- Email: [email protected]
Lightlike rays in stationary spacetimes with critical growth
Nonlinear Analysis 69 (2008) 304–313 www.elsevier.com/locate/na
Lightlike rays in stationary spacetimes with critical growth Valeria Luisi Dipartimento di Matematica, Universit`a degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy Received 27 February 2007; accepted 24 May 2007
Abstract The aim of this paper is to extend some previous results on the existence of lightlike geodesics joining a point to a line to the case of stationary Lorentzian manifolds whose metric coefficients have an optimal growth. c 2007 Elsevier Ltd. All rights reserved.
MSC: 53C50; 58E05; 58E10 Keywords: Stationary Lorentzian manifold; Lightlike geodesic; Asymptotic behavior; Fermat principle; Ljusternik–Schnirelman category
1. Introduction and the main result In order to introduce the main result of this paper, let us recall some useful definitions. A Lorentzian manifold is a couple (M, h·, ·iz ) where M is a smooth connected finite dimensional manifold and h·, ·iz is a Lorentzian metric, that is a smooth symmetric (0, 2) tensor field which induces a bilinear form of index 1 on the tangent space of each point of M. In a Lorentzian manifold (M, h·, ·iz ), a geodesic is a smooth curve z : I → M which solves the equation Ds z˙ (s) = 0
for all s ∈ I,
where Ds denotes the covariant derivative along z induced by the Levi-Civita connection of h·, ·iz and I is a real interval. It is known that, if z = z(s) is a geodesic on M, then a constant E(z) ∈ R exists such that E(z) ≡ h˙z (s), z˙ (s)iz
for all s ∈ I.
The sign of E(z) determines the causal character of z; more precisely, z is called timelike, lightlike or spacelike if E(z) is negative, null or positive, respectively. From a physical point of view, lightlike geodesics play an important role in the study of gravitational fields. In fact, they represent light rays in gravitational fields so it is useful to investigate their existence. On the other hand, from a variational point of view, this means that we should look for information on the zero critical level of the action functional E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.05.019
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
f (z) =
Z
305
h˙z , z˙ iz ds I
on a suitable manifold of curves. But, in general, this is quite hard. Anyway, some results can be obtained in some model spacetimes. Definition 1.1. A Lorentzian manifold M, h·, ·iz is a (standard) stationary spacetime if there exists a finite dimensional connected Riemannian manifold (M0 , h·, ·i) such that M = M0 × R and h·, ·iz is given by hζ, ζ iz = hξ, ξ i + 2 hδ (x) , ξ i τ − β (x) τ 2
(1.1)
for any z = (x, t) ∈ M0 × R and ζ = (ξ, τ ) ∈ Tz M ≡ Tx M0 × R, where δ : M0 → T M0 is a smooth vector field and β : M0 → R is a smooth and strictly positive scalar field. If δ ≡ 0, M is called (standard) static. A classical example of such a spacetime is the Kerr one which represents the gravitational field outside a rotating object (for more details, see [5] or [12]). Here, for M = M0 × R a stationary spacetime, our purpose is to study the problem of the existence of lightlike geodesics joining a point z 0 = (x0 , t0 ) to a line γ = {x1 } × R, fixing x0 , x1 ∈ M0 and t0 ∈ R, i.e., to solve Ds z˙ (s) = 0 ∀s ∈ I, h˙z (s) , z˙ (s)iz = 0 ∀s ∈ I, (P1 ) x (0) = x0 , x (1) = x1 , t (0) = t0 . The following result can be stated. Theorem 1.2. Let M = M0 × R be a stationary spacetime endowed with metric (1.1). Suppose that: (H1 ) (M0 , h·, ·i) is a complete smooth (at least C 3 ) n-dimensional Riemannian manifold; (H2 ) β has an asymptotic quadratic behavior while δ has a linear growth, that is there exist λ, µ ≥ 0, k1 , k2 ∈ R and a point x¯ ∈ M0 such that for all x ∈ M0 we have 0 < β (x) ≤ λd 2 (x, x) ¯ + k1 , p hδ (x) , δ (x)i ≤ µd (x, x) ¯ + k2 ,
(1.2) (1.3)
where d (·, ·) is the distance induced on M0 by its Riemannian metric h·, ·i. Then, fixing an event z 0 = (x0 , t0 ) and a point x1 ∈ M0 , there exist at least two non-trivial lightlike geodesics joining z 0 to the vertical line γ = {x1 } × R if x0 6= x1 . Furthermore, in itself, then there exist two sequences of such lightlike geodesics z n+ = − if M−0 is− non-contractible + + + xn , tn , z n = xn , tn such that tn (1) % +∞ and tn− (1) & −∞ as n % +∞. The problem of searching for lightlike geodesics joining a point to a line was first studied by Fortunato, Giannoni and Masiello in [8] by introducing a variational principle similar to the Fermat one which allowed them to solve problem (P1 ) under the hypotheses 0 < ν ≤ β (x) ≤ M for all x ∈ M0 , sup {hδ (x) , δ (x)i : x ∈ M0 } < +∞, for some ν, M > 0. Then, this result was extended to the search for light rays joining two submanifolds in stationary spacetimes (see [3, Theorem 1.1]). Later on, this outcome was improved by Candela and Salvatore in [4] by assuming that coefficient β grows at most subquadratically and δ at most sublinearly. These assumptions were bettered in [10] at least for in static spacetimes. In fact, in such a manifold even if coefficient β grows quadratically light rays exist. Here, we want to extend this result to the stationary case as growth conditions (1.2) and (1.3) allow one to prove geodesic connectedness (see [1, Theorem 1.2]). Let us point out that in this paper, as in [1], there has been removed the hypothesis of β being far away from zero.
306
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
2. Variational tools Let M = M0 × R be a stationary Lorentzian manifold equipped with metric (1.1) and fix z 0 = x0, t0 ∈ M, γ = {x1 } × R with x0 , x1 ∈ M0 , t0 ∈ R. By the Nash Embedding Theorem, if M0 is a C 3 n-dimensional Riemannian manifold, we can consider it as a submanifold of a suitable Euclidean space R N so that h·, ·i is the restriction to M0 of its Euclidean metric. Since geodesics are independent by affine reparametrizations, without loss of generality, we can take I = [0, 1] and t0 = 0. Obviously, in assumption (H2 ) we can take x¯ = x0 , so that (1.2) and (1.3) can be replaced by β (x) ≤ λd 2 (x, x0 ) + 1
for all x ∈ M0
(2.1)
and p
hδ (x) , δ (x)i ≤
√
λd (x, x0 ) + 1
for all x ∈ M0 .
(2.2)
H 1 (I, M0 ) can be easily identified with the Sobolev space of the absolutely continuous curves
Space x : I → RN whose derivative is square summable and such that x (I ) ⊂ M0 . Such a space can be endowed with the norm Z 1 Z 1 kxk2 = hx, hx, xi ds. ˙ xi ˙ ds + 0
0
Ω 1 (x
0 , x 1 ),
the set of H 1 -curves in M0 joining x0 to x1 and defined in I ; it can be proved that o Ω 1 (x0 , x1 ) ≡ x ∈ H 1 (I, R N ) : x (I ) ⊂ M0 , x (0) = x0 , x (1) = x1 .
Let us consider
n
If M0 is complete, then Ω 1 (x0 , x1 ) is a complete Riemannian manifold (see [13]) and its tangent space is given by Tx Ω 1 (x0 , x1 ) = {ξ ∈ H 1 (I, T M0 ) : ξ(s) ∈ Tx(s) M0 ∀ s ∈ I, ξ(0) = ξ(1) = 0}. Finally, we can define W0 , the subspace of H 1 (I, R) of those curves t = t (s) such that t (0) = 0. Clearly, for all t ∈ W0 we have Tt W0 = W0 . As already remarked, problem (P1 ) has a variational structure and it is easy to prove that z = x, ¯ t¯ is a lightlike geodesic joining z 0 = x0, 0 to γ if and only if it is a critical point of the C 1 functional Z 1 Z Z 1 1 1 1 ˙ hδ (x) , xi hx, (2.3) β (x) t˙2 ds, ˙ xi ˙ ds + ˙ t ds − f (z) = 2 0 2 0 0 with critical level f (z) = 0, where f 0 (z) [ζ ] = f 0 (z) [(ξ, τ )] Z 1 Z = hx, ˙ ξ˙ ids + 0
1
δ 0 (x) [ξ ] , x˙ t˙ds +
0 1
Z + 0
hδ (x) , xi ˙ τ˙ ds −
1
Z
hδ (x) , ξ˙ it˙ds
0
1 2
1
Z 0
β 0 (x) [ξ ] t˙2 ds −
1
Z
β (x) t˙τ˙ ds
(2.4)
0
for all z = (x, t) ∈ Z ≡ Ω 1 (x0 , x1 ) × W0 , ζ = (ξ, τ ) ∈ Tz Z ≡ Tx Ω 1 (x0 , x1 ) × W0 . But, as seen in the static case, the functional defined in (2.3) is unbounded both from above and from below. For this reason, Fortunato, Giannoni and Masiello in [8] stated a new variational principle similar to the Fermat one so as to introduce a new functional, arrival time T = T (x), which is bounded from below on Riemannian manifold Ω 1 (x0 , x1 ). Defining F : Ω 1 (x0 , x1 ) → R as v !Z u Z 1 Z 1 Z 1 1 1 u hδ(x), xi ˙ hδ(x), xi ˙ 2 t F(x) = ds + hx, ˙ xids ˙ + ds ds, β(x) β(x) 0 0 0 β(x) 0 the following variational principle can be stated.
307
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
Theorem 2.1 (Fermat Principle). Let z ∗ : I → M, z ∗ = z ∗ (s), be a smooth curve such that z ∗ = (x ∗ , t ∗ ). Then, the following statements are equivalent: (a) z ∗ is a solution of problem (P1 ) with arrival time t ∗ (1) = T > 0; (b) x ∗ is a critical point of functional F with critical level T = F (x ∗ ) > 0 and, for all s ∈ I , we have t ∗ (s) = ΦT (x ∗ ) (s) with Z s hδ (x ∗ (σ )) , x˙ ∗ (σ )i dσ ΦT x ∗ (s) = β (x ∗ (σ )) 0 !Z !−1 Z 1 Z 1 s hδ (x ∗ ) , x˙ ∗ i 1 1 ds dσ ds . + T− ∗ ∗ β (x ∗ ) 0 β (x (σ )) 0 β (x ) 0 Proof. Its proof is in [8, Theorem 2.3] (and also in [11, Theorem 6.2.2]). As they are useful in the following, we outline its main tools. Fixed T ∈ R, define WT = {t ∈ W0 | t (1) = T } , which is an affine submanifold of H 1 (I, R) whose tangent space is H01 (I, R) = τ ∈ H 1 (I, R) | τ (0) = τ (1) = 0 . If f T is the restriction of functional f to Z T = Ω 1 (x0 , x1 ) × WT , then, it can be proved that z ∗ = (x ∗ , t ∗ ) is a critical point of functional f T in Z T , i.e. it solves (P1 ) with arrival time T , if and only if x ∗ is a critical point of JT (x) = f T (x, ΦT (x)) ,
x ∈ Ω 1 (x0 , x1 ) ,
(2.5)
with JT (x ∗ ) = 0 and t ∗ = ΦT (x ∗ ), where ΦT : Ω 1 (x0 , x1 ) → WT is defined above (see [9] or [11, Section 3.3]). Thus, our problem is reduced to the search for a couple (x, T ) ∈ Ω 1 (x0 , x1 ) × R, for example with T > 0, a solution of the problem (∂ H (x, T ) = 0, ∂x H (x, T ) = 0, with H (x, T ) = 2JT (x); hence, it has to be an x critical point of F with critical level T = F (x).
Remark 2.2. By differentiating the map x ∈ Ω 1 (x0 , x1 ) 7−→ H (x, F (x)) ∈ R we have ∂H ∂H (x, F (x)) + (x, F (x)) F 0 (x) = 0 ∂x ∂t
for all x ∈ Ω 1 (x0 , x1 ).
(2.6)
Remark 2.3. In the proof of Theorem 2.1, the choice of T > 0 can be replaced with T < 0 obtaining that z ∗ = (x ∗ , t ∗ ) is a solution of problem (P1 ) with arrival time t ∗ (1) = T < 0 if and only if x ∗ is a critical point of the functional v !Z u Z 1 Z 1 Z 1 1 1 u hδ (x) , xi hδ (x) , xi ˙ ˙ 2 t ˜ hx, F (x) = ds − ˙ xi ˙ ds + ds ds β (x) β (x) 0 0 0 β (x) 0 with critical level T = F˜ (x ∗ ) < 0. In order to prove Theorem 1.2, the following abstract theory is useful (for more details, see [14]). Definition 2.4. Let X be a topological space and A ⊆ X . The Ljusternik–Schnirelman category of A in X (cat X A) is the least number of closed and contractible subsets of X covering A. If this is not possible we say that cat X A = +∞. We define cat X = cat X X .
308
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
Definition 2.5. Let Ω be a Riemannian manifold and f a C 1 functional on Ω . f is said to satisfy the Palais–Smale condition if any (xn )n ⊂ Ω such that ( f (xn ))n
is bounded and
lim
n→+∞
f 0 (xn ) = 0
converges in Ω up to subsequences. Theorem 2.6 (Ljusternik–Schnirelman). Let Ω be a complete Riemannian manifold and f : Ω → R a C 1 functional. If f satisfies the Palais–Smale condition and is bounded from below, then f attains its infimum and has at least cat Ω critical points. Furthermore, if supΩ F = +∞ and cat Ω = +∞ there exists a sequence of critical points (xn )n ⊂ Ω such that F (xn ) % +∞. Finally, in order to estimate the Ljusternik–Schnirelman category of Ω 1 (x0 , x1 ), we need the following result (see [7]). Proposition 2.7 (Fadell–Husseini). If M0 is a manifold not contractible in itself, then for all x0 , x1 ∈ M0 the manifold of curves Ω 1 (x0 , x1 ) has infinite category and possesses compact subsets of arbitrarily high category. 3. Proof of Theorem 1.2 To prove the main theorem we want to check that functional F satisfies the hypotheses of Theorem 2.6. To this end, the following lemmas are needed. Lemma 3.1. Let β satisfy assumption (H2 ). If (xk )k ⊂ Ω 1 (x0 , x1 ) is such that kx˙k k → +∞, then Z 1 kx˙k k2 ds → +∞ as k → +∞, 0 β (x k ) R1 where kx˙k k2 = 0 hx˙k , x˙k i ds. Proof. See [2, Proposition 4.1].
(3.1)
Lemma 3.2. Let ( f k )k be a sequence of non-negative functions, 0 6= f k ∈ L 2 ([0, ak ]), such that c > 0 exists which satisfies 1/2 Z ak Z ak √ f k2 (r ) dr c+ f k (r ) dr > ak for all k ∈ N. (3.2) 0
0
Define ak
Z mk =
f k (r ) dr
and
0
mk . Dk = r ∈ [0, ak ] : f k (r ) ≥ 2ak
If the measure of Dk , |Dk |, is written as |Dk | = k ak , then it results that k > 1 −
8c 4c2 − 2 mk mk
Proof. See [1, Lemma 2.5].
for all k ∈ N.
Lemma 3.3. Under the hypotheses of Theorem 1.2 there cannot exist a sequence (xk )k ⊂ Ω 1 (x0 , x1 ) satisfying both (F (xk ))k
is bounded from above
(3.3)
and kx˙k k → +∞ as k → +∞.
(3.4)
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
309
Proof. Arguing by contradiction, assume the existence of a sequence (xk )k ⊂ Ω 1 (x0 , x1 ) such that both (3.3) and (3.4) hold. We denote by Z 1 Z 1 |hδ (xk ) , x˙k i| 1 ds and mk = ds. ak = β β (xk ) (x ) k 0 0 If hδ (xk ) , x˙k i ds ≥ 0 β (xk )
1
Z 0
for all k large enough,
then for such k’s we have s Z Z 1 hx˙k , x˙k i ds F (xk ) ≥
1 0
0
1 ds = β (xk )
s
1
Z 0
kx˙k k2 ds β (xk )
which goes to infinity for Lemma 3.1 in contradiction with (3.3). Similarly, if (m k )k is bounded from above, then Lemma 3.1 contradicts (3.3). So, let us consider the case in which lim m k = +∞
(3.5)
k→+∞
(up to subsequences). Thus, (3.3) implies that a constant c > 0 exists such that s Z 1 hδ (xk ) , x˙k i2 2 c ≥ F (xk ) ≥ −m k + ak kx˙k k + ak ds, β (xk ) 0
(3.6)
which implies s
1
Z
c + mk ≥ 0
hδ (xk ) , x˙k i2 √ ds ak . β (xk )
(3.7)
For the Cauchy–Schwarz inequality we obtain !1/2 Z 1 Z 1 |hδ (xk ) , x˙k i| hδ (xk ) , x˙k i2 √ ak , ds ≤ ds β β ) ) (x (x k k 0 0 and thus, Z ak 0
1
hδ (xk ) , x˙k i2 ds ≥ m 2k ; β (xk )
hence, by (3.6) we have q c + m k ≥ ak kx˙k k2 + m 2k from which we have c2 + 2 cm k ≥ ak kx˙k k2 ; whence, mk ak for a suitable A > 0. On the other hand, defining dxk f k (r ) = δ (xk (s (r ))) , (s (r )) 6= 0 ds kx˙k k2 ≤ A
(3.7) and (3.9) imply (3.2).
(3.8)
with dr =
ds . β (xk (s))
(3.9)
310
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
Then, Lemma 3.2 gives k > 1 −
8c 4c2 − 2, mk mk
(3.10)
where k is such that |Dk | = k ak
and
dxk mk ; Dk = r ∈ [0, ak ] : δ (xk (s (r ))) , (s (r )) ≥ ds 2ak
or, equivalently, Z ds = k a k D˜ k β (x k (s))
(3.11)
with mk . D˜ k = s ∈ I : |hδ (xk (s)) , x˙k (s)i| ≥ 2ak
(3.12)
Therefore, from (2.2) and (3.12) we have on D˜ k √ √ λ length (xk (s)) + 1 |x˙k (s)| ≥ λd (xk (s) , x0 ) + 1 |x˙k (s)| ≥ |δ (xk (s))| |x˙k (s)| ≥ |hδ (xk (s)) , x˙k (s)i| ≥
mk , 2ak
where length (xk (s)) is the length of the portion of the curve xk ([0, s]). From the last inequality and (3.8) for all k we have Z Z ds 4ak mk |x˙k (s)|2 ds ≤ 4A. 2 ≤ ak D˜ k √ m k D˜ k λ length (xk (s)) + 1
(3.13)
On the other hand, for the Cauchy–Schwarz inequality and (3.8) we have r mk √ length (xk (s)) ≤ A s for all s ∈ I, ak and thus mk ak
Z 0
1
ds √
2 ≥
λ length (xk (s)) + 1
mk ak
1
Z 0
=
2 lg λA
ds q
2
λA makk s + 1
r
λA
which goes to infinity for (3.4) and (3.8). And so, for (3.13) we must have Z ds mk lim 2 = +∞, k→+∞ ak D˜ c √ k λ length (xk (s)) + 1 as D˜ kc = I \ D˜ k . On the other hand, defining Z ds mk dk = 2 , ak D˜ kc √ λ length (xk (s)) + 1
mk 1 +1 + q − 1 , mk ak λA ak + 1
(3.14)
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
(2.1) and (3.11) imply Z (1 − k ) ak =
D˜ kc
311
ds ak ≥ dk , β (xk ) mk
from which we have dk 1 − k ≥ . mk Thus, (3.10) implies dk ≤ (1 − k ) m k < 8c +
4c2 mk
in contradiction with (3.14) (recall (3.5)).
Proposition 3.4. Under the hypotheses of Theorem 1.2 functional F satisfies the Palais–Smale condition in Ω 1 (x0 , x1 ). Proof. Let (xk )k ⊂ Ω 1 (x0 , x1 ) be such that (F (xk ))k
is bounded and
lim F 0 (xk ) = 0.
(3.15)
k→+∞
From Lemma 3.3 and (3.15) we have that (kx˙k k)k is bounded. It is also easy to see that sup {d (xk (s) , x0 ) | s ∈ I, k ∈ N} < +∞. H 1 (I, R N ).
(3.16) H 1 (I, R N )
Hence, there exists x ∈ such that, up to subsequences, we Furthermore, (xk )k is bounded in have xk * x weakly in H 1 I, R N and xk → x uniformly in I . Since M0 is complete, x ∈ Ω 1 (x0 , x1 ). What remains to be proved is that this convergence is also strong in Ω 1 (x0 , x1 ). For simplicity consider z k = (xk , tk ) and Tk = F (xk ) with tk = ΦTk (xk ). Since, by the definition of ΦTk and (3.6), sequence (tk )k is bounded in H 1 (I, R), it converges weakly to a certain t ∈ WT . So, if we set τk = tk − t, clearly we have τk * 0
weakly in H01 (I, R) ,
(3.17)
up to subsequences. By [6, Lemma 2.1] there exist two sequences (ξk )k , (νk )k ⊂ H 1 I, R N such that ξk ∈ Txk Ω 1 (x0 , x1 ) , ξk * 0
weakly in H 1
xk − x = ξk + νk for all k ∈ R, and νk → 0 strongly in H 1 I, R N . I, R N
(3.18)
On the other hand, by (3.15) we have F 0 (xk ) [ξk ] = o (1) , and Remark 2.2 implies ∂H ∂H (xk , Tk ) [ξk ] + (xk , Tk ) [τk ] F 0 (xk ) [ξk ] = 0, ∂x ∂t so by (3.17) and (3.19) it follows that ∂H (xk , Tk ) [ξk ] = o (1) . ∂x Thus, as we have ∂H (xk , Tk ) [ξk ] = 2JT0 k (xk ) [ξk ] , ∂x we also have JT0 k (xk ) [ξk ] = o (1) .
(3.19)
312
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
Hence, reasoning as in [11, Lemma 3.4.1], as JT0 k (xk ) [ξk ] = f T0 k (xk , tk ) [(ξk , 0)] , we obtain o (1) = f T0 k (xk , tk ) [(ξk , 0)] Z 1 Z 1 Z 1 Z
0 1 1 0 hδ (xk ) , ξ˙k it˙k ds − δ (xk ) [ξk ] , x˙k t˙k ds + hx˙k , ξ˙k ids + = β (xk ) [ξk ] t˙k2 ds. 2 0 0 0 0
Since sequence t˙k is bounded, from (3.18) it follows that k
1
Z
δ (xk ) [ξk ] , x˙k t˙k ds − 0
0
1 2
1
Z 0
β 0 (xk ) [ξk ] t˙k2 ds = o (1) ,
so we have Z 1 Z 1 hx˙k , ξ˙k ids + hδ (xk ) , ξ˙k it˙k ds = o (1) . 0
0
But xk → x uniformly in I and so, for (3.16), we have that δ(xk ) → δ(x) uniformly in I and then Z 1 Z 1 Z 1 ˙ ˙ hδ (xk ) − δ (x) , ξk it˙k ds + hδ (x) , ξ˙k it˙k ds = o (1) . hδ (xk ) , ξk it˙k ds = 0
0
0
From which we obtain Z 1 hx˙k , ξ˙k ids = o (1) , 0
and, by applying (3.18) again, Z 1 hξ˙k , ξ˙k ids = o (1) , 0
so ξk → 0 strongly in H 1 I, R N . Hence, sequence (xk )k strongly converges to x.
Proof of Theorem 1.2. In order to prove Theorem 1.2 we have to verify the hypotheses of Theorem 2.6. Lemma 3.3 ensures that F is bounded from below. In fact, if this does not hold, a sequence (xk )k exists such that F (xk ) → −∞. Then, (3.4) must hold in order not to contradict Lemma 3.3. So, by Proposition 3.4 we can apply Theorem 2.6 to functional F in the complete Riemannian manifold Ω 1 (x0 , x1 ) obtaining that F has at least a critical point. Moreover, if M0 is not contractible in itself, then Proposition 2.7 implies that functional F has infinitely many critical points (xk )k ⊂ Ω 1 (x0 , x1 ) such that lim F (xk ) = +∞;
k→+∞
hence by Theorem 2.1 there exists a sequence of geodesics (z n = (xn , tn ))n such that tn (1) % +∞ as n % +∞. Furthermore, by reasoning in the same way, Remark 2.3 allows us to complete the proof by finding geodesics with negative arrival time. Acknowledgments The author wishes to thank Prof. Anna Maria Candela for her very valuable hints and support in discussing this problem. References [1] R. Bartolo, A.M. Candela, J.L. Flores, Geodesic connectedness of stationary spacetimes with optimal growth, J. Geom. Phys. 56 (2006) 2025–2038. [2] R. Bartolo, A.M. Candela, J.L. Flores, M. S´anchez, Geodesics in static Lorentzian manifolds with critical quadratic behavior, Adv. Nonlinear Stud. 3 (2003) 471–494.
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
313
[3] A.M. Candela, A. Salvatore, Light rays joining two submanifolds in space-times, J. Geom. Phys. 22 (1997) 281–297. [4] A.M. Candela, A. Salvatore, Normal geodesics in stationary Lorentzian manifolds with unbounded coefficients, J. Geom. Phys. 44 (2002) 171–195. [5] J.K. Beem, P.E. Ehrlich, K.L. Easley, Global Lorentzian Geometry, in: Monographs Textbooks Pure Appl. Math., vol. 202, Dekker Inc., New York, 1996. [6] V. Benci, D. Fortunato, On the existence of infinitely many geodesics on space-time manifolds, Adv. Math. 105 (1994) 1–25. [7] E. Fadell, S. Husseini, Category of loop spaces of open subsets in Euclidean space, Nonlinear Anal. TMA 17 (1991) 1153–1161. [8] 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. [9] F. Giannoni, A. Masiello, On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal. 101 (1991) 340–369. [10] V. Luisi, Light rays in static spacetimes with critical asymptotic behavior. A variational approach, Electron. J. Differential Equations 114 (2006) 1–9. [11] A. Masiello, Variational Methods in Lorentzian Geometry, in: Pitman Res. Notes Math. Ser., vol. 309, Longman Sci. Tech., Harlow, 1994. [12] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Inc., New York, 1983. [13] R.S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963) 299–340. [14] M. Struwe, Variational Methods and Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edition, Springer, Berlin, 1990.
Lightlike rays in stationary spacetimes with critical growth Valeria Luisi Dipartimento di Matematica, Universit`a degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy Received 27 February 2007; accepted 24 May 2007
Abstract The aim of this paper is to extend some previous results on the existence of lightlike geodesics joining a point to a line to the case of stationary Lorentzian manifolds whose metric coefficients have an optimal growth. c 2007 Elsevier Ltd. All rights reserved.
MSC: 53C50; 58E05; 58E10 Keywords: Stationary Lorentzian manifold; Lightlike geodesic; Asymptotic behavior; Fermat principle; Ljusternik–Schnirelman category
1. Introduction and the main result In order to introduce the main result of this paper, let us recall some useful definitions. A Lorentzian manifold is a couple (M, h·, ·iz ) where M is a smooth connected finite dimensional manifold and h·, ·iz is a Lorentzian metric, that is a smooth symmetric (0, 2) tensor field which induces a bilinear form of index 1 on the tangent space of each point of M. In a Lorentzian manifold (M, h·, ·iz ), a geodesic is a smooth curve z : I → M which solves the equation Ds z˙ (s) = 0
for all s ∈ I,
where Ds denotes the covariant derivative along z induced by the Levi-Civita connection of h·, ·iz and I is a real interval. It is known that, if z = z(s) is a geodesic on M, then a constant E(z) ∈ R exists such that E(z) ≡ h˙z (s), z˙ (s)iz
for all s ∈ I.
The sign of E(z) determines the causal character of z; more precisely, z is called timelike, lightlike or spacelike if E(z) is negative, null or positive, respectively. From a physical point of view, lightlike geodesics play an important role in the study of gravitational fields. In fact, they represent light rays in gravitational fields so it is useful to investigate their existence. On the other hand, from a variational point of view, this means that we should look for information on the zero critical level of the action functional E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.05.019
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
f (z) =
Z
305
h˙z , z˙ iz ds I
on a suitable manifold of curves. But, in general, this is quite hard. Anyway, some results can be obtained in some model spacetimes. Definition 1.1. A Lorentzian manifold M, h·, ·iz is a (standard) stationary spacetime if there exists a finite dimensional connected Riemannian manifold (M0 , h·, ·i) such that M = M0 × R and h·, ·iz is given by hζ, ζ iz = hξ, ξ i + 2 hδ (x) , ξ i τ − β (x) τ 2
(1.1)
for any z = (x, t) ∈ M0 × R and ζ = (ξ, τ ) ∈ Tz M ≡ Tx M0 × R, where δ : M0 → T M0 is a smooth vector field and β : M0 → R is a smooth and strictly positive scalar field. If δ ≡ 0, M is called (standard) static. A classical example of such a spacetime is the Kerr one which represents the gravitational field outside a rotating object (for more details, see [5] or [12]). Here, for M = M0 × R a stationary spacetime, our purpose is to study the problem of the existence of lightlike geodesics joining a point z 0 = (x0 , t0 ) to a line γ = {x1 } × R, fixing x0 , x1 ∈ M0 and t0 ∈ R, i.e., to solve Ds z˙ (s) = 0 ∀s ∈ I, h˙z (s) , z˙ (s)iz = 0 ∀s ∈ I, (P1 ) x (0) = x0 , x (1) = x1 , t (0) = t0 . The following result can be stated. Theorem 1.2. Let M = M0 × R be a stationary spacetime endowed with metric (1.1). Suppose that: (H1 ) (M0 , h·, ·i) is a complete smooth (at least C 3 ) n-dimensional Riemannian manifold; (H2 ) β has an asymptotic quadratic behavior while δ has a linear growth, that is there exist λ, µ ≥ 0, k1 , k2 ∈ R and a point x¯ ∈ M0 such that for all x ∈ M0 we have 0 < β (x) ≤ λd 2 (x, x) ¯ + k1 , p hδ (x) , δ (x)i ≤ µd (x, x) ¯ + k2 ,
(1.2) (1.3)
where d (·, ·) is the distance induced on M0 by its Riemannian metric h·, ·i. Then, fixing an event z 0 = (x0 , t0 ) and a point x1 ∈ M0 , there exist at least two non-trivial lightlike geodesics joining z 0 to the vertical line γ = {x1 } × R if x0 6= x1 . Furthermore, in itself, then there exist two sequences of such lightlike geodesics z n+ = − if M−0 is− non-contractible + + + xn , tn , z n = xn , tn such that tn (1) % +∞ and tn− (1) & −∞ as n % +∞. The problem of searching for lightlike geodesics joining a point to a line was first studied by Fortunato, Giannoni and Masiello in [8] by introducing a variational principle similar to the Fermat one which allowed them to solve problem (P1 ) under the hypotheses 0 < ν ≤ β (x) ≤ M for all x ∈ M0 , sup {hδ (x) , δ (x)i : x ∈ M0 } < +∞, for some ν, M > 0. Then, this result was extended to the search for light rays joining two submanifolds in stationary spacetimes (see [3, Theorem 1.1]). Later on, this outcome was improved by Candela and Salvatore in [4] by assuming that coefficient β grows at most subquadratically and δ at most sublinearly. These assumptions were bettered in [10] at least for in static spacetimes. In fact, in such a manifold even if coefficient β grows quadratically light rays exist. Here, we want to extend this result to the stationary case as growth conditions (1.2) and (1.3) allow one to prove geodesic connectedness (see [1, Theorem 1.2]). Let us point out that in this paper, as in [1], there has been removed the hypothesis of β being far away from zero.
306
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
2. Variational tools Let M = M0 × R be a stationary Lorentzian manifold equipped with metric (1.1) and fix z 0 = x0, t0 ∈ M, γ = {x1 } × R with x0 , x1 ∈ M0 , t0 ∈ R. By the Nash Embedding Theorem, if M0 is a C 3 n-dimensional Riemannian manifold, we can consider it as a submanifold of a suitable Euclidean space R N so that h·, ·i is the restriction to M0 of its Euclidean metric. Since geodesics are independent by affine reparametrizations, without loss of generality, we can take I = [0, 1] and t0 = 0. Obviously, in assumption (H2 ) we can take x¯ = x0 , so that (1.2) and (1.3) can be replaced by β (x) ≤ λd 2 (x, x0 ) + 1
for all x ∈ M0
(2.1)
and p
hδ (x) , δ (x)i ≤
√
λd (x, x0 ) + 1
for all x ∈ M0 .
(2.2)
H 1 (I, M0 ) can be easily identified with the Sobolev space of the absolutely continuous curves
Space x : I → RN whose derivative is square summable and such that x (I ) ⊂ M0 . Such a space can be endowed with the norm Z 1 Z 1 kxk2 = hx, hx, xi ds. ˙ xi ˙ ds + 0
0
Ω 1 (x
0 , x 1 ),
the set of H 1 -curves in M0 joining x0 to x1 and defined in I ; it can be proved that o Ω 1 (x0 , x1 ) ≡ x ∈ H 1 (I, R N ) : x (I ) ⊂ M0 , x (0) = x0 , x (1) = x1 .
Let us consider
n
If M0 is complete, then Ω 1 (x0 , x1 ) is a complete Riemannian manifold (see [13]) and its tangent space is given by Tx Ω 1 (x0 , x1 ) = {ξ ∈ H 1 (I, T M0 ) : ξ(s) ∈ Tx(s) M0 ∀ s ∈ I, ξ(0) = ξ(1) = 0}. Finally, we can define W0 , the subspace of H 1 (I, R) of those curves t = t (s) such that t (0) = 0. Clearly, for all t ∈ W0 we have Tt W0 = W0 . As already remarked, problem (P1 ) has a variational structure and it is easy to prove that z = x, ¯ t¯ is a lightlike geodesic joining z 0 = x0, 0 to γ if and only if it is a critical point of the C 1 functional Z 1 Z Z 1 1 1 1 ˙ hδ (x) , xi hx, (2.3) β (x) t˙2 ds, ˙ xi ˙ ds + ˙ t ds − f (z) = 2 0 2 0 0 with critical level f (z) = 0, where f 0 (z) [ζ ] = f 0 (z) [(ξ, τ )] Z 1 Z = hx, ˙ ξ˙ ids + 0
1
δ 0 (x) [ξ ] , x˙ t˙ds +
0 1
Z + 0
hδ (x) , xi ˙ τ˙ ds −
1
Z
hδ (x) , ξ˙ it˙ds
0
1 2
1
Z 0
β 0 (x) [ξ ] t˙2 ds −
1
Z
β (x) t˙τ˙ ds
(2.4)
0
for all z = (x, t) ∈ Z ≡ Ω 1 (x0 , x1 ) × W0 , ζ = (ξ, τ ) ∈ Tz Z ≡ Tx Ω 1 (x0 , x1 ) × W0 . But, as seen in the static case, the functional defined in (2.3) is unbounded both from above and from below. For this reason, Fortunato, Giannoni and Masiello in [8] stated a new variational principle similar to the Fermat one so as to introduce a new functional, arrival time T = T (x), which is bounded from below on Riemannian manifold Ω 1 (x0 , x1 ). Defining F : Ω 1 (x0 , x1 ) → R as v !Z u Z 1 Z 1 Z 1 1 1 u hδ(x), xi ˙ hδ(x), xi ˙ 2 t F(x) = ds + hx, ˙ xids ˙ + ds ds, β(x) β(x) 0 0 0 β(x) 0 the following variational principle can be stated.
307
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
Theorem 2.1 (Fermat Principle). Let z ∗ : I → M, z ∗ = z ∗ (s), be a smooth curve such that z ∗ = (x ∗ , t ∗ ). Then, the following statements are equivalent: (a) z ∗ is a solution of problem (P1 ) with arrival time t ∗ (1) = T > 0; (b) x ∗ is a critical point of functional F with critical level T = F (x ∗ ) > 0 and, for all s ∈ I , we have t ∗ (s) = ΦT (x ∗ ) (s) with Z s hδ (x ∗ (σ )) , x˙ ∗ (σ )i dσ ΦT x ∗ (s) = β (x ∗ (σ )) 0 !Z !−1 Z 1 Z 1 s hδ (x ∗ ) , x˙ ∗ i 1 1 ds dσ ds . + T− ∗ ∗ β (x ∗ ) 0 β (x (σ )) 0 β (x ) 0 Proof. Its proof is in [8, Theorem 2.3] (and also in [11, Theorem 6.2.2]). As they are useful in the following, we outline its main tools. Fixed T ∈ R, define WT = {t ∈ W0 | t (1) = T } , which is an affine submanifold of H 1 (I, R) whose tangent space is H01 (I, R) = τ ∈ H 1 (I, R) | τ (0) = τ (1) = 0 . If f T is the restriction of functional f to Z T = Ω 1 (x0 , x1 ) × WT , then, it can be proved that z ∗ = (x ∗ , t ∗ ) is a critical point of functional f T in Z T , i.e. it solves (P1 ) with arrival time T , if and only if x ∗ is a critical point of JT (x) = f T (x, ΦT (x)) ,
x ∈ Ω 1 (x0 , x1 ) ,
(2.5)
with JT (x ∗ ) = 0 and t ∗ = ΦT (x ∗ ), where ΦT : Ω 1 (x0 , x1 ) → WT is defined above (see [9] or [11, Section 3.3]). Thus, our problem is reduced to the search for a couple (x, T ) ∈ Ω 1 (x0 , x1 ) × R, for example with T > 0, a solution of the problem (∂ H (x, T ) = 0, ∂x H (x, T ) = 0, with H (x, T ) = 2JT (x); hence, it has to be an x critical point of F with critical level T = F (x).
Remark 2.2. By differentiating the map x ∈ Ω 1 (x0 , x1 ) 7−→ H (x, F (x)) ∈ R we have ∂H ∂H (x, F (x)) + (x, F (x)) F 0 (x) = 0 ∂x ∂t
for all x ∈ Ω 1 (x0 , x1 ).
(2.6)
Remark 2.3. In the proof of Theorem 2.1, the choice of T > 0 can be replaced with T < 0 obtaining that z ∗ = (x ∗ , t ∗ ) is a solution of problem (P1 ) with arrival time t ∗ (1) = T < 0 if and only if x ∗ is a critical point of the functional v !Z u Z 1 Z 1 Z 1 1 1 u hδ (x) , xi hδ (x) , xi ˙ ˙ 2 t ˜ hx, F (x) = ds − ˙ xi ˙ ds + ds ds β (x) β (x) 0 0 0 β (x) 0 with critical level T = F˜ (x ∗ ) < 0. In order to prove Theorem 1.2, the following abstract theory is useful (for more details, see [14]). Definition 2.4. Let X be a topological space and A ⊆ X . The Ljusternik–Schnirelman category of A in X (cat X A) is the least number of closed and contractible subsets of X covering A. If this is not possible we say that cat X A = +∞. We define cat X = cat X X .
308
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
Definition 2.5. Let Ω be a Riemannian manifold and f a C 1 functional on Ω . f is said to satisfy the Palais–Smale condition if any (xn )n ⊂ Ω such that ( f (xn ))n
is bounded and
lim
n→+∞
f 0 (xn ) = 0
converges in Ω up to subsequences. Theorem 2.6 (Ljusternik–Schnirelman). Let Ω be a complete Riemannian manifold and f : Ω → R a C 1 functional. If f satisfies the Palais–Smale condition and is bounded from below, then f attains its infimum and has at least cat Ω critical points. Furthermore, if supΩ F = +∞ and cat Ω = +∞ there exists a sequence of critical points (xn )n ⊂ Ω such that F (xn ) % +∞. Finally, in order to estimate the Ljusternik–Schnirelman category of Ω 1 (x0 , x1 ), we need the following result (see [7]). Proposition 2.7 (Fadell–Husseini). If M0 is a manifold not contractible in itself, then for all x0 , x1 ∈ M0 the manifold of curves Ω 1 (x0 , x1 ) has infinite category and possesses compact subsets of arbitrarily high category. 3. Proof of Theorem 1.2 To prove the main theorem we want to check that functional F satisfies the hypotheses of Theorem 2.6. To this end, the following lemmas are needed. Lemma 3.1. Let β satisfy assumption (H2 ). If (xk )k ⊂ Ω 1 (x0 , x1 ) is such that kx˙k k → +∞, then Z 1 kx˙k k2 ds → +∞ as k → +∞, 0 β (x k ) R1 where kx˙k k2 = 0 hx˙k , x˙k i ds. Proof. See [2, Proposition 4.1].
(3.1)
Lemma 3.2. Let ( f k )k be a sequence of non-negative functions, 0 6= f k ∈ L 2 ([0, ak ]), such that c > 0 exists which satisfies 1/2 Z ak Z ak √ f k2 (r ) dr c+ f k (r ) dr > ak for all k ∈ N. (3.2) 0
0
Define ak
Z mk =
f k (r ) dr
and
0
mk . Dk = r ∈ [0, ak ] : f k (r ) ≥ 2ak
If the measure of Dk , |Dk |, is written as |Dk | = k ak , then it results that k > 1 −
8c 4c2 − 2 mk mk
Proof. See [1, Lemma 2.5].
for all k ∈ N.
Lemma 3.3. Under the hypotheses of Theorem 1.2 there cannot exist a sequence (xk )k ⊂ Ω 1 (x0 , x1 ) satisfying both (F (xk ))k
is bounded from above
(3.3)
and kx˙k k → +∞ as k → +∞.
(3.4)
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
309
Proof. Arguing by contradiction, assume the existence of a sequence (xk )k ⊂ Ω 1 (x0 , x1 ) such that both (3.3) and (3.4) hold. We denote by Z 1 Z 1 |hδ (xk ) , x˙k i| 1 ds and mk = ds. ak = β β (xk ) (x ) k 0 0 If hδ (xk ) , x˙k i ds ≥ 0 β (xk )
1
Z 0
for all k large enough,
then for such k’s we have s Z Z 1 hx˙k , x˙k i ds F (xk ) ≥
1 0
0
1 ds = β (xk )
s
1
Z 0
kx˙k k2 ds β (xk )
which goes to infinity for Lemma 3.1 in contradiction with (3.3). Similarly, if (m k )k is bounded from above, then Lemma 3.1 contradicts (3.3). So, let us consider the case in which lim m k = +∞
(3.5)
k→+∞
(up to subsequences). Thus, (3.3) implies that a constant c > 0 exists such that s Z 1 hδ (xk ) , x˙k i2 2 c ≥ F (xk ) ≥ −m k + ak kx˙k k + ak ds, β (xk ) 0
(3.6)
which implies s
1
Z
c + mk ≥ 0
hδ (xk ) , x˙k i2 √ ds ak . β (xk )
(3.7)
For the Cauchy–Schwarz inequality we obtain !1/2 Z 1 Z 1 |hδ (xk ) , x˙k i| hδ (xk ) , x˙k i2 √ ak , ds ≤ ds β β ) ) (x (x k k 0 0 and thus, Z ak 0
1
hδ (xk ) , x˙k i2 ds ≥ m 2k ; β (xk )
hence, by (3.6) we have q c + m k ≥ ak kx˙k k2 + m 2k from which we have c2 + 2 cm k ≥ ak kx˙k k2 ; whence, mk ak for a suitable A > 0. On the other hand, defining dxk f k (r ) = δ (xk (s (r ))) , (s (r )) 6= 0 ds kx˙k k2 ≤ A
(3.7) and (3.9) imply (3.2).
(3.8)
with dr =
ds . β (xk (s))
(3.9)
310
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
Then, Lemma 3.2 gives k > 1 −
8c 4c2 − 2, mk mk
(3.10)
where k is such that |Dk | = k ak
and
dxk mk ; Dk = r ∈ [0, ak ] : δ (xk (s (r ))) , (s (r )) ≥ ds 2ak
or, equivalently, Z ds = k a k D˜ k β (x k (s))
(3.11)
with mk . D˜ k = s ∈ I : |hδ (xk (s)) , x˙k (s)i| ≥ 2ak
(3.12)
Therefore, from (2.2) and (3.12) we have on D˜ k √ √ λ length (xk (s)) + 1 |x˙k (s)| ≥ λd (xk (s) , x0 ) + 1 |x˙k (s)| ≥ |δ (xk (s))| |x˙k (s)| ≥ |hδ (xk (s)) , x˙k (s)i| ≥
mk , 2ak
where length (xk (s)) is the length of the portion of the curve xk ([0, s]). From the last inequality and (3.8) for all k we have Z Z ds 4ak mk |x˙k (s)|2 ds ≤ 4A. 2 ≤ ak D˜ k √ m k D˜ k λ length (xk (s)) + 1
(3.13)
On the other hand, for the Cauchy–Schwarz inequality and (3.8) we have r mk √ length (xk (s)) ≤ A s for all s ∈ I, ak and thus mk ak
Z 0
1
ds √
2 ≥
λ length (xk (s)) + 1
mk ak
1
Z 0
=
2 lg λA
ds q
2
λA makk s + 1
r
λA
which goes to infinity for (3.4) and (3.8). And so, for (3.13) we must have Z ds mk lim 2 = +∞, k→+∞ ak D˜ c √ k λ length (xk (s)) + 1 as D˜ kc = I \ D˜ k . On the other hand, defining Z ds mk dk = 2 , ak D˜ kc √ λ length (xk (s)) + 1
mk 1 +1 + q − 1 , mk ak λA ak + 1
(3.14)
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
(2.1) and (3.11) imply Z (1 − k ) ak =
D˜ kc
311
ds ak ≥ dk , β (xk ) mk
from which we have dk 1 − k ≥ . mk Thus, (3.10) implies dk ≤ (1 − k ) m k < 8c +
4c2 mk
in contradiction with (3.14) (recall (3.5)).
Proposition 3.4. Under the hypotheses of Theorem 1.2 functional F satisfies the Palais–Smale condition in Ω 1 (x0 , x1 ). Proof. Let (xk )k ⊂ Ω 1 (x0 , x1 ) be such that (F (xk ))k
is bounded and
lim F 0 (xk ) = 0.
(3.15)
k→+∞
From Lemma 3.3 and (3.15) we have that (kx˙k k)k is bounded. It is also easy to see that sup {d (xk (s) , x0 ) | s ∈ I, k ∈ N} < +∞. H 1 (I, R N ).
(3.16) H 1 (I, R N )
Hence, there exists x ∈ such that, up to subsequences, we Furthermore, (xk )k is bounded in have xk * x weakly in H 1 I, R N and xk → x uniformly in I . Since M0 is complete, x ∈ Ω 1 (x0 , x1 ). What remains to be proved is that this convergence is also strong in Ω 1 (x0 , x1 ). For simplicity consider z k = (xk , tk ) and Tk = F (xk ) with tk = ΦTk (xk ). Since, by the definition of ΦTk and (3.6), sequence (tk )k is bounded in H 1 (I, R), it converges weakly to a certain t ∈ WT . So, if we set τk = tk − t, clearly we have τk * 0
weakly in H01 (I, R) ,
(3.17)
up to subsequences. By [6, Lemma 2.1] there exist two sequences (ξk )k , (νk )k ⊂ H 1 I, R N such that ξk ∈ Txk Ω 1 (x0 , x1 ) , ξk * 0
weakly in H 1
xk − x = ξk + νk for all k ∈ R, and νk → 0 strongly in H 1 I, R N . I, R N
(3.18)
On the other hand, by (3.15) we have F 0 (xk ) [ξk ] = o (1) , and Remark 2.2 implies ∂H ∂H (xk , Tk ) [ξk ] + (xk , Tk ) [τk ] F 0 (xk ) [ξk ] = 0, ∂x ∂t so by (3.17) and (3.19) it follows that ∂H (xk , Tk ) [ξk ] = o (1) . ∂x Thus, as we have ∂H (xk , Tk ) [ξk ] = 2JT0 k (xk ) [ξk ] , ∂x we also have JT0 k (xk ) [ξk ] = o (1) .
(3.19)
312
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
Hence, reasoning as in [11, Lemma 3.4.1], as JT0 k (xk ) [ξk ] = f T0 k (xk , tk ) [(ξk , 0)] , we obtain o (1) = f T0 k (xk , tk ) [(ξk , 0)] Z 1 Z 1 Z 1 Z
0 1 1 0 hδ (xk ) , ξ˙k it˙k ds − δ (xk ) [ξk ] , x˙k t˙k ds + hx˙k , ξ˙k ids + = β (xk ) [ξk ] t˙k2 ds. 2 0 0 0 0
Since sequence t˙k is bounded, from (3.18) it follows that k
1
Z
δ (xk ) [ξk ] , x˙k t˙k ds − 0
0
1 2
1
Z 0
β 0 (xk ) [ξk ] t˙k2 ds = o (1) ,
so we have Z 1 Z 1 hx˙k , ξ˙k ids + hδ (xk ) , ξ˙k it˙k ds = o (1) . 0
0
But xk → x uniformly in I and so, for (3.16), we have that δ(xk ) → δ(x) uniformly in I and then Z 1 Z 1 Z 1 ˙ ˙ hδ (xk ) − δ (x) , ξk it˙k ds + hδ (x) , ξ˙k it˙k ds = o (1) . hδ (xk ) , ξk it˙k ds = 0
0
0
From which we obtain Z 1 hx˙k , ξ˙k ids = o (1) , 0
and, by applying (3.18) again, Z 1 hξ˙k , ξ˙k ids = o (1) , 0
so ξk → 0 strongly in H 1 I, R N . Hence, sequence (xk )k strongly converges to x.
Proof of Theorem 1.2. In order to prove Theorem 1.2 we have to verify the hypotheses of Theorem 2.6. Lemma 3.3 ensures that F is bounded from below. In fact, if this does not hold, a sequence (xk )k exists such that F (xk ) → −∞. Then, (3.4) must hold in order not to contradict Lemma 3.3. So, by Proposition 3.4 we can apply Theorem 2.6 to functional F in the complete Riemannian manifold Ω 1 (x0 , x1 ) obtaining that F has at least a critical point. Moreover, if M0 is not contractible in itself, then Proposition 2.7 implies that functional F has infinitely many critical points (xk )k ⊂ Ω 1 (x0 , x1 ) such that lim F (xk ) = +∞;
k→+∞
hence by Theorem 2.1 there exists a sequence of geodesics (z n = (xn , tn ))n such that tn (1) % +∞ as n % +∞. Furthermore, by reasoning in the same way, Remark 2.3 allows us to complete the proof by finding geodesics with negative arrival time. Acknowledgments The author wishes to thank Prof. Anna Maria Candela for her very valuable hints and support in discussing this problem. References [1] R. Bartolo, A.M. Candela, J.L. Flores, Geodesic connectedness of stationary spacetimes with optimal growth, J. Geom. Phys. 56 (2006) 2025–2038. [2] R. Bartolo, A.M. Candela, J.L. Flores, M. S´anchez, Geodesics in static Lorentzian manifolds with critical quadratic behavior, Adv. Nonlinear Stud. 3 (2003) 471–494.
V. Luisi / Nonlinear Analysis 69 (2008) 304–313
313
[3] A.M. Candela, A. Salvatore, Light rays joining two submanifolds in space-times, J. Geom. Phys. 22 (1997) 281–297. [4] A.M. Candela, A. Salvatore, Normal geodesics in stationary Lorentzian manifolds with unbounded coefficients, J. Geom. Phys. 44 (2002) 171–195. [5] J.K. Beem, P.E. Ehrlich, K.L. Easley, Global Lorentzian Geometry, in: Monographs Textbooks Pure Appl. Math., vol. 202, Dekker Inc., New York, 1996. [6] V. Benci, D. Fortunato, On the existence of infinitely many geodesics on space-time manifolds, Adv. Math. 105 (1994) 1–25. [7] E. Fadell, S. Husseini, Category of loop spaces of open subsets in Euclidean space, Nonlinear Anal. TMA 17 (1991) 1153–1161. [8] 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. [9] F. Giannoni, A. Masiello, On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal. 101 (1991) 340–369. [10] V. Luisi, Light rays in static spacetimes with critical asymptotic behavior. A variational approach, Electron. J. Differential Equations 114 (2006) 1–9. [11] A. Masiello, Variational Methods in Lorentzian Geometry, in: Pitman Res. Notes Math. Ser., vol. 309, Longman Sci. Tech., Harlow, 1994. [12] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Inc., New York, 1983. [13] R.S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963) 299–340. [14] M. Struwe, Variational Methods and Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edition, Springer, Berlin, 1990.