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According this paper:

Morris, Michael S. and Thorne, Kip S. and Yurtsever, Ulvi (1988) Wormholes, time machines, and the weak energy condition. Physical Review Letters, 61 (13). pp. 1446-1449. ISSN 0031-9007 (https://authors.library.caltech.edu/9262/)

in the section named "Conversion of wormhole into time machine", the authors said:

This motion causes the right mouth to "age" less than the left as seen from exterior. Consequently, at late times by traversing the wormhole from right mouth to left one can travel backward in time

My question is: since the proper time is less than the time measured by an observer at rest, crossing the wormhole from right to left isn't go actually to the future?

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I read the article in the section indicated.

Yes, I think your observation is correct. The right mouth first accelerates away from the left mouth, assumed to remain at rest in that spacetime reference frame, and then reverses its motion to return to its original location. It is the classical configuration of the twins paradox. The right mouth shows younger than the left mouth.

Consequently, to travel back in time you have to traverse the wormhole from left mouth to right mouth.

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The article is correct.

You're getting confused by the concept of "aging", which I don't think is very useful for this scenario.

Let's put a clock into each mouth. Now, when we send the right mouth off on its journey, it comes back with its clock reading 1 year, while the left mouth's clock reads 100 years.

Now, somebody who stays inside the wormhole will see the clocks ticking at the same rate. Thus, if we go into the right mouth when its clock reads one year, we should come out of the left mouth at the time its clock reads 1 year, which is 99 years before we went into the right mouth.

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  • $\begingroup$ The twins paradox treatment of the right mouth induces a time shift negative compared to the reference frame time, which is also the time of the left mouth. To describe the wormhole the left and right mouths points in spacetime are identified; so to travel back in the past you enter the left mouth and exit the right mouth. $\endgroup$ – Michele Grosso Feb 24 at 18:14

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