In an inertial frame without gravity, the fastest-running clock is a stationary one. In a...

In an inertial frame without gravity, the fastest-running clock is a stationary one. In a freely falling frame, it should be a clock accelerating with the frame. All observers agree on the proper time passing on a clock, so observers on Earth should find that the path actually followed by an object/clock that is subject only to gravity is one that maximizes the proper time from starting point to ending point. Here we verify that this is true. Assuming v «c, regular (speed-dependent) time dilation tells us that the time interval dr we see on a moving clock is related to our interval dt on Earth's surface by

Equation (2-30) tells us that the time dT passing on a clock ticking higher in the gravitational field is larger than the time dt on the surface according to

Taking both effects into account at once, we have