This tutorial explains how the frequency and wavelength of light from a distant star or galaxy changes if the galaxy or star is moving relative to the observer.
Furthermore, we observe that the further away the galaxy is from us, the faster it is moving away from us.
Light from most galaxies we observe is 'red shifted' - the wavelength of the light has increased towards the red end of the spectrum. A 'stretched' wavelength means that the light source most be travelling away from us - an example of the doppler effect.
We can compare the absorption spectra (pattern of missing wavelengths) of an element on Earth with that of the element in the galaxy and see that the dark bands are all slightly moved by the same amount. Usually the bands will move towards the red end of the spectrum, indicating that the galaxy is moving away from us (called 'recession').
Galaxies that move towards us exhibit 'blue shift', but there are not many of these galaxies.
These observations provide evidence for the 'Big Bang' theory.
So...
There is an observable increase in the wavelength of light from most distant galaxies. The further away the galaxies are, the greater the observed increase in wavelength, and so the faster the are moving away from us. This effect is called 'Red Shift' because the wavelengths are stretched towards the red end of the spectrum.
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