Photon's wave period extends by the same factor as its length. Time dilation is the expansion of time. The problem with FLRW metric is that it doesn't take cosmic time dilation into account. For null geodesic, that is for a photon with no proper time the metric is [math]0 = -(cdt)^2+(a(t)dr)^2[/math]. The second term [math]a(t)dr[/math] is the equivalent of the expanding wavelength. [math]dt[/math] has to be multiplied by the same factor [math]a(t)[/math] to apply the cosmic time dilation. [math]a(t)dt[/math] is the equivalent of the extending wave period. As a consequence we have [math]0 = -(a(t)cdt)^2+(a(t)dr)^2[/math] which is the Minkowski metric for a photon after dividing the equation by [math]a(t)[/math], so [math]0 = -(cdt)^2+(dr)^2[/math] gives [math]cdt=dr[/math] that gives [math]ct=r[/math]. That's Hubble radius for the universe age. How do you like it as the observable universe radius as a result of the appliance of cosmic time dilation to FLRW metric?
