Can the reference frames in the H-F and M-G-P be explained in everyday language?

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Offline sciconoclast

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Slightly Puzzled.

   It seems that the reference frame in the Hafele-Keating experiment is not moving with the rotation of the earth and its gravitational field but that in the Michelson-Gale-Pearson experiment it is.   I can not follow Einstein's explanations of these experiments.   But then Einstein himself made a mistake in predicting the outcome of the M-G-P experiment and later made a correction.   Can the difference in reference frames, if there is one, be explained in ordinary language that anyone can follow?     Does it have anything to do with one experiment measuring the relative speed of light and the other using a physical process to measure relative time?

                                                                Looking forward to an explanation.