Time dilation, as predicted by the theory of relativity, does not directly affect the perceived speed of light in a medium. The speed of light in a vacuum, denoted by "c," is a fundamental constant and is considered the same for all observers, regardless of their relative motion or gravitational fields.
However, the perception of the speed of light can be affected by time dilation when light travels through different gravitational fields or relative velocities. According to general relativity, the presence of a gravitational field or high relative velocities can cause time to dilate, meaning that the flow of time is experienced differently by observers in different reference frames.
In a strong gravitational field or near a massive object, time dilation occurs, and clocks closer to the source of gravity tick slower compared to clocks further away. As a result, if light passes through a gravitational field, it can be subject to gravitational time dilation, causing it to appear to travel slower or faster when measured by observers in different gravitational potentials.
Similarly, relative motion between the source of light and an observer can lead to time dilation effects due to special relativity. When an observer is in motion relative to a light source, their perception of the frequency of light can be altered. This effect is known as the Doppler effect, and it causes a shift in the wavelength (and therefore frequency) of light as observed by the moving observer.
It's important to note that the speed of light in a vacuum remains constant, but the perceived frequency or wavelength of light can be influenced by time dilation effects caused by relative motion or gravitational fields.