The cause of the redshift for cosmological distances and time dilation is the expansion of the universe, as described by the theory of general relativity. This phenomenon is known as cosmological redshift.
When light travels through an expanding universe, the space between the source of light and the observer stretches, causing the wavelength of the light to increase. As a result, the light is "stretched" to longer wavelengths, shifting it towards the red end of the electromagnetic spectrum. This is what we observe as redshift.
The expansion of the universe also affects the perception of time. As the universe expands, the distances between objects increase, and the space itself expands. This expansion of space causes a stretching of time as well. Therefore, for distant objects in an expanding universe, the passage of time appears to slow down from the perspective of an observer in a different reference frame.
The relationship between cosmological redshift and time dilation is interconnected. As light travels through the expanding universe and experiences redshift, the time it takes for the light to reach an observer is also affected. This is because the expansion of space stretches out the wavelength of the light and increases its travel time.
Both cosmological redshift and time dilation are consequences of the dynamic nature of the universe and the expansion of spacetime. They are fundamental aspects of our understanding of the large-scale structure and evolution of the cosmos.