Navy,
You’re describing the “Hafele-Keating” jet experiment done in 1971.
Here’s the answer to the conundrum. It’s a straight Special Theory of Relativity situation. The Earth rotates counter-clockwise - eastward - as viewed from above the North Pole looking down at the Earth. At the equator, the angular velocity at the surface is approximately 1,000 mph.
The rest frame is the center of gravity - the center of the Earth’s core.
First let’s assume that both planes travel at the same altitude and velocity. We can discount any gravity issues because at the same altitude the effect is the same for both aircraft. We can plug in 500 mph as their velocity. They makle their journey along the equator. Last, we put an observer at the Earth’s core in the rest frame and have that observer report what s/he sees:
Jet #1: Jet #1 travels east at 500 mph. But the Earth is turning to the east at 1000 mph - 500 mph faster than Jet #1. Our observer sees Jet #1 traveling 500 mph to the west. From the observer’s point of view Jet #1 can’t keep up with the speed of the rotation of the Earth.
Jet #2: Jet #2 travels west at 500 mph. Again, the Earth is turning to the east at 1000 mph. Jet #2 gets a 1,000 mph boost in velocity from the POV of the observer. Our observer sees Jet #1 traveling 1,500 mph to the west.
This explains why the clocks end up being a few nanoseconds different at the end of the flight. Jet #2 was traveling 1,000 mph faster than Jet #1 with respect to the rest frame. Its clock will tick more slowly than the clock on Jet #1.
If we apply the Lorentz Transformation to each airplane we discover that Jet #2’s clock ran ~2.23^-12 seconds slower than Jet #1’s clock (2.23 picoseconds) during the ~25,000 mile trip.
You might thnk that this situation sounds illogical. But remember, it is from the point of view of the rotating observer at the center of gravity. S/he turns through the same angle in the same time that the surface turns through. That’s why s/he can add or subtract from each plane’s velocity (Remember “velocity” - not “speed”.. Velocity is a vector - it has both magnitude and direction. That’s a central issue in this problem.)
In the case of Jet #1 s/he is turning in the same direcction as the aircraft. It loses 1,000 mph from her POV no matter how fast is it moving. She sees it moving 1,000 mph less than its indicated ground speed. Jet #2 is moving opposite her angular velocity. She sees it moving 1,000 mph faster than its indicated ground speed.
If she correctly reported their velocities with reference to her angular motion she should say that Jet #1 was traveling at minus 500 mph and Jet #2 was traveling at minus 1,500 mph.
I had to throw that last paragraph in for the purists and mathematicians.
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