Infinitely many spacelike periodic trajectories on a class of Lorentz manifolds
type: Item
instance of: scholarly article
title: Infinitely many spacelike periodic trajectories on a class of Lorentz manifolds
zbMATH Open document ID: 0807.53055
author: Carlo Greco
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1994-08-31T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108322
review text: 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 \(\dot z^ t\).
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 627433
zbMATH Keywords: geodesics
zbMATH Keywords: static spacetimes
zbMATH Keywords: spacelike geodesics
zbMATH Keywords: Killing field
zbMATH Keywords: periodic
zbMATH Keywords: variational methods
reviewed by: Marcus Kriele
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 627433