Open Access
Research Article

General Relativity and Gravitation

, Volume 44, Issue 5, pp 1267-1283

On the twin paradox in static spacetimes: I. Schwarzschild metric

Authors

  • Leszek M. Sokołowski
    • Astronomical ObservatoryJagiellonian University
    • Copernicus Center for Interdisciplinary Studies

DOI: 10.1007/s10714-012-1337-4

Abstract

Motivated by a conjecture put forward by Abramowicz and Bajtlik we reconsider the twin paradox in static spacetimes. According to a well known theorem in Lorentzian geometry the longest timelike worldline between two given points is the unique geodesic line without points conjugate to the initial point on the segment joining the two points. We calculate the proper times for static twins, for twins moving on a circular orbit (if it is a geodesic) around a centre of symmetry and for twins travelling on outgoing and ingoing radial timelike geodesics. We show that the twins on the radial geodesic worldlines are always the oldest ones and we explicitly find the the conjugate points (if they exist) outside the relevant segments. As it is of its own mathematical interest, we find general Jacobi vector fields on the geodesic lines under consideration. In the first part of the work we investigate Schwarzschild geometry.

Keywords

Twin paradox Static spacetimes Jacobi fields Conjugate points

Acknowledgments

I am deeply indebted to Zdzisław Golda for solving a differential equation with the aid of Mathematica and to Sebastian Szybka for making some analytic and numerical computations. This work was supported by a grant from the John Templeton Foundation.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Copyright information

© The Author(s) 2012