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Proper Time of a Clock Moving in a Uniform Gravitational Field

What is time-flow rate of a clock moving in (a uniform) gravitational field? We can get the solution by using only the basic facts of special relativity. Just like a circle is invariant under a rotational transformation, a hyperbola is invariant under a Lorentz transformation. Based on the Lorentz invariance of the hyperbola, we can obtain the coordinates of a constantly acceleration observer, the Kottler-Moller-Rindler coordinate. The coordinate is now that of (a uniform) gravitational field, due to the equivalence principle. We show that the time-flow rate of a clock moving in (a uniform) gravitational field we can derived by using the Kottler-Moller-Rindler coordinate and a space-time diagram.