Infinitely many spacelike periodic trajectories on a class of Lorentz manifolds (Q1332508)

The author studies global properties of geodesics in a certain class of static spacetimes. Let \(x : = (x^ 1, x^ 2, x^ 3)\) and \(g = \alpha_{ij} (x) dx^ i dx^ j + \beta (x) dt^ 2\). Then there exist infinitely many spacelike geodesics whose projection along the integral curves of the Killing field \(\partial_ t\) are periodic, if some conditions on \(\alpha\) and \(\beta\) are satisfied. Most notably, it is assumed that \(\beta\) has a positive minimum at \(x = 0\) and is radially increasing far enough from the origin \((p \beta (x) \leq d \beta_ x (x)\) for \(| x | \geq R\), where \(R > 0\) and \(p > 2)\). There are more conditions assumed (incl. conditions on \(\alpha\) and its radial derivative) but I feel that the above is the main one (it rules out Minkowski space). The proof uses variational methods. A key rôle is played by the fact that \(\int g_{ab} \dot z^ a \dot z^ b = \int \alpha_{ij} \dot z^ i \dot z^ j - \int \beta (\dot z^ t)^ 2\) is convex with respect to \(\dot z^ t\).