The Local and Global Geometrical Aspects of the Twin Paradox in Static Spacetimes: I.\nobreakspace {}Three Spherically Symmetric Spacetimes

Abstract

We investigate local and global properties of timelike geodesics in three static spherically symmetric spacetimes. These properties are of its own mathematical relevance and provide a solution of the physical 'twin paradox' problem. The latter means that we focus our studies on the search of the longest timelike geodesics between two given points. Due to problems with solving the geodesic deviation equation we restrict our investigations to radial and circular (if exist) geodesics. On these curves we find general Jacobi vector fields, determine by means of them sequences of conjugate points and with the aid of the comoving coordinate system and the spherical symmetry we determine the cut points. These notions identify segments of radial and circular gepdesics which are locally or globally of maximal length. In de Sitter spacetime all geodesics are globally maximal. In CAdS and BertottiRobinson spacetimes the radial geodesics which infinitely many times oscillate between antipodal points in the space contain infinite number of equally separated conjugate points and there are no other cut points. Yet in these two spacetimes each outgoing or ingoing radial geodesic which does not cross the centre is globally of maximal length. Circular geodesics exist only in CAdS spacetime and contain an infinite sequence of equally separated conjugate points. The geodesic curves which intersect the circular ones at these points may either belong to the two-surface θ = π/2 or lie outside it.1

On the geometric nature of the twin paradox in curved spacetimes

“…The reader interested in these technicalities is referred to our papers [11][12][13][14] containing also many references. These methods turn out relatively effective, at least in the case of high symmetry.…”

This Year's Nobel Prize (2022) in Physics for Entanglement and Quantum Information: the New Revolution in Quantum Mechanics and Science

“…However, if one grants the latter viewpoint (to the reference frame of the space-traveler), the homebody should be younger 20 There are many papers (e.g. Ben-Ya'acov 2016;Franson 2016;Fung, Clark, Lewis, Wu 2016;de Wolf 2016;Garcia-Escartin, Chamorro-Posada 2015;Gasperini 2014;Lindkvist, Sabín, Fuentes, Dragan, Svensson, Delsing, Johansson 2014;Moreau, Devaux, Lantz 2014;Sokołowski, Golda 2014;2014a;Grøn 2013;Carvalho 2012;Friebe 2012;Ord 2012;Sfarti 2012;Sokołowski 2012;Boblest, Müller, Wunner 2011;Grøn, Braeck 2011;Lichtenegger, Iorio 2011;Székely 2010;Grandou, Rubin 2009;Müller, King, Adis 2008;Roukema, Bajtlik 2008;Abramowicz, Bajtlik, Kluźniak 2007;Boccaletti, Catoni, Catoni 2007;2007a;Grøn 2006;Kohler 2006;McCall 2006;Madarász, Németi, Székely 2006;Ghosal, Nepal, Das 2005;Pesic 2003;Soni 2002;Uzan, Luminet, Lehoucq, Peter 2002;Barrow, Levin 2001;Dolby, Gull 2001;Weeks 2001;Cranor, Heider, Price 2000;Schön 1998;…”

Jacobi Fields and Conjugate Points on Timelike Geodesics in Special Spacetimes

“…We assume F (r) = r for a generic SSS metric and postpone discussing the case F (r) = const = a to the next section (the special case of the Bertotti-Robinson spacetime has been studied separately [11]). The timelike Killing vector is K α = κδ α 0 and κ = const is a normalization factor (chosen either at r = 0 or at spatial infinity).…”

“…Section 4 contains the main result found by the procedure presented in Section 2: the GDEs on circular geodesics (if these exist) are the same for all SSS spacetimes and the solutions found earlier for Schwarzschild metric universally apply. Actually, a preliminary version of the present work had first appeared as arXiv:1402.3976v1 [gr-qc] and preceded two published papers [11,12] which contained detailed results for a few simplest spacetimes found by applying the general method developed here. Therefore, to avoid repetitions, the presentation of the method here is not fully complete and we refer the reader to those two papers for a discussion of some of its aspects.…”