The equations for time dilation in general relativity (GR) and special relativity (SR) describe distinct phenomena related to the curvature of spacetime and relative motion, respectively. While both effects can be understood in terms of the behavior of clocks and the passage of time, they arise from different underlying principles and concepts.
In general relativity, the equation you mentioned ([2GM/c^2] /R) is an expression for gravitational time dilation near a massive object. It describes how the strength of the gravitational field, characterized by the mass (M) and the distance (R) from the object, affects the flow of time. As clocks move through regions of different gravitational potential, they experience a difference in the rate at which time elapses.
On the other hand, in special relativity, time dilation arises from the relative motion between two observers. The Lorentz factor accounts for the effects of velocity on the passage of time. As an object moves at relativistic speeds relative to an observer, its internal processes, including its clocks, appear to slow down from the observer's perspective.
It's important to note that these effects are not simply an artificial separation or a hidden vibration function, but rather fundamental consequences of the theories of relativity. General relativity provides a comprehensive framework for understanding gravity as the curvature of spacetime caused by mass and energy. Special relativity deals with the behavior of objects moving at high speeds, where space and time become intertwined.
While there may be underlying physical processes that contribute to the observed time dilation, the equations themselves are derived from well-established principles and have been experimentally confirmed. They provide accurate descriptions of the behavior of clocks in the presence of gravity or relative motion.