Special relativity can still be used to calculate time dilation for a satellite even when the observer is not in an inertial frame of reference. Special relativity deals with the effects of relative motion and allows for the calculation of time dilation in various scenarios.
In the case of a satellite, the observer on Earth can be considered in a non-inertial frame of reference due to the rotation and orbital motion of the Earth. However, by considering the satellite's motion relative to the Earth and applying the principles of special relativity, we can still calculate the time dilation experienced by the satellite.
The equation for time dilation in special relativity is given by:
t' = t / √(1 - v^2/c^2)
Where: t' is the time experienced by the satellite (moving object) t is the time experienced by an observer on Earth (stationary observer) v is the relative velocity between the satellite and the observer on Earth c is the speed of light in a vacuum
To calculate the time dilation of the satellite, you would need to know the relative velocity between the satellite and the observer on Earth. This velocity can be determined by considering the satellite's orbital motion, taking into account factors such as the satellite's altitude, orbital speed, and the rotational speed of the Earth.
By plugging the appropriate values into the time dilation equation, you can calculate the time dilation factor experienced by the satellite relative to an observer on Earth. This factor represents the ratio of the satellite's proper time (t') to the time experienced by the observer on Earth (t). The time dilation factor will be greater than 1, indicating that time passes more slowly for the satellite relative to the observer on Earth.
It's important to note that the specific calculations can be complex, involving considerations of relativistic effects, gravitational fields, and the satellite's trajectory. Advanced models and formulas are used in practice to accurately account for these factors when calculating time dilation for satellites in the context of special relativity.