Consider the Hafele-Keating experiment discussed in section R4.5. In this experiment, two atomic clocks were synchronized (event A), one was put on a jet and flown around the world, and then the clocks were compared (event B). Our task in this problem is to make a reasonably realistic prediction of the discrepancy that we would expect between the clocks. Suppose the plane starts at a point on the equator and flies around the world at a speed of 290 m/s relative to the ground. Assume the plane cruises at an altitude of about 35,000 ft (10.7 km) above the earth's surface.

(A) Make a prediction of how much the clocks will disagree when they are compared at the end of the experiment if the plane flies east around the equator (do not ignore the earth's rotation). Describe any assumptions or approximations you are compelled to make. (Hint: Do your analysis in an inertial reference frame fixed to the earth's center but which does not rotate with the earth. You can use the coordinate time measured in this frame between the initial synchronization and the final comparison events as a reference to compare earth clocks and plane clocks. It is tricky to calculate an exact number for this coordinate time, but in your final calculation, not that this time will be equal to the time measured on the earth's surface to much better than +-1%.)

(B) Repeat your calculation, assuming the plane flies west. Why is your answer different from the one you found for part (a)?

(C) General relativity predicts that a clock that is a distance h higher in a gravitational field than the second clock will run faster than the lower clock by the factor 1+gh/c^2 (in SI units), where g is the earth's gravitational field strength (9.8 m/s^2). How does including this information change your answer to part (A)?

I would greatly appreciate an explanation for this problem.

The answers should be...

(A) 269 ns

(B) -143 ns

(C) 107 ns, -304 ns