Clock A sits at rest on the earth, and clock B circles the earth in an orbit

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Clock A sits at rest on the earth, and clock B circles the earth in an orbit that skims along the ground. Both A and B are essentially at the same radius, so the GR time-dilation effect yields no difference in their times. But B is moving relative to A, so A will see B running slow, due to the usual SR time-dilation effect. The orbiting clock, B, will therefore show a smaller elapsed proper time each time it passes A. In other words, the clock under the influence of only gravity (B) does not show the maximal proper time, in conflict with what we have been calling the maximal-proper-time principle. Explain.
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