The twin paradox in a cosmological context

Abstract

Recently Abramowicz and Bajtlik [ArXiv: 0905.2428 (2009)] have studied the twin paradox in Schwarzschild spacetime. Considering circular motion they showed that the twin with a nonvanishing 4-acceleration is older than his brother at the reunion and argued that in spaces that are asymptotically Minkowskian there exists an absolute standard of rest determining which twin is oldest at the reunion. Here we show that with vertical motion in Schwarzschild spacetime the result is opposite: The twin with a non-vanishing 4-acceleration is younger. We also deduce the existence of a new relativistic time effect, that there is either a time dilation or an increased rate of time associated with a clock moving in a rotating frame. This is in fact a first order effect in the velocity of the clock, and must be taken into account if the situation presented by Abramowicz and Bajtlik is described from the rotating rest frame of one of the twins. Our analysis shows that this effect has a Machian character since the rotating state of a frame depends upon the motion of the cosmic matter due to the inertial dragging it causes. We argue that a consistent formulation and resolution of the twin paradox makes use of the general principle of relativity and requires the introduction of an extended model of the Minkowski spacetime. In the extended model Minkowski spacetime is supplied with a cosmic shell of matter with radius equal to its own Schwarzschild radius, so that there is perfect inertial dragging inside the shell. * Electronic address: Oyvind.Gron@iu.hio.no

Uniform acceleration in general relativity

“…The time dilation is a product of the time dilation due to the potential energy at the position of the clock and the time dilation due to the clock's velocity. The power series expansion (66) contains all of the usual terms plus a new term which has only been obtained in Schwarzschild spacetime [10].…”

“…Now (7), (10) and (6) imply thatλ (2) · λ (0) = 0, while (7), (13) and (6) imply thaṫ λ (2) · λ (1) = τ 1 . Let τ 2 = −(v (3) ) 2 > 0, and define…”

The twin paradox and Mach’s principle

“…However, in the case described by [1] it is again the twin traveling the longer space path who records less proper time. On the other hand, [11] presented an example which appears to be opposite: in the case of one twin staying fixed in the vicinity of a massive object and the other twin moving radially upward and downward, it is the latter twin who is older upon reunion.…”

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

“…As it should be expected, the farther the point r 0 is from the event horizon the smaller the ratio s C /s A is and always the radial geodesic C is longer than the non-geodesic curve A. That s C > s A was previously found in a special case in [12]. Both the geodesic curves B and C satisfy the conditions of Proposition 4, hence there do exist conjugate points and to identify them we now find Jacobi fields on these worldlines.…”