidad de Granada, 18071 Granada, Spain E-mail: [email protected] Abstract The notion of uniform circular motion in a general spacetime is introduced as a particular case of a planar motion. The initial value problem of the corresponding differential equation is analysed in detail. Geometrically, an observer which obeys a uniform circular motion is characterized as a Lorentzian helix. The completeness of its inextensible trajectories is studied in Generalized Robertson-Walker spacetimes and in a relevant family of pp-wave spacetimes. The results may be physically interpreted saying that, under reasonable assumptions, a uniformly circular observer lives forever in these spacetimes, providing the absence of the singularities defined by these timelike curves. 2010 MSC: 83C10, 83C75, 53C50. Keywords: Uniform circular motion; Fermi-Walker covariant derivative; Completeness of inextensible trajectories; Plane Wave spacetime; General Relativity. 1 Introduction Uniform circular motion has been widely studied in Special Relativity (see, for instance, [8]). Relevant physical phenomena and paradoxes, usually related with the Thomas precession, have motivated its study, and its interest is still present (see, for instance [20] for an intuitive introduction). The usual approach consists in setting a family of inertial observers, one of them is considered ‘the center’. Thus, an observer is said that describes a uniform circular motion with respect to the fixed ‘center’ if the trajectory measured by that family of inertial observers is circular and its angular velocity is constant for them. Others approaches have been done using suitable Frenet equations [10], [11]. ∗The first and third authors are partially supported by Spanish MINECO and ERDF project MTM201452232-P. The second author by Spanish MINECO and ERDF project MTM2013-47828-C2-1-P. Specific motions which may be seen as very particular cases of uniform circular motions have been previously considered in relevant relativistic models with some rotational symmetry, as Schwarzschild, Reissner-Nordstr¨om and Kerr spacetimes [14, Ch. 25]. Each of these spacetimes has a remarkable family of observers with a similar role that the inertial observers in Minkowski spacetime. An observer that is placed on the surface of the star or the black hole (according the case) and at rest with respect to it, is considered the center of the circular trajectories. The uniform circular observer describes a circle with respect to the special fixed family of observers and the measured angular velocity is constant. The analysis of this kind of motions has a recognized physical and technological interest because they correspond to the orbits of some artificial satellites, planets or stars (see, for instance, [12]). We are interested here in introducing a definition of uniform circular motion in a general spacetime, involving only the physical observable quantities measured by the proper observer. In order to determine the inherent kinematic state of an observer we will focus on its proper acceleration. In other words, we will give an ‘intrinsic’ definition of uniform circular motion, without considering an external family of distinguished observers for which the motion describes has a circular trajectory or not. Note, in addition, that the existence of such a family of observers is not guaranteed in a generic spacetime. Of course, our definition will agree with the standard notion in previously quoted cases. Intuitively, an observer is able to detect its proper acceleration by using a giroscope or, more generally, an accelerometer. An accelerometer may be thought as a sphere in whose center there is a small ball which is supported from elastic radii to the sphere surface. If a free falling observer carries such an accelerometer, then it will notice that the small ball remains just at the center. Whereas the ball will be displaced if the observer obeys an accelerated motion. For instance, a uniform accelerated motion may be recognised from a constant displacement of the small ball [4]. This idea has the advantage that may be used independently if the spacetime is relativistic or not. Thus, it would be natural that if an observer checks that the small ball describes a plane uniform rotation, then it believes that it obeys a uniform circular motion. The first challenge we have to face is to state a notion of ‘planar’ motion in an arbitraty spacetime. Classically, a motion is said to be planar when the projection of its space-time trajectory on the absolute Euclidean space is contained in a plane. Equivalently, its proper acceleration is contained in the same plane at any instant. This alternative notion may be extended to any spacetime. Obviously, as it happens in Classical Mechanics, a uniform circular motion should be a planar motion. The subtle problem in Relativity consists in to give sense to the sentence the same plane forever. This is well done making use of the Fermi-Walker connection of each observer (Definition 1). Our procedure lies in the realm of modern Lorentzian geometry and, as far as we know, is new in our approach (compare with [11]). In order to do that, recall that a particle of mass m > 0 in a spacetime (M, h,i) is a curve γ:I−→ M, such that its velocity γ0satisfies hγ0, γ 0i=−m2and it is future pointing. A particle with m= 1 is called an observer (see, for instance, [17]). The covariant derivative of γ0,Dγ 0 dt , is its (proper) acceleration, which may be seen as a mathematical translation of the values measured by the accelerometer. Assume the particle γobeys a planar motion. In order to arrive to a suitable notion of uniform circular motion, we will require that the modulus of its acceleration remains 2 unchanged, i.e., Dγ 0 dt 2= constant. On the other hand, we need a connection along γthat permits to compare spatial directions at different instants of the life of γ, i.e., we need such a connection to compute how the proper acceleration of γchanges. In General Relativity this connection is known as the Fermi-Walker connection of γ(see Section 2 for more details). Thus, using the corresponding Fermi-Walker covariant derivative b D dt, if a particle obeys a uniform circular (UC) motion, it is also necessary, b D dt Dγ 0 dt 2= constant, i.e., the modulus of the change of its acceleration should be constant (Section 2). We will arrive to the main notion of Definition 3 collecting suitable the three previous conditions. On the other hand, UC motions appear naturally in any spacetime, for instance they arise from a dynamical point of view. Considering an electrically charged particle with nonzero rest mass (γ(t), m, q) in presence of an electromagnetic field F, then the dynamics of the particle is completely described by the well-known Lorentz force equation (see, for instance, [17, Def. 3.8.1]), mDγ 0 dt =qe F(γ0), where e Fis the (1,1)-tensor field metrically equivalent to the closed 2-form F. Now, let us consider in Minkowski spacetime L4the particular electromagnetic field, F= 2B0dx ∧dy, where B0>0 is a constant and (t, x, y, z) are the standard coordinates of L4. The family of inertial observers ∂/∂t measures a uniform magnetic field with modulus B0and pointing towards ∂/∂z (and zero electric field) for F. Now, the particle γobeys a UC motion, and its trajectory is expressed as [17, Prop. 3.8.2], γ(τ) = p+p1 + R2w2mτ, R cos(wmτ +ϑ), R sin(wmτ +ϑ),0, with w=qB0 m∈R,p∈L4,R > 0 and ϑ∈R, whenever the initial velocity of particle with respect to the family of inertial observers lies in the plane xy [17, p. 88]. Next, the paper is organized as follows. In Section 2 several mathematical preliminaries are introduced to arrive to the notion of UC observer (Definition 3). Section 3 is devoted to expose how a UC observer can be seen as a Lorentzian helix in a general spacetime (Equation 3.1). The corresponding differential system and the associated initial value problem are analysed in detail in Section 4. Later, we use this result to characterize each UC observer as a solution of a fourth-order differential equation (Proposition 19). A representation of any UC observer is given in Section 5. Section 6 is dedicated to characterize geometrically UC observers as the projections on the spacetime of the integral curves of a certain vector field G defined on a suitable fiber bundle over the spacetime (Lemma 6.1). Using G, the completeness of inextensible UC motions is analysed in the search of geometric assumptions which assure 3 that inextensible UC observers do not disappear in a finite proper time. This technique is applied to spacetime which admit a certain timelike symmetry. In fact, we obtain that any UC observer in a Generalized Robertson-Walker spacetime can be extended whenever its worldline lies in a compact subset of the spacetime (Theorem 6.4). Finally, in Section 7, we prove the completeness of inextensible UC observers in an important class of pp-wave spacetimes. In this case we make use of a different and more analytical approach (Theorem 7.3). In both cases, the absence of singularities of this kind is found. 2 The notion of uniform circular motion Consider a spacetime M, i.e., an n(≥2)−dimensional manifold endowed with a time orientable Lorentzian metric h,iwhich we agree to have signature (−,+, ..., +), and with a fixed time orientation. Points of Mas called events and an observer in Mis a (smooth) curve γ:I−→ M,Ian open interval of R(0 ∈I), such that hγ0(t), γ 0(t)i=−1 and γ0(t) lies in the future time cone in Tγ(t)Mfor all t∈I. In brief, γ0(t) is a future pointing unit timelike vector for any proper time tof γ. At each event γ(t), the tangent space Tγ(t)Mlinearly splits as Tγ(t)M=Tt⊕Rt, where Tt:= Span{γ0(t)}and Rt:= T⊥ t. Clearly, Ttis a negative definite line in Tγ(t)M and Rtis a spacelike hyperplane of Tγ(t)M. For n= 4, the 3-dimensional subspace Rt may be interpreted as the instantaneous physical space observed by γat the instant tof its clock. Consequently, vectors in Rtrepresent observable quantities for γat t. Note that the acceleration vector field Dγ 0 dt satisfies Dγ 0 dt (t)∈Rt, for any t. In fact, it is observed by γ whereas the velocity vector field γ0is not observable by γ. In order γcompares v1∈Rt1with v2∈Rt2, for t1< t2and |v1|=|v2|, it could use the parallel transport defined by the Levi-Civita covariant derivative along γ, Pγ t1,t2:Tγ(t1)M−→ Tγ(t2)M. However, this linear isometry does not satisfy Pγ t1,t2(Rt1) = Rt2, in general. This is a serious inconvenience. To avoid it, the observer can use a more subtle mathematical tool. In fact, recall that γpossesses a (private) connection, called its Fermi-Walker connection, which is defined as follows [17, p. 51]. Consider the Levi-Civita connection ∇associated to the Lorentzian metric of spacetime. It induces a connection along each γ:I−→ Msuch that the corresponding covariant derivative is the usual covariant derivative of vector fields Y∈X(γ), namely, DY dt (t) = ∇γ0(t)e Y, where e Yis a local extension of Yon an open neighbourhood of each event experimented by γ(t) in M. For Y∈X(γ), denote by YT t, Y R tthe orthogonal projections of Yton Ttand Rt, respectively, i.e., YT t=−hYt, γ 0(t)iγ0(t) and YR t=Yt−YT t. Clearly, we have YT, Y R∈X(γ). According to [17, Prop. 2.2.1] the Fermi-Walker connection of γis the unique connection b ∇along γwhich satisfies b ∇XY=∇XYTT+∇XYRR, 4 for any X∈X(I) and Y∈X(γ), being ∇the induced connection on γfrom the Levi-Civita connection of M. Now denote by b D/dt the covariant derivative corresponding to b ∇. Then, it is not difficult to prove the following relationship with the Levi-Civita covariant derivative [17, Prop. 2.2.2], b DY dt =DY dt +hγ0, Y iDγ 0 dt −Dγ 0 dt , Y γ0,(1) for any Y∈X(γ). Clearly, we have b D dt =D dt if and only if γis a geodesic, i.e., the observer is free falling. In addition, the following Leibnitz type rule holds, d dthX , Y i=Db DX dt , Y E+DX , b DY dt E,(2) for any X, Y ∈X(γ). Associated to the Fermi-Walker covariant derivative along γthere exists a parallel transport b Pγ t1,t2:Tγ(t1)M−→ Tγ(t2)M, which is a linear isometry and satisfies b Pγ t1,t2(Rt1) = Rt2. Therefore, given v1∈Rt1and v2∈Rt2, with t1< t2and |v1|=|v2|, the observer γmay consider b Pγ t1,t2(v1) instead of v1, with the advantage that b Pγ t1,t2(v1) may be compared with v2 (see also [14, Sec. 6.5]). Now we introduce the crucial concept of ‘planar motion’ to make precise when an observer considers that it is moving along a plane. Intuitively, an observer will say that its motion is planar when the small ball of its accelerometer moves along a constant plane. In the mathematical translation of this intuitive idea, the main difficulty lies in what is the meaning of a ‘constant plane’ relative to the observer. For this purpose we will use the Fermi-Walker connection exposed above. Definition 1 An observer γ:I−→ Mobeys a planar motion if for some t0∈I, there exists an observable plane Πt0⊂Rt0⊂Tγ(t0)M, such that b Pγ t,t0Dγ 0 dt (t)∈Πt0(3) for any t∈I. Intuitively, Dγ 0 dt corresponds to the displacement of the small ball of the accelerometer, and b D dt Dγ 0 dt may be seen as the velocity of the ball. Whenever both vectors are linearly independent, both directions define the observable 2-plane Πt0. 5 As direct consequence of the definition, using the equality b D dt Dγ 0 dt (t0) = lim ε→0 1 εb Pγ t0+ε,t0Dγ 0 dt (t0+ε)−Dγ 0 dt (t0), gives that the vector b D dt Dγ 0 dt (t0) is also in Πt0. Indeed, if γis not in an unchanged direction motion at a neighbourhood of the instant t0, in the terminology of [5], then the plane Πt0is generated by the proper acceleration of the observer and the variation which it measures, i.e., Πt0= span (Dγ dt (t0),b D dt Dγ 0 dt (t0)). In this case, we may define the following family of 2-planes along γ Πt:= span (b Pγ t0,t Dγ dt (t0),b Pγ t0,t b D dt Dγ dt (t0)!)⊂Rt0.(4) Observe that this family of planes is Fermi-Walker parallel in the sense of the following definition. Definition 2 Given an observer γ:I−→ Min the spacetime M, a family of planes along γ,{Πt}t∈I, is said to be Fermi-Walker parallel if for any t1, t2∈Iand for any vector v∈Πt1, the following relation holds b Pγ t1,t2(v)∈Πt2. In addition, the previous family of planes (4) satisfies the following property. Lemma 2.1 For any t, t1∈I, we have b Pγ t,t1Dγ 0 dt (t)∈Πt1. Proof. Taking the inverse mapping of b Pγ t,t0in (3), we have that there exist a, b ∈Rsuch that Dγ dt (t) = ab Pγ t0,t Dγ dt (t0)+bb Pγ t0,t b D dt Dγ dt (t0)!. Now, the desired relation follows by taking b Pγ t,t1in both members of the previous equality.  It should be pointing out that the family {Πt}t∈Isatisfies the previous property, but it is not unique in general (a generically planar motion may be a free falling motion from some instant). However, if the observer γis not an unchanged direction observer [5], i.e., if nDγ dt (t),b D dt Dγ dt (t)oare linearly independent for any t∈I, then the only family of 2-planes satisfying Lemma 2.1 is {Πt}t∈I. 6 Now we will introduce a uniform circular (UC) motion as a very particular case of planar motion. Intuitively, a UC observer will see that the small ball of its accelerometer is rotating with constant angular velocity, describing a circular trajectory. Hence, the velocity of the ball for the observer, b D dt Dγ 0 dt , will have a constant modulus. Motivated by these intuitive ideas, we are in a position to give an accurately definition. Definition 3 An observer γ:I−→ Mwhich satisfies a planar motion is said to obey a UC motion if Dγ 0 dt 2=a2and b D dt Dγ 0 dt 2=a2w2,(5) where the constants a, w satisfy a, w > 0 and a<w. Here ais de modulus of the acceleration, and wcorresponds to the angular velocity that the observer perceives. Therefore, motivated from the classical relation between the radius R, the angular velocity wand the centripetal acceleration aon a circular motion, a=w2R, a UC observer will measure a uniform rotation with frequency w 2πand ‘radius’ equal to R:= a/w2. We emphasize that this quantity Rdoes not represent a real observable distance in general. It is only the radius of the trajectory which the UC observer assumes, using the classical intuition, from the evolution of its acceleration. The assumption a<wis imposed to exclude other different kind of motions, as we will discuss it at the end of Section 5. Remark 2.2 Note that if a= 0 is permited, we would recover the definition of a free falling observer. Moreover, when a > 0, if w= 0 is permited, we would obtain the definition of a uniform accelerated observer [4]. Observe that, in this case, the trajectory measured by the observer may be thought with infinite ‘radius’, i.e., the observer obeys a rectilinear motion [5]. Naturally, in order to determine a UC observer trajectory, it is necessary to know the initial observable 2-plane, the initial spin sense and the initial values of the position, 4velocity and proper acceleration. In an n(≥3)-dimensional spacetime, the initial 2-plane can be determined by means of n−3 observable directions u4,··· , un∈γ0(0)⊥, orthogonal to the initial acceleration Dγ 0 dt (0). So, the vector b D dt Dγ 0 dt (0) will point towards the unique observable direction which is orthogonal to Dγ 0 dt (0) and u4,··· , un. From equation (5), the modulus of the vector b D dt Dγ 0 dt (0) is also known, and it is equal to aw. However, the initial spin sense is needed to determine the sense of that vector. The initial plane Π0is given by Π0= span (Dγ 0 dt (0),b D dt Dγ 0 dt (0)). 7 We consider the following unit vectors, related with the initial values of the problem, u1=γ0(0), u2=1 a Dγ 0 dt (0), u3=1 aw b D dt Dγ 0 dt (0), and consider the following Fermi-Walker parallel vector fields along γ, e4(t),··· , en(t) satisfying the initial conditions ei(0) = ui, for each 4 ≤i≤n. Now, the family of FermiWalker parallel planes (4), corresponding to the UC observer γis given by Πt= span{b Pγ 0,t(u2),b Pγ 0,t(u3)}=span{b Pγ 0,t(u1), e4(t),··· , en(t)}⊥⊂Rt. Remark 2.3 Note that, in the physically relevant case n= 4, we have e4(t) = 1 a2w Dγ 0 dt ×b D dt Dγ 0 dt , where ×denotes the natural cross product defined in Rt. From the previous discussion we can state the following initial value equations for a UC observer with ‘frequency’ w 2πand ‘radius’ R=a w2, which is expressed as hγ0, γ 0i=−1,(6) Dγ 0 dt 2=a2,(7) b D dt Dγ 0 dt 2=a2w2,(8) b Dei dt = 0 for 4 ≤i≤n, (9) Dγ 0 dt , ei= 0 for 4 ≤i≤n, (10) under the initial conditions γ(0) = p, γ 0(0) = u1,Dγ 0 dt (0) = a u2,b D dt Dγ 0 dt (0) = aw u3,(11) ei(0) = uifor 4 ≤i≤n, where, pis an event in the n-dimensional spacetime M. Note that (10) automatically implies that *b D dt Dγ 0 dt (t), ei(t)+= 0 for 4 ≤i≤n, t ∈I. 8 The local existence and uniqueness of this initial problem is not yet guaranteed because it is not possible write it in the normal form (therefore the classical Picard-Lindeloff Theorem can not be applied). On the other hand, the initial condition are imposed to the third derivative, in spite of the system is of third order. However, in Section 4 we will prove that system (6)-(11) has a unique inextensible solution. 3 UC motion as a Lorentzian helix In this section we analyse the UC motion from a more geometric viewpoint. First, we proceed to find the Frenet equations of each UC observer. Let γ:I−→ Mbe a UC observer with angular velocity wand radius R=a w2. We define the following three vector fields along γ, which are orthonormal from equations (2) and (5), e1(t) = γ0(t), e2(t) = 1 a Dγ 0 dt (t), e3(t) = 1 aw b D dt Dγ 0 dt (t). Let {u4,··· , un}be n−3 orthonormal vectors in Tγ(0)M, such that, {e1(0), e2(0), e3(0), u4, . . . , un} is an orthogonal basis of Tγ(0)M. Consider the Fermi-Walker parallel vector fields along γ starting at ui, ei(t) = b Pγ 0,t(ui),for 4 ≤i≤n. Since a UC motion is a planar motion, the 2-plane Πtis orthogonal to the subspace generated by {b Pγ 0,t(ui)}4≤i≤n. So vector fields {ej(t)}1≤j≤nare orthonormal at every instant t∈I. Now, we are in a position to obtain the Frenet equations. A direct computation give us De1 dt =a e2. On the other hand, De2 dt =1 a"b D dt Dγ 0 dt +a2γ0#=a e1+w e3. Taking into account that γis a UC observer, we obtain DDe3 dt , e1E=1 awDb D dt b D dt Dγ 0 dt !, γ 0E= 0, and DDe3 dt , e2E=1 a2w"d dthb D dt Dγ 0 dt ,Dγ 0 dt i−b D dt Dγ 0 dt 2#=−w. 9 Lemma 6.1 There exists a unique vector field Gon Va,w n,3(M)such that the curves t7−→ γ(t), γ 0(t),Dγ 0 dt (t), b D dt Dγ 0 dt (t)are the integral curves of G, for any solution γof equation (19). Once defined G, we will look for assumptions which assert its completeness. Recall that an integral curve αof a vector field defined on some interval [0, b), b < +∞, can be extended to b(as an integral curve) if and only if there exists a sequence {tm}m, tm%b, such that {α(tm)}mconverges (see for instance [15, Lemma 1.56]). The following technical result directly follows from this fact and Lemma 6.1. Lemma 6.2 Let γ: [0, b)−→ Mbe a solution of equation (19) with 0<b<∞. The curve γcan be extended to bas a solution of (19) if and only if there exists a sequence nγ(tm), γ 0(tm),Dγ 0 dt (tm), b D dt Dγ 0 dt (tm)omwhich is convergent in Va,w n,3(M)when tm→b. Although we know that |γ0(t)|2=−1, this is not enough to apply Lemma 6.2 even in the geometrically relevant case of Mcompact. The reason is similar to the possible geodesic incompleteness of a compact Lorentzian manifold (see for instance [15, Ex. 7.16],[18]). However, it is relevant that if a compact Lorentzian manifold admits a timelike conformal vector field, then it must be geodesically complete [18]. Therefore, from a geometric viewpoint, it is natural to assume the existence of such infinitesimal conformal symmetry to deal with the extendibility of the solutions of (19)-(20). Recall that a vector field Kon Mis called conformal if the Lie derivative of the metric with respect to Ksatisfies LKh,i= 2hh,i,(26) for some h∈C∞(M), equivalently, the local flows of Kare conformal maps. In particular, if (26) holds with h= 0, Kis called Killing vector field. On the other hand, if a vector field Ksatisfies ∇XK=hX for all X∈X(M),(27) then clearly we get (26). Moreover, for the 1-form Kbmetrically equivalent to K, we have dKb(X, Y ) = h∇XK, Y i−h∇YK, Xi= 0, for all X, Y ∈X(M), i.e., Kbis closed. We will call to Kwhich satisfies (27) a conformal and closed vector field. A Lorentzian manifold which admits a timelike conformal and closed vector field is locally a Generalized Robertson-Walker spacetime [3], [19]. The following result, inspired from [1, Lemma 9], will be decisive to assure that the image of the curve in Va,w n,3(M), associated to a UC observer γ, is contained in a compact subset. Lemma 6.3 Let Mbe a spacetime and let Qbe a unit timelike vector field. If γ:I−→ Mis a solution of (19)-(20) such that γ(I)lies in a compact subset of Mand hQ, γ 0iis bounded on I, then the image of t7−→ γ(t), γ 0(t),Dγ 0 dt (t), b D dt Dσ0 dt (t)is contained in a compact subset of Va,w n,3(M). 16 Proof. Consider the 1-form Qbmetrically equivalent to Qand the auxiliary Riemannian metric gR:= h,i+ 2 Qb⊗Qb. We have, gR(γ0, γ 0) = hγ0, γ 0i+ 2 hQ, γ 0i2, which, by hypothesis, is bounded on I. Hence, there exists a constant c > 0 such that γ(I), γ 0(I),Dγ 0 dt (I),b D dtDσ0 dt (I)⊂C, C:= (p, u1, au2, awu3)∈Va,w n,3(M) : p∈C1, gR(u1, u1)≤c, where C1is a compact set on Msuch that γ(I)⊂C1. Hence, Cis a compact in Va,w n,3(M).  Now, we are in a position to state the following completeness result (compare with [1, Th. 1] and [2, Th. 1]), Theorem 6.4 Let Mbe a spacetime which admits a timelike conformal and closed vector field K. If InfMp−hK, Ki>0then, each solution γ:I−→ Mof (19)-(20) such that γ(I) lies in a compact subset of Mcan be extended. Proof. Let I= [0, b), 0 <b<+∞, be the domain of a solution γof equation (19)-(20). Multiplying γ0by the vector field Kand making use of the representation (25, we obtain, hK , γ 0i=w √w2−a2hK , Li+a √w2−a2hcos pw2−a2thK , Mi+ sin pw2−a2thK , Nii. On the other hand, taking into account that Lis Levi-Civita parallel and (27), d dthK , Li=DDK dt , LE=hhγ0, Li(h◦γ) = −w(h◦γ) √w2−a2. Analogously, d dthK , Mi=a(h◦γ) √w2−a2cos pw2−a2t, and d dthK , Ni=a(h◦γ) √w2−a2sin pw2−a2t. Using now that γ(I) is contained in a compact of M, the function h◦γis bounded on I. Therefore, since Iis assumed bounded, the functions hK , Li,hK , Miand hK , Niare bounded on Iand, as consequence, there exists a constant c1>0 such that |hK, γ 0i| < c1.(28) Now, if we put Q:= K |K|, where |K|2=−hK, Ki>0, then Qis a unit timelike vector field such that, by (28), |hQ, γ 0i| ≤ m c1on I, where m= SupM|K|−1<∞. The proof ends making use of Lemmas 6.2 and 6.3.  Remark 6.5 Note that the previous theorem implies the following result of mathematical interest: Let Mbe a compact spacetime which admits a timelike conformal and closed vector field K. Then, each inextensible solution of (19)-(20) must be complete. Note that the Lorentzian universal covering of Minherits the completeness of inextensible UC observers form the same fact on M. 17 7 Completeness of UC trajectories in a Plane Wave spacetime In this section, we study the completeness of the inextensible UC trajectories with positive prescribed acceleration, but working in a more analytical way. Let us consider a spacetime Madmitting a global chart x0, x1,··· , xn. In these coordinates, we can write Equation (26) as follows γ0 k(t) = w √w2−a2Lk(t) + a √w2−a2hcos √w2−a2tMk(t) + sin √w2−a2tNk(t)i, L0 k(t) = Pi,j −Γk ij √w2−a2hwLiLj+acos √w2−a2tLiMj+asin √w2−a2tLiNji, M0 k(t) =Pi,j −Γk ij √w2−a2hwMiLj+acos √w2−a2tMiMj+asin √w2−a2tMiNji,(29) N0 k(t) =Pi,j −Γk ij √w2−a2hwNiLj+acos √w2−a2tNiMj+asin √w2−a2tNiNji, γk(0) = pk, Lk(0) = 1 √w2−a2(wu1k+au3k), Mk(0) = −1 √w2−a2(au1k+wu3k), Nk(0) = u2k. Here, u1k,u2kand u3kare the coordinates of the vectors u1,u2and u3respectively, and satisfy u1iu1jgij(0) = −1, u2iu2jgij(0) = u2iu2jgij(0) = 1, u1iu2jgij(0) = u1iu3jgij(0) = u2iu3jgij(0) = 0, being gij(0) the coefficients of the metric at the point γ(0) in these coordinates. Moreover, all the Christoffel symbols are evaluated on γ, i.e., Γk ij(t) := Γk ijγ0(t),··· , γn(t). A (four dimensional) Plane Wave is a spacetime (M, g) which admits a Brinkmann coordinate system [9], i.e., a coordinate system in which the metric has the form g=H(u, x, y)du2+ 2dudv +dx2+dy2, where H(u, x, y) is a quadratic function in the coordinates x and y with coefficients depending on u, that is, H(u, x, y) = A(u)x2+B(u)y2+C(u)xy +D(u)x+E(u)y+F(u).(30) From now on, it is assumed that Madmits a global Brinkmann coordinate system, which we will denote by (u, v, x, y). We also identify Mwith R4. In these coordinates, the Christoffel symbols of gare easily computed as follows Γ1 i,j = 0 for all i, j = 1,...,4 (31) Γ2 1,1=1 2 ∂H ∂u ,Γ2 1,3= Γ2 3,1=1 2 ∂H ∂x ,Γ2 1,4= Γ2 4,1=1 2 ∂H ∂y (32) Γ3 1,1=−1 2 ∂H ∂x ,Γ4 1,1=−1 2 ∂H ∂y ,(33) and the remaining symbols are zero. 18 Now, let us consider a UC observer γ:I→R4satisfying the initial conditions γ(0) = p, γ 0(0) = u1,Dγ 0 dt (0) = a u2,b D dt Dγ 0 dt (0) = aw u3, with p, u1, u2, u3∈R4. Our final objective is to prove that such trajectory is extensible to the whole real line, i.e., that the maximal interval of definition of γis I=R. By Proposition 5.1, we can write γ0(t) = w √w2−a2L(t) + a √w2−a2hcos pw2−a2tM(t) + sin pw2−a2tN(t)i where L, M, N :I→R4are solutions of system (29) with initial conditions L(0) = 1 √w2−a2(wu1+au3), M(0) = −1 √w2−a2(au1+wu3), N(0) = u2. Writing in coordinates L= (L1, L2, L3, L4), M = (M1, M2, M3, M4), N = (N1, N2, N3, N4), we have a simple but important fact. Lemma 7.1 The first components of L, M, N are constant with value L1=1 √w2−a2(wu11 +au31), M1=−1 √w2−a2(au11 +wu31), N1=u21 Proof. It follows trivially from (31) and (29) that L0 1=M0 1=N0 1= 0, then L1, M1, N1are constants and equal to the respective initial condition.  A direct consequence of the latter lemma is that γ0 1(t) = w √w2−a2L1+a √w2−a2hcos pw2−a2tM1+ sin pw2−a2tN1i and we have an explicit expression for γ1(t) as γ1(t) = p1+w √w2−a2L1+ (34) +a √w2−a2hM1Rt 0cos √w2−a2tdt +N1Rt 0sin √w2−a2tdti. Lemma 7.2 As solutions of system (29), the functions L3, M3, N3, L4, M4, N4are extensible to the whole real line. Proof. The equations from (29) for k= 3,4 are L0 k(t) = −Γk 11 √w2−a2hwL2 1+acos pw2−a2tL1M1+asin pw2−a2tL1N1i, M0 k(t) = −Γk 11 √w2−a2hwM1L1+acos pw2−a2tM2 1+asin pw2−a2tM1N1i, N0 k(t) = −Γk 11 √w2−a2hwN1L1+acos pw2−a2tN1M1+asin pw2−a2tN2 1i. 19 The expressions between brackets are trigonometric functions. We define f(t) = 1 √w2−a2hwL2 1+acos √w2−a2tL1M1+asin √w2−a2tL1N1i, g(t) = 1 √w2−a2hwM1L1+acos √w2−a2tM2 1+asin √w2−a2tM1N1i, h(t) = 1 √w2−a2hwN1L1+acos √w2−a2tN1M1+asin √w2−a2tN2 1i. Then, the previous system is written as L0 k(t) = −f(t)Γk 11(γ(t)), M0 k(t) = −g(t)Γk 11(γ(t)), N0 k(t) = −h(t)Γk 11(γ(t)). (35) The key point is to analyse the particular form of the Christoffel symbols Γk 11(γ(t)), k= 3,4. Considering that His defined by (30), we have Γ3 11(γ) = −1 2 ∂H ∂x (γ(t)) = 2A(γ1)γ3+C(γ1)γ4+D(γ1), and Γ4 11(γ) = −∂H ∂y (γ(t)) = 2B(γ1)γ4+C(γ1)γ3+D(γ1), where γ1(t) is explicitly given by (34). Since γk(t) = pk+ Rt 0hw √w2−a2Lk(s) + a √w2−a2hcos √w2−a2tMk(s) + sin √w2−a2tNk(s)iids, then system (35) (with k= 3,4) can be seen as an integro-differential system of six equations. To pass to a standard system of differential equations, we define the new variables Lk=w √w2−a2Zt 0 Lk(s)ds, Mk=a √w2−a2Zt 0 cos pw2−a2sMk(s)ds, Nk=a √w2−a2Zt 0 sin pw2−a2sNk(s)ds, for k= 3,4. With the new variables, L0 k(t) = w √w2−a2Lk(t), M0 k(t) = a √w2−a2cos √w2−a2tMk(t), N0 k(t) = a √w2−a2sin √w2−a2tNk(t), (36) for k= 3,4. Besides, γk(t) = Lk+Mk+Nk+pk(k= 3,4). 20 Remember that γ1(t) is known explicitly, see (34). Therefore, attending to the expression of the Christoffel symbols computed before, equations (35) are linear on the variables Lk,Mk,Nk. Summing up, equations (35)-(36) compose a linear system of 12 equations on the involved variables Lk, Mk, Nk,Lk,Mk,Nk(k= 3,4). The basic theory of linear systems states that any solution of a linear is globally defined on the whole real line, closing the proof.  We end the manuscript with the following result. Theorem 7.3 Every UC inextensible trajectory in a Plane Wave spacetime admitting a global Brinkmann chart is complete. Proof. Up to now, we have proved that Lk, Mk, Nkwith k= 1,3,4 are defined on the whole R. To finish the proof, it remains to prove the completeness of L2(t), M2(t), N2(t). The equations (29) for L2is L0 2(t) = X i,j −Γ2 ij √w2−a2hwLiLj+acos pw2−a2tLiMj+asin pw2−a2tLiNji, but note that Γ2 ij = 0 if i= 2 or j= 2, and moreover Hdoes not depend on the second variable. This implies that the right-hand side part of the latter equation depends only on functions Lk(t), Mk(t), Nk(t) (k=1,3,4), which we have proved that are globally defined, but not on L2, M2, N2. Thus, L0 2(t) is defined for every t, and a simple integration leads to the conclusion. An analogous argument serves for M2(t), N2(t).  Acknowledgments The authors would like to thank Miguel S´anchez for his useful comments and suggestions to improve the manuscript. References [1] A.M. Candela, A. Romero and M. S´anchez, Completeness of the trajectories of particles coupled to a general force field, Arch. Rational Mech. 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