The notion of a uniformly accelerated rectilinear motion of an observer in a general spacetime is analysed in detail. From a geometric viewpoint, a uniformly accelerated observer may be seen as a Lorentzian circle. Finally, we find geometric assumptions to ensure that an inextensible uniformly accelerated
rectilinear observer does not disappear in a finite proper time

Carriazo Rubio, Alfonso (Coordinador)

Fuente Benito, Daniel de la

Fuente Benito, Daniel de la (Coordinador)

Márquez García, María del Carmen (Coordinador)

Romero Sarabia, Alfonso (Coordinador)

idUS. Depósito de Investigación Universidad de Sevilla

XI Encuentro Andaluz de GeometríaIMUS (Universidad de Sevilla), 15 de mayo de 2015, págs. 2934The uniformly accelerated motion in General Re-lativity from a geometric point of viewDaniel de la FuenteAbstract. The notion of a uniformly accelerated rectilinear motion of an ob-server in a general spacetime is analysed in detail. From a geometric viewpoint,a uniformly accelerated observer may be seen as a Lorentzian circle. Finally, weﬁnd geometric assumptions to ensure that an inextensible uniformly acceleratedrectilinear observer does not disappear in a ﬁnite proper time.1. IntroductionThe deﬁnition of uniformly accelerated motion in General Relativity hasbeen discussed many times over the last 50 years. In the pioneering work byRindler [4], it was partially motivated by some aspects of intergalactic rockettravel by use of the special relativistic formulas for hyperbolic motion.The relation between uniformly accelerated motion and Lorentzian circlesin Lorentz-Minkowski spacetime was used by Rindler in [4] to deﬁne what henamed hyperbolic motion in General Relativity, extending uniformly acceleratedmotion in Lorentz-Minkowski spacetime.First, we present an approach to the study of uniformly accelerated motionin General Relativity in the realm of modern Lorentzian geometry. In orderto do that, let γ : I −→ M be an observer in the spacetime M . Its (proper)acceleration is given by the covariant derivative of its velocity γ ′, i.e., Dγ′dt .Intuitively, the particle obeys a uniformly accelerated motion if its accelerationremains to be unchanged. Mathematically, we need a connection along γ whichpermits to compare spatial directions at diﬀerent instants of the life of γ. InGeneral Relativity this connection is known as the Fermi-Walker connectionof γ. Thus, using the Fermi-Walker covariant derivative D̂dt , we will say that aparticle obeys a uniformly accelerated (UA) motion if,D̂dt(Dγ ′dt)= 0. (1)The family of UA observers in the Lorentz-Minkowski spacetime Ln wasDaniel de la Fuente, delafuente@ugr.esDepartamento de Matemática Aplicada, Universidad de Granada30 D. de la Fuentecompletely determined long time ago [4]. It consists of timelike geodesics andLorentzian circles. For instance, in L2, using the usual coordinates (x, t), theUA observer γ(τ) =(x(τ), t(τ))throughout (0, 0) with zero velocity relative tocertain family of inertial observers (the integral curves of vector ﬁeld ∂t) andproper acceleration a is given by,x(τ) = c2a[cosh(aτc)− 1] , t(τ) = casinh(aτc),where τ ∈ R is the proper time of γ, and c is the light speed in vacuum.In this paper we expose how UA observers can be seen as Lorentzian cir-cles in any general spacetime. After that, we characterize UA observers as theprojection on the spacetime of the integral curves of a vector ﬁeld deﬁned on acertain ﬁber bundle over the spacetime. Using this vector ﬁeld, the completenessof inextensible UA motions is analysed in the search of geometric assumptionswhich assure that inextensible UA observers do not disappear in a ﬁnite pro-per time (in particular, the absence of timelike singularities). In particular, anyinextensible UA observer is complete under the assumption of compactness ofthe spacetime and that it admits a conformal and closed timelike vector ﬁeld.2. Uniformly accelerated observer as a LorentziancircleA spacetime is a time orientable n(≥ 2)− dimensional Lorentzian manifold(M, 〈 , 〉), endowed with a ﬁxed time orientation. As usual, we will consider anobserver in M as a (smooth) curve γ : I −→ M , I an open interval of R, suchthat 〈γ ′(t), γ ′(t)〉 = −1 and γ ′(t) is future pointing for any proper time t of γ.At each event γ(t) the tangent space Tγ(t)M splits asTγ(t)M = Tt ⊕Rt,where Tt = Span{γ ′(t)} and Rt = T⊥t . Rt is interpreted as the instantaneousphysical space observed by γ at t. Clearly, the observer γ is able to comparespatial directions at t, but in order to compare v1 ∈ Rt1 with v2 ∈ Rt2 , theobserver γ must use a suitable connection. We proceed to describe it.For each Y ∈ X(γ) put Y Tt , Y Rt the orthogonal projections of Yt on Tt andRt, respectively, i.e., Y Tt = −〈Yt, γ ′(t)〉 γ ′(t) and Y Rt = Yt − Y Tt . In this way, if∇ denotes the Levi-Civita connection, we have, [5, Prop. 2.2.1],Along this paper the signature of a Lorentzian metric is (−,+, ...,+).Uniformly accelerated motion in GR 31Proposition 2.1. There exists a unique connection ∇̂ along γ such that∇̂XY =(∇XY T )T + (∇XY R)R ,for any X ∈ X(I) and Y ∈ X(γ).This connection ∇̂ is called the Fermi-Walker connection of γ. It shows thesuggestive property that if Y ∈ X(γ) satisﬁes Y = Y R (i.e., Yt may be observedby γ at any t) then(∇̂XY)t∈ Rt for any t.Denote by D̂/dt the covariant derivative corresponding to ∇̂. Then, we have[5, Prop. 2.2.2],D̂Ydt=DYdt+ 〈γ ′, Y 〉 Dγ′dt− 〈Dγ′dt, Y 〉 γ ′, (2)for any Y ∈ X(γ). Note that D̂dt = Ddt if and only if γ is free falling.The acceleration Dγ′dt satisﬁesDγ ′dt (t) ∈ Rt, for any t. Therefore, it may beobserved by γ whereas the velocity γ ′ is not observable by γ.Now, we are in a position to give rigorously the notion of UA observer. Anobserver γ : I −→M is said to obey a uniformly accelerated motion ifP̂ γt1,t2(Dγ ′dt(t1))=Dγ ′dt(t2), (3)for any t1, t2 ∈ I with t1 < t2, equivalently, if the equation (1) holds everywhereon I, i.e., Dγ′dt is Fermi-Walker parallel along γ. Clearly, if γ is free falling, thenit is a UA observer.Since we deal with a third-order ordinary diﬀerential equation, the followinginitial value problem has a unique local solution,D̂dt(Dγ ′dt)= 0, (4)γ(0) = p, γ ′(0) = v, Dγ′dt (0) = w,where p ∈ M and v, w ∈ TpM such that |v|2 = −1, 〈v, w〉 = 0, |w|2 = a2, anda is a positive constant. In addition, from deﬁnition it is clear that∣∣Dσ ′dt∣∣2(t) isconstant on I.Taking into account formula (2), an observer γ satisﬁes equation (1) if andonly ifD2γ ′dt2=〈Dγ ′dt,Dγ ′dt〉γ ′, (5)32 D. de la Fuentewhich is a third order equation.Consider a UA observer γ : I −→ M with a = ∣∣Dγ ′dt ∣∣ > 0 and put e1(t) =γ ′(t), e2(t) = 1aDγ ′dt (t). Then, from (5) we haveDe1dt= ae2(t) andDe2dt= ae1(t).Conversely, assume this system holds true for an observer γ with a > 0 constant.Then, a (non free falling) UA observer may be seen as a Lorentzian circle ofconstant curvature a and identically zero torsion.The previous results can be summarized as follows,Proposition 2.2. γ : I −→M is a UA observer iﬀ one of the following asser-tions holds:(a) γ is a solution of third-order diﬀerential equation (5).(b) γ is a Lorentzian circle or it is free falling.(c) γ has constant curvature and the remaining curvatures equal to zero.(d) γ, viewed as an isometric immersion from (I,−dt2) to M , is totally um-bilical with parallel mean curvature vector.3. Completeness of the inextensible UA trajecto-riesFirst of all, we are going to relate the solutions of equation (5) with theintegral curves of a certain vector ﬁeld on a Stiefel bundle type on M .Given a Lorentzian linear space E and a ∈ R, a > 0, denote by V an,2(E) the(n,2)-Stiefel manifold over E, deﬁned byV an,2(E) ={(v, w) ∈ E2 : |v|2 = −1, |w|2 = a2, 〈v, w〉 = 0}.We will call (n,2)-Stiefel bundle, V an,2(M), to the bundle over M with ﬁberV an,2(TpM).A key tool in the study of completeness is contained in the following lemma,which is proved in detail in [2].Lemma 3.1. There exists a unique vector ﬁeld G on V an,2(M) such that thecurves t 7−→ (γ(t), γ ′(t), Dγ ′dt (t)) are the integral curves of G, for any solutionγ of equation (4).Uniformly accelerated motion in GR 33Once deﬁned G, we will look for assumptions which assert its completeness.The following result directly follows from Lemma 3.1.Lemma 3.2. Let γ : [0, b) −→M be a solution of equation (4) with 0 < b <∞.The curve γ can be extended to b as a solution of (4) if and only if there existsa sequence{γ(tn), γ′(tn), Dγ′dt (tn)}nwhich is convergent in V an,2(M).Although we know that |γ ′(t)|2 = −1, this is not enough to apply Lemma3.2 even in the geometrically relevant case of M compact.Recall that a vector ﬁeld K on M is said closed and conformal if satisﬁes∇XK = hX for all X ∈ X(M). (6)Note that for any curve γ : I −→M , if K is closed and conformal, we haveddt〈K, γ ′〉 = 〈K, Dγ ′dt〉+ h(γ)| γ ′|2. (7)The following result, inspired from [1, Lemma 9], will be decisive to assurethat the image of the curve in V an,2(M), associated to a UA observer γ, iscontained in a compact subset.Lemma 3.3. Let M be a spacetime and let Q be a unitary timelike vector ﬁeld.If γ : I −→ M is a solution of (4) such that γ(I) lies in a compact subset ofM and 〈Q, γ ′〉 is bounded on I, then the image of t 7−→ (γ(t), γ ′(t), Dγ ′dt ) iscontained in a compact subset of V an,2(M) where a is the constant |Dγ′dt |.Proof. Consider the 1-form Qb metrically equivalent to Q and the associatedRiemannian metric gR := 〈 , 〉+ 2Qb ⊗Qb. We have,gR(γ′, γ ′) = 〈γ ′, γ ′〉+ 2 〈Q, γ ′〉2,which, by hypothesis, is bounded on I. Hence, there exists a constant c > 0 suchthat (γ(I), γ ′(I), Dγ′dt (I)) ⊂ C,C :={(p, v, w) ∈ V an,2(M) : p ∈ C1, gR(v, v) ≤ c},where C1 is a compact set on M such that γ(I) ⊂ C1. Hence, C is a compactin V an,2(M). 2Now, we are in a position to state the following completeness result.34 D. de la FuenteTheorem 3.1. Let M be a spacetime which admits a timelike conformal andclosed vector ﬁeld K. If ı´nfM√−〈K,K〉 > 0 then, each solution γ : I −→M of(4) such that γ(I) lies in a compact subset of M can be extended.Proof. Let I = [0, b), 0 < b < +∞, be the domain of a solution γ of equation(4). Derivating (7), it followsd2dt2〈K, γ ′〉 = 〈DKdt,Dγ ′dt〉+ 〈K, D2γ ′dt2〉 − ddt(h ◦ γ).The ﬁrst right term vanishes becauseK is conformal and closed. On the otherhand, the second right term equals to a2〈K, γ ′〉. Thus, the function t 7→ 〈K, γ ′〉satisﬁes the following diﬀerential equation,d2dt2〈K, γ ′〉 − a2〈K, γ ′〉 = (h ◦ γ)′(t). (8)Using now that γ(I) is contained in a compact of M , the function h ◦ γ isbounded on I. Moreover, since I is assumed bounded, using (8) there exists aconstant c1 > 0 such that|〈K, γ ′〉| < c1. (9)Now, if we put Q := K|K| , where |K|2 = −〈K,K〉 > 0, then Q is a unitarytimelike vector ﬁeld such that, by (9),|〈Q, γ ′〉| ≤ mc1 on I,where m = supM |K|−1 < ∞. The proof ends making use of Lemmas 3.2 and3.3. 2References[1] A.M. Candela, A. Romero and M. Sánchez, Completeness of the trajec-tories of particles coupled to a general force ﬁeld, Arch.Rational Mech.Anal., 208 (2013), 255274.[2] De la Fuente, A. Romero, Uniformly accelerated motion in General Rela-tivity: completeness of inextensible trajectories, Gen.Rel.Grav.,47 (2015).[3] K. Nomizu, K. Yano, On circles and spheres in Riemannian geometry,Math. Ann., 210 (1974), 163170.[4] W. Rindler, Hyperbolic motion in curved space time, Phys. Rev., 119(1960), 20822089 .[5] R.K. Sachs, H. Wu, General Relativity for Mathematicians, Grad. Textsin Math., 48, Springer-Verlag, 1977.

The uniformly accelerated motion in General Relativity from a geometric point of view

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The uniformly accelerated motion in General Relativity from a geometric point of view

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