\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 114, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/114\hfil Light rays in static spacetimes]
{Light rays in static spacetimes
with critical asymptotic behavior: A variational approach}
\author[V. Luisi\hfil EJDE-2006/114\hfilneg]
{Valeria Luisi}

\address{Valeria Luisi \hfill\break
Dipartimento di Matematica,
Universit\`a degli Studi di Bari \\
Via E. Orabona 4, 70125 Bari, Italy}
\email{vluisi@dm.uniba.it}

\date{}
\thanks{Submitted November 28, 2005. Published September 21, 2006.}
\subjclass[2000]{53C50, 58E05, 58E10}
\keywords{Static spacetime; light ray;
 quadratic asymptotic behavior; \hfill\break\indent
 Fermat principle; Ljusternik-Schnirelman category}

\begin{abstract}
 Let $\mathcal{M}=\mathcal{M}_{0}\times \mathbb{R}$ be a Lorentzian
 manifold equipped with the static metric
 $\langle \cdot ,\cdot \rangle _{z}=\langle \cdot ,\cdot \rangle
 -\beta (x)dt^{2}$.
 The aim of this paper is investigating the existence of lightlike
 geodesics joining a point $z_{0}=(x_{0},t_{0})$ to a line
 $\gamma =\{ x_{1}\} \times \mathbb{R}$ when coefficient
 $\beta $ has a quadratic asymptotic behavior by means of a variational
 approach.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction and main result}

The aim of this paper is investigating the existence of lightlike geodesics
in suitable semi-Riemannian manifolds by using variational tools and
topological methods.
First of all, we recall the main definitions.

A \emph{Lorentzian manifold} is a couple $(\mathcal{M},\langle
\cdot ,\cdot \rangle _{z})$ where $\mathcal{M}$ is a smooth
connected finite-dimensional manifold and $\langle \cdot ,\cdot
\rangle _{z}$ is a Lorentzian metric, that is a smooth symmetric
$(0,2)$ tensor field which induces on the tangent space of each
point of $\mathcal{M}$ a bilinear form of index $1$.

The importance of the study of these manifolds comes from General
Relativity, since some 4-dimensional Lorentzian manifolds are solutions of
Einstein's equations. Differently from a Riemannian metric, a Lorentzian one
does not induce a positive definite bilinear form on its tangent space, thus
each one of its tangent vectors $v\neq 0$ can be \emph{timelike},
 \emph{lightlike} or \emph{spacelike} if the scalar product $\langle
v,v\rangle _{z}$ is negative, null or positive, respectively, while
$v=0$ is always spacelike.

In order to have informations on the geometry of a Lorentzian manifold, it
is important to look for its geodesics and, similarly to the definition
given for a geodesic in a Riemannian manifold, the following definition can
be stated.

\begin{definition} \rm
Let $\mathcal{M}$ be a smooth finite dimensional manifold equipped with a
Lorentzian metric $\langle \cdot ,\cdot \rangle _{z}$. A geodesic
in $\mathcal{M}$ is a smooth curve $z:I\to \mathcal{M}$ which solves
the equation
\begin{equation*}
D_{s}\dot{z}(s)=0\quad \text{for all }s\in I,
\end{equation*}
where $D_{s}$ denotes the covariant derivative along $z$ induced by the
Levi-Civita connection of $\langle \cdot ,\cdot \rangle _{z}$ and
$I$ is a real interval.
\end{definition}

Let $z=z(s)$ be a geodesic on $\mathcal{M}$. It is easy to
check that there exists a constant $E(z)\in \mathbb{R}$ such
that
\begin{equation*}
\langle \dot{z}(s),\dot{z}(s)\rangle
_{z}\equiv E(z)\quad \text{for all }s\in I.
\end{equation*}
So, all the tangent vectors $\dot{z}(s)$, $s\in I$, have the
same causal character and a geodesic $z=z(s)$ is
\emph{timelike}, \emph{lightlike} or \emph{spacelike} if $E(z)$
is negative, null or positive, respectively.

From a physical point of view the most significant geodesics are the
timelike and the lightlike ones, named \emph{causal geodesics}. In
particular, in General Relativity gravitational fields can be descrived by
means of suitable Lorentzian manifolds in which lightlike geodesics allow
one to represent light rays.

In general, the study of geodesics in Lorentzian manifolds is not possible
up to consider special models. Here, we deal with (standard) static
manifolds defined as follows.

\begin{definition} \rm
A Lorentzian manifold $(\mathcal{M},\langle \cdot ,\cdot \rangle _{z})$
is called (standard) static if there exists a
finite dimensional Riemannian manifold
$(\mathcal{M}_{0},\langle \cdot ,\cdot \rangle )$ such that
$\mathcal{M}=\mathcal{M} _{0}\times \mathbb{R}$ and
$\langle \cdot ,\cdot \rangle _{z}$ is given by
\begin{equation}
\langle \zeta ,\zeta \rangle _{z}=\langle \xi ,\xi
\rangle -\beta (x)\tau ^{2}  \label{metric}
\end{equation}
for any $z=(x,t)\in \mathcal{M}_{0}\times \mathbb{R}$ and
$\zeta =(\xi ,\tau )\in T_{z}\mathcal{M}\equiv T_{x}\mathcal{M}
_{0}\times \mathbb{R}$, where $\beta :\mathcal{M}_{0}\to \mathbb{R}$
is a smooth and strictly positive scalar field.
\end{definition}

Physically interesting examples of static spacetimes are anti-de Sitter,
Schwar-zschild or Reissner-Nordstr\"{o}m ones; in the two-dimensional case,
static spacetimes are essentially equivalent to Generalized Robertson-Walker
ones (for more details, see \cite{Beem}).

Let $\mathcal{M}=\mathcal{M}_{0}\times \mathbb{R}$ be a
Lorentzian manifold endowed with the static metric defined in
\eqref{metric}. Our main result is stated as under the following
hypotheses:
\begin{itemize}
\item[(H1)] $(\mathcal{M}_{0},\langle \cdot ,\cdot
\rangle )$ is a complete $\mathcal{C}^{3}$ $n$-dimensional
Riemannian manifold;

\item[(H2)] $\beta $ has an asymptotic quadratic behaviour, that is
there exist $\lambda \geq 0$, $\mu _{1},\mu _{2}\in \mathbb{R}$ and a point
$\bar{x}\in \mathcal{M}_{0}$ such that
\begin{equation}
\beta (x)\leq \lambda d^{2}(x,\bar{x})+\mu
_{1}d^{p}(x,\bar{x})+\mu _{2}  \label{quadratic}
\end{equation}
where $d(\cdot ,\cdot )$ is the distance induced on $\mathcal{M}
_{0}$ by its Riemannian metric $\langle \cdot ,\cdot \rangle $
and $0\leq p<2$.
\end{itemize}

\begin{theorem}\label{main theorem}
 Suppose that {\rm (H1), (H2)} are satisfied.
Then, there exist at least two non trivial lightlike geodesics
joining the point $z_{0}=(x_{0,}t_{0})$ to the vertical line
$\gamma =\{x_{1}\}\times \mathbb{R}$ if $x_{0}\neq x_{1}$.
Furthermore, if $\mathcal{M}_{0}$ is non contractible in itself, then there
exist two sequences of such lightlike geodesics
$z_{n}^{+}=(x_{n}^{+},t_{n}^{+})$, $z_{n}^{-}=(x_{n}^{-},t_{n}^{-})$
such that $t_{n}^{+}(1)\nearrow +\infty $ and
$t_{n}^{-}(1)\searrow -\infty $ as $n\nearrow +\infty $.
\end{theorem}

In the previous years, the existence of causal geodesics in a
static spacetime has been widely exploited. A first result follows
from Avez-Seifert Theorem
(see \cite[Theorem 2.14]{Beem}), in which it is proved that in a globally
hyperbolic spacetime two causally related points can be joined by a causal
geodesic. In fact, if $\mathcal{M}_{0}$ is a complete Riemannian manifold
and $\beta $ has an (at most) quadratic growth the corresponding static
spacetime is globally hyperbolic and Avez-Seifert Theorem can be applied
(see \cite[Corollary 3.4]{Sanchez2}).

Furthermore, the existence of lightlike geodesics joining a point
$z_{0}=(x_{0},t_{0})$ to a line $\gamma =\{x_{1}\}
\times \mathbb{R}$ has been obtained by means of geometrical tools in
assumptions (H1) and (H2) (see Remark \ref{geometric}).

On the other hand, by using variational methods, the existence and the
multiplicity of such lightlike geodesics have been proved when $\beta $ is
bounded (see \cite[Subsection 6.3]{Mas}) and then when $\beta $ has a
subquadratic growth (see \cite{Capmaspic}). But in \cite{BCFS} the geodesic
connectedness has been guaranteed if $\beta $ grows quadratically and this
condition is optimal as showed by a family of explicit counterexamples (for
more details, see \cite[Section 7]{BCFS}).

Here, we want to improve both the variational results, by using hypothesis
(H2), and the geometric existence one, obtaining a
multiplicity theorem.

\section{Variational tools}

Let $\mathcal{M}=\mathcal{M}_{0}\times \mathbb{R}$ be a static Lorentzian
manifold equipped with metric \eqref{metric}. Furthermore, fix
$z_{0}=(x_{0,}t_{0})\in \mathcal{M}$ and
$\gamma =\{x_{1}\}\times \mathbb{R}\subset \mathcal{M}$.

If $(\mathcal{M}_{0},\langle \cdot ,\cdot \rangle )$
is a $\mathcal{C}^{3}$ $n$-dimensional Riemannian manifold, by the Nash
Embedding Theorem we can assume that $\mathcal{M}_{0}$ is a submanifold of
$\mathbb{R}^{N}$ and $\langle \cdot ,\cdot \rangle $ is the
restriction to $\mathcal{M}_{0}$ of its Euclidean metric, still denoted by
$\langle \cdot ,\cdot \rangle $.

On the other hand, without loss of generality, we can take $I=[0,1
] $, as geodesics are independent by affine reparametrizations, and
$t_{0}=0$, since the coefficient $\beta $ of metric \eqref{metric} does not
depend on the coordinate $t$ (in fact, if $z(s)=(x(
s),t(s))$ is a geodesic and $T>0$ is a real
number, the curve $z_{T}(s)=(x(s),t(
s)+T)$ is still a geodesic).

Hence, our problem is reduced to look for solutions of
\begin{equation} \label{P1}
\begin{gathered}
D_{s}\dot{z}(s)=0\quad \forall \text{ }s\in I, \\
\langle \dot{z}(s),\dot{z}(s)\rangle
_{z}=0\quad \forall \, s\in I, \\
x(0)=x_{0},\quad x(1)=x_{1},\quad t(0)=0.
\end{gathered}
\end{equation}
Let $H^{1}(I,\mathbb{R}^{N})$ be the Sobolev space of the
absolutely continuous curves $x=x(s)$ whose derivative is
square summable. Such a space can be endowed with the norm
\begin{equation*}
\|x\|^{2}=\int_{0}^{1}\langle \dot{x},\dot{x}
\rangle ds+\int_{0}^{1}\langle x,x\rangle ds.
\end{equation*}
If $\Omega ^{1}(x_{0},x_{1})$ is the set of $H^{1}$-curves in
$\mathcal{M}_{0}$ joining $x_{0}$ to $x_{1}$ and defined in $I$, then it is
\begin{equation*}
\Omega ^{1}(x_{0},x_{1})\equiv \{x\in H^{1}(I,
\mathbb{R}^{N}): x(I)\subset \mathcal{M}_{0},\;
x(0)=x_{0},\; x(1)=x_{1}\}.
\end{equation*}
If $\mathcal{M}_{0}$ is complete, $\Omega ^{1}(x_{0},x_{1})$ is
a complete Riemannian manifold (see \cite{PAL}) and its tangent space is
\begin{equation*}
T_{x}\Omega ^{1}(x_{0},x_{1})=\{\xi \in H^{1}(I,T
\mathcal{M}_{0}): \xi (s)\in T_{x(s)}
\mathcal{M}_{0}\; \forall s\in I,\xi (0)=\xi
(1)=0\}.
\end{equation*}
Furthermore, we can define $H^{1}(0)$, the subspace of
$H^{1}(I,\mathbb{R})$ of those curves $t=t(s)$ such
that $t(0)=0$.

It is well known that problem \eqref{P1} has a variational
structure; so it is quite standard to prove that $\overline{z}=\overline{z}
(s)$ is a lightlike geodesic joining $z_{0}=(
x_{0,}0)$ to $\gamma $ if and only if it is a critical point of the
$\mathcal{C}^{1}$ functional
\begin{equation}
f(z)=\frac{1}{2}\int_{0}^{1}\langle \dot{z},\dot{z}
\rangle _{z}ds\text{,\quad in }Z=\Omega ^{1}(x_{0},x_{1})
\times H^{1}(0),  \label{functional}
\end{equation}
with critical level $f(\overline{z})=0$. But a direct
investigation of the zero critical level of $f$ is not easy as the
functional in (\ref{functional}) is unbounded both from below and from
above. In order to overcome this difficulty, Fortunato, Giannoni and
Masiello in \cite{FGM} \ stated a new variational principle similar to the
Fermat one so to introduce a new functional, arrival time $T=T(
x)$, which is bounded from below on Riemannian manifold $\Omega
^{1}(x_{0},x_{1})$.

\begin{theorem}[Fermat principle]
\label{Fermat principle}Let $\overline{z}:I\to \mathcal{M}$,
$\overline{z}=\overline{z}(s)$, be a smooth curve such that
$\overline{z}=(\overline{x},\overline{t})$. Then, the following
statements are equivalent:
\begin{enumerate}
\item[(a)] $\overline{z}$ is a solution of problem \eqref{P1}
with arrival time $\overline{t}(1)=T>0$;

\item[(b)] $\overline{x}$ is a critical point of functional
\begin{equation*}
F(x)=\sqrt{\int_{0}^{1}\langle \dot{x},\dot{x}
\rangle ds\cdot \int_{0}^{1}\frac{1}{\beta (x)}ds}\quad
\text{in }\Omega ^{1}(x_{0},x_{1}),
\end{equation*}
with critical level $T=F(\overline{x})>0$ and
\begin{equation}
\overline{t}(s)=T\Big(\int_{0}^{1}\frac{1}{\beta (
\overline{x})}ds\Big)^{-1}\int_{0}^{s}\frac{1}{\beta (
\overline{x})}d\sigma \quad \text{for all }s\in I.
\label{arrivaltime}
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
The proof can be found, for example, in \cite{Mas}. Anyway, here, for
completeness, we outline its main arguments.
Fixed $T\in \mathbb{R}$, let us define
\begin{equation*}
W_{T}=\{t\in H^{1}(0): t(1)=T\} .
\end{equation*}
It is easy to see that $W_{T}$ is an affine submanifold of $H^{1}(I,
\mathbb{R})$ whose tangent space is given by $H_{0}^{1}(I,
\mathbb{R})=\{\tau \in H^{1}(I,\mathbb{R})\mid
\tau (0)=\tau (1)=0\}$.
Thus, the space of curves joining $z_{0}$ to $(x_{1},T)$ is
\begin{equation*}
Z_{T}=\Omega ^{1}(x_{0},x_{1})\times W_{T}
\end{equation*}
with tangent space given by
\begin{equation*}
T_{z}Z_{T}\equiv T_{x}\Omega ^{1}(x_{0},x_{1})\times
H_{0}^{1}(I,\mathbb{R})\quad \text{for any }z=(
x,t)\in Z_{T}.
\end{equation*}
Let $f_{T}$ be the restriction of functional $f$ to $Z_{T}$, so, by (\ref
{metric}), for each $z=(x,t)\in Z_{T}$ it is
\begin{equation}
f_{T}(z)=\frac{1}{2}\int_{0}^{1}\langle \dot{x},\dot{x}
\rangle ds-\frac{1}{2}\int_{0}^{1}\beta (x)\dot{t}^{2}ds
\label{expression}
\end{equation}
whose differential is given by
\begin{equation}
f_{T}'(z)[\zeta] =\int_{0}^{1}\langle
\dot{x},\dot{\xi}\rangle ds-\frac{1}{2}\int_{0}^{1}\beta '(
x)[\xi] \dot{t}^{2}ds-\int_{0}^{1}\beta (x)
\dot{t}\dot{\tau}ds  \label{derivative}
\end{equation}
for all $\zeta =(\xi ,\tau )\in T_{x}\Omega ^{1}(
x_{0},x_{1})\times H_{0}^{1}(I,\mathbb{R})$.
If, for simplicity, we assume
\begin{gather*}
\frac{\partial f_{T}}{\partial x}(z)[\xi]
=f_{T}'(z)[(\xi ,0)] \quad
\forall \text{ }\xi \in T_{x}\Omega ^{1}(x_{0},x_{1}), \\
\frac{\partial f_{T}}{\partial t}(z)[\tau]
=f_{T}'(z)[(0,\tau )]
\quad \forall \text{ }\tau \in H_{0}^{1}(I,\mathbb{R}),
\end{gather*}
then, $z$ is a critical point of functional $f_{T}$ in $Z_{T}$ if and only
if $z\in N_{T}$ and $\frac{\partial f_{T}}{\partial x}(z)[
\xi] =0$ for all $\xi \in T_{x}\Omega ^{1}(x_{0},x_{1})
$, where $N_{T}=\{z\in Z_{T}\mid \frac{\partial f_{T}}{\partial t}
(z)\equiv 0\}$ is the kernel of $\frac{\partial f_{T}}{
\partial t}$ in $Z_{T}$.

Let us remark that, by (\ref{derivative}) and simple calculations it follows
that $N_{T}$ is the graph of the $\mathcal{C}^{1}$ map $\Phi _{T}:\Omega
^{1}(x_{0},x_{1})\to W_{T}$ such that
\begin{equation}
\Phi _{T}(x)(s)=T\Big(\int_{0}^{1}\frac{1}{\beta
(x(\sigma ))}d\sigma \Big)^{-1}\int_{0}^{s}
\frac{1}{\beta (x(\sigma ))}d\sigma \text{\quad
for all }s\in I.  \label{tie2}
\end{equation}
So, considered the restriction of $f_{T}$ to $N_{T}$, we can define a new
$\mathcal{C}^{1}$ functional
\begin{equation}
J_{T}(x)=f_{T}(x,\Phi _{T}(x)),
\quad x\in \Omega ^{1}(x_{0},x_{1}),  \label{relation}
\end{equation}
which can be explicitely written as
\begin{equation*}
J_{T}(x)=\frac{1}{2}\int_{0}^{1}\langle \dot{x},\dot{x}
\rangle ds-\frac{1}{2}T^{2}\Big(\int_{0}^{1}\frac{1}{\beta (
x)}ds\Big)^{-1},\quad x\in \Omega ^{1}(
x_{0},x_{1}).
\end{equation*}
Now, let us point out that
$\overline{z}=(\overline{x},\overline{t})$ solves \eqref{P1}
with $\overline{t}(1)=T$
if and only if $\overline{z}\in Z_{T}$ is such that
$f_{T}'(\overline{z})=0$ and $f_{T}(\overline{z})=0$; hence, if
and only if $\overline{x}\in \Omega ^{1}(x_{0},x_{1})$ is such
that $J_{T}'(\overline{x})=0$, $J_{T}(\overline{x})=0$ and
$\overline{t}=\Phi _{T}(\overline{x})$.

Therefore, our problem \eqref{P1} is reduced to search for $T>0$
and $x\in \Omega ^{1}(x_{0},x_{1})$ such that
\begin{equation} \label{P2}
\begin{gathered}
J_{T}'(x)=0, \\
J_{T}(x)=0,
\end{gathered}
\end{equation}
i.e., to search for a couple $(x,T)\in \Omega ^{1}(
x_{0},x_{1})\times \mathbb{R}_{+}^{\ast }$ solution of the problem
\begin{equation} \label{P3}
\begin{gathered}
\frac{\partial H}{\partial x}(x,T)=0, \\
H(x,T)=0,
\end{gathered}
\end{equation}
with $H(x,T)=2J_{T}(x)$.
By solving the equation $H(x,T)=0$ we obtain
\begin{equation*}
T^{2}=\int_{0}^{1}\langle \dot{x},\dot{x}\rangle ds\cdot
\int_{0}^{1}\frac{1}{\beta (x)}ds\text{;}
\end{equation*}
thus, in order to have a positive $T$, we can consider
\begin{equation}
F(x)=\sqrt{\int_{0}^{1}\langle \dot{x},\dot{x}
\rangle ds\cdot \int_{0}^{1}\frac{1}{\beta (x)}ds}.
\label{finalfunctional}
\end{equation}
We can easily see that $\mathcal{G=}\{(x,t)\in \Omega
^{1}(x_{0},x_{1})\times \mathbb{R}_{+}\mid H(x,T)
=0\}$ is the graph of $F$.
So, by applying the abstract theorem in \cite[Theorem 2.3]{FGM}
 it follows that, if $(\overline{x},T)$ is a solution of problem
\eqref{P3} with $T>0$, then $\overline{x}$ is a critical point of $F$
such that $T=F(\overline{x})>0$ and vice versa.
\end{proof}

\begin{remark} \rm \label{derivativetie}
By differentiating the map $x\in \Omega
^{1}(x_{0},x_{1})\longmapsto H(x,F(x))
\in \mathbb{R}$ we have
\begin{equation}
\frac{\partial H}{\partial x}(x,F(x))+\frac{
\partial H}{\partial T}(x,F(x))F'(
x)=0\quad \text{for all } x\in \Omega ^{1}(x_{0},x_{1}).  \label{tie}
\end{equation}
\end{remark}

\begin{remark} \rm
Functional $F$ is continuous in all
$\Omega ^{1}(x_{0},x_{1})$, but eventually it is not differentiable
only at the zero level, i.e. on constant curves. Thus, if
$x_{0}\neq x_{1}$ the manifold $\Omega ^{1}(x_{0},x_{1})$ does not
contain any constant curve, so $F$ is a smooth functional on all
$\Omega ^{1}(x_{0},x_{1})$.
\end{remark}

\begin{remark} \rm
\label{timeequivalence}
By reasoning as outlined in the previous proof
it is also possible to prove that:
\begin{enumerate}
\item[(a)] $\overline{z}=(\overline{x},\overline{t})$ is
a solution of problem \eqref{P1} with arrival time
$\overline{t}(1)=T<0$;
\end{enumerate}
if and only if
\begin{enumerate}
\item[(b)] $\overline{x}$ is a critical point of functional
\begin{equation*}
F_{-}(x)=-\sqrt{\int_{0}^{1}\langle \dot{x},\dot{x}
\rangle ds\cdot \int_{0}^{1}\frac{1}{\beta (x)}ds}
=-F(x)\quad \text{\emph{on} }\Omega ^{1}(
x_{0},x_{1}),
\end{equation*}
with critical level $T=F_{-}(\overline{x})<0$,
and $\overline{t}=\overline{t}(s)$ is given by
(\ref{arrivaltime}).
\end{enumerate}
\end{remark}

Now, our aim is reduced to look for critical points of $F$ in
$\Omega^{1}(x_{0},x_{1})$. Thus, we need the Ljusternik-Schnirelmann
Theory (for more details, see, e.g., \cite{STR}).

\begin{definition} \rm
Let $X$ be a topological space and $A\subseteq X$. The
Ljusternik-Schnirelman category of $A$ in $X$ ($\mathop{\rm cat}_{X}A$)
is the least number of closed and contractible subsets of $X$ covering $A$.
If this is not possible we say that $\mathop{\rm cat}_{X}A=+\infty $.
We denote $\mathop{\rm cat}X=\mathop{\rm cat}_{X}X$.
\end{definition}

\begin{definition} \rm
Let $\Omega $ be a Riemannian manifold and $f$ a $\mathcal{C}^{1}$
functional on $\Omega $.
$f$ is said to satisfy the Palais-Smale condition if any $(
x_{n})_{n}\subset \Omega $ such that
\begin{equation*}
(f(x_{n}))_{n}\text{ is bounded and }\underset{
n\to +\infty }{\lim }f'(x_{n})=0
\end{equation*}
converges in $\Omega $ up to subsequences.
\end{definition}

\begin{theorem}[Ljusternik-Schnirelmann]
\label{Lju-Sch}Let $\Omega $ be a complete Riemannian manifold and $f$ a
$\mathcal{C}^{1}$ functional on $\Omega $.
If $f$ satisfies the Palais-Smale condition and is bounded from below, then
$f$ attains its infimum and has at least $cat\Omega $ critical points.
Furthermore, if $\underset{\Omega }{\sup }F=+\infty $ and $cat\Omega
=+\infty $ there exists a sequence of critical points $(x_{n})
_{n}\subset \Omega $ such that $F(x_{n})\nearrow +\infty $.
\end{theorem}

\begin{remark} \rm
If $1\leq k\leq cat\Omega $, then each critical level $c_{k}$
of the previous theorem is given by
\begin{equation*}
c_{k}=\underset{A\in \Gamma _{k}}{\inf }\underset{x\in A}{\sup }f(
x)
\end{equation*}
where $\Gamma _{k}=\{A\subseteq X : \mathop{\rm cat}_{\Omega }A\geq
k\}$.
\end{remark}

At last, in order to estimate the Ljusternik-Schnirelmann category of
$\Omega ^{1}(x_{0},x_{1})$, we need the following result (see
\cite{FaHus}).

\begin{proposition}[Fadell-Husseini]
\label{fadell-husseini} If $\mathcal{M}_{0}$ is a manifold not contractible
in itself, then for all $x_{0},x_{1}\in \mathcal{M}_{0}$ the manifold of
curves $\Omega ^{1}(x_{0},x_{1})$ has infinite category and
possesses compact subsets of arbitrarily high category.
\end{proposition}

\section{Proof of the main theorem}

Obviously, functional $F$ is bounded from below as it is
\begin{equation*}
F(x)\geq 0\quad \text{for all }x\in \Omega ^{1}(x_{0},x_{1}).
\end{equation*}
Anyway, in order to prove the Palais-Smale condition, a stronger property is
needed.
For this aim, we recall a technical lemma (for more details on the proof,
see \cite[Proposition 4.1]{BCFS} or also \cite[Lemma 2.4]{BCF}).

\begin{lemma}
\label{procoercivity}
Let $\beta $ satisfy assumption {\rm (H2)}.
If $(x_{k})_{k}\subset \Omega ^{1}(x_{0},x_{1})$ is such that
$\|\dot{x}_{k}\|\to +\infty $, then
\begin{equation*}
\int_{0}^{1}\frac{\|\dot{x}_{k}\|^{2}}{\beta (
x_{k})}ds\to +\infty \quad \text{as }k\to +\infty,
\end{equation*}
where $\|\dot{x}_{k}\|^{2}=\int_{0}^{1}\langle \dot{x
}_{k},\dot{x}_{k}\rangle ds$.
\end{lemma}

An easy consequence of Lemma \ref{procoercivity} is the following result.

\begin{lemma}
\label{coercivity}Functional $F$ is coercive in $\Omega ^{1}(
x_{0},x_{1})$, i.e.
\begin{equation*}
F(x)\to +\infty \quad \text{if }\|\dot{x}
\|\to +\infty .
\end{equation*}
\end{lemma}

\begin{proposition}
\label{palais-smale}Assume that $\mathcal{M}_{0}$ is complete and $\beta $
has a quadratic growth as in (\ref{quadratic}). Then functional $F$
satisfies the Palais-Smale condition in $\Omega ^{1}(
x_{0},x_{1})$.
\end{proposition}

\begin{proof}
Let $(x_{k})_{k}\subset \Omega ^{1}(x_{0},x_{1})$
be such that
\begin{equation}
(F(x_{k}))_{k}\text{ is bounded and }\underset{
k\to +\infty }{\lim }F'(x_{k})=0.
\label{P-S}
\end{equation}
 From Lemma \ref{coercivity} and (\ref{P-S}) we have that $(\|
\dot{x}_{k}\|)_{k}$ is bounded. Furthermore, it is also
easy to see that
\begin{equation*}
\sup \{d(x_{k}(s),x_{0})\mid s\in I,
k\in \mathbb{N}\}<+\infty .
\end{equation*}
 Thus, there exists $R>0$ such that the family
$\{x_{k}(s) :  s\in I,k\in \mathbb{R}\}$ is contained in
$B_{R}(0)=\{x\in \mathcal{M}_{0}: d(x,x_{0})\leq R\}$ which is compact,
so there exist $M,\nu >0$ such that
\begin{equation}
\nu \leq \beta (x_{k}(s))\leq M\text{\quad for
all }s\in I,k\in \mathbb{N}.  \label{limitation}
\end{equation}
Furthermore, $(x_{k})_{k}$ is bounded in
$H^{1}(I,\mathbb{R}^{N})$. Hence, there exists
$x\in H^{1}(I,\mathbb{R}^{N})$ such that, up to subsequences,
it is $x_{k}\rightharpoonup x$
weakly in $H^{1}(I,\mathbb{R}^{N})$ and $x_{k}\to x$
uniformly in $I$. Since $\mathcal{M}_{0}$ is complete,
$x\in \Omega^{1}(x_{0},x_{1})$. What remains to prove is that this
convergence is also strong in $\Omega ^{1}(x_{0},x_{1})$. For
simplicity, consider $z_{k}=(x_{k},t_{k})$ and
$T_{k}=F(x_{k})$ with $t_{k}=\Phi _{T_{k}}(x_{k})$. Hence,
by (\ref{tie2}) and (\ref{limitation}) sequence $(t_{k})_{k}$ is
bounded in $H^{1}(I,\mathbb{R})$.

On the other hand, by \cite[Lemma 2.1]{B-F} there exist two sequences
$(\xi _{k})_{k},(\nu _{k})_{k}\subset H^{1}(I,
\mathbb{R}^{N})$ such that
\begin{gather}
\xi _{k}\in T_{x_{k}}\Omega ^{1}(x_{0},x_{1})\text{,\quad }
x_{k}-x=\xi _{k}+\nu _{k}\quad \text{for all }k\in \mathbb{R},  \notag
\\
\xi _{k}\rightharpoonup 0\text{ weakly in }H^{1}(I,\mathbb{R}
^{N})\text{ and }\nu _{k}\to 0\text{ strongly in }H^{1}(
I,\mathbb{R}^{N}).  \label{splitting}
\end{gather}
By (\ref{P-S}) it is
\begin{equation}
F'(x_{k})[\xi _{k}] =o(1),  \label{PS0}
\end{equation}
while by Remark \ref{derivativetie} it follows
\begin{equation*}
\frac{\partial H}{\partial x}(x_{k},T_{k})+\frac{\partial H}{
\partial T}(x_{k},T_{k})F'(x_{k})=0,
\end{equation*}
where $(\frac{\partial H}{\partial T}(x_{k},T_{k}))
_{k}$ is bounded.
Evaluating this operator on 
$\xi _{k}\in T_{x_{k}}\Omega ^{1}(x_{0},x_{1})$, we obtain
\begin{equation*}
\frac{\partial H}{\partial x}(x_{k},T_{k})[\xi _{k}]
+\frac{\partial H}{\partial T}(x_{k},T_{k})F'(
x_{k})[\xi _{k}] =0,
\end{equation*}
so, by (\ref{PS0}) it is
\begin{equation*}
\frac{\partial H}{\partial x}(x_{k},T_{k})[\xi _{k}]
=o(1).
\end{equation*}
Hence, being
\begin{equation*}
\frac{\partial H}{\partial x}(x_{k},T_{k})[\xi _{k}]
=2J_{T_{k}}'(x_{k})[\xi _{k}] ,
\end{equation*}
it follows
\begin{equation*}
J_{T_{k}}'(x_{k})[\xi _{k}] =o(1).
\end{equation*}
So, reasoning as in \cite[Lemma 3.4.1]{Mas}, since
\begin{equation*}
J_{T_{k}}'(x_{k})[\xi _{k}]
=f_{T_{k}}'(x_{k},t_{k})[(\xi_{k},0)] ,
\end{equation*}
we have that
\begin{equation*}
o(1)=f_{T_{k}}'(x_{k},t_{k})[(
\xi _{k},0)] =\int_{0}^{1}\langle \dot{x}_{k},\dot{\xi}
_{k}\rangle ds-\frac{1}{2}\int_{0}^{1}\beta '(x_{k})
[\xi _{k}] \dot{t}_{k}^{2}ds.
\end{equation*}
Whence, being the sequence $(\|\dot{t}_{k}\|)
_{k}$ bounded, (\ref{splitting}) implies
\begin{equation*}
\int_{0}^{1}\beta '(x_{k})[\xi _{k}] \dot{t
}_{k}^{2}ds=o(1),
\end{equation*}
so it is
\begin{equation*}
\int_{0}^{1}\langle \dot{x}_{k},\dot{\xi}_{k}\rangle ds=o(1)
,
\end{equation*}
and, by applying again (\ref{splitting}), we obtain
\begin{equation*}
\int_{0}^{1}\langle \dot{\xi}_{k},\dot{\xi}_{k}\rangle ds=o(1)
,
\end{equation*}
so $\xi _{k}\to 0$ strongly in $H^{1}(I,\mathbb{R}^{N})$
. Hence, sequence $(x_{k})_{k}$ converges strongly to $x$.
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{main theorem}]
By Lemma \ref{coercivity} and Proposition \ref{palais-smale} we can apply
Theorem \ref{Lju-Sch} to functional $F$ in the complete Riemannian manifold
$\Omega ^{1}(x_{0},x_{1})$ obtaining that $F$ has at least a
critical point. Moreover, if $\mathcal{M}_{0}$ is non contractible in
itself, then for Proposition \ref{fadell-husseini}, functional $F$ has
infinitely many critical points $(x_{k})_{k}\subset \Omega
^{1}(x_{0},x_{1})$ such that
\begin{equation*}
\underset{k\to +\infty }{\lim }F(x_{k})=+\infty \text{;}
\end{equation*}
whence, by Theorem \ref{Fermat principle} there exists a sequence of
geodesics $(z_{n}=(x_{n},t_{n}))_{n}$ such that
$t_{n}(1)\nearrow +\infty $ as $n\nearrow +\infty $.
Furthermore, by reasoning in the same way, Remark \ref{timeequivalence}
allows us to complete the proof by finding geodesics with negative arrival
time.
\end{proof}

\begin{remark} \rm
\label{geometric} The existence of lightlike geodesics joining a point
to a line, in a static spacetime satisfying assumptions
(H1)and (H2), has already been
proved by means of geometrical tools. In fact, by using a correspondence
between the trajectories of particles under a potential and the geodesics on
a static spacetime, in \cite[Section 4]{Sanchez}, S\'{a}nchez proved that a
point and a line can be joined by a lightlike geodesic if and only if the
conformal Riemannian metric
$g^{\ast }=\beta ^{-1}\langle \cdot ,\cdot\rangle $
 is geodesically connected. On the other hand, by
Hopf-Rinow Theorem, $g^{\ast }$ is geodesically connected if and
only if it is complete. But, by \cite[Theorem 3.1]{Sanchez3}, the metric
$g^{\ast }$ is complete when $\langle \cdot ,\cdot \rangle$ is complete
and $\beta $ grows at most quadratically, as in
our hypotheses.
\end{remark}

\subsection*{Acknowledgments}
The author wish to thank Professor Anna
Maria Candela for her precious help and support and to the
Departamento de Geometria y Topologia of Universidad de Granada
 for its kind hospitality. A special thanks to Professot
 Miguel S\'{a}nchez for his
suggestions about the geometrical point of view of this problem.

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\end{document}