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Geodesics in static spacetimes and t-periodic trajectories
Nonlinear Analysis 35 (1999) 677 – 686
Geodesics in static spacetimes and t -periodic trajectories Miguel Sanchez ∗ Departamento de Geometra y Topologa, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain Received 25 February 1997; accepted 25 March 1997
Keywords: Lorentz metrics; Geodesics; Static spacetimes; Autonomous Lagrangian systems; Periodic trajectories
1. Introduction and set-up In this article we consider geodesics in a standard static spacetime. Some problems regarding these geodesics which have been studied previously deal with geodesic connectedness [6], geodesic completeness [23] and the existence of t-periodic trajectories [4, 6, 1, 13]. Our purpose is to show a relation between geodesics in a static spacetime and the classical problem of particles accelerated by a potential in a Riemannian manifold. This relation is then exploited to study t-periodic trajectories. In order to be precise, we now de ne some concepts. A (standard) static spacetime (M; g) is a product manifold, M = R × M where M is any connected manifold
(1.1a)
endowed with a Lorentzian metric g which can be written as ∗ gR ≡ − dt 2 + gR : g = −( ◦M )R∗ (dt 2 ) + M
(1.1b)
Here dt 2 is the usual metric on R, and gR , are, respectively, a Riemannian metric and a positive function on M ; the natural projections R , M of M on R, M , resp., will be omitted without possibility of confusion. For simplicity, all the objects are assumed to be C ∞ , even though for the fundamental correspondence in Section 2 it ∗
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suces to assume they are just C 1 . A tangent vector v ∈ T M will be called timelike (resp. lightlike; spacelike) if g(v; v)¡0 (resp. g(v; v) = 0; g(v; v)¿0); as a geodesic has constant energy E = 12 ·g( 0 ; 0 ), it will be called timelike, lightlike or spacelike accordingly. It is easy to check (see Section 2) that the slices {t0 } × M ⊂ M are totally geodesic; so, any geodesic of (M; gR ) yields (spacelike) geodesics on (M; g) of the form (t0 ; ) for any t0 ∈ R; these geodesics will be called basic. Any geodesic (which will be assumed inextendible) yields new geodesics by means of ane reparameterizations; we will say that two geodesics are geometrically distinct if one of them cannot be obtained from the other in this way (or, equivalently, if their ranges are di erent). Moreover, if (s) = (t(s); x(s)) is a geodesic, then (s) = (t(s) + t0 ; x(s)) is a geodesic too, for any xed t0 ∈ R. We will consider as equivalent to all the geodesics which can be obtained from a xed one by using these trivial ways; that is, two non-constant geodesics and , (s) = (t(s); x(s)) are equivalent when there are a; a0 ; t0 ∈ R such that (s) = (t(as + a0 ) + t0 ; x(as + a0 )) for all s. The name trajectories of particles under the potential V (or simply particles) will be used for the inextendible solutions of the classical (Riemannian) problem on (M; gR ): ∇Rx0 x0 = −∇R V ◦ x;
with V = −1= ;
(1.2)
where ∇R denotes the Levi–Civita connection for gR . The constant E = ( 12 )gR (x0 ; x0 ) + V ◦ x will be called the total energy (kinetic plus potential) of the particle. Now, consider two such particles, x: I → M; y: J → M; (I; J intervals); we will say that they are geometrically distinct if there are no s0 ∈ I; r0 ∈ J such that x(s0 ) = y(r0 ); x0 (s0 ) = y0 (r0 ), otherwise, they will be equivalent. Our two concepts of geometric distinctness (for geodesics and for particles) are a bit di erent, even though natural in our approach. Observe that even if the particles x and y are geometrically distinct, their ranges may coincide; if they are equivalent, they have equal total energy. This paper is laid out as follows. In Section 2, the following result is developed: 1.1. Fundamental Correspondence (FC) Consider the quotient sets P and G whose elements are, respectively, the classes of equivalent trajectories of particles under potential V, given in Eq. (1.2), and the classes of non-basic and equivalent geodesics of (M; g). Then, there exists a natural bijection B : P → G: In this bijection, each particle x has a total energy E equal to the energy of certain canonical representatives of B([x]), where [x] is the class of equivalent trajectories of x. So, the study of (non-basic, non-equivalent) geodesics of each causal character, timelike, lightlike or spacelike, correspond to the study of particles under V with E¡0; E = 0 or E¿0, resp. As a simple application of this correspondence, we will study t-periodic trajectories in Section 3 (see De nition 3.1). These trajectories were introduced by Benci and
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Fortunato [4] for static spacetimes with M = R3 . In [4, 6], the existence of certain timelike t-periodic trajectories is shown when M = Rn , under suitable conditions. For compact M , the results previously obtained are as follows: (1) Greco [13] showed the existence of timelike t-periodic trajectories for each suciently large universal period, if 1 (M ) 6= 0 or if the maximum of is an isolated critical level (see also [24]). (2) Benci et al. [6], proved the existence of spacelike t-periodic trajectories for all universal periods, if 1 (M ) is nite, and (3) Candela [8] proved the existence of a lightlike t-periodic trajectory, in the more general setting of stationary spacetimes. More results on t-periodic trajectories in static as well as in other spacetimes, can be found in [1, 7, 8, 14, 16–18]. At the beginning of Section 3, we rede ne t-periodic trajectories and discuss brie y the concept of geometric distinctness, which will be used in a slightly stronger sense than in some of these references. Then, applying FC, we will see that the study of t-periodic trajectories corresponds with the study of closed trajectories for particles under V . As a consequence, all the results of this classical problem can now be used, yielding the existence of t-periodic trajectories under very general conditions, Corollaries 3.4, 3.5, 3.6, 3.10, and Theorem 3.9. The variational method underlying this approach is quite di erent to those in the quoted references, and our concept of t-periodic trajectories is a bit more subtle; so, we conclude in Section 4 with a discussion on the scope and limitations of this new point of view. Broadly speaking, it is argued that, after a suitable normalization, it is natural to look for t-periodic trajectories with a xed energy, proper period or universal period. Our approach using FC easily solves how to look for such trajectories with a xed energy or proper period, but becomes more complicated for a xed universal period. Other problems regarding geodesics in static spacetimes are also discussed under this approach.
2. Fundamental correspondence The static spacetime√(M; g) is a warped product having (M; gR ) as the base, (R; −dt 2 ) as the ber, and f = as the warping function (we follow the notation in [20, Ch. 7]). So, the equation of its geodesics is well known, and a curve (s) = (t(s); x(s)) in M is a geodesic if and only if it satis es: t 00 = (log(1= ◦ x))0 t 0 ;
(2.1a)
∇Rx0 x0 = − 12 t 02 ∇R ◦ x
(2.1b)
(see, for instance, [20, Proposition 7.38]). The rst equation can be easily integrated, yielding: t0 ◦ x ≡
(2.2a)
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for a constant ∈ R. This relation has a clear geometrical meaning: the vector eld @t on M is a Killing vector eld for g; then, its product by a geodesic must be a constant, and Eq. (2.2a) can be rewritten as −g( 0 ; @t ) = :
(2.2b)
Substitute t 0 in Eq. (2.1b) from its expression in Eq. (2.2a); then, each geodesic satisfying Eq. (2.2b) for a value of yields a solution to the equation: ∇Rx0 x0 = −∇R V ◦ x;
with V = −2 =2 :
(2.3)
The case = 0 represents the basic geodesics (with geodesic equations t 0 ≡ 0; ∇Rx0 x0 ≡ 0), which can be regarded as trivial. For cases 6= 0, there is a di erent Eq. (2.3) each time a value of is chosen. Nevertheless, if we are interested in the study of geometrically distinct geodesics, we can x a value for ; say, for con√ venience, = 2. Each (non-basic) geodesic can be reparameterized in precisely one √ way by using a homothety to satisfy Eq. (2.2b) with = 2. Summarizing, we can consider the next system of equations: √ (2.4a) t 0 = 2= ◦ x; ∇Rx0 x0 = − ∇R V ◦ x
with V ( = V√2 ) = − 1= :
Each non-basic geodesic of (M; g) can be reparametrized to satisfy √ −g( 0 ; @t ) = 2
(2.4b)
(2.5)
and, then, it is a solution to Eq. (2.4); reciprocally, each solution to Eq. (2.4) yields a (non-basic) geodesic satisfying Eq. (2.5). Note that Eq. (2.4b) is the equation of a (Riemannian) particle under potential V, and can be integrated for each initial condition independently of Eq. (2.4a). Once this equation has been integrated, the solution can be put in Eq. (2.4a) and, so, Eq. (2.4a) is also solved. Moreover, if satis es Eq. (2.5) its energy is E = 12 (gR (x0 ; x0 ) − ( ◦ x)t 02 ) = 12 gR (x0 ; x0 ) + V ◦ x:
(2.6)
That is, the (kinetic and total) energy of is equal to the total (kinetic plus potential) energy of the corresponding particle under V . So, this particle will have a total energy less than, equal to or greater than 0 according to the causal character of . Now, it is straightforward to check that the fundamental correspondence stated in Section 1 holds. In fact, note that the canonical representatives of B([x]) are just the geodesics
= (t; z) satisfying Eq. (2.5) and z ∈ [x] (one canonical representative of [x] can be obtained from the other one reparameterizing by a translation, and adding a constant t0 to the component t). Remark 2.1. The normalization (2:5) has been possible because basic geodesics have been neglected. Therefore; this normalization selects the geometrically distinct geodesics better than the more usual normalization g( 0 ; 0 ) ∈ {−1; 0; 1}.
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3. T-periodic trajectories De nition 3.1. A geodesic : R → M; (s) = (t(s); x(s)); will be called t-periodic (or a t-periodic trajectory) of universal period T ∈ R and proper period b¿0 if t(b) = t(0) + T; x(b) = x(0);
t 0 (b) = t 0 (0);
x0 (b) = x0 (0):
(3.1a) (3.1b)
In this case; we can always assume T ≥ 0; with equality just for basic geodesics; in particular; if satis es Eq. (2.5) then T ¿0. If x is constant; say x(s) ≡ x0 for all s; then the t-periodic trajectory will be called trivial; in this case, it implies ∇R (x0 ) = 0. When there is no possibility of confusion, will be called T -periodic to mean that it is t-periodic with universal period T . In the quoted references on this topic, it is usually assumed that b = 1. In this case, we will say that the t-periodic trajectory is normalized in the proper period. We will maintain our concepts of equivalence and geometrical distinctness for t-periodic trajectories; so, two much trajectories ; will be geometrically distinct if (R) 6= (R). This is a more restrictive assumption than the more usual one in previous literature: according to that de nitions and are rst assumed to be normalized in the proper period, and then they are geometrically distinct if ([0; 1]) 6= ([0; 1]) (even though perhaps (R) = (R); see, for example [6, De nition 1.4] or [1, Remark 1.5]). Moreover, note that we also consider as equivalent two t-periodic trajectories with di erent ranges, if their components on R di er by a constant. Remark 3.2. (A) The universal period can be computed from Z b (1= (x(s))) ds T = 0
(3.2)
with given by Eq. (2:2). Denote by E a geodesic of energy E. If E = (t(s); x(s)) is T -periodic of proper period b; we can construct a t-periodic trajectory bE normalized in the proper period just by putting bE (s) = (t(bs); x(bs)). Of course; the universal period of bE is again T; and −g( 0bE ; @t ) = b. As we will assume that Eq. (2:5) holds; the proper period b will be allowed to vary freely. (B) Obviously; the T -periodic trajectory E of proper period b is also a nT -periodic trajectory of proper period nb and energy E; for all n ∈ N. From each such t-periodic trajectory; we can construct nbE normalized in the proper period (of universal period nT ). Of course; all these trajectories are equivalent in our de nition; even though; if T 6= 0; nbE ([0; 1]) 6= mbE ([0; 1]) for any distinct n; m ∈ N. Clearly; if there exists ¿0 such that there are t-periodic trajectories all of universal period T ∈]0; [; then we can construct (not necessarily geometrically distinct) t-periodic trajectories for every universal period T ¿0. On the other hand; the minimum universal period for a non-trivial t-periodic trajectory is independent of ane reparameterizations; as Eq. (3:2) shows (cf.; for example; with [6; Theorem 5:1]; [8; Theorem 1:3]; [1; Theorem 1:6; Corollary 1:10] and [16; Remark 1:7]).
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Convention 3.3. In the remainder of this paper, we will consider geodesics of (M; g) satisfying Eq. (2.5). So, two geodesics of di erent energy will be necessarily nonequivalent (in particular; geometrically distinct): Consider a geodesic = (t; x) of (M; g): By using Eq. (2.4a), x is closed (that is, it satis es Eq. (3.1b)) if and only if is t-periodic. Thus, the problem of nding a t-periodic trajectory of energy E is equivalent to the problem of nding a closed trajectory of a particle under V with energy E in the Riemannian manifold (M; gR ). Of course, this is a classical problem in Lagrangian (Riemannian) systems, and all the known results on this vast topic will have their translations for t-periodic trajectories. As an example, we will consider the translations of three Lagrangian results. So, the next theorem is a consequence of Benci’s theorems 1.3 and 1.4 in [2]. Corollary 3.4. Consider a static spacetime (M; g); for any E ∈ R such that: (i) E = {q ∈ M=V (q) ≤ E} is compact; and (ii) ∇V (p) 6= 0 for all p in the boundary of E ; there exists a non-trivial t-periodic trajectory of energy E. (The hypotheses can be relaxed some times; see, for instance, the Theorem in [11] or the local version of the Theorem in [3].) For condition (ii), take into account that the set of critical value of has measure 0; so one can obtain as a straightforward consequence: Corollary 3.5. There exists a non-numerable set of non-trivial t-periodic trajectories with energy smaller than Sup V in (M; g) provided that has a strict local minimum; or if the values of on the boundary of a compact subset K are greater than the minimum of at K. Note that, as E¡Sup V; all the trajectories obtained in this way are timelike. As a second Lagrangian result, we can use Birkho –Lewis-type results to obtain again timelike trajectories { n = (tn ; xn )} but now with diverging proper periods bn and the range of the projections xn converging to a minimum of . In this case, equality (3.2) can be used to check that the universal periods are not bounded. For example, using [5] we have: Corollary 3.6. Assume that (M; gR ) is equal to Rn with its canonical metric h·; ·i; and V can be written; in the neighborhood U of a minimum point x0 = 0 as: V (x) = V (0) + 12 hA(x); xi + W (x); where A is the Hessian of V at 0 and W is a function satisfying h∇Wx ; xi ≥ pW (x) ≥ c|x|p for some c¿0 and p¿2; ∀x∈U . Then; there exists a sequence { n = (tn ; xn )} of Tn -periodic trajectories with {Tn } → ∞ and {xn } → 0 in C 1 .
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In the case of E¿V (which includes all the lightlike or spacelike geodesics, and, if is bounded, it also includes some timelike ones) a natural way of nding closed trajectories of particles is to consider the Jacobi metric gE = 2(E − V )gR :
(3.3)
But, rst, we consider the next concept: De nition 3.7. A t-periodic trajectory = (t; x) is a rotation if x0 does not vanish at any point; otherwise, it is a libration. That is, is a libration or a rotation when the particle x is in the classical sense (see, for instance, [15]). Of course, all t-periodic trajectories with E¿V are rotations. Remark 3.8. All the t-periodic rotations with energy E can be obtained from the Jacobi metric gE in Eq. (3:3) de ned on those points of M where E¿V; as follows. with Let be a gE -geodesic of unit speed; and reparameterize it as (s) = (r(s)) r 0 (s) = 2(E − V ((s))). Then; is a particle under potential V with total energy E; all such particles can be obtained in this way. Moreover; is closed if and only if is also closed (see; for instance, [12]). On the other hand; take the unit speed gE for all s. Even though and ∗ are equivalent geodesic ; and de ne ∗ (s) = (−s) as geodesics; they yield trajectories of particles under V; say; and ∗ ; which are not equivalent (according to our de nition in Section 1). Thus; from FC; any class of equivalent closed geodesics for gE yields two non-equivalent t-periodic rotations. Now, concentrate on the case E¿V . Consider the set of geodesics which can be obtained from by an ane reparameterization which preserves the orientation of R, and call this set the class of equivalent oriented geodesics of . Let M(gE ) be the set of all such classes. Then, we can state: Theorem 3.9. Consider the bijection B : P → G of FC; and put GE = {B([xE ]): [xE ] is any class of a particle with energy E} for each E¿V: Then; there exists a natural bijection M(gE ) → GE . Thus, as an immediate consequence of this theorem and the existence of closed geodesics in compact manifolds, we have: Corollary 3.10. Assume that M in (M; g) is compact; and x any non-trivial free homotopy class H (if M has just the trivial homotopy class; x this one). Then; for each energy E¿V there exist two non-equivalent t-periodic trajectories E = (tE ; xE );
E = (tE ; x E ) with xE ; x E ∈ H: In particular; there exist non-numerable sets of spacelike and timelike (nonequivalent and non-trivial) t-periodic trajectories; and there are at least two lightlike ones.
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By taking a product manifold it is easy to check that one cannot obtain more trajectories in the lightlike case, except when a multiplicity result for closed geodesics in (M; gR ) can be used. Corollary 3.10 also holds if H is taken as a conjugacy class of the fundamental group. 4. Conclusion The fundamental correspondence in Section 2 yields very general conditions for the existence of t-periodic trajectories. In fact, the theorem and corollaries in Section 3 should be compared with those appearing in previous references. Nevertheless, it is worth pointing out the kind of problems which can or cannot be studied easily with our method. Classically, when closed trajectories for particles under a potential are studied (as a problem in an autonomous Lagrangian system), one can choose between xing the energy or xing the period; see, for example [22]. When t-periodic trajectories are considered, we have three parameters: the proper period b, the universal period T and the energy E. Due to the existence of equivalent trajectories, one of the parameters b or E can be regarded as meaningless (recall that T is independent of reparameterizations, Remark 3.2). Thus, we can x b = 1 or E ∈ {−1; 0; 1} – even though these normalizations do not suppress all the equivalent trajectories, as explained in Remarks 2.1 and 3.2(B). Moreover, a new parameter de ned in Eq. (2.2) appears naturally. This parameter is related with b and E, and one of them can be substituted by . As the best way to study geometrically distinct geodesics is to x the value of , it is natural to consider the next three problems: Find a t-periodic trajectory satisfying Convention 3.3 and with: (1) a xed energy E, (2) a xed proper period b, (3) a xed universal period T. These problems can be interpreted from a physical point of view: timelike and lightlike t-periodic trajectories are the relativistic versions of the periodic motions (under the gravitational force) in classical Lagrangian mechanics. In √ fact, the proper period of such a trajectory is related (in fact, it is equal up to a factor −E) with the “proper time” in which the particle completes a turn in M , and the universal period with the corresponding “universal time”; in classical mechanics there is no distinction between them. The fundamental correspondence makes it possible to use directly the well-known results on particles for the problems (1) and (2) above. In fact, in Section 3 we saw several results for a xed energy due to its easier physical meaning; of course, we can nd similar results for a xed proper period. Nevertheless, the study of the problem (3) is more complicated in our approach. Observe that once a t-periodic trajectory with, say, xed energy E, has been obtained, one can compute its universal period T from Eq. (3.2). Varying E one can obtain t-periodic trajectories with (in general) di erent universal periods, but we cannot impose a priori the size of T . The conclusion in Corollary 3.6 shows an example of what can be said directly on universal periods with this technique. Analogously, some conclusions can be obtained
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under the hypothesis of the other results. For example, consider in Corollary 3.10 a non-trivial homotopy class H. Take for each E¿V , the particle xE which is a reparameterization of a gE -geodesic in H, shortest and with speed 1. One can check that the corresponding TE -periodic trajectories of proper periods bE satisfy limE→∞ bE = 0. Thus, from Eq. (3.2) and the compactness of M we also have limE→∞ TE = 0, that is If M is compact, for any non-trivial free homotopy class H there exists a sequence
n = (tn ; xn ) of spacelike Tn -periodic trajectories with xn ∈ H and limn→∞ Tn = 0. On the other hand, the explicit formula for the periods in Benci’s result [2, Remark V] can also be used to obtain certain control of the universal periods in Corollaries 3.4 and 3.5. Similar goals and limitations are obtained when applying FC on geodesic connectedness. Take two points x1 ; x2 ∈ M joined by the trajectory of the particle xE : [0; bE ] → M . We can then construct a geodesic E = (tE ; xE ) joining the point (0; x1 ) ∈ M with the line L[x2 ] = {(t; x2 )=t ∈ R}. The arrival time tE (bE ) can also be computed from Eq. (3.2). It is known [6, Theorem 1.1] that if gR is complete and 0¡inf ≤ sup ¡∞
(4.1)
then (M; g) is geodesically connected. That is, if all the possible xE joining x1 and x2 are considered, then geodesics joining (0; x1 ) with all the points of L[x2 ] are obtained (taking into account the basic geodesics too). But, unfortunately, this cannot be checked directly from Eq. (3.2). Nevertheless, if Eq. (4.1) does not hold, geodesic connectedness is not guaranteed (anti-de Sitter spacetime is a counterexample, see [21]). The most one can obtain in this case is the existence of geodesics joining any point and any line. It has been studied in [10] for lightlike geodesics in stationary spacetimes; for related results in the static case see [19], and in the stationary one, [9]. FC can easily be applied to this problem; so, using the Jacobi metric to check the existence of particles joining x1 and x2 with E ≥ 0, we can state: Any (t1 ; x1 ) ∈ M and any line L[x2 ] can be joined by a lightlike geodesic if and only if the metric (1= )gR is geodesically connected; moreover; if gR is complete then (t1 ; x1 ) and L[x2 ] can be joined by (non-basic) spacelike geodesics of all energies E¿0. Finally, it is worth pointing out that the separate study of the particle under a potential becomes the natural approach for solving the problem of geodesic completeness. In fact, this is an underlying idea in the study of geodesic completeness of general warped products developed in [23]. Acknowledgements This work was carried out during the author’s stay at the University of Bari. The author would like to acknowledge Prof. L. Pisani, Prof. A.M. Candela and everyone at the Dipartimento di Matematica of this University for their kind hospitality. The author is especially very grateful to Prof. D. Fortunato for his attention, helpful suggestions
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and discussions. This work has been partially supported by an EC Contract (Human Capital and Mobility) ERBCHRXCT940494. References [1] R. Bartolo, A. Masiello, On the existence of in nitely many trajectories for a class of Lorentzian manifolds like Schwarzschild and Reissner-Nordstrom spacetimes, J. Math. Anal. Appl. 199 (1996) 14 –38. [2] V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, Ann. Inst. Henri Poincare 1 (5) (1984) 401– 412. [3] M.S. Berger, Periodic solutions of second order dynamical systems and isoperimetric variational problems, Amer. J. Math. 93 (1971) 1–10. [4] V. Benci, D. Fortunato, Periodic trajectories for the Lorentz metric of a static gravitational eld, in: H. Berestycki, J.M. Coron, I. Ekeland (Eds.), Proc. on “Variational Methods”, Paris, 1988, pp. 413– 429. [5] V. Benci, D. Fortunato, A “Birkho -Lewis” type result for a class of Hamiltonian systems, Manuscripta Math. 59 (1987) 441– 456. [6] V. Benci, D. Fortunato, F. Giannoni, On the existence of multiple geodesics in static space–times, Ann. Inst. Henri Poincare 8 (1991) 79 –102. [7] V. Benci, D. Fortunato, F. Giannoni, On the existence of periodic trajectories in static Lorentzian manifolds with singular boundary, Nonlinear Analysis, a tribute in honour of Giovanni Prodi, Pisa (1991) 109 –133. [8] A.M. Candela, Lightlike periodic trajectories in space–times, Ann. Mat. Pura Appl. CLXXI (1996) 131–158. [9] A.M. Candela, A. Salvatore, Light rays joining two submanifolds in space–times, J. Geom. Phys. 22 (1997) 281–297. [10] D. Fortunato, F. Giannoni, A. Masiello, A Fermat principle for stationary space–times and applications to light rays, J. Geom. Phys. 15 (1995) 159 –188. [11] W.B. Gordon, A theorem of periodic solutions to Hamiltonian systems with convex potential, J. Di erential Equations 10 (1971) 324 –335. [12] W.B. Gordon, On the equivalence of second order systems occurring in the calculus of variations, Arch. Rational Mech. Anal. 50 (1973) 118–126. [13] C. Greco, Periodic trajectories in static space–times, Proc. Roy. Soc. Edinburgh 113A (1989) 99 –103. [14] C. Greco, Multiple periodic trajectories for a class of Lorentz metrics of a time-dependent gravitational eld, Math. Ann. 287 (1990) 515 –521. [15] V.V. Kozlov, Calculus of variations in the large and classical mechanics, Russian Math. Surveys 40 (2) (1985) 37–71. [16] A. Masiello, Timelike periodic trajectories in stationary Lorentz manifolds, Nonlinear Anal. TMA 19 (1992) 531–545. [17] A. Masiello, On the existence of a timelike trajectory for a Lorentzian metric, Proc. Roy. Soc. Edinburgh 125A (1995) 807–815. [18] A. Masiello, L. Pisani, Existence of a time-like periodic trajectory for a time-dependent Lorentz metric, Ann. Univ. Ferrara – Sci. Mat. XXXVI (1990) 207–222. [19] J. Molina, Alcuni applicazioni della Teoria di Morse a varieta di Lorentz, Tesi di Dottorato, Pisa, 1996. [20] B. O’Neill, Semi-Riemannian Geometry, Academic Press, San Diego, CA, 1983. [21] R. Penrose, Techniques of Di erential Topology in Relativity, Conf. Board Math. Sci., vol. 7, SIAM, Philadelphia, 1972. [22] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978) 157–184. [23] A. Romero, M. Sanchez, On the completeness of certain families of semi-Riemannian manifolds, Geometric Dedicata 53 (1994) 103–117. [24] M. Sanchez, Timelike periodic trajectories in spatially compact Lorentz manifolds, Proc. Am. Math. Soc., to appear.
Geodesics in static spacetimes and t -periodic trajectories Miguel Sanchez ∗ Departamento de Geometra y Topologa, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain Received 25 February 1997; accepted 25 March 1997
Keywords: Lorentz metrics; Geodesics; Static spacetimes; Autonomous Lagrangian systems; Periodic trajectories
1. Introduction and set-up In this article we consider geodesics in a standard static spacetime. Some problems regarding these geodesics which have been studied previously deal with geodesic connectedness [6], geodesic completeness [23] and the existence of t-periodic trajectories [4, 6, 1, 13]. Our purpose is to show a relation between geodesics in a static spacetime and the classical problem of particles accelerated by a potential in a Riemannian manifold. This relation is then exploited to study t-periodic trajectories. In order to be precise, we now de ne some concepts. A (standard) static spacetime (M; g) is a product manifold, M = R × M where M is any connected manifold
(1.1a)
endowed with a Lorentzian metric g which can be written as ∗ gR ≡ − dt 2 + gR : g = −( ◦M )R∗ (dt 2 ) + M
(1.1b)
Here dt 2 is the usual metric on R, and gR , are, respectively, a Riemannian metric and a positive function on M ; the natural projections R , M of M on R, M , resp., will be omitted without possibility of confusion. For simplicity, all the objects are assumed to be C ∞ , even though for the fundamental correspondence in Section 2 it ∗
E-mail: [email protected].
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suces to assume they are just C 1 . A tangent vector v ∈ T M will be called timelike (resp. lightlike; spacelike) if g(v; v)¡0 (resp. g(v; v) = 0; g(v; v)¿0); as a geodesic has constant energy E = 12 ·g( 0 ; 0 ), it will be called timelike, lightlike or spacelike accordingly. It is easy to check (see Section 2) that the slices {t0 } × M ⊂ M are totally geodesic; so, any geodesic of (M; gR ) yields (spacelike) geodesics on (M; g) of the form (t0 ; ) for any t0 ∈ R; these geodesics will be called basic. Any geodesic (which will be assumed inextendible) yields new geodesics by means of ane reparameterizations; we will say that two geodesics are geometrically distinct if one of them cannot be obtained from the other in this way (or, equivalently, if their ranges are di erent). Moreover, if (s) = (t(s); x(s)) is a geodesic, then (s) = (t(s) + t0 ; x(s)) is a geodesic too, for any xed t0 ∈ R. We will consider as equivalent to all the geodesics which can be obtained from a xed one by using these trivial ways; that is, two non-constant geodesics and , (s) = (t(s); x(s)) are equivalent when there are a; a0 ; t0 ∈ R such that (s) = (t(as + a0 ) + t0 ; x(as + a0 )) for all s. The name trajectories of particles under the potential V (or simply particles) will be used for the inextendible solutions of the classical (Riemannian) problem on (M; gR ): ∇Rx0 x0 = −∇R V ◦ x;
with V = −1= ;
(1.2)
where ∇R denotes the Levi–Civita connection for gR . The constant E = ( 12 )gR (x0 ; x0 ) + V ◦ x will be called the total energy (kinetic plus potential) of the particle. Now, consider two such particles, x: I → M; y: J → M; (I; J intervals); we will say that they are geometrically distinct if there are no s0 ∈ I; r0 ∈ J such that x(s0 ) = y(r0 ); x0 (s0 ) = y0 (r0 ), otherwise, they will be equivalent. Our two concepts of geometric distinctness (for geodesics and for particles) are a bit di erent, even though natural in our approach. Observe that even if the particles x and y are geometrically distinct, their ranges may coincide; if they are equivalent, they have equal total energy. This paper is laid out as follows. In Section 2, the following result is developed: 1.1. Fundamental Correspondence (FC) Consider the quotient sets P and G whose elements are, respectively, the classes of equivalent trajectories of particles under potential V, given in Eq. (1.2), and the classes of non-basic and equivalent geodesics of (M; g). Then, there exists a natural bijection B : P → G: In this bijection, each particle x has a total energy E equal to the energy of certain canonical representatives of B([x]), where [x] is the class of equivalent trajectories of x. So, the study of (non-basic, non-equivalent) geodesics of each causal character, timelike, lightlike or spacelike, correspond to the study of particles under V with E¡0; E = 0 or E¿0, resp. As a simple application of this correspondence, we will study t-periodic trajectories in Section 3 (see De nition 3.1). These trajectories were introduced by Benci and
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Fortunato [4] for static spacetimes with M = R3 . In [4, 6], the existence of certain timelike t-periodic trajectories is shown when M = Rn , under suitable conditions. For compact M , the results previously obtained are as follows: (1) Greco [13] showed the existence of timelike t-periodic trajectories for each suciently large universal period, if 1 (M ) 6= 0 or if the maximum of is an isolated critical level (see also [24]). (2) Benci et al. [6], proved the existence of spacelike t-periodic trajectories for all universal periods, if 1 (M ) is nite, and (3) Candela [8] proved the existence of a lightlike t-periodic trajectory, in the more general setting of stationary spacetimes. More results on t-periodic trajectories in static as well as in other spacetimes, can be found in [1, 7, 8, 14, 16–18]. At the beginning of Section 3, we rede ne t-periodic trajectories and discuss brie y the concept of geometric distinctness, which will be used in a slightly stronger sense than in some of these references. Then, applying FC, we will see that the study of t-periodic trajectories corresponds with the study of closed trajectories for particles under V . As a consequence, all the results of this classical problem can now be used, yielding the existence of t-periodic trajectories under very general conditions, Corollaries 3.4, 3.5, 3.6, 3.10, and Theorem 3.9. The variational method underlying this approach is quite di erent to those in the quoted references, and our concept of t-periodic trajectories is a bit more subtle; so, we conclude in Section 4 with a discussion on the scope and limitations of this new point of view. Broadly speaking, it is argued that, after a suitable normalization, it is natural to look for t-periodic trajectories with a xed energy, proper period or universal period. Our approach using FC easily solves how to look for such trajectories with a xed energy or proper period, but becomes more complicated for a xed universal period. Other problems regarding geodesics in static spacetimes are also discussed under this approach.
2. Fundamental correspondence The static spacetime√(M; g) is a warped product having (M; gR ) as the base, (R; −dt 2 ) as the ber, and f = as the warping function (we follow the notation in [20, Ch. 7]). So, the equation of its geodesics is well known, and a curve (s) = (t(s); x(s)) in M is a geodesic if and only if it satis es: t 00 = (log(1= ◦ x))0 t 0 ;
(2.1a)
∇Rx0 x0 = − 12 t 02 ∇R ◦ x
(2.1b)
(see, for instance, [20, Proposition 7.38]). The rst equation can be easily integrated, yielding: t0 ◦ x ≡
(2.2a)
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for a constant ∈ R. This relation has a clear geometrical meaning: the vector eld @t on M is a Killing vector eld for g; then, its product by a geodesic must be a constant, and Eq. (2.2a) can be rewritten as −g( 0 ; @t ) = :
(2.2b)
Substitute t 0 in Eq. (2.1b) from its expression in Eq. (2.2a); then, each geodesic satisfying Eq. (2.2b) for a value of yields a solution to the equation: ∇Rx0 x0 = −∇R V ◦ x;
with V = −2 =2 :
(2.3)
The case = 0 represents the basic geodesics (with geodesic equations t 0 ≡ 0; ∇Rx0 x0 ≡ 0), which can be regarded as trivial. For cases 6= 0, there is a di erent Eq. (2.3) each time a value of is chosen. Nevertheless, if we are interested in the study of geometrically distinct geodesics, we can x a value for ; say, for con√ venience, = 2. Each (non-basic) geodesic can be reparameterized in precisely one √ way by using a homothety to satisfy Eq. (2.2b) with = 2. Summarizing, we can consider the next system of equations: √ (2.4a) t 0 = 2= ◦ x; ∇Rx0 x0 = − ∇R V ◦ x
with V ( = V√2 ) = − 1= :
Each non-basic geodesic of (M; g) can be reparametrized to satisfy √ −g( 0 ; @t ) = 2
(2.4b)
(2.5)
and, then, it is a solution to Eq. (2.4); reciprocally, each solution to Eq. (2.4) yields a (non-basic) geodesic satisfying Eq. (2.5). Note that Eq. (2.4b) is the equation of a (Riemannian) particle under potential V, and can be integrated for each initial condition independently of Eq. (2.4a). Once this equation has been integrated, the solution can be put in Eq. (2.4a) and, so, Eq. (2.4a) is also solved. Moreover, if satis es Eq. (2.5) its energy is E = 12 (gR (x0 ; x0 ) − ( ◦ x)t 02 ) = 12 gR (x0 ; x0 ) + V ◦ x:
(2.6)
That is, the (kinetic and total) energy of is equal to the total (kinetic plus potential) energy of the corresponding particle under V . So, this particle will have a total energy less than, equal to or greater than 0 according to the causal character of . Now, it is straightforward to check that the fundamental correspondence stated in Section 1 holds. In fact, note that the canonical representatives of B([x]) are just the geodesics
= (t; z) satisfying Eq. (2.5) and z ∈ [x] (one canonical representative of [x] can be obtained from the other one reparameterizing by a translation, and adding a constant t0 to the component t). Remark 2.1. The normalization (2:5) has been possible because basic geodesics have been neglected. Therefore; this normalization selects the geometrically distinct geodesics better than the more usual normalization g( 0 ; 0 ) ∈ {−1; 0; 1}.
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3. T-periodic trajectories De nition 3.1. A geodesic : R → M; (s) = (t(s); x(s)); will be called t-periodic (or a t-periodic trajectory) of universal period T ∈ R and proper period b¿0 if t(b) = t(0) + T; x(b) = x(0);
t 0 (b) = t 0 (0);
x0 (b) = x0 (0):
(3.1a) (3.1b)
In this case; we can always assume T ≥ 0; with equality just for basic geodesics; in particular; if satis es Eq. (2.5) then T ¿0. If x is constant; say x(s) ≡ x0 for all s; then the t-periodic trajectory will be called trivial; in this case, it implies ∇R (x0 ) = 0. When there is no possibility of confusion, will be called T -periodic to mean that it is t-periodic with universal period T . In the quoted references on this topic, it is usually assumed that b = 1. In this case, we will say that the t-periodic trajectory is normalized in the proper period. We will maintain our concepts of equivalence and geometrical distinctness for t-periodic trajectories; so, two much trajectories ; will be geometrically distinct if (R) 6= (R). This is a more restrictive assumption than the more usual one in previous literature: according to that de nitions and are rst assumed to be normalized in the proper period, and then they are geometrically distinct if ([0; 1]) 6= ([0; 1]) (even though perhaps (R) = (R); see, for example [6, De nition 1.4] or [1, Remark 1.5]). Moreover, note that we also consider as equivalent two t-periodic trajectories with di erent ranges, if their components on R di er by a constant. Remark 3.2. (A) The universal period can be computed from Z b (1= (x(s))) ds T = 0
(3.2)
with given by Eq. (2:2). Denote by E a geodesic of energy E. If E = (t(s); x(s)) is T -periodic of proper period b; we can construct a t-periodic trajectory bE normalized in the proper period just by putting bE (s) = (t(bs); x(bs)). Of course; the universal period of bE is again T; and −g( 0bE ; @t ) = b. As we will assume that Eq. (2:5) holds; the proper period b will be allowed to vary freely. (B) Obviously; the T -periodic trajectory E of proper period b is also a nT -periodic trajectory of proper period nb and energy E; for all n ∈ N. From each such t-periodic trajectory; we can construct nbE normalized in the proper period (of universal period nT ). Of course; all these trajectories are equivalent in our de nition; even though; if T 6= 0; nbE ([0; 1]) 6= mbE ([0; 1]) for any distinct n; m ∈ N. Clearly; if there exists ¿0 such that there are t-periodic trajectories all of universal period T ∈]0; [; then we can construct (not necessarily geometrically distinct) t-periodic trajectories for every universal period T ¿0. On the other hand; the minimum universal period for a non-trivial t-periodic trajectory is independent of ane reparameterizations; as Eq. (3:2) shows (cf.; for example; with [6; Theorem 5:1]; [8; Theorem 1:3]; [1; Theorem 1:6; Corollary 1:10] and [16; Remark 1:7]).
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Convention 3.3. In the remainder of this paper, we will consider geodesics of (M; g) satisfying Eq. (2.5). So, two geodesics of di erent energy will be necessarily nonequivalent (in particular; geometrically distinct): Consider a geodesic = (t; x) of (M; g): By using Eq. (2.4a), x is closed (that is, it satis es Eq. (3.1b)) if and only if is t-periodic. Thus, the problem of nding a t-periodic trajectory of energy E is equivalent to the problem of nding a closed trajectory of a particle under V with energy E in the Riemannian manifold (M; gR ). Of course, this is a classical problem in Lagrangian (Riemannian) systems, and all the known results on this vast topic will have their translations for t-periodic trajectories. As an example, we will consider the translations of three Lagrangian results. So, the next theorem is a consequence of Benci’s theorems 1.3 and 1.4 in [2]. Corollary 3.4. Consider a static spacetime (M; g); for any E ∈ R such that: (i) E = {q ∈ M=V (q) ≤ E} is compact; and (ii) ∇V (p) 6= 0 for all p in the boundary of E ; there exists a non-trivial t-periodic trajectory of energy E. (The hypotheses can be relaxed some times; see, for instance, the Theorem in [11] or the local version of the Theorem in [3].) For condition (ii), take into account that the set of critical value of has measure 0; so one can obtain as a straightforward consequence: Corollary 3.5. There exists a non-numerable set of non-trivial t-periodic trajectories with energy smaller than Sup V in (M; g) provided that has a strict local minimum; or if the values of on the boundary of a compact subset K are greater than the minimum of at K. Note that, as E¡Sup V; all the trajectories obtained in this way are timelike. As a second Lagrangian result, we can use Birkho –Lewis-type results to obtain again timelike trajectories { n = (tn ; xn )} but now with diverging proper periods bn and the range of the projections xn converging to a minimum of . In this case, equality (3.2) can be used to check that the universal periods are not bounded. For example, using [5] we have: Corollary 3.6. Assume that (M; gR ) is equal to Rn with its canonical metric h·; ·i; and V can be written; in the neighborhood U of a minimum point x0 = 0 as: V (x) = V (0) + 12 hA(x); xi + W (x); where A is the Hessian of V at 0 and W is a function satisfying h∇Wx ; xi ≥ pW (x) ≥ c|x|p for some c¿0 and p¿2; ∀x∈U . Then; there exists a sequence { n = (tn ; xn )} of Tn -periodic trajectories with {Tn } → ∞ and {xn } → 0 in C 1 .
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In the case of E¿V (which includes all the lightlike or spacelike geodesics, and, if is bounded, it also includes some timelike ones) a natural way of nding closed trajectories of particles is to consider the Jacobi metric gE = 2(E − V )gR :
(3.3)
But, rst, we consider the next concept: De nition 3.7. A t-periodic trajectory = (t; x) is a rotation if x0 does not vanish at any point; otherwise, it is a libration. That is, is a libration or a rotation when the particle x is in the classical sense (see, for instance, [15]). Of course, all t-periodic trajectories with E¿V are rotations. Remark 3.8. All the t-periodic rotations with energy E can be obtained from the Jacobi metric gE in Eq. (3:3) de ned on those points of M where E¿V; as follows. with Let be a gE -geodesic of unit speed; and reparameterize it as (s) = (r(s)) r 0 (s) = 2(E − V ((s))). Then; is a particle under potential V with total energy E; all such particles can be obtained in this way. Moreover; is closed if and only if is also closed (see; for instance, [12]). On the other hand; take the unit speed gE for all s. Even though and ∗ are equivalent geodesic ; and de ne ∗ (s) = (−s) as geodesics; they yield trajectories of particles under V; say; and ∗ ; which are not equivalent (according to our de nition in Section 1). Thus; from FC; any class of equivalent closed geodesics for gE yields two non-equivalent t-periodic rotations. Now, concentrate on the case E¿V . Consider the set of geodesics which can be obtained from by an ane reparameterization which preserves the orientation of R, and call this set the class of equivalent oriented geodesics of . Let M(gE ) be the set of all such classes. Then, we can state: Theorem 3.9. Consider the bijection B : P → G of FC; and put GE = {B([xE ]): [xE ] is any class of a particle with energy E} for each E¿V: Then; there exists a natural bijection M(gE ) → GE . Thus, as an immediate consequence of this theorem and the existence of closed geodesics in compact manifolds, we have: Corollary 3.10. Assume that M in (M; g) is compact; and x any non-trivial free homotopy class H (if M has just the trivial homotopy class; x this one). Then; for each energy E¿V there exist two non-equivalent t-periodic trajectories E = (tE ; xE );
E = (tE ; x E ) with xE ; x E ∈ H: In particular; there exist non-numerable sets of spacelike and timelike (nonequivalent and non-trivial) t-periodic trajectories; and there are at least two lightlike ones.
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By taking a product manifold it is easy to check that one cannot obtain more trajectories in the lightlike case, except when a multiplicity result for closed geodesics in (M; gR ) can be used. Corollary 3.10 also holds if H is taken as a conjugacy class of the fundamental group. 4. Conclusion The fundamental correspondence in Section 2 yields very general conditions for the existence of t-periodic trajectories. In fact, the theorem and corollaries in Section 3 should be compared with those appearing in previous references. Nevertheless, it is worth pointing out the kind of problems which can or cannot be studied easily with our method. Classically, when closed trajectories for particles under a potential are studied (as a problem in an autonomous Lagrangian system), one can choose between xing the energy or xing the period; see, for example [22]. When t-periodic trajectories are considered, we have three parameters: the proper period b, the universal period T and the energy E. Due to the existence of equivalent trajectories, one of the parameters b or E can be regarded as meaningless (recall that T is independent of reparameterizations, Remark 3.2). Thus, we can x b = 1 or E ∈ {−1; 0; 1} – even though these normalizations do not suppress all the equivalent trajectories, as explained in Remarks 2.1 and 3.2(B). Moreover, a new parameter de ned in Eq. (2.2) appears naturally. This parameter is related with b and E, and one of them can be substituted by . As the best way to study geometrically distinct geodesics is to x the value of , it is natural to consider the next three problems: Find a t-periodic trajectory satisfying Convention 3.3 and with: (1) a xed energy E, (2) a xed proper period b, (3) a xed universal period T. These problems can be interpreted from a physical point of view: timelike and lightlike t-periodic trajectories are the relativistic versions of the periodic motions (under the gravitational force) in classical Lagrangian mechanics. In √ fact, the proper period of such a trajectory is related (in fact, it is equal up to a factor −E) with the “proper time” in which the particle completes a turn in M , and the universal period with the corresponding “universal time”; in classical mechanics there is no distinction between them. The fundamental correspondence makes it possible to use directly the well-known results on particles for the problems (1) and (2) above. In fact, in Section 3 we saw several results for a xed energy due to its easier physical meaning; of course, we can nd similar results for a xed proper period. Nevertheless, the study of the problem (3) is more complicated in our approach. Observe that once a t-periodic trajectory with, say, xed energy E, has been obtained, one can compute its universal period T from Eq. (3.2). Varying E one can obtain t-periodic trajectories with (in general) di erent universal periods, but we cannot impose a priori the size of T . The conclusion in Corollary 3.6 shows an example of what can be said directly on universal periods with this technique. Analogously, some conclusions can be obtained
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under the hypothesis of the other results. For example, consider in Corollary 3.10 a non-trivial homotopy class H. Take for each E¿V , the particle xE which is a reparameterization of a gE -geodesic in H, shortest and with speed 1. One can check that the corresponding TE -periodic trajectories of proper periods bE satisfy limE→∞ bE = 0. Thus, from Eq. (3.2) and the compactness of M we also have limE→∞ TE = 0, that is If M is compact, for any non-trivial free homotopy class H there exists a sequence
n = (tn ; xn ) of spacelike Tn -periodic trajectories with xn ∈ H and limn→∞ Tn = 0. On the other hand, the explicit formula for the periods in Benci’s result [2, Remark V] can also be used to obtain certain control of the universal periods in Corollaries 3.4 and 3.5. Similar goals and limitations are obtained when applying FC on geodesic connectedness. Take two points x1 ; x2 ∈ M joined by the trajectory of the particle xE : [0; bE ] → M . We can then construct a geodesic E = (tE ; xE ) joining the point (0; x1 ) ∈ M with the line L[x2 ] = {(t; x2 )=t ∈ R}. The arrival time tE (bE ) can also be computed from Eq. (3.2). It is known [6, Theorem 1.1] that if gR is complete and 0¡inf ≤ sup ¡∞
(4.1)
then (M; g) is geodesically connected. That is, if all the possible xE joining x1 and x2 are considered, then geodesics joining (0; x1 ) with all the points of L[x2 ] are obtained (taking into account the basic geodesics too). But, unfortunately, this cannot be checked directly from Eq. (3.2). Nevertheless, if Eq. (4.1) does not hold, geodesic connectedness is not guaranteed (anti-de Sitter spacetime is a counterexample, see [21]). The most one can obtain in this case is the existence of geodesics joining any point and any line. It has been studied in [10] for lightlike geodesics in stationary spacetimes; for related results in the static case see [19], and in the stationary one, [9]. FC can easily be applied to this problem; so, using the Jacobi metric to check the existence of particles joining x1 and x2 with E ≥ 0, we can state: Any (t1 ; x1 ) ∈ M and any line L[x2 ] can be joined by a lightlike geodesic if and only if the metric (1= )gR is geodesically connected; moreover; if gR is complete then (t1 ; x1 ) and L[x2 ] can be joined by (non-basic) spacelike geodesics of all energies E¿0. Finally, it is worth pointing out that the separate study of the particle under a potential becomes the natural approach for solving the problem of geodesic completeness. In fact, this is an underlying idea in the study of geodesic completeness of general warped products developed in [23]. Acknowledgements This work was carried out during the author’s stay at the University of Bari. The author would like to acknowledge Prof. L. Pisani, Prof. A.M. Candela and everyone at the Dipartimento di Matematica of this University for their kind hospitality. The author is especially very grateful to Prof. D. Fortunato for his attention, helpful suggestions
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and discussions. This work has been partially supported by an EC Contract (Human Capital and Mobility) ERBCHRXCT940494. References [1] R. Bartolo, A. Masiello, On the existence of in nitely many trajectories for a class of Lorentzian manifolds like Schwarzschild and Reissner-Nordstrom spacetimes, J. Math. Anal. Appl. 199 (1996) 14 –38. [2] V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, Ann. Inst. Henri Poincare 1 (5) (1984) 401– 412. [3] M.S. Berger, Periodic solutions of second order dynamical systems and isoperimetric variational problems, Amer. J. Math. 93 (1971) 1–10. [4] V. Benci, D. Fortunato, Periodic trajectories for the Lorentz metric of a static gravitational eld, in: H. Berestycki, J.M. Coron, I. Ekeland (Eds.), Proc. on “Variational Methods”, Paris, 1988, pp. 413– 429. [5] V. Benci, D. Fortunato, A “Birkho -Lewis” type result for a class of Hamiltonian systems, Manuscripta Math. 59 (1987) 441– 456. [6] V. Benci, D. Fortunato, F. Giannoni, On the existence of multiple geodesics in static space–times, Ann. Inst. Henri Poincare 8 (1991) 79 –102. [7] V. Benci, D. Fortunato, F. Giannoni, On the existence of periodic trajectories in static Lorentzian manifolds with singular boundary, Nonlinear Analysis, a tribute in honour of Giovanni Prodi, Pisa (1991) 109 –133. [8] A.M. Candela, Lightlike periodic trajectories in space–times, Ann. Mat. Pura Appl. CLXXI (1996) 131–158. [9] A.M. Candela, A. Salvatore, Light rays joining two submanifolds in space–times, J. Geom. Phys. 22 (1997) 281–297. [10] D. Fortunato, F. Giannoni, A. Masiello, A Fermat principle for stationary space–times and applications to light rays, J. Geom. Phys. 15 (1995) 159 –188. [11] W.B. Gordon, A theorem of periodic solutions to Hamiltonian systems with convex potential, J. Di erential Equations 10 (1971) 324 –335. [12] W.B. Gordon, On the equivalence of second order systems occurring in the calculus of variations, Arch. Rational Mech. Anal. 50 (1973) 118–126. [13] C. Greco, Periodic trajectories in static space–times, Proc. Roy. Soc. Edinburgh 113A (1989) 99 –103. [14] C. Greco, Multiple periodic trajectories for a class of Lorentz metrics of a time-dependent gravitational eld, Math. Ann. 287 (1990) 515 –521. [15] V.V. Kozlov, Calculus of variations in the large and classical mechanics, Russian Math. Surveys 40 (2) (1985) 37–71. [16] A. Masiello, Timelike periodic trajectories in stationary Lorentz manifolds, Nonlinear Anal. TMA 19 (1992) 531–545. [17] A. Masiello, On the existence of a timelike trajectory for a Lorentzian metric, Proc. Roy. Soc. Edinburgh 125A (1995) 807–815. [18] A. Masiello, L. Pisani, Existence of a time-like periodic trajectory for a time-dependent Lorentz metric, Ann. Univ. Ferrara – Sci. Mat. XXXVI (1990) 207–222. [19] J. Molina, Alcuni applicazioni della Teoria di Morse a varieta di Lorentz, Tesi di Dottorato, Pisa, 1996. [20] B. O’Neill, Semi-Riemannian Geometry, Academic Press, San Diego, CA, 1983. [21] R. Penrose, Techniques of Di erential Topology in Relativity, Conf. Board Math. Sci., vol. 7, SIAM, Philadelphia, 1972. [22] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978) 157–184. [23] A. Romero, M. Sanchez, On the completeness of certain families of semi-Riemannian manifolds, Geometric Dedicata 53 (1994) 103–117. [24] M. Sanchez, Timelike periodic trajectories in spatially compact Lorentz manifolds, Proc. Am. Math. Soc., to appear.