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Original by Michael Weiss.

The Twin Paradox: The Distance Dependence Objection

With our "standard example" (see the Introduction), Stella's accounting of Terence's ageing runs like this: one-seventh of a year on the Outbound Leg, one-seventh of a year on the Inbound Leg, and the rest — 14 years minus two-sevenths — during the turnaround.  You may recall she does the turnaround in a day, according to Terence, or about 15 hours by her own clock.  (Let's just say 15, and hang the minutes; the exact figure won't matter.)

Say Stella takes a longer journey, spending 2 years on both the Inbound and Outbound Legs, for a total of 4 years of her time, or 28 years according to Terence.  But she still takes the same 15 hours for the turnaround.

So when Stella and Terence have their joyous reunion, Terence is 28 years older (plus a day).  This time Stella's accounting of Terence's ageing runs like so: Terence aged two-sevenths years on the Outbound Leg, ditto for the Inbound Leg, and so Terence must have aged over 27 years during the turnaround.

Summing up: according to Stella, Terence ages around 13 years 7 months on the turnaround for the shorter trip, but over 27 years on the turnaround for the longer trip.  Yet Stella says the two turnarounds took the same time.  And Terence agrees.

The resolutions are similar to those given for the Time Gap Objection.  How much Terence ages during the turnaround is not something you can directly observe, according to SR.  The Doppler Shift Analysis focuses on what Terence and Stella actually see through their telescopes, which avoids the difficulty.  Stella's accounting is just that: accounting, dependent on particular reference frames, and in particular on switching from one inertial reference frame to another.  No wonder that accounting tricks can produce surprising results.  See the Time Gap Objection for more details.

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