The concept of time dilation in special and general relativity does not involve a conscious or deliberate "knowledge" that determines how time should dilate. It is a natural consequence of the geometry of spacetime and the relative motion between observers.
In the Hafele-Keating experiment, atomic clocks were flown on airplanes that traveled in opposite directions around the Earth. The experiment aimed to test the predictions of time dilation resulting from differences in gravitational potential and relative velocities.
According to general relativity, the presence of mass causes a curvature of spacetime, which results in gravitational time dilation. Clocks in stronger gravitational fields, such as near the Earth's surface, will appear to tick slower when observed from a region with weaker gravity. This effect was taken into account in the experiment.
In addition to gravitational time dilation, special relativity predicts time dilation due to relative velocities. When objects move relative to each other at high speeds, their observed times can appear to run slower compared to a stationary observer. This effect is known as time dilation.
In the Hafele-Keating experiment, the airplanes traveled in opposite directions around the Earth, which meant they experienced different velocities relative to a stationary observer on the ground. As a result, the clocks on the airplanes experienced different amounts of time dilation due to their relative velocities.
It is important to note that time dilation is not a conscious process or a result of "knowing" which direction the clocks are traveling. It is a fundamental aspect of the spacetime geometry and relative motion between observers, as described by the equations of special and general relativity. The predictions of time dilation have been confirmed through numerous experiments and observations, providing strong evidence for the validity of these theories.