Coupled with the recession of the distant galaxies is the Doppler/gravitational shift of their spectral radiation. This is used to determine the velocity of a receding galaxy. Because the recession velocities being considered may be a significant fraction of the Terminal Velocity in this Domain, ( ~ the velocity of light), the derivation of redshift presented here will include the effects of relativistic velocity correction. However, by virtue of the definition of the Relativistic Space-Time Domain D₁, the derivation of spectral redshift in this Domain will differ slightly from that classically presented in the literature. The derivation of redshift in D₁ is presented below, combining the methods in both [5] and [1], Appendix D.

Let the velocity of a distant spectral emitter, (galaxy), as a function of the time at its location, with reference to some distant point of observation within the Universe be u_G. If this source galaxy, within its spectral output generates a particular wave with frequency f₁^/ it will be given by

n₁^/ is the number of cycles generated by the source in a time t₁^/.

t₁^/ is the local time of the source galaxy.

When this wave is emitted and emerges into the gravitational field of the source galaxy, (D₁), due to the resulting temporal rate change, its frequency will become

u₁ is the temporal rate at the location of the source galaxy.

t₁ is the local time of free space within D₁ at the source galaxy.

This wavelength incorporates the Doppler shift. The wave then travels to the distant point of observation, where, if the temporal rate is u₂, then the apparent wavelength will become

Now f₁^// is the frequency of the incident wave from the source galaxy at the location of the observer and is given by

This apparent wavelength now contains a variation due to two causes. The Doppler shift due to u_G, and a gravitational shift represented by the ratio of the temporal rates at the two locations, u₂/u₁. If another wave, of frequency f₂, is generated from an identical source, at the location of the observer, its wavelength will be

for (A.13), and is clearly a rearrangement of the so called "gross" Doppler effect.

It is important to note that u_G as determined in (A.11) to (A.13) above, is slower than the classical result due to the presence of the multiplier temporal rate term u₁.

Eq.(A.12) gives the recession velocity of the distant galaxy from the point of observation. To determine specific values, in addition to knowledge of l"₁ and l₂, it is obviously necessary to know the values of u₁ and u₂. These can be obtained from [1], Eqs. (4.6) and (4.7), wherein a reasonable estimate of the masses of the source galaxy and the galaxy containing the point of observation is required. The value of s in [1], Eq.(4.7) will be the estimated effective radius of the source galaxy, and, for the point of observation, the distance of that point from the centre of the galaxy in which it resides. Clearly the accuracy with which u_G is determined, will depend entirely upon the accuracy with which these gravitational parameters can be estimated.

In addition to the above, allowance should be made for the temporal rate of the position of each location relative to the centre of the Universe. This allowance would normally be incorporated into the temporal rates at the source galaxy and point of observation. However, this data is currently unknown and so any computed u_G will contain a degree of error. It is however, considered that this error will be fairly small because of the extreme size, and age of the Universe as estimated in the main text, ( ~ 45 x 10⁹ years). The error should only approach significant levels for the most distant galaxies that lie on the radius vector from the centre of the Universe through the point of observation.

One further point concerning the velocity of light. In general, in the Relativistic Domain representing the entire Universe, it is given by the term cu, thereby containing the temporal rate. The temporal rate is given by () in phase I and () in phase II. In both of these equations the parameter s_u appears in the denominator. Because this term, the radius of the physical Universe, varies with time, so then will the velocity of light vary according to the contraction of s_u in phase I and its expansion in phase II. This is in addition to the variation that occurs due to gravitation. The temporal variation is again, due to the magnitude of the parameters involved, a very slow one.

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