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Normal geodesics in static spacetimes with critical asymptotic behavior
Nonlinear Analysis 63 (2005) e357 – e367 www.elsevier.com/locate/na
Normal geodesics in static spacetimes with critical asymptotic behavior Anna Maria Candela∗,1 Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy
Abstract This paper deals with the existence of normal geodesics joining two given submanifolds in a static spacetime when the coefficient of its metric has a quadratic growth. A suitable variational approach allows one to use the classical Ljusternik–Schnirelman theory. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 53C50; 58E05; 58E10; 49J40 Keywords: Static spacetimes; Normal geodesics; Quadratic growth; Variational principle; Ljusternik–Schnirelman category
1. Introduction In the last years an increasing amount of interest has been devoted to the study of problems which arise in general relativity and some results have been obtained by using a variational approach and topological methods. In order to set the problem we want to study in this paper, let us recall the main definitions in the theory of Lorentzian manifolds (for more details, see, e.g., [3,14,18]). If (M , ·, ·L ) is a semi-Riemannian manifold, a smooth curve z : I → M (I real interval) is a geodesic in M if DsL z˙ (s) = 0
for all s ∈ I,
∗ Tel.: 39 0805442711; fax: +39 0805963612.
E-mail address: [email protected] (A.M. Candela). 1 Supported by M.I.U.R. (research funds ex 40% and 60%).
0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.09.032
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where z˙ is the tangent field along z and DsL z˙ is the covariant derivative of z˙ along z induced by the Levi–Civita connection of ·, ·L . Since the set of geodesics is invariant by affine reparametrizations, from now on without loss of generality we can choose I = [0, 1]. A semi-Riemannian manifold (M , ·, ·L ) is named Lorentzian if its metric ·, ·L is of index 1 on each tangent space Tz M , z ∈ M . Let us remark that, for simplicity, some Lorentzian manifolds are also called spacetimes since, from a physical point of view, some four-dimensional ones are models of trajectories in gravitational fields. It is well known that, if z=z(s) is a geodesic in a spacetime (M , ·, ·L ), then there exists a constant E(z) ∈ R such that ˙z(s), z˙ (s)L ≡ E(z) for all s ∈ I . A geodesic is named timelike, lightlike or spacelike, respectively, if E(z) < 0, E(z) = 0 or E(z) > 0, respectively. Here, we want to investigate the existence of geodesics joining two given submanifolds in a special class of Lorentzian manifolds and our aim is to show that a suitable variational principle allows one to use the classical Ljusternik–Schnirelman Theory. Definition 1.1. Let (M , ·, ·L ) be a Lorentzian manifold and let N0 , N1 be two submanifolds of M . A geodesic z : I → M joining N0 to N1 is normal if
z(0) ∈ N0 , z˙ (0) ∈ Tz(0) N⊥ 0,
z(1) ∈ N1 , z˙ (1) ∈ Tz(1) N⊥ 1,
where, for i ∈ {0, 1}, Tz(i) N⊥ i denotes the orthogonal space of Tz(i) Ni in Tz(i) M with respect to Lorentzian metric ·, ·L . Given two submanifolds N0 and N1 in M , as in the Riemannian case (cf. [15]) it is possible to prove that z = z(s) is a normal geodesic joining N0 to N1 if and only if it is a critical point of the action functional
1
f (z) = 0
˙z, z˙ L ds
(1.1)
on a suitable manifold of curves Z (for more details, see Section 2). But, unlike the Riemannian action functional, Lorentzian f is unbounded both from above and below and its critical points have infinite Morse index. So, in general, the existence of critical levels of f cannot be directly investigated by means of classical topological methods. Nevertheless, such problems can be got over in some “good” Lorentzian manifolds for a suitable choice of submanifolds. Definition 1.2. A Lorentzian manifold (M , ·, ·L ) is a standard static spacetime if there exists a connected Riemannian manifold (M0 , ·, ·) such that M = M0 × R and ·, ·L = ·, · − (x) dt 2
(1.2)
on any tangent space Tz M ≡ Tx M0 × R, z = (x, t) ∈ M , where : M0 → R is smooth and strictly positive.
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More in general, a Lorentzian manifold (M , ·, ·L ) is said to be static if it admits an irrotational timelike Killing vector field K. In this case, the spacetime splits locally as a standard static one, M = M0 × R with a metric ·, ·L as in (1.2), being K = *t , = −*t , *t L > 0 and M0 any integral manifold of K ⊥ , the orthogonal distribution to K (for more details, see [22 or also 2, Section 2]). On the other hand, in particular, Definition 1.2 includes also some classical examples of spacetimes which arise in general relativity such as anti-de Sitter, Schwarzschild or Reissner–Nordström spacetime while in the two-dimensional case static manifolds are essentially equivalent to Generalized Robertson–Walker spacetimes (see, for example, [13, Subsection 6.1]). Our main results can be stated as follows: Theorem 1.3. Let M = M0 × R be a Lorentzian manifold equipped with static metric ·, ·L defined in (1.2). Suppose that (H1 ) (M0 , ·, ·) is a complete C 3 n-dimensional Riemannian manifold; (H2 ) there exist 0, > 0 and a point x0 ∈ M0 such that 0 < (x) d 2 (x, x0 ) +
for all x ∈ M0 ,
where d(·, ·) is the distance induced on M0 by its Riemannian metric ·, ·. Moreover, let N0 = P0 × {t0 }, N1 = P1 × {t1 } be two submanifolds of M with t0 , t1 ∈ R and P0 , P1 ⊂ M0 . Assume (H3 ) P0 and P1 are closed submanifolds of M0 such that at least one of them is compact. Then, if N0 ∩ N1 = ∅, a non-trivial normal geodesic joining N0 to N1 exists, while they are at least cat(P0 × P1 ) if both P0 and P1 are contractible in M0 . Furthermore, if not only both P0 and P1 are contractible in M0 but also M0 is noncontractible in itself, then a sequence of such geodesics (zn )n exists such that E(zn ) +∞ if n +∞. Remark 1.4. Clearly, N0 ∩ N1 = ∅ if and only if it is t0 = t1 and P0 ∩ P1 = ∅. In this case, if x¯ ∈ P0 ∩ P1 the constant curve z¯ = (x, ¯ t0 ) is a trivial normal geodesic joining N0 to N1 . So, in order to prove the existence of non-trivial normal geodesics, it is enough to assume P0 ∩ P1 = ∅ if t0 = t1 . On the contrary, such a hypothesis is not necessary for proving the existence of infinitely many normal geodesics since surely there exist infinitely many ones which are spacelike. From one hand, if is bounded and P0 = {x¯0 } (x¯0 ∈ M0 ), then Theorem 1.3 implies [17, Theorems 1.1, 1.3]. Furthermore, if it is also P1 = {x¯1 } (x¯1 ∈ M0 ), assumption (H3 ) is obvious while N0 ∩ N1 = ∅ means (x¯0 , t0 ) = (x¯1 , t1 ). Then, the previous theorem implies the geodesic connectedness of M stated in [2, Theorem 1.1] (see also [5] or, more in general, [16] and references therein if is bounded while [11] if has a subquadratic growth) and, surely, its assumption (H2 ) cannot be improved (for the existence of a family of geodesically disconnected static spacetimes with superquadratic, but arbitrarily close to quadratic, see [2, Section 7]).
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On the other hand, the main theorems in [9] imply some results on the existence of normal geodesics joining two submanifolds in a static manifold whose coefficient grows at most subquadratically. Remark 1.5. In assumptions (H1 ) and (H2 ) manifold M is globally hyperbolic (cf. [20, Corollary 3.4]) but this is not enough to imply that M is geodesically connected (for a counterexample, see [21]).
2. Variational tools Let (M , ·, ·L ) be a static spacetime with M = M0 × R and ·, ·L as in (1.2), where (M0 , ·, ·) is a Riemannian manifold such that (H1 ) holds. Moreover, let P0 and P1 be two submanifolds of M0 and fix t1 ∈ R. Since zT = (x, t + T ) (T ∈ R) is a geodesic when z = (x, t) is a geodesic in M , without loss of generality we can assume t0 = 0. Whence, our aim is to study the existence of normal geodesics z : I → M joining N0 = P0 × {0} to N1 = P1 × {t1 } by means of variational tools (for more details, see, e.g., [16]). By Nash Embedding Theorem we can assume that M0 is a submanifold of an Euclidean space RN and ·, · is the restriction to M0 of the Euclidean metric of RN while d(·, ·) is the corresponding distance, i.e., d(x1 , x2 ) = inf if x1 , x2 ∈ M0 , ˙, ˙ ds : ∈ Ax1 ,x2 L
where ∈ Ax1 ,x2 if : L → M0 is a piecewise smooth curve joining x1 to x2 . Hence, it can be proved that manifold H 1 (I, M0 ) can be identified with the set of the absolutely continuous curves x : I → RN with square summable derivative such that x(I ) ⊂ M0 . Furthermore, since M0 is a complete Riemannian manifold with respect to ·, ·, also H 1 (I, M0 ) equipped with its standard Riemannian structure is a complete Riemannian manifold. Let Z be the smooth manifold of all the H 1 (I, M )-curves joining N0 to N1 , while (P0 , P1 ) denotes the smooth submanifold of H 1 (I, M0 ) which contains all the curves joining P0 to P1 with Tx (P0 , P1 ) = { ∈ Tx H 1 (I, M0 ) : (0) ∈ Tx(0) P0 , (1) ∈ Tx(1) P1 } for all x ∈ (P0 , P1 (cf. [15]). By the product structure of M , it follows that Z ≡ (P0 , P1 ) × W,
where W = {t ∈ H 1 (I, R) : t (0) = 0, t (1) = t1 }.
Clearly, it is W = H01 + T ∗ with T ∗ : s ∈ I → t1 s ∈ R, H01 = {t ∈ H 1 (I, R) : t (0) = t (1) = 0}.
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Whence, W is a closed affine submanifold of H 1 (I, R) with tangent space Tt W = H01 for all t ∈ W ; so, taken any curve z = (x, t) ∈ Z it is Tz Z ≡ Tx (P0 , P1 ) × H01 . Let us remark that, if assumption (H3 ) holds, submanifold Z of H 1 (I, M ) can be equipped with the Riemannian structure , H = (, ), (, )H =
1
1
Ds , Ds ds +
0
˙ 2 ds
0
for any z = (x, t) ∈ Z, = (, ) ∈ Tz Z. Moreover, submanifold (P0 , P1 ), hence Z, is complete and a constant K > 0 exists such that ˙ +K d(x(s), x0 )x
for all s ∈ I, x ∈ (P0 , P1 ).
(2.1)
By (1.2) it follows that action integral f in (1.1) becomes f (z) =
1
(x, ˙ x ˙ − (x)t˙2 ) ds,
z = (x, t) ∈ Z.
(2.2)
0
It is easy to prove that f is a C 1 functional with Fréchet differential f (z)[] = 2
1
1
x, ˙ Ds ds −
0
(x)[] t˙2 ds − 2
0
1
(x)t˙˙ ds,
(2.3)
0
for all z = (x, t) ∈ Z and = (, ) ∈ Tz Z, where denote the derivative of with respect to the Riemannian structure on M0 . It can be proved that Definition 1.1 implies the following variational principle (for more details see, e.g., [9, Proposition 2.1]). Proposition 2.1. A curve z : I → M is a normal geodesic joining N0 to N1 if and only if z ∈ Z is a critical point of functional f in Z. A way to get over the lack of boundedness of f on Z can be by introducing a new functional which depends only on Riemannian variable x and, hopefully, to apply the classical topological theories, such as the Ljusternik–Schnirelman one, to this new map. Clearly, such an approach is allowed since coefficient in metric (1.2) is time-independent and a Killing vector field exists on M ; thus, reasoning as in [5, Theorem 2.1] the following result can be stated. Proposition 2.2. Let z¯ = (x, ¯ t¯) ∈ Z. The following propositions are equivalent: (i) z¯ is a critical point of functional f in Z as defined in (2.2); (ii) x¯ is a critical point of the C 1 functional
1
J (x) = 0
x, ˙ x ˙ ds
− t12
1 0
1 ds (x)
−1 in (P0 , P1 ),
(2.4)
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and t¯ = (x), ¯ with
s
(x)(s) ¯ = K(x) ¯ 0
1 d , (x) ¯
1
K(x) ¯ = t1 0
1 ds (x) ¯
−1 .
(2.5)
In both these cases it is J (x) ¯ = f (x, ¯ (x)). ¯ Remark 2.3. In the proof of Proposition 2.2 taken x ∈ (P0 , P1 ) it is t = (x) if and only if f (x, t)[(0, )] = 0 for all ∈ H01 ; so, J (x)[] = f (x, (x))[(, )] for all (, ) ∈ Tx (P0 , P1 ) × H01 . 3. Ljusternik–Schnirelman theory First of all, let us recall the main tools of the Ljusternik–Schnirelman Theory (for more details, see e.g., [1,16,19]). Definition 3.1. Let X be a topological space. Given A ⊆ X, the Ljusternik–Schnirelman category of A in X, briefly cat X (A), is the least number of closed and contractible subsets of X covering A. If it is not possible to cover A with a finite number of such sets, it is catX (A) = +∞. We denote cat(X) = cat X (X). Definition 3.2. Let be a Riemannian manifold. A C 1 functional g : → R satisfies the Palais–Smale condition if any (xn )n ⊂ such that (g(xn ))n is bounded
and
lim g (xn ) = 0
n→+∞
converges in up to subsequences. Theorem 3.3. Let be a complete Riemannian manifold. If g is a C 1 functional on which satisfies the Palais–Smale condition and is bounded from below, then g attains its infimum and has at least cat() critical points. Remark 3.4. If 1k cat(), then each critical level ck in Theorem 3.3 is characterized as ck = inf sup g(x) where k = {A ⊆ : cat (A) k}. A∈ k x∈A
In order to apply Theorem 3.3 to functional J defined in (2.4), we need evaluating the Ljusternik–Schnirelman category of the manifold of curves (P0 , P1 ) introduced in the previous section. Proposition 3.5. Let (M0 , ·, ·) be a smooth complete connected finite-dimensional Riemannian manifold and let P0 , P1 be closed submanifolds both contractible in M0 . Then cat((P0 , P1 ))cat(P0 × P1 ). (For the proof, see [8, Theorem 3.7]).
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Proposition 3.6. Let (M0 , ·, ·) be a smooth complete connected finite-dimensional Riemannian manifold and let P0 , P1 be two of its closed submanifolds. If M0 is not contractible in itself while both P0 and P1 are contractible in M0 , then (P0 , P1 ) has infinite category and possesses compact subsets of arbitrary high category. (For the proof, see [10,12]).
4. Proof of the main theorem Here and in the following, let (M, ·, ·L ) be a static spacetime with M = M0 × R and ·, ·L as in (1.2). Suppose that (H1 ) and (H2 ) hold. Moreover, let P0 and P1 be two submanifolds of M0 which satisfy (H3 ). Fixed any t1 ∈ R, by Section 2 our aim is to look for the critical points of functional f defined in (2.2) on the manifold Z ≡ (P0 , P1 ) × W or, equivalently, those ones of J defined in (2.4) on the manifold (P0 , P1 ) (see Proposition 2.2). First of all, let us point out some properties of functional J in (P0 , P1 ). Proposition 4.1. In (P0 , P1 ) functional J is bounded from below and coercive, i.e., J (x) −→ +∞
if
x ˙ → +∞,
with · usual L2 -norm. Proof. Since by assumption (H2 ) and definition (2.4) it is J (x)x ˙ 2 − t12
1 0
1 ds 2 d (x, x0 ) +
−1
for all x ∈ (P0 , P1 ), the proof is trivial either if t1 = 0 or if = 0. Now, assume t1 = 0 and , > 0. Clearly, it is enough to prove that G(x) = x ˙
2
− t12
1 0
1 ds 1 d 2 (x, x0 ) + 1
−1
is bounded from below and coercive in (P0 , P1 ) (here, 1 = /). Fixed x ∈ (P0 , P1 ) and reasoning as in [6, Lemma 3.4], a one-dimensional function y : I → R can be defined as s d(x(0), x0 ) + 0 x ˙ d if s ∈ [0, s0 ], 1 y(s) = ˙ d if s ∈ (s0 , 1] d(x(1), x0 ) + s x for a suitable s0 ∈ I such that y ∈ H 1 (I, R). Clearly, by definition it is y(0) = d(x(0), x0 ), y(1) = d(x(1), x0 ) and y ˙ 2 = x ˙ 2,
d(x(s), x0 ) y(s) for all s ∈ I.
(4.1)
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Hence, it is G(x)y ˙
2
− t12
1
−1
ds 1 y 2 + 1 1 −1 2 y ˙ 1 − t12 . ds 2 0 1 y + 1
y ˙ 2
1 0
(4.2)
In order to complete the proof, let us remark that hypothesis (H3 ) holds so, for example, P0 is a compact submanifold of M0 (similar arguments can be used in the proof if P1 is compact). Reasoning as in the proof of [2, Proposition 4.1] for each s¯ ∈ I it is
1 0
y ˙ 2 ds 1 y 2 + 1
1/2
1
d
1
√
ds arcsinh( 1 y) ds 1 0
s¯
d
1
√
ds arcsinh( 1 y) ds; 1 0
whence, 1
2 1 y ˙ 2 ds arcsinh( y(¯ s )) − arcsinh( d(x(0), x )) (4.3) 1 1 0 2 1 0 1 y + 1 √ with x(0) ∈ P0 , where the set {arcsinh( 1 d(x(0), x0 )) : x ∈ (P0 , P1 )} is bounded as P0 is compact. Now, let (xk )k be a sequence in (P0 , P1 ) and consider the corresponding sequence (yk )k in H 1 (I, R). If we suppose
1 0
yk2 ds → +∞,
(4.4)
then y˙k → +∞ and there exists (sk )k ⊂ I such that yk (sk ) → +∞; thus, (4.3) gives
1
0
y˙k 2 ds → +∞ 1 yk2 + 1
as k → +∞;
hence, necessarily, lim G(xk ) = +∞.
k→+∞
So, if G is not bounded from below, (xk )k ⊂ (P0 , P1 ) exists such that G(xk ) → −∞. Then, necessarily, (4.2) implies 0
1
1 1 yk2 + 1
−1 ds
→ +∞.
(4.5)
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But by Cauchy–Schwarz inequality it is −1 1 1 1 (1 yk2 + 1) ds; ds 2 y + 1 0 0 1 k thus, (4.5) implies (4.4) and the previous arguments yield a contradiction. On the other hand, if G is not coercive, (xk )k ⊂ (P0 , P1 ) exists such that x˙k → +∞ and (G(xk ))k is bounded. Thus, by (4.1) it is also y˙k → +∞; hence, by (4.2) it follows (4.5) which gives again (4.4) and a contradiction is obtained, too. Proposition 4.2. Functional J satisfies the Palais–Smale condition on (P0 , P1 ). Proof. Let (xk )k ⊂ (P0 , P1 ) and M > 0 be such that lim J (xk ) = 0.
sup |J (xk )| M,
(4.6)
k→+∞
k∈N
Clearly, Proposition 4.1 and (4.6) imply that (x˙k )k is bounded; thus, (2.1) gives sup{d(xk (s), x0 ) : s ∈ I, k ∈ N} < + ∞, which implies two constants a1 , a2 > 0 exist such that 1 inf (xk (s))a1 and |K(xk )||t1 | (d 2 (xk , x0 ) + ) ds a2 for all k ∈ N, s∈I
0
with K(·) as in (2.5). Whence, taken tk = (xk ) ∈ W ( (·) as in (2.5)) it is |t˙k (s)| a2 /a1 a.e. in I; so, (t˙k )k is bounded, too. Then, defined zk = (xk , tk ), since P0 or P1 is compact, some comments in Section 2 and the above results imply that (zk )k is bounded in Z, even better, there exists z = (x, t) such that, up to a subsequence, zk z weakly in H 1 (I, RN ) × H 1 (I, R). In particular, it is also zk → z uniformly in I; so z ∈ Z, since both P0 and P1 are closed in the complete manifold M0 . By [4, Lemma 2.1] there exist two sequences (k )k , ( k )k ⊂ H 1 (I, RN ) such that k ∈ Txk (P0 , P1 ), k 0 weakly
xk − x = k + k
for all k ∈ N,
and k → 0 strongly in H 1 (I, RN ).
So, taking k = t − tk ∈ H01 , it is k 0 weakly in H 1 (I, R) and by Remark 2.3, (2.3) and (4.6) it follows: lim f (zk )[(k , k )] = 0, 1 which implies ˙ k , ˙ k ds +
k→+∞
0
1 0
(xk )˙2k ds = o(1),
as ( (xk ))k∈N is bounded, (k )k∈N and (k )k∈N converge uniformly to 0 in I and tk =t −k , xk = x + k + k . Whence, zk → z strongly in Z.
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Proof of Theorem 1.3. By Theorem 3.3 and Propositions 4.1 and 4.2 it follows that J attains its infimum and has at least cat((P0 , P1 )) critical points. Hence, Propositions 3.5 and 3.6 complete the proof. Remark 4.3. If M is equipped with a metric similar to (1.2) but with a coefficient which depends also on time variable t, a so-called orthogonal splitting Lorentzian manifold, the variational principle stated in Proposition 2.2 does not hold, so we cannot introduce a new functional bounded from below in order to apply the Ljusternik–Schnirelman theory. Anyway, it is possible to look directly for the critical points of action functional f on Z by means of the relative category theory but, in this setting, up to now no result known if is unbounded (if is bounded, see [7]). References [1] A. Ambrosetti, Critical points and nonlinear variational problems, Mém. Soc. Math. France (N.S.) 49 (1992). [2] R. Bartolo, A.M. Candela, J.L. Flores, M. Sánchez, Geodesics in static Lorentzian manifolds with critical quadratic behavior, Adv. Nonlinear Stud. 3 (2003) 471–494. [3] J.K. Beem, P.E. Ehrlich, K.L. Easley, Global Lorentzian geometry, Monogr. Textbooks Pure Appl. Math., 202, Dekker Inc., New York, 1996. [4] V. Benci, D. Fortunato, On the existence of infinitely many geodesics on space–time manifolds, Adv. Math. 105 (1994) 1–25. [5] V. Benci, D. Fortunato, F. Giannoni, On the existence of multiple geodesics in static space–times, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 79–102. [6] A.M. Candela, J.L. Flores, M. Sánchez, A quadratic Bolza-type problem in a Riemannian manifold, J. Differential Equations 193 (2003) 196–211. [7] A.M. Candela, A. Masiello, A. Salvatore, Existence and multiplicity of normal geodesics in Lorentzian manifolds, J. Geom. Anal. 10 (2000) 623–651. [8] A.M. Candela, A. Salvatore, Light rays joining two submanifolds in space–times, J. Geom. Phys. 22 (1997) 281–297. [9] A.M. Candela, A. Salvatore, Normal geodesics in stationary Lorentzian manifolds with unbounded coefficients, J. Geom. Phys. 44 (2002) 171–195. [10] A.M. Canino, On p-convex sets and geodesics, J. Differential Equations 75 (1988) 118–157. [11] E. Caponio, A. Masiello, P. Piccione, Some global properties of static spacetimes, Math. Z. 244 (2003) 457–468. [12] E. Fadell, S. Husseini, Category of loop spaces of open subsets in Euclidean space, Nonlinear Anal. 17 (1991) 1153–1161. [13] J.L. Flores, M. Sánchez, Geodesic connectedness and conjugate points in GRW spacetimes, J. Geom. Phys. 36 (2000) 285–314. [14] S.W. Hawking, G.F.R. Ellis, The large scale structure of Space–Time, Cambridge University Press, London, 1973. [15] W. Klingenberg, Riemannian Geometry, de Gruyter, Berlino, 1982. [16] A. Masiello, Variational methods in Lorentzian geometry, Pitman Res. Notes Math. Ser., vol. 309, Longman Sci. Tech., Harlow, 1994. [17] J. Molina, Existence and multiplicity of normal geodesics on static space–time manifolds, Boll. Unione Mat. Ital. A 10 (1996) 85–98. [18] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Inc., New York, 1983. [19] R.S. Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology 5 (1966) 115–132. [20] M. Sánchez, Some remarks on causality theory and variational methods in Lorentzian manifolds, Confer. Sem. Mat. Univ. Bari 265 (1997).
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[21] M. Sánchez, On the geometry of generalized Robertson–Walker Spacetimes: geodesics, Gen. Relativity Gravitation 30 (1998) 915–932. [22] M. Sánchez, On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes, Differential Geom. Appl., to appear.
Normal geodesics in static spacetimes with critical asymptotic behavior Anna Maria Candela∗,1 Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy
Abstract This paper deals with the existence of normal geodesics joining two given submanifolds in a static spacetime when the coefficient of its metric has a quadratic growth. A suitable variational approach allows one to use the classical Ljusternik–Schnirelman theory. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 53C50; 58E05; 58E10; 49J40 Keywords: Static spacetimes; Normal geodesics; Quadratic growth; Variational principle; Ljusternik–Schnirelman category
1. Introduction In the last years an increasing amount of interest has been devoted to the study of problems which arise in general relativity and some results have been obtained by using a variational approach and topological methods. In order to set the problem we want to study in this paper, let us recall the main definitions in the theory of Lorentzian manifolds (for more details, see, e.g., [3,14,18]). If (M , ·, ·L ) is a semi-Riemannian manifold, a smooth curve z : I → M (I real interval) is a geodesic in M if DsL z˙ (s) = 0
for all s ∈ I,
∗ Tel.: 39 0805442711; fax: +39 0805963612.
E-mail address: [email protected] (A.M. Candela). 1 Supported by M.I.U.R. (research funds ex 40% and 60%).
0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.09.032
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where z˙ is the tangent field along z and DsL z˙ is the covariant derivative of z˙ along z induced by the Levi–Civita connection of ·, ·L . Since the set of geodesics is invariant by affine reparametrizations, from now on without loss of generality we can choose I = [0, 1]. A semi-Riemannian manifold (M , ·, ·L ) is named Lorentzian if its metric ·, ·L is of index 1 on each tangent space Tz M , z ∈ M . Let us remark that, for simplicity, some Lorentzian manifolds are also called spacetimes since, from a physical point of view, some four-dimensional ones are models of trajectories in gravitational fields. It is well known that, if z=z(s) is a geodesic in a spacetime (M , ·, ·L ), then there exists a constant E(z) ∈ R such that ˙z(s), z˙ (s)L ≡ E(z) for all s ∈ I . A geodesic is named timelike, lightlike or spacelike, respectively, if E(z) < 0, E(z) = 0 or E(z) > 0, respectively. Here, we want to investigate the existence of geodesics joining two given submanifolds in a special class of Lorentzian manifolds and our aim is to show that a suitable variational principle allows one to use the classical Ljusternik–Schnirelman Theory. Definition 1.1. Let (M , ·, ·L ) be a Lorentzian manifold and let N0 , N1 be two submanifolds of M . A geodesic z : I → M joining N0 to N1 is normal if
z(0) ∈ N0 , z˙ (0) ∈ Tz(0) N⊥ 0,
z(1) ∈ N1 , z˙ (1) ∈ Tz(1) N⊥ 1,
where, for i ∈ {0, 1}, Tz(i) N⊥ i denotes the orthogonal space of Tz(i) Ni in Tz(i) M with respect to Lorentzian metric ·, ·L . Given two submanifolds N0 and N1 in M , as in the Riemannian case (cf. [15]) it is possible to prove that z = z(s) is a normal geodesic joining N0 to N1 if and only if it is a critical point of the action functional
1
f (z) = 0
˙z, z˙ L ds
(1.1)
on a suitable manifold of curves Z (for more details, see Section 2). But, unlike the Riemannian action functional, Lorentzian f is unbounded both from above and below and its critical points have infinite Morse index. So, in general, the existence of critical levels of f cannot be directly investigated by means of classical topological methods. Nevertheless, such problems can be got over in some “good” Lorentzian manifolds for a suitable choice of submanifolds. Definition 1.2. A Lorentzian manifold (M , ·, ·L ) is a standard static spacetime if there exists a connected Riemannian manifold (M0 , ·, ·) such that M = M0 × R and ·, ·L = ·, · − (x) dt 2
(1.2)
on any tangent space Tz M ≡ Tx M0 × R, z = (x, t) ∈ M , where : M0 → R is smooth and strictly positive.
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More in general, a Lorentzian manifold (M , ·, ·L ) is said to be static if it admits an irrotational timelike Killing vector field K. In this case, the spacetime splits locally as a standard static one, M = M0 × R with a metric ·, ·L as in (1.2), being K = *t , = −*t , *t L > 0 and M0 any integral manifold of K ⊥ , the orthogonal distribution to K (for more details, see [22 or also 2, Section 2]). On the other hand, in particular, Definition 1.2 includes also some classical examples of spacetimes which arise in general relativity such as anti-de Sitter, Schwarzschild or Reissner–Nordström spacetime while in the two-dimensional case static manifolds are essentially equivalent to Generalized Robertson–Walker spacetimes (see, for example, [13, Subsection 6.1]). Our main results can be stated as follows: Theorem 1.3. Let M = M0 × R be a Lorentzian manifold equipped with static metric ·, ·L defined in (1.2). Suppose that (H1 ) (M0 , ·, ·) is a complete C 3 n-dimensional Riemannian manifold; (H2 ) there exist 0, > 0 and a point x0 ∈ M0 such that 0 < (x) d 2 (x, x0 ) +
for all x ∈ M0 ,
where d(·, ·) is the distance induced on M0 by its Riemannian metric ·, ·. Moreover, let N0 = P0 × {t0 }, N1 = P1 × {t1 } be two submanifolds of M with t0 , t1 ∈ R and P0 , P1 ⊂ M0 . Assume (H3 ) P0 and P1 are closed submanifolds of M0 such that at least one of them is compact. Then, if N0 ∩ N1 = ∅, a non-trivial normal geodesic joining N0 to N1 exists, while they are at least cat(P0 × P1 ) if both P0 and P1 are contractible in M0 . Furthermore, if not only both P0 and P1 are contractible in M0 but also M0 is noncontractible in itself, then a sequence of such geodesics (zn )n exists such that E(zn ) +∞ if n +∞. Remark 1.4. Clearly, N0 ∩ N1 = ∅ if and only if it is t0 = t1 and P0 ∩ P1 = ∅. In this case, if x¯ ∈ P0 ∩ P1 the constant curve z¯ = (x, ¯ t0 ) is a trivial normal geodesic joining N0 to N1 . So, in order to prove the existence of non-trivial normal geodesics, it is enough to assume P0 ∩ P1 = ∅ if t0 = t1 . On the contrary, such a hypothesis is not necessary for proving the existence of infinitely many normal geodesics since surely there exist infinitely many ones which are spacelike. From one hand, if is bounded and P0 = {x¯0 } (x¯0 ∈ M0 ), then Theorem 1.3 implies [17, Theorems 1.1, 1.3]. Furthermore, if it is also P1 = {x¯1 } (x¯1 ∈ M0 ), assumption (H3 ) is obvious while N0 ∩ N1 = ∅ means (x¯0 , t0 ) = (x¯1 , t1 ). Then, the previous theorem implies the geodesic connectedness of M stated in [2, Theorem 1.1] (see also [5] or, more in general, [16] and references therein if is bounded while [11] if has a subquadratic growth) and, surely, its assumption (H2 ) cannot be improved (for the existence of a family of geodesically disconnected static spacetimes with superquadratic, but arbitrarily close to quadratic, see [2, Section 7]).
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On the other hand, the main theorems in [9] imply some results on the existence of normal geodesics joining two submanifolds in a static manifold whose coefficient grows at most subquadratically. Remark 1.5. In assumptions (H1 ) and (H2 ) manifold M is globally hyperbolic (cf. [20, Corollary 3.4]) but this is not enough to imply that M is geodesically connected (for a counterexample, see [21]).
2. Variational tools Let (M , ·, ·L ) be a static spacetime with M = M0 × R and ·, ·L as in (1.2), where (M0 , ·, ·) is a Riemannian manifold such that (H1 ) holds. Moreover, let P0 and P1 be two submanifolds of M0 and fix t1 ∈ R. Since zT = (x, t + T ) (T ∈ R) is a geodesic when z = (x, t) is a geodesic in M , without loss of generality we can assume t0 = 0. Whence, our aim is to study the existence of normal geodesics z : I → M joining N0 = P0 × {0} to N1 = P1 × {t1 } by means of variational tools (for more details, see, e.g., [16]). By Nash Embedding Theorem we can assume that M0 is a submanifold of an Euclidean space RN and ·, · is the restriction to M0 of the Euclidean metric of RN while d(·, ·) is the corresponding distance, i.e., d(x1 , x2 ) = inf if x1 , x2 ∈ M0 , ˙, ˙ ds : ∈ Ax1 ,x2 L
where ∈ Ax1 ,x2 if : L → M0 is a piecewise smooth curve joining x1 to x2 . Hence, it can be proved that manifold H 1 (I, M0 ) can be identified with the set of the absolutely continuous curves x : I → RN with square summable derivative such that x(I ) ⊂ M0 . Furthermore, since M0 is a complete Riemannian manifold with respect to ·, ·, also H 1 (I, M0 ) equipped with its standard Riemannian structure is a complete Riemannian manifold. Let Z be the smooth manifold of all the H 1 (I, M )-curves joining N0 to N1 , while (P0 , P1 ) denotes the smooth submanifold of H 1 (I, M0 ) which contains all the curves joining P0 to P1 with Tx (P0 , P1 ) = { ∈ Tx H 1 (I, M0 ) : (0) ∈ Tx(0) P0 , (1) ∈ Tx(1) P1 } for all x ∈ (P0 , P1 (cf. [15]). By the product structure of M , it follows that Z ≡ (P0 , P1 ) × W,
where W = {t ∈ H 1 (I, R) : t (0) = 0, t (1) = t1 }.
Clearly, it is W = H01 + T ∗ with T ∗ : s ∈ I → t1 s ∈ R, H01 = {t ∈ H 1 (I, R) : t (0) = t (1) = 0}.
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Whence, W is a closed affine submanifold of H 1 (I, R) with tangent space Tt W = H01 for all t ∈ W ; so, taken any curve z = (x, t) ∈ Z it is Tz Z ≡ Tx (P0 , P1 ) × H01 . Let us remark that, if assumption (H3 ) holds, submanifold Z of H 1 (I, M ) can be equipped with the Riemannian structure , H = (, ), (, )H =
1
1
Ds , Ds ds +
0
˙ 2 ds
0
for any z = (x, t) ∈ Z, = (, ) ∈ Tz Z. Moreover, submanifold (P0 , P1 ), hence Z, is complete and a constant K > 0 exists such that ˙ +K d(x(s), x0 )x
for all s ∈ I, x ∈ (P0 , P1 ).
(2.1)
By (1.2) it follows that action integral f in (1.1) becomes f (z) =
1
(x, ˙ x ˙ − (x)t˙2 ) ds,
z = (x, t) ∈ Z.
(2.2)
0
It is easy to prove that f is a C 1 functional with Fréchet differential f (z)[] = 2
1
1
x, ˙ Ds ds −
0
(x)[] t˙2 ds − 2
0
1
(x)t˙˙ ds,
(2.3)
0
for all z = (x, t) ∈ Z and = (, ) ∈ Tz Z, where denote the derivative of with respect to the Riemannian structure on M0 . It can be proved that Definition 1.1 implies the following variational principle (for more details see, e.g., [9, Proposition 2.1]). Proposition 2.1. A curve z : I → M is a normal geodesic joining N0 to N1 if and only if z ∈ Z is a critical point of functional f in Z. A way to get over the lack of boundedness of f on Z can be by introducing a new functional which depends only on Riemannian variable x and, hopefully, to apply the classical topological theories, such as the Ljusternik–Schnirelman one, to this new map. Clearly, such an approach is allowed since coefficient in metric (1.2) is time-independent and a Killing vector field exists on M ; thus, reasoning as in [5, Theorem 2.1] the following result can be stated. Proposition 2.2. Let z¯ = (x, ¯ t¯) ∈ Z. The following propositions are equivalent: (i) z¯ is a critical point of functional f in Z as defined in (2.2); (ii) x¯ is a critical point of the C 1 functional
1
J (x) = 0
x, ˙ x ˙ ds
− t12
1 0
1 ds (x)
−1 in (P0 , P1 ),
(2.4)
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and t¯ = (x), ¯ with
s
(x)(s) ¯ = K(x) ¯ 0
1 d , (x) ¯
1
K(x) ¯ = t1 0
1 ds (x) ¯
−1 .
(2.5)
In both these cases it is J (x) ¯ = f (x, ¯ (x)). ¯ Remark 2.3. In the proof of Proposition 2.2 taken x ∈ (P0 , P1 ) it is t = (x) if and only if f (x, t)[(0, )] = 0 for all ∈ H01 ; so, J (x)[] = f (x, (x))[(, )] for all (, ) ∈ Tx (P0 , P1 ) × H01 . 3. Ljusternik–Schnirelman theory First of all, let us recall the main tools of the Ljusternik–Schnirelman Theory (for more details, see e.g., [1,16,19]). Definition 3.1. Let X be a topological space. Given A ⊆ X, the Ljusternik–Schnirelman category of A in X, briefly cat X (A), is the least number of closed and contractible subsets of X covering A. If it is not possible to cover A with a finite number of such sets, it is catX (A) = +∞. We denote cat(X) = cat X (X). Definition 3.2. Let be a Riemannian manifold. A C 1 functional g : → R satisfies the Palais–Smale condition if any (xn )n ⊂ such that (g(xn ))n is bounded
and
lim g (xn ) = 0
n→+∞
converges in up to subsequences. Theorem 3.3. Let be a complete Riemannian manifold. If g is a C 1 functional on which satisfies the Palais–Smale condition and is bounded from below, then g attains its infimum and has at least cat() critical points. Remark 3.4. If 1k cat(), then each critical level ck in Theorem 3.3 is characterized as ck = inf sup g(x) where k = {A ⊆ : cat (A) k}. A∈ k x∈A
In order to apply Theorem 3.3 to functional J defined in (2.4), we need evaluating the Ljusternik–Schnirelman category of the manifold of curves (P0 , P1 ) introduced in the previous section. Proposition 3.5. Let (M0 , ·, ·) be a smooth complete connected finite-dimensional Riemannian manifold and let P0 , P1 be closed submanifolds both contractible in M0 . Then cat((P0 , P1 ))cat(P0 × P1 ). (For the proof, see [8, Theorem 3.7]).
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Proposition 3.6. Let (M0 , ·, ·) be a smooth complete connected finite-dimensional Riemannian manifold and let P0 , P1 be two of its closed submanifolds. If M0 is not contractible in itself while both P0 and P1 are contractible in M0 , then (P0 , P1 ) has infinite category and possesses compact subsets of arbitrary high category. (For the proof, see [10,12]).
4. Proof of the main theorem Here and in the following, let (M, ·, ·L ) be a static spacetime with M = M0 × R and ·, ·L as in (1.2). Suppose that (H1 ) and (H2 ) hold. Moreover, let P0 and P1 be two submanifolds of M0 which satisfy (H3 ). Fixed any t1 ∈ R, by Section 2 our aim is to look for the critical points of functional f defined in (2.2) on the manifold Z ≡ (P0 , P1 ) × W or, equivalently, those ones of J defined in (2.4) on the manifold (P0 , P1 ) (see Proposition 2.2). First of all, let us point out some properties of functional J in (P0 , P1 ). Proposition 4.1. In (P0 , P1 ) functional J is bounded from below and coercive, i.e., J (x) −→ +∞
if
x ˙ → +∞,
with · usual L2 -norm. Proof. Since by assumption (H2 ) and definition (2.4) it is J (x)x ˙ 2 − t12
1 0
1 ds 2 d (x, x0 ) +
−1
for all x ∈ (P0 , P1 ), the proof is trivial either if t1 = 0 or if = 0. Now, assume t1 = 0 and , > 0. Clearly, it is enough to prove that G(x) = x ˙
2
− t12
1 0
1 ds 1 d 2 (x, x0 ) + 1
−1
is bounded from below and coercive in (P0 , P1 ) (here, 1 = /). Fixed x ∈ (P0 , P1 ) and reasoning as in [6, Lemma 3.4], a one-dimensional function y : I → R can be defined as s d(x(0), x0 ) + 0 x ˙ d if s ∈ [0, s0 ], 1 y(s) = ˙ d if s ∈ (s0 , 1] d(x(1), x0 ) + s x for a suitable s0 ∈ I such that y ∈ H 1 (I, R). Clearly, by definition it is y(0) = d(x(0), x0 ), y(1) = d(x(1), x0 ) and y ˙ 2 = x ˙ 2,
d(x(s), x0 ) y(s) for all s ∈ I.
(4.1)
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Hence, it is G(x)y ˙
2
− t12
1
−1
ds 1 y 2 + 1 1 −1 2 y ˙ 1 − t12 . ds 2 0 1 y + 1
y ˙ 2
1 0
(4.2)
In order to complete the proof, let us remark that hypothesis (H3 ) holds so, for example, P0 is a compact submanifold of M0 (similar arguments can be used in the proof if P1 is compact). Reasoning as in the proof of [2, Proposition 4.1] for each s¯ ∈ I it is
1 0
y ˙ 2 ds 1 y 2 + 1
1/2
1
d
1
√
ds arcsinh( 1 y) ds 1 0
s¯
d
1
√
ds arcsinh( 1 y) ds; 1 0
whence, 1
2 1 y ˙ 2 ds arcsinh( y(¯ s )) − arcsinh( d(x(0), x )) (4.3) 1 1 0 2 1 0 1 y + 1 √ with x(0) ∈ P0 , where the set {arcsinh( 1 d(x(0), x0 )) : x ∈ (P0 , P1 )} is bounded as P0 is compact. Now, let (xk )k be a sequence in (P0 , P1 ) and consider the corresponding sequence (yk )k in H 1 (I, R). If we suppose
1 0
yk2 ds → +∞,
(4.4)
then y˙k → +∞ and there exists (sk )k ⊂ I such that yk (sk ) → +∞; thus, (4.3) gives
1
0
y˙k 2 ds → +∞ 1 yk2 + 1
as k → +∞;
hence, necessarily, lim G(xk ) = +∞.
k→+∞
So, if G is not bounded from below, (xk )k ⊂ (P0 , P1 ) exists such that G(xk ) → −∞. Then, necessarily, (4.2) implies 0
1
1 1 yk2 + 1
−1 ds
→ +∞.
(4.5)
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But by Cauchy–Schwarz inequality it is −1 1 1 1 (1 yk2 + 1) ds; ds 2 y + 1 0 0 1 k thus, (4.5) implies (4.4) and the previous arguments yield a contradiction. On the other hand, if G is not coercive, (xk )k ⊂ (P0 , P1 ) exists such that x˙k → +∞ and (G(xk ))k is bounded. Thus, by (4.1) it is also y˙k → +∞; hence, by (4.2) it follows (4.5) which gives again (4.4) and a contradiction is obtained, too. Proposition 4.2. Functional J satisfies the Palais–Smale condition on (P0 , P1 ). Proof. Let (xk )k ⊂ (P0 , P1 ) and M > 0 be such that lim J (xk ) = 0.
sup |J (xk )| M,
(4.6)
k→+∞
k∈N
Clearly, Proposition 4.1 and (4.6) imply that (x˙k )k is bounded; thus, (2.1) gives sup{d(xk (s), x0 ) : s ∈ I, k ∈ N} < + ∞, which implies two constants a1 , a2 > 0 exist such that 1 inf (xk (s))a1 and |K(xk )||t1 | (d 2 (xk , x0 ) + ) ds a2 for all k ∈ N, s∈I
0
with K(·) as in (2.5). Whence, taken tk = (xk ) ∈ W ( (·) as in (2.5)) it is |t˙k (s)| a2 /a1 a.e. in I; so, (t˙k )k is bounded, too. Then, defined zk = (xk , tk ), since P0 or P1 is compact, some comments in Section 2 and the above results imply that (zk )k is bounded in Z, even better, there exists z = (x, t) such that, up to a subsequence, zk z weakly in H 1 (I, RN ) × H 1 (I, R). In particular, it is also zk → z uniformly in I; so z ∈ Z, since both P0 and P1 are closed in the complete manifold M0 . By [4, Lemma 2.1] there exist two sequences (k )k , ( k )k ⊂ H 1 (I, RN ) such that k ∈ Txk (P0 , P1 ), k 0 weakly
xk − x = k + k
for all k ∈ N,
and k → 0 strongly in H 1 (I, RN ).
So, taking k = t − tk ∈ H01 , it is k 0 weakly in H 1 (I, R) and by Remark 2.3, (2.3) and (4.6) it follows: lim f (zk )[(k , k )] = 0, 1 which implies ˙ k , ˙ k ds +
k→+∞
0
1 0
(xk )˙2k ds = o(1),
as ( (xk ))k∈N is bounded, (k )k∈N and (k )k∈N converge uniformly to 0 in I and tk =t −k , xk = x + k + k . Whence, zk → z strongly in Z.
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Proof of Theorem 1.3. By Theorem 3.3 and Propositions 4.1 and 4.2 it follows that J attains its infimum and has at least cat((P0 , P1 )) critical points. Hence, Propositions 3.5 and 3.6 complete the proof. Remark 4.3. If M is equipped with a metric similar to (1.2) but with a coefficient which depends also on time variable t, a so-called orthogonal splitting Lorentzian manifold, the variational principle stated in Proposition 2.2 does not hold, so we cannot introduce a new functional bounded from below in order to apply the Ljusternik–Schnirelman theory. Anyway, it is possible to look directly for the critical points of action functional f on Z by means of the relative category theory but, in this setting, up to now no result known if is unbounded (if is bounded, see [7]). References [1] A. Ambrosetti, Critical points and nonlinear variational problems, Mém. Soc. Math. France (N.S.) 49 (1992). [2] R. Bartolo, A.M. Candela, J.L. Flores, M. Sánchez, Geodesics in static Lorentzian manifolds with critical quadratic behavior, Adv. Nonlinear Stud. 3 (2003) 471–494. [3] J.K. Beem, P.E. Ehrlich, K.L. Easley, Global Lorentzian geometry, Monogr. Textbooks Pure Appl. Math., 202, Dekker Inc., New York, 1996. [4] V. Benci, D. Fortunato, On the existence of infinitely many geodesics on space–time manifolds, Adv. Math. 105 (1994) 1–25. [5] V. Benci, D. Fortunato, F. Giannoni, On the existence of multiple geodesics in static space–times, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 79–102. [6] A.M. Candela, J.L. Flores, M. Sánchez, A quadratic Bolza-type problem in a Riemannian manifold, J. Differential Equations 193 (2003) 196–211. [7] A.M. Candela, A. Masiello, A. Salvatore, Existence and multiplicity of normal geodesics in Lorentzian manifolds, J. Geom. Anal. 10 (2000) 623–651. [8] A.M. Candela, A. Salvatore, Light rays joining two submanifolds in space–times, J. Geom. Phys. 22 (1997) 281–297. [9] A.M. Candela, A. Salvatore, Normal geodesics in stationary Lorentzian manifolds with unbounded coefficients, J. Geom. Phys. 44 (2002) 171–195. [10] A.M. Canino, On p-convex sets and geodesics, J. Differential Equations 75 (1988) 118–157. [11] E. Caponio, A. Masiello, P. Piccione, Some global properties of static spacetimes, Math. Z. 244 (2003) 457–468. [12] E. Fadell, S. Husseini, Category of loop spaces of open subsets in Euclidean space, Nonlinear Anal. 17 (1991) 1153–1161. [13] J.L. Flores, M. Sánchez, Geodesic connectedness and conjugate points in GRW spacetimes, J. Geom. Phys. 36 (2000) 285–314. [14] S.W. Hawking, G.F.R. Ellis, The large scale structure of Space–Time, Cambridge University Press, London, 1973. [15] W. Klingenberg, Riemannian Geometry, de Gruyter, Berlino, 1982. [16] A. Masiello, Variational methods in Lorentzian geometry, Pitman Res. Notes Math. Ser., vol. 309, Longman Sci. Tech., Harlow, 1994. [17] J. Molina, Existence and multiplicity of normal geodesics on static space–time manifolds, Boll. Unione Mat. Ital. A 10 (1996) 85–98. [18] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Inc., New York, 1983. [19] R.S. Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology 5 (1966) 115–132. [20] M. Sánchez, Some remarks on causality theory and variational methods in Lorentzian manifolds, Confer. Sem. Mat. Univ. Bari 265 (1997).
A.M. Candela / Nonlinear Analysis 63 (2005) e357 – e367
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[21] M. Sánchez, On the geometry of generalized Robertson–Walker Spacetimes: geodesics, Gen. Relativity Gravitation 30 (1998) 915–932. [22] M. Sánchez, On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes, Differential Geom. Appl., to appear.