Some conundrums such as the Ehrenfest paradox have been raised in relation to the relativistic approach to circular motions. Different tangential velocities along the radius of a rotating frame bring about a nonuniform scalar field of potential. Based on the Schwarzschild metric for the potential field, we consistently and comprehensively deal with the problems of relativistic circular motion including the Ehrenfest paradox and the Sagnac effect. The Ehrenfest paradox is readily resolved via a visualization of wave propagation in the field, which shows that the length of radius in the rotating frame is different from the corresponding one seen in the laboratory frame. From the visualization, the anisotropy of the speed of light in inertial frames is also clearly shown. Moreover, a coordinate transformation for the circular motion at a fixed radius is developed based on the metric, which enables us to derive a transformation between inertial frames through limiting operations. The derived inertial transformation becomes the same as the Lorentz transformation if the standard synchronization is introduced. With the transformations, the Michelson-Morley experiment result and the generalized Sagnac effect are consistently explained.