TITLE: If an astronaut travels through a wormhole, what is proper time in this situation?
QUESTION [0 upvotes]: Is it:

The time measured by a clock held by the astronaut throughout the journey
The time measured by observers watching from the initial entry to the wormhole
The time measured by observers awaiting the astronaut at the exit side of the wormhole

I'm thinking that it's the first one because according to Wikipedia, it is the time measured by a clock following the world line (in this case, the tunnel of the wormhole). Am I right?

REPLY [0 votes]: As @СимонТыран says, proper time is always measured by the travelling clock.
However, there is an interesting effect regarding the so-called Twin Paradox in the presence of a wormhole through which one twin travels while the other does not. Assume both twins are at the same location half way between two wormhole mouths but one is moving at constant velocity and will pass through the wormhole and meet their twin again "on the other side".
Normally, the situation in which there are two observers in constant relative motion (i.e. when there are two inertial frames) is perfectly symmetric, meaning that both frames of reference are equivalent - each sees the other's clock ticking more slowly. 
But, motion relative to the wormhole breaks the symmetry in such a way that the twin who passes through the wormhole has genuinely experienced less proper time than the stationary twin, even though they never accelerate - as happens in the "classic" Twin Paradox. When they next meet the travelling twin is absolute younger than the stationary twin, because the stationary twin really is at rest in a preferred frame.
It is the presence of the wormhole that breaks Lorentz symmetry and creates this preferred frame so that the measured proper time is always longer for the twin who stays in one place. (Technically speaking the travelling twin cannot achieve what is called Einstein clock synchronisation, whereas the stationary twin can.)
These papers discuss the issue (in great technical detail)
D. Bansal, J. Laing, and A. Sriharan, “On the twin paradox in a universe with a compact dimension,” arXiv preprint gr-qc/0503070, 2005.
J. D. Barrow and J. J. Levin, “The Twin paradox in compact spaces,” Phys. Rev. A, vol. 63, no. 4, p. 44104, 2001.
C. H. Brans and D. R. Stewart, “Unaccelerated-Returning-Twin Paradox in Flat Space-Time,” Phys. Rev. D, vol. 8, p. 1662, 1973.
J. P. Uzan, J. P. Luminet, R. Lehoucq, and P. Peter, “The twin paradox and space topology,” Eur. J. Phys., vol. 23, no. 3, p. 277, 2002.
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