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Cosmological time dilation (or cosmic time dilation) is the phenomenon that events observed at cosmological distances (i.e., with a significant cosmological redshift) appear to us to take longer than if the event was nearby. Given the expanding distance between us and the event (Hubble expansion), light takes increasing amounts of time to reach us over the course of the event. The lengthening ratio is 1 to 1+z, the same ratio as redshifted wavelengths from the same distance, and the two can be considered the same phenomenon. Cosmological time dilation would occur from the increasing distance irrespective of the time dilation associated with relativity (just as a musical performance would be audible to you for a longer time to you if you are moving away from it), but relativity does affect the amount of time dilation, significantly at cosmological distances (at which the receding motion is at a relativistic speed); special relativity is relevant and the Lorentz transformation applies: the adjustment identical to the method used for a calculating a redshifted wave period.
Such time dilation is observed: observations of distant events such as supernova light curves do last longer than those of corresponding nearer events, the ratio in proportion to the redshift(s) determined from recognizable spectral lines.