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I am interested in knowing what are the differences between gravitational and cosmological redshifts?

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Gravitational red shift is due to the energy of a photon in a gravitational field. So the photon needs energy to escape the gravitational force and therefore the wavelength increases (redshift).

The cosmological redshift is due to the expansion of the universe and thus is not connected to a gravtiational field. The wavelength of the photon stretches like the spacetime does because of the Hubble expansion of the universe.

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