What is it about?

This is about the famous "twin paradox" which originally first appeared in special relativity. It has a purely geometrical nature: two observers (twins) start from the same point in space and time, travel along different worldlines and finally meet at some point. Their physical and biological clocks have measured different times of their travel. In special relativity explanation of this apparent paradox is simple: physical time is the length of the clock's worldline and as in Euclidean geometry, if two points are connected by two distinct paths, their lengths are different. In curved spacetime the same effect takes place, but now the quantitative description is far more complicated. Due to peculiarity of Lorentz geometry one cannot ask of which worldline connecting two given points in a spacetime is the shortest one, but which is the longest one. The paper discusses the mathematical framework for searching for longest worldlines.

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Why is it important?

I present the relevant theorems in global Lorentzian geometry concerning the search for longest timelike geodesic lines and briefly show results on studies of conjugate points and cut points in a number of simple and physically important spacetimes having spherical symmetry

Perspectives

This paper is merely an introduction to investigations of locally and globally longest timelike geodesics. Our current study show that there is great wealth of features of these geodesics and that even quite similar spacetimes may significantly differ in these features. It is necessary to investigate a large number of spacetimes.

Leszek SOKOŁOWSKI
Uniwersytet Jagiellonski w Krakowie

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This page is a summary of: On the geometric nature of the twin paradox in curved spacetimes, Demonstratio Mathematica, April 2017, De Gruyter,
DOI: 10.1515/dema-2017-0006.
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