2.0 The Relativistic Characteristics of Rapidly Spinning Spherical Bodies.2.1 MassTo determine the relativistic effects upon the mass of a spherical body spinning at a rapid angular rate, consider Fig. 2.1 below. 
In Fig. 2.1 the volume of the elemental is
The rest mass of the elemental is then
The velocity of the elemental due to the spin of the body is
The energy mass of the elemental is therefore
The toroidal energy mass is obtained by integrating (2.4) with respect to f over the limits 0 £ f £ 2p to give
To complete the exercise of obtaining the energy mass of the spinning body it is first necessary to integrate (2.5) with respect to j between the limits 0 £ j £ p. Then it is necessary to integrate the result with respect to r over the limits 0 £ r £G. While these integrations can be performed in an exact and rigorous manner it is a somewhat lengthy and involved process. It is simpler to make a relativistic approximation at this point which is then "reversed" after the integrations are completed. This process is valid because the expansions are absolutely convergent series and the result is accurate to about 1.12% as w Þ c. The approximation is therefore quite adequate to illustrate the significance of the result. Thus binomially expanding the square root in (2.5), the first integral specified above becomes
Eq.(2.6) integrates to
Integrating (2.7) with respect to r as specified then yields
for the energy mass of the spinning sphere. Performing the reverse expansion on (2.8), using just the first relativistic term gives
where the classical expression for the rest mass has also been inserted. Eq.(2.9) can be written
where Ggyr is the classical radius of gyration of a stationary spherical homogeneous mass,
Eq.(2.10) is the energy mass of the spinning body wherein the relativistic mass increase is the stored kinetic energy induced by the applied accelerating torque. The important feature about this result is apparent in (2.9) where it can be seen that if wG = c then the mass becomes
and is a maximum. Thus the surface of the sphere at the spin circumference, can be accelerated to the terminal velocity of D0( ~ the speed of light), while the energy mass of the sphere remains finite. This is solely due to the distributed nature of the mass and that each toroidal element between the centre and the spin circumference is spinning at a velocity lower than wG. Because the energy mass of the sphere is finite under this condition, only a finite amount of energy has been applied to reach this state, and it would therefore be possible to apply additional energy in the form of an accelerating torque, to further increase the spin rate. The consequence of this would be that the surface of the sphere at the spin circumference would tend to exceed the terminal velocity of D0, and thus contravene its primary criterion of existence, i.e. that the maximum spatial velocity attainable in that Domain is the velocity constant c. To avoid this the mass must therefore lose energy. In a future paper concerning the existence of de Broglie matter waves in D0, it will be shown that matter can only exist at the terminal velocity of D0 as pure kinetic energy, possibly in the form of photons/electromagnetic radiation. It is therefore proposed that a spherical mass induced to spin such that the surface at the spin circumference tends to exceed the terminal velocity in D0, avoids that anomaly by losing kinetic energy at that surface, by converting it to radiant energy at some frequency and wavelength proportional to the energy applied to increase the angular rate. The emission of the spectra would be in accordance with the quantum laws of Planck and de Broglie. This phenomenon may have significant implications in Cosmology and Atomic Structure Theory, and is discussed further along those lines in the concluding remarks.
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P.G.Bass, June 2006
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