On the geometric nature of the twin paradox in curved spacetimes

• Autorzy: Sokołowski L. M.

Identyfikatory

DOI

10.1515/dema-2017-0006

• Języki publikacji: EN

Abstrakty

EN

The famous „twin paradox” of special relativity is of purely geometric nature and formulated in curved spacetimes of general relativity motivates investigations of the timelike geodesic structure of these manifolds. Except for the maximally symmetric spacetimes the search for the longest timelike curves is hard, complicated and requires both advanced methods of global Lorentzian geometry and solving the intricate geodesic deviation equation. This article is a theoretical introduction to the problem. First we describe the procedure of determining the locally longest curves; it is algorithmic in the sense of consisting of a small number of definite steps and is effective if the geodesic deviation equation may be solved. Then we discuss the problem of globally maximal timelike curves; due to its nonlocal nature there is no prescription of how to solve it in finite number of steps. In the case of sufficiently high symmetry of the manifold also the globally longest curves may be found. Finally we briefly present some results recently found.

Słowa kluczowe

EN

local and global Lorentzian geometry; Jacobi fields; maximal timelike curves; conjugate points; cut points

PL

geometria Lorentziana; krzywe czasopodobne; punkty sprzężone; punkty przecięcia

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2017
• Tom: Vol. 50, nr 1
• Strony: 56--65
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Sokołowski L. M.

• Astronomical Observatory, Jagiellonian University, Orla 171, Kraków 30-244, Poland

Bibliografia

• [1] Stephani H., Relativity, an introduction to special and general relativity, third edition, par. 3.4, Cambridge University Press, Cambridge, 2004
• [2] Hawking S. W., Ellis G. F. R., The large scale structure of space-time, Cambridge University Press, Cambridge, 1973
• [3] Wald R. M., General relativity, University of Chicago Press, Chicago, 1984
• [4] Podolsky J., Švarc R., Interpreting spacetimes of any dimension using geodesic deviation, Phys. Rev. D85:044057, 2012 [arXiv:1201.4790v2 [gr-qc]]
• [5] Fuchs H., Solutions of the equations of geodesic deviation for static spherical symmetric space-times, Ann. d. Physik, 1983, 40, 231-233
• [6] Bażański S. L., Hamilton-Jacobi formalism for geodesics and geodesic deviations, J. Math. Phys., 1989, 30, 1018-1029
• [7] Bażański S. L., Jaranowski P., Geodesic deviation in the Schwarzschild spacetime, J. Math. Phys., 1989, 30, 1794-1803
• [8] Fuchs H., Parallel transport and geodesic deviation in static spherically symmetric space-times, Astron. Nachr., 1990, 311, 219-222
• [9] Fuchs H., Deviation of circular geodesics in static spherically symmetric space-times, Astron. Nachr., 1990, 311, 271-276
• [10] Beem J. K., Ehrlich P. E., Easley K. L., Global Lorentzian geometry, second edition. Marcel Dekker, New York, 1996
• [11] Sokołowski L. M., On the twin paradox in static spacetimes: I. Schwarzschild metric, Gen. Rel. Grav., 2012, 44, 1267-1283 [arXiv:1203.0748 [gr-qc]]
• [12] Sokołowski L. M., Golda Z. A., The local and global geometrical aspects of the twin paradox in static spacetimes: I. Three spherically symmetric spacetimes, Acta Phys. Polon. B, 2014, 45, 1051-1075 [arXiv:1402.6511v2 [gr-qc]]
• [13] Sokołowski L. M., Golda Z. A., The local and global geometrical aspects of the twin paradox in static spacetimes: II. Reissner-Nordström and ultrastatic metrics, Acta Phys. Polon. B, 2014, 45, 1713-1741 [arXiv:1404.5808 [gr-qc]]
• [14] Sokołowski L. M., Golda Z. A., Jacobi fields and conjugate points on timelike geodesics in special spacetimes, Acta Phys. Polon. B, 2015, 46, 773-787
• [15] Sokołowski L. M., Golda Z. A., Every timelike geodesic in anti-de Sitter spacetime is a circle of the same radius, Internat. J. Mod. Phys. D, 2016, 25, 1650007 (6 pgs) [arXiv:1602.07111 [gr-qc]]

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).