geodesic equation for a light wave, which is where is the spacetime interval is the time interval is the spatial interval is the speed of light is the time-dependent cosmic scale factor is the curvature per unit area. For an observer observing the crest of a light wave at a position and time , the crest of the light wave was emitted at a time in the past and a distant position . Integrating over the path in both space and time that the light wave travels yields: In general, the wavelength of light is not the same for the two positions and times considered due to the changing properties of the metric. When the wave was emitted, it had a wavelength . The next crest of the light wave was emitted at a time The observer sees the next crest of the observed light wave with a wavelength to arrive at a time Since the subsequent crest is again emitted from and is observed at , the following equation can be written: The right-hand side of the two integral equations above are identical which means Using the following manipulation: we find that: For very small variations in time (over the period of one cycle of a light wave) the scale factor is essentially a constant ( today and previously). This yields which can be rewritten as Using the definition of redshift provided above, the equation is obtained. In an expanding universe such as the one we inhabit, the scale factor is monotonically increasing as time passes, thus, is positive and distant galaxies appear redshifted. Using a model of the expansion of the universe, redshift can be related to the age of an observed object, the so-called cosmic time–redshift relation. Denote a density ratio as : with the critical density demarcating a universe that eventually crunches from one that simply expands. This density is about three hydrogen atoms per cubic meter of space. At large redshifts, , one finds: where is the present-day Hubble constant, and is the redshift. Distinguishing between cosmological and local effects For cosmological redshifts of additional Doppler redshifts and blueshifts due to the peculiar motions of the galaxies relative to one another cause a wide scatter from the standard Hubble Law. The resulting situation can be illustrated by the Expanding Rubber Sheet Universe, a common cosmological analogy used to describe the expansion of space. If two