Is Time Relative in a Synchronized Clock Experiment?

[Sleepycoaster]

Homework Statement

Imagine two synchronized atomic clocks with hands that turn at the same rate. Put one of these on a plane starting in NYC and fly it around the world once, and leave the other at NYC. Because the one that flew supposedly took a more convoluted path, it should be behind the clock that stayed stationary once the trip is over.

Homework Equations

If this is true, couldn't we say that the clock that stayed in NYC is "moving" relative to the flying clock? From the perspective of the pilot of the plane, shouldn't the clock that stayed in NYC be behind? Since it's impossible for the pilot and a person staying in NYC to disagree on which clock is behind when the clocks are finally compared, what happens?

The Attempt at a Solution

At first it seemed like the clock that flew should be behind because its circular path is still convoluted in any inertial frame. But then I realized that so is the one in NYC from the perspective of the pilot.

 

[TSny]

This is kind of a tricky question. Neither clock is in an inertial frame, so you might want to introduce an inertial frame moving with the center of the earth.

You should consider two cases: (1) the clock on the plane flies eastward around the earth; (2) the plane flies westward around the earth

 

[Sleepycoaster]

Thank you, I'll definitely consider that detail.