On the existence of infinitely many geodesics on space-time manifolds
Let \(N_ 0\) be a connected, compact \(C^ k\) \((k \geq 3)\) \(n\)- dimensional manifold and set \({N} = N_ 0 \times \mathbb{R}\). Let \(g = g(P, t)\) \((P \in {N}_ 0, t \in \mathbb{R})\) be a \(C^ k\) pseudo-Riemannian metric on \(N\) satisfying the following assumptions: (i) \(g\) does not depend on \(t\), (ii) \(g\) induces a Riemannian metric on \(N_ 0\), (iii) \(g(P) \left[\left(\begin{smallmatrix} 0\\ h\end{smallmatrix} \right), \left( \begin{smallmatrix} 0 \\ h \end{smallmatrix} \right) \right] < 0\) for any \(P \in {N}_ 0\) and \(h \in \mathbb{R} \setminus \{0\}\). Let \(a_ 1 = (P_ 1, t_ 1)\), \(a_ 2 = (P_ 2, t_ 2)\) be two events (i.e., two points in \(N\)), and consider the functional \[ f(\nu) = 2^{-1} \cdot \int^ 1_ 0 g(\nu (s)) [\dot \nu (s), \dot \nu(s)] ds, \] where \(\nu : [0,1] \to N\) is a smooth curve on \(N\) joining \((P_ 1, t_ 1)\) and \((P_ 2, t_ 2)\) and \(\dot \nu(s)\) denotes the tangent vector \(t_ 0 \nu\) at \(\nu(s)\). The geodesics for the metric \(g\) joining \(a_ 1\) and \(a_ 2\) are critical points of the functional \(f\). The main result of the paper is the following theorem. Theorem. Let \(g\) satisfy assumptions (i), (ii), (iii) and \(a_ 1\), \(a_ 2\) be any two points in \(N\). Then there exists a sequence \(\{ \nu_ n\}\) of geodesics joining \(a_ 1\), \(a_ 2\) with the property \(\lim_{n \to \infty} f(\nu_ n) = + \infty\). Moreover, if we assume also that all the geodesics for \(g\) are nonsingular, only finitely many of these geodesics \(\nu_ n\) are time-like.
scientific article; zbMATH DE number 611197
scientific article