The Red Shift of Atomic Spectra in D₁.

This is the third classic test to which the General Theory was subjected to verify its applicability within the Solar System. It is therefore necessary that the existence of atomic spectra within D₁ meet the same criteria.
Note that in this Appendix the value of Spatial Terminal Velocity is used for the velocity of electromagnetic radiation in a vacuum. This has been done solely to enable comparison of the results of this Appendix with similar effects in the General Theory. It may not be strictly correct however, because such radiation must possess a mass, however small, by virtue of Einstein’s universal energy-mass relationship, and it is not possible, within a finite time to accelerate any mass to the Spatial Terminal Velocity of a Relativistic Domain. Accordingly, it should also be noted that the Spatial Terminal Velocity of D₁ is by virtue of the function u, a variable dependent upon s and so therefore is the velocity of electromagnetic radiation as defined in this Appendix.

Accordingly, the wavelength of an atomic spectra at the point of emission in D₁, the surface of a gravitational source, a distance of s ₁ from its centre, is given by

(D1)

where f₁ is the frequency of the spectra, and cu₁ its velocity of propagation at the point of emission.

The solution of the rectilinear version of (3.17), (with w = 0), for an initial condition of is

(D2)

so that after travelling directly away from the source to a point of observation, a distance of s ₂ from the centre of the source, the wavelength of the spectra will be

(D3)

Another spectra of an identical atom emitted at the point of observation will possess a wavelength of

(D4)

so that from (D3) and (D4)

(D5)

is the frequency of the first spectra after travelling to the point of observation and is given by

(D6)

where n₁ is an integral number of cycles and dt ₁ and dt ₂ are elements of time at the points of emission and observation respectively. u₁ and u₂ are the temporal rates at these locations. Therefore

(D7)

where E₁ and E₂ are the energies imparted to the two respective waves by the process of emission. Because this process is an internal function of the atom concerned, the energy of emission is independent of the location within the Domain in which it occurs. Therefore

(D8)

and so

(D9)

Because u₂ >u₁, exhibits an apparent red shift compared to l ₂. Also note from (D7) and (D8) it is clear that f₁ = f₂.

If the point of observation is sufficiently far from the point of emission, it may, (as in the literature), be approximated to free space, i.e. u₂ ® 1, as in D₀ and then

(D10)

which after insertion of (4.7) may be further approximated to

(D11)

This is effectively the result most often quoted in the literature, [4], [5].

It should be noted that a comparison of the wavelength of the first wave upon reaching the point of observation with its wavelength at the point of emission produces the result

(D12)

showing that the true red shift of the travelling wave is greater than when simply compared to a wave emitted at the point of observation. Substitution of (D12) into (D9) then gives

(D13)

as would be expected.

The mechanism behind the shift is that as the wave moves away from the source, it continuously moves through an increasing temporal rate, which causes its frequency to decrease. This produces a corresponding increase in spectral wavelength.

Note that from (D1), if the geometrical radius of the gravitational source is equal to, (or less than), twice its gravitational radius, the propogation velocity of emission is zero. Hence electromagnetic radiation from such a physical body is impossible. This indicates that "Black Holes" are at least mathematically permissible within the Domain D₁, as they are in the General Theory. However, it will be shown in a future paper that there are other constraints which prohibit the formation of Black Holes within the relativistic Domain D₁.

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