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

TL;DR

This paper analyzes the twin paradox within Schwarzschild spacetime, calculating proper times for different worldlines, identifying conjugate points, and exploring geodesic properties to deepen understanding of relativistic effects in static spacetimes.

Contribution

It provides explicit calculations of proper times for various twins and identifies conjugate points in Schwarzschild geometry, advancing the mathematical understanding of the twin paradox.

Findings

Radial geodesic twins are always the oldest.

Conjugate points are explicitly found outside relevant segments.

General Jacobi vector fields are derived for the geodesics.

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 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.