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. However, the calculations become more complex as they involve dealing with accelerated frames of reference.
To calculate the time dilation of a satellite in a non-inertial frame of reference, such as an orbiting satellite, you would need to consider the effects of both special relativity and general relativity. General relativity accounts for the gravitational time dilation caused by the curvature of spacetime near massive objects.
The basic steps involved in calculating the time dilation of a satellite in a non-inertial frame are as follows:
Determine the satellite's velocity: Measure the satellite's velocity relative to an observer on Earth or with respect to a local inertial frame of reference.
Account for special relativity effects: Use the Lorentz transformation equations of special relativity to calculate the time dilation caused by the relative velocity between the satellite and the observer on Earth.
Account for general relativity effects: Consider the gravitational time dilation caused by the difference in gravitational potential between the satellite and the observer on Earth. This involves using the gravitational time dilation equation derived from general relativity, which incorporates the mass and radius of the Earth.
Combine the effects: Combine the time dilation factors calculated from special relativity and general relativity to obtain the overall time dilation experienced by the satellite.
It's worth noting that the precise calculations would depend on the specific scenario, such as the orbit of the satellite and its distance from Earth. The calculations may involve approximations and mathematical models to account for the complexities of the system.
Overall, while special relativity provides a framework for calculating time dilation in non-inertial frames, the full analysis requires incorporating the effects of general relativity to accurately determine the time dilation experienced by a satellite in orbit.