Jacobi fields, conjugate points and cut points on timelike geodesics in special spacetimes
Leszek M. Sokołowski, Zdzisław A. Golda
备注：
A revised and expanded version, self-contained and written in an expository style. 36 pages, 0 figures. A substantially abridged version appeared in Acta Physica Polonica B
Several physical problems such as the `twin paradox' in curved spacetimes have purely geometrical nature and may be reduced to studying properties of bundles of timelike geodesics. The paper is a general introduction to systematic investigations of the geodesic structure of physically relevant spacetimes. The investigations are focussed on the search of locally and globally maximal timelike geodesics. The method of dealing with the local problem is in a sense algorithmic and is based on the geodesic deviation equation. Yet the search for globally maximal geodesics is non-algorithmic and cannot be treated analytically by solving a differential equation. Here one must apply a mixture of methods: spacetime symmetries (we have effectively employed the spherical symmetry), the use of the comoving coordinates adapted to the given congruence of timelike geodesics and the conjugate points on these geodesics. All these methods have been effectively applied in both the local and global problems in a number of simple and important spacetimes and their outcomes have already been published in three papers. Our approach shows that even in Schwarzschild spacetime (as well as in other static spherically symetric ones) one can find a new unexpected geometrical feature: instead of one there are three different infinite sets of conjugate points on each stable circular timelike geodesic curve. Due to problems with solving differential equations we are dealing solely with radial and circular geodesics.
In this paper we return to the subject of Jacobi metrics for timelike and null geodsics in stationary spactimes, correcting some previous misconceptions. We show that not only null geodesics, but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulae. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behaviour of Maxwell's equations in the curved spa…
Geodesics and shadows of the spindle-deformed Kerr black hole
Key Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of Frontier and Interdisciplinary Science, Shandong University（粒子物理与粒子辐照教育部重点实验室，前沿交叉科学研究院，山东大学）；School of Mathematics and Physics, Qingdao University of Science and Technology（青岛科技大学数理学院）；Department of Physics, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Institute of Interdisciplinary Studies, Hunan Normal University（湖南师范大学物理系，教育部低维量子结构与量子控制重点实验室，交叉科学研究院）；Hunan Research Center of the Basic Discipline for Quantum Effects and Quantum Technologies, Hunan Normal University（湖南师范大学量子效应与量子技术基础学科研究中心）
Virtual Institute of Astroparticle Physics（虚拟天体粒子物理研究所）；Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago（伊利诺伊大学芝加哥分校数学、统计与计算机科学系）
Gravitational redshift/blueshift of light emitted by geodesic test particles, frame-dragging and pericentre-shift effects, in the Kerr-Newman-de Sitter and Kerr-Newman black hole geometries
