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Original by Michael Weiss.

# The Twin Paradox: The Time Gap Objection

Try this on for size.

Make the turnaround instantaneous.  Relativity puts an upper on speed, but no upper limit on acceleration.  An instantaneous Turnaround Event is the limiting case of shorter and shorter turnarounds, and so the theory should handle it.

During the Outbound Leg, Terence ages less than two months, according to Stella.  (12 Stella-months / time dilation factor of 7.)  During the Inbound Leg, Terence also ages less than two months, according to Stella, by the same computation.  The Turnaround Event is instantaneous.  Total Terence ageing: less than 4 months, it would seem.  Yet Terence is supposed to be over 14 years older when Stella returns!  Where did the missing time go?

The Doppler Shift Analysis makes short work of this.  Stella sees (through her telescope) Terence age hardly at all during her Outbound Leg, and nearly the full 14 years during her Inbound Leg.  No gap.

Of course, she calculates something different, taking into account Doppler shifts and the finite speed of light.  These calculations must be based on reference frames.  In fact, for Stella to get two months for the Outbound Leg and two months for the Inbound Leg (as in the paragraph above), she has to switch inertial reference frames midway through her journey.

Now different inertial reference frames have different notions of simultaneity.  The Outbound reference frame says: "At the same time that Stella makes her turnaround, Terence's clock reads about two months."  The Inbound reference frame says: "At the same time that Stella makes her turnaround, Terence's clock reads about 13 years and 10 months."  The apparent "gap" is just an accounting error, caused by switching from one frame to another.

The Spacetime Analysis offers the same elucidation, graphically expressed.

Finally, the Equivalence Principle Analysis reply can be found in that entry.

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