Ie. Segal, GEOMETRIC DERIVATION OF THE CHRONOMETRIC REDSHIFT, Proceedings of the National Academy of Sciences of the United Statesof America, 90(23), 1993, pp. 11114-11116
The chronometric redshift-distance relation z = tan2(1/2rho), where rh
o is the distance in radians in the Einstein metric, is derived by an
elementary geometric analysis comparable to that in traditional analys
is of the expanding universe model. The differential dT(t) of Einstein
time evolution T(t) through time t, as applied to the local Minkowski
coordinates x(mu), takes the form sec2(1/2t). At the point of observa
tion t = rho, implying that for a sufficiently localized source, obser
ved wave lengths are a factor of sec2(1/2rho) greater than the corresp
onding emitted wave lengths.