The cancellation of gravitational time dilation by kinematic time dilation at a specific orbital height, such as the geostationary orbit, is not a coincidence but a result of the specific conditions at that altitude.
In the case of the geostationary orbit, which is approximately 35,786 kilometers above the Earth's equator, two effects come into play: gravitational time dilation and kinematic time dilation.
Gravitational time dilation arises from the influence of gravity on the flow of time. The closer an object is to a massive body, the stronger the gravitational field and the slower time runs for that object. Conversely, the farther away an object is from a massive body, the weaker the gravitational field and the faster time runs.
Kinematic time dilation, on the other hand, results from relative motion between observers. According to special relativity, time runs slower for a moving object relative to a stationary observer. This effect becomes significant as the object's velocity increases, approaching the speed of light.
In the case of the geostationary orbit, the orbital height is chosen such that the gravitational time dilation due to the Earth's gravity is precisely balanced by the kinematic time dilation caused by the orbital velocity of the satellite.
At an altitude of approximately 3,000 kilometers, the gravitational time dilation due to Earth's gravity is weaker compared to the surface. As a result, time runs faster at this altitude compared to the surface of the Earth. However, the orbital velocity of a satellite at that height is high enough to induce a significant kinematic time dilation, which causes time to run slower.
By carefully selecting an orbital height of around 3,000 kilometers, the gravitational time dilation and the kinematic time dilation can cancel out, resulting in a net effect where time runs at a similar rate as it does on the surface of the Earth. This is desirable for applications such as communication satellites in geostationary orbit, as it allows for synchronization with ground-based systems.
It's important to note that this cancellation of time dilation effects is specific to the geostationary orbit and the conditions associated with it. At different altitudes or in different gravitational environments, the balance between gravitational and kinematic time dilation would be different.